https://kbwiki.ercoftac.org/w/api.php?action=feedcontributions&user=Kassinos&feedformat=atom KBwiki - User contributions [en] 2024-03-28T20:26:52Z User contributions MediaWiki 1.39.2 https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38864 CFD Simulations AC7-02 2020-06-15T18:27:19Z <p>Kassinos: /* Mesh convergence analysis */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Mesh_analysis_LES2 |Mesh convergence analysis]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see section [[#Comp_domain_meshes_LES1 | Computational domain and meshes]]). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical_Accuracy_LES3|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_Accuracy_LES3&quot;&gt;&lt;/div&gt;<br /> <br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38863 CFD Simulations AC7-02 2020-06-15T18:26:19Z <p>Kassinos: /* Mesh convergence analysis */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Mesh_analysis_LES2 |Mesh convergence analysis]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see section [[#Comp_domain_meshes| Computational domain and meshes]]). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical_Accuracy_LES3|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_Accuracy_LES3&quot;&gt;&lt;/div&gt;<br /> <br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38862 CFD Simulations AC7-02 2020-06-15T18:12:18Z <p>Kassinos: /* Numerical accuracy */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Mesh_analysis_LES2 |Mesh convergence analysis]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical_Accuracy_LES3|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_Accuracy_LES3&quot;&gt;&lt;/div&gt;<br /> <br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38861 CFD Simulations AC7-02 2020-06-15T18:06:08Z <p>Kassinos: /* Large Eddy Simulations &amp;mdash; Case LES3 */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Mesh_analysis_LES2 |Mesh convergence analysis]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical_Accuracy_LES3|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES3&quot;&gt;&lt;/div&gt;<br /> <br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38860 CFD Simulations AC7-02 2020-06-15T13:38:21Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Mesh_analysis_LES2 |Mesh convergence analysis]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38859 CFD Simulations AC7-02 2020-06-15T13:36:45Z <p>Kassinos: /* Comparison of different LES subgrid-scale models */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical_Accuracy_LES2|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> &lt;div id=&quot;Comparison_LES_models&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38858 CFD Simulations AC7-02 2020-06-15T13:36:16Z <p>Kassinos: /* Mesh convergence analysis */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical_Accuracy_LES2|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> &lt;div id=&quot;Mesh_analysis_LES2&quot;&gt;&lt;/div&gt;<br /> <br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38857 CFD Simulations AC7-02 2020-06-15T13:35:10Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical_Accuracy_LES2|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison_LES_models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38856 CFD Simulations AC7-02 2020-06-15T13:33:50Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> &lt;div id=&quot;Comp_domain_meshes_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38855 CFD Simulations AC7-02 2020-06-15T13:33:14Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Comp_domain_meshes_LES1| Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38854 CFD Simulations AC7-02 2020-06-15T13:31:32Z <p>Kassinos: /* Numerical accuracy */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_LES1&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38853 CFD Simulations AC7-02 2020-06-15T13:30:57Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical_accuracy_LES1|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38852 Evaluation AC7-02 2020-06-15T13:29:59Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#Numerical_accuracy_RANS |Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38851 CFD Simulations AC7-02 2020-06-15T13:29:31Z <p>Kassinos: /* Numerical accuracy */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38850 Evaluation AC7-02 2020-06-15T13:28:07Z <p>Kassinos: /* Airflow in the human upper airways */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#Numerical Accuracy|Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38849 Evaluation AC7-02 2020-06-15T13:26:16Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#RANS Simulations#Numerical Accuracy|Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38848 Evaluation AC7-02 2020-06-15T13:25:16Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#RANS Simulations#Numerical Accuracy|Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38847 CFD Simulations AC7-02 2020-06-15T13:08:09Z <p>Kassinos: /* Numerical accuracy */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> ===Numerical accuracy===<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38846 Evaluation AC7-02 2020-06-15T13:07:37Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#RANS Simulations#Numerical_Accuracy_RANS|Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38845 CFD Simulations AC7-02 2020-06-15T13:04:44Z <p>Kassinos: /* Numerical accuracy */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> &lt;div id=&quot;Numerical_accuracy_RANS&quot;&gt;&lt;/div&gt;<br /> <br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38844 Evaluation AC7-02 2020-06-15T12:20:15Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#RANS Simulations#Numerical Accuracy|Numerical Accuracy]] for RANS). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Best_Practice_Advice_AC7-02&diff=38843 Best Practice Advice AC7-02 2020-06-15T12:18:58Z <p>Kassinos: /* Turbulence Models */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> =Best Practice Advice=<br /> ==Key Fluid Physics==<br /> &lt;!--<br /> Briefly describe the key fluid physics/flow regimes which exert an influence on the DOAPs. Ideally this should draw together into a coherent picture the associated UFR descriptions together with any important interactions which are AC specific. Mention the UFRs associated with this AC that you have considered in drafting your best practice advice. ''Access the Knowledge Base to find the UFRs associated with your AC''.<br /> --&gt;<br /> <br /> In the present AC, experiments and simulations were conducted at a flowrate of 60 L/min through an upper airway geometry. <br /> At this flow conditions, the Reynolds number for air in the trachea is 4920, which is well within the turbulent regime. <br /> Geometric effects, such as the bent in the oropharyngeal region and the constriction at the laryngeal glottis (just upstream of the trachea, see [[#Figure25|figure 25]]<br /> ) enhance turbulence levels as the air moves from the inlet to the region of the trachea. <br /> Turbulent kinetic energy levels reach a peak in the shear layer formed between the high speed laryngeal jet and the surrounding (low speed) air (see [[#Figure25|figure 25]]). <br /> The characteristics of the laryngeal jet formation bear a resemblance to the flow through a constricted pipe, which can be classified as a free shear flow where the wall serves to confine the spreading of the jet rather than producing turbulence ([[Lib:Best Practice Advice AC7-02#Tawhai2011|Tawhai &amp; Lin, 2011]]). <br /> High turbulence levels persist in the region of the first bifurcation (stations H1-H2 &amp; J1-J2 in [[Lib:CFD Simulations AC7-02#Figure12|figure 12]](b)).<br /> <br /> ==Application Uncertainties==<br /> &lt;!--<br /> List any uncertainties which make a high fidelity CFD model difficult to assemble. Typical examples might include:<br /> *a gas leakage between two components which is difficult to resolve on practical meshes, andeven if it is resolved, the leakage flow conditions may not be known.<br /> *flow conditions at inlet to the AC (or indeed other boundaries) which may be complex and not precisely known.<br /> *fine details of the geometry are imprecise.<br /> Briefly discuss the sensitivity of the DOAP predictions to these uncertainties and their impact on the BPA. In particular, can clear, unequivocal BPA be given or is it necessary to introduce appropriate caveats.<br /> --&gt;<br /> <br /> The differences between measurements and simulations can result from several uncertainties involved in the tests. <br /> A first source of uncertainty are the inlet conditions, which are not perfectly matched between the measurements and the computations. <br /> In the experiments, the lung model was placed in an open liquid tank with a piston diaphragm pump attached to a linear actuator to achieve a quasi-stationary inspiratory flow. <br /> The stroke of the piston followed a cyclic triangular function with an adjustable falling constant slope and thus constant velocity to match different flow rates during inspiration. <br /> The measured mean velocity at the inlet of the model, shown in [[Lib:Test Data AC7-02#Figure7|figure 7]], is asymmetric, probably due to the action of the piston diaphragm pump. <br /> In the computations, instead of reproducing the measured inlet conditions, either uniform or turbulent inlet velocity profiles were prescribed. <br /> Due to a leakage flow between the upper and lower part of the model in the experiments, the achieved flowrate within the main bifurcation and bronchi region was about 10% lower than in the upper part of the model. <br /> As a result, a maximum flowrate of &lt;math&gt; Q_{w/g} &lt;/math&gt; = 28.56 L/min could be achieved in the measurements. <br /> This value is slightly lower than &lt;math&gt; Q_{w/g} &lt;/math&gt; = 31.75 L/min, which is the target value for an equivalent air flowrate of 60L/min through the model. <br /> Although the flow is well within the turbulent regime, the theoretical maximum Reynolds number decreases from 4921 to 4286.<br /> <br /> ==Computational Domain and Boundary Conditions==<br /> <br /> The geometry of the extrathoracic airways must be included because turbulence is generated in this region that propagates in the first airway generations. <br /> Concerning the boundary conditions, the inlet velocity profile is important and thus realistic inlet conditions should be used. <br /> At the outlets, it is important to apply correct pressures such that the ventilation of the airway tree is physiologically realistic ([[Lib:Best Practice Advice AC7-02#Yin2010|Yin et al., 2010]]). <br /> In the present AC, in order to simplify the experimental setup and be able to perform the flow measurements, uniform pressures were prescribed at all outlets.<br /> <br /> Concerning the inlet conditions for the turbulent variables in RANS calculations, the application of a turbulence intensity of 5% for the k-ω SST model at the extended inlet (10xDinlet) yielded higher turbulent kinetic energy values close to the inlet of the model compared to the mapped inlet condition (Inlet 1). <br /> The k-ε models were found to provide overall higher turbulence levels than the k-ω SST model, especially at the near-wall regions.<br /> <br /> ==Discretisation and Grid Resolution==<br /> <br /> Since it is not possible to generate a structured hexahedral grid for the present geometry due to its complexity, a higher refinement ratio should be applied to avoid numerical diffusion. <br /> In addition to that, layers of prismatic elements should be added near the wall boundaries for a better prediction of this region, not only with regard to flow properties itself, but the flow conditions seen by the particles, i.e. mean velocity and turbulence properties. <br /> Airflow through the glottis constriction at the larynx, illustrated in [[#Figure25|figure 25]], bears a resemblance to flow through a constricted pipe. <br /> This type of flow may be classified as a free shear flow in which the wall serves to confine the spreading of the jet rather than producing turbulence. <br /> In this case, turbulence is most active at the interface between two free streams (high and low speed) and along the jet core and therefore, fine mesh resolution should be placed accordingly to capture strong turbulence activities. <br /> Use of a strict y+ = 1 condition in the generation of the near-wall mesh but extremely coarse mesh in the core region of the airway model is conceptually wrong ([[Lib:Best Practice Advice AC7-02#Tawhai2011|Tawhai &amp; Lin, 2011]]).<br /> Recommended values for the parameters involved in mesh generation (initial cell height, average expansion ratio, number of near-wall prism layers, average cell volume in the domain, number of computational cells etc.) can be found in Tables 4,5 (LES) and 6 (RANS).<br /> <br /> LES were found to give similar results independent of the discretisation method used (Finite Volume or Finite Element).<br /> <br /> &lt;div id=&quot;Figure25&quot;&gt;&lt;/div&gt;<br /> [[File:jet.png|center|thumb|500px|'''Figure 25''': Laryngeal jet formed at the glottis constriction. (a) Isosurfaces of &lt;math&gt; |&lt;u&gt;| &lt;/math&gt; = 6.7m/s ; (b) Mean velocity contours; (c) Turbulent kinetic energy contours.]]<br /> <br /> ==Turbulence Models==<br /> <br /> <br /> The ability of four LES subgrid-scale models was assessed by comparing their predictions to the PIV data. <br /> These models were the WALE, the Smagorinsky, the variational multiscale (VMS) WALE and the QR eddy-viscosity model from Verstappen. <br /> All the models were found to provide very similar results. <br /> It is concluded that the influence of the subgrid scales on the airflow in the human upper airways is small, and the choice of subgrid-scale turbulence model is not as important as in cases with higher Reynolds numbers.<br /> <br /> In general the RANS prediction for the velocity magnitude follow trends similar to those of the measured data. <br /> RANS however show poor performance at locations with shear layers, recirculation and flow separation. <br /> Moreover, RANS simulations consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This is a direct consequence of discarding many of the elements of the underlying turbulence physics when solving only for the mean flow (see discussion in section [[Lib:Evaluation AC7-02|Evaluation]]). <br /> The k-ε models provides overall higher turbulence levels than the k-ω SST turbulence model.<br /> <br /> Results presented in AC7-01 showed that the deposition fractions obtained with RANS models achieved a quite good agreement with LES and measurements. <br /> This might seem surprising given the tendency of RANS to underpredict turbulence intensities at several stations downstream of the glottis constriction. <br /> This apparent paradox probably relates to the fact that the dominant deposition mechanism in the upper airways is inertial impaction. <br /> While inertial impaction tends to be dominated by mean flow effects, turbulent dispersion still plays an important role, especially in regions where there is significant large-scale anisotropy in the turbulence. <br /> Hence, it is known that using only the time-averaged air velocity field for deposition studies (this is the case in RANS) leads to deposition overpredictions ([[Lib:Best Practice Advice AC7-02#Matida2004|Matida et al., 2004]]).<br /> Still, RANS deposition predictions can be improved when they are used together with a turbulent dispersion model, as was done in AC7-01. <br /> When using a turbulent dispersion model, individual particles are allowed to interact successively with discrete eddies, each eddy having length, velocity and lifetime characteristic scales obtained from the primary flow calculation results. <br /> It is therefore important to correctly select the turbulent dispersion model and it’s parameters for accurate deposition predictions. <br /> LES on the other hand, since they resolve the large scale eddies, do not need a model to account for these in particle-laden flows. <br /> The study of [[#armenio1999|Armenio et al. (1999)]] has shown that the motion of inertial particles in low to moderate Reynolds number flows, is not influenced from the unresolved sub-grid scales in LES.<br /> <br /> In conclusion, LES are more capable than RANS in predictions of airflow in the human upper airways, since they can account better for the physics of the turbulent flow without the need to adjust model parameters.<br /> <br /> ==Recommendations for Future Work==<br /> <br /> The present application challenge focuses on the airflow that develops in the human upper airways. <br /> The airway geometry has been considered rigid. <br /> In reality, the lung expands and contracts and the airway walls deform during inhalation and exhalation ([[Lib:Best Practice Advice AC7-02#Mead-Hunter2013|Mead-Hunter et al., 2013]]). <br /> In addition, there is periodic movement of the glottal aperture during tidal breathing that regulates the respiratory airflow dynamics ([[Lib:Best Practice Advice AC7-02#Xi2018|Xi et al., 2018]]).<br /> It is therefore important to assess the effect of wall deformation on the developed flow features inside the airways.<br /> <br /> The higher temperature and humidity of the human body compared to the inhaled ambient air results in heat and water vapor transfer as the air is transported in the airways <br /> ([[Lib:Best Practice Advice AC7-02#Wu2014|Wu et al., 2014]]).<br /> During inhalation, air is heated and humidified by the airway walls until it reaches the body temperature and approximately 100% relative humidity. <br /> Such effects can play a role in air and inhaled aerosols transport inside the airways and should be further examined in the future studies.<br /> <br /> ==Acknowledgements==<br /> <br /> The present application challenge is based upon work from COST Action MP1404 SimInhale &quot;Simulation and pharmaceutical technologies for advanced patient-tailored inhaled medicines&quot;, supported by COST (European Cooperation in Science and Technology - www.cost.eu).<br /> <br /> ==References==<br /> &lt;div id=&quot;Adrian2011&quot;&gt;&lt;/div&gt;<br /> Adrian, R.J. and Westerweel, J. 2011 <br /> :Particle Image Velocimetry. ''Cambridge University Press, Cambridge.''<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;armenio1999&quot;&gt;&lt;/div&gt;<br /> Armenio, V., Piomelli, U. &amp; Fiorotto, V. 1999<br /> :Effect of the subgrid scales on particle motion. ''Physics of Fluids'' '''11''' (10), 3030&amp;nbsp;&amp;ndash;&amp;nbsp;3042.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Charnyi2017&quot;&gt;&lt;/div&gt;<br /> Charnyi, Sergey, Heister, Timo, Olshanskii, Maxim A. &amp; Rebholz, Leo G. 2017 <br /> :On conservation laws of Navier-Stokes Galerkin discretizations. ''Journal of Computational Physics'' '''337''', 289&amp;nbsp;&amp;ndash;&amp;nbsp;308.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Codina2001&quot;&gt;&lt;/div&gt;<br /> Codina, R. 2001 <br /> :Pressure stability in fractional step finite element methods for incompressible flows. ''Journal of Computational Physics'' '''170''', 112&amp;nbsp;&amp;ndash;&amp;nbsp;140.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Hughes2000&quot;&gt;&lt;/div&gt;<br /> Hughes, Thomas J.R., Mazzei, Luca &amp; Jansen, Kenneth E. 2000 <br /> :Large eddy simulation and the variational multiscale method. ''Computing and Visualization in Science'' '''3''' (1), 47&amp;nbsp;&amp;ndash;&amp;nbsp;59.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Janke2019&quot;&gt;&lt;/div&gt;<br /> Janke, T., Koullapis, P., Kassinos, S.C. &amp; Bauer, K. 2019 <br /> :PIV measurements of the Siminhale benchmark case. ''European Journal of Pharmaceutical Sciences'' '''133''', 183&amp;nbsp;&amp;ndash;&amp;nbsp;189.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Jasak1996&quot;&gt;&lt;/div&gt;<br /> Jasak, H. 1996 <br /> :Error analysis and estimation for the finite volume method with applications to fluid flows. ''PhD thesis, Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, UK''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Jofre2014&quot;&gt;&lt;/div&gt;<br /> Jofre, L., Lehmkuhl, O., Ventosa, J., Trias, F.X. &amp; Oliva, A. 2014 <br /> :Conservation properties of unstructured finite-volume mesh schemes for the Navier-Stokes equations. ''Numerical Heat Transfer, Part B: Fundamentals'' '''54''' (1), 53&amp;nbsp;&amp;ndash;&amp;nbsp;79.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Kim1985&quot;&gt;&lt;/div&gt;<br /> Kim, J. &amp; Moin, P. 1985 <br /> :Application of a fractional-step method to incompressible Navier-Stokes equations. ''Journal of Computational Physics'' '''59''' (2), 308&amp;nbsp;&amp;ndash;&amp;nbsp;323.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Koullapis2018&quot;&gt;&lt;/div&gt;<br /> Koullapis, P., Kassinos, S. C., Muela, J., Perez-Segarra, C., Rigola, J., Lehmkuhl, O., Cui, Y., Sommerfeld, M., Elcner, J., Jicha, M., Saveljic, I., Filipovic, N., Lizal, F. &amp; Nicolaou, L. 2018 <br /> :Regional aerosol deposition in the human airways: The SimInhale benchmark case and a critical assessment of ''in silico'' methods. ''European Journal of Pharmaceutical Sciences'' '''113''', 77&amp;nbsp;&amp;ndash;&amp;nbsp;94.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Koullapis2016&quot;&gt;&lt;/div&gt;<br /> Koullapis, P. G., Kassinos, S.C., Bivolarova, M. P. &amp; Melikov, A. K. 2016 <br /> :Particle deposition in a realistic geometry of the human conducting airways: Effects of inlet velocity profile, inhalation flowrate and electrostatic charge. ''Journal of Biomechanics'' '''49''', 2201&amp;nbsp;&amp;ndash;&amp;nbsp;2212.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lehmkuhl2019&quot;&gt;&lt;/div&gt;<br /> Lehmkuhl, O., Houzeaux, G., Owen, H., Chrysokentis, G. &amp; Rodriguez, I. 2019 <br /> :A low-dissipation finite element scheme for scale resolving simulations of turbulent flows. ''Journal of Computational Physics'' '''390''', 51&amp;nbsp;&amp;ndash;&amp;nbsp;65.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lehmkuhl2007&quot;&gt;&lt;/div&gt;<br /> Lehmkuhl, O., Perez Segarra, C.D., Borrell, R., Soria, M. &amp; Oliva, A. 2007 <br /> :Termofluids: A new Parallel unstructured CFD code for the simulation of turbulent industrial problems on low cost PC cluster. ''Proceedings of the Parallel CFD Conference'' pp. 1&amp;nbsp;&amp;ndash;&amp;nbsp;8.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lilly1992&quot;&gt;&lt;/div&gt;<br /> Lilly, D. K. 1992 <br /> :A proposed modification of the Germano subgrid-scale closure method. ''Physics of Fluids A'' '''4''' (3), 633&amp;nbsp;&amp;ndash;&amp;nbsp;635.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lin2007&quot;&gt;&lt;/div&gt;<br /> Lin, C.-L., Tawhai, M.H., Mclennan, G. &amp; Hoffman, E.A. 2007 <br /> :Characteristics of the turbulent laryngeal jet and its effect on airflow in the human intra-thoracic airways. ''Respiratory Physiology and Neurobiology'' '''157''', 295&amp;nbsp;&amp;ndash;&amp;nbsp;309.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lizal2015&quot;&gt;&lt;/div&gt;<br /> Lizal, Frantisek, Belka, Miloslav, Adam, Jan, Jedelsky, Jan &amp; Jicha, Miroslav 2015 <br /> :A method for ''in vitro'' regional aerosol deposition measurement in a model of the human tracheobronchial tree by the positron emission tomography. ''Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine'' '''229''' (10), 750&amp;nbsp;&amp;ndash;&amp;nbsp;757.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lizal2012&quot;&gt;&lt;/div&gt;<br /> Lizal, Frantisek, Elcner, Jakub, Hopke, Philip K, Jedelsky, Jan &amp; Jicha, Miroslav 2012 <br /> :Development of a realistic human airway model. ''Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine'' '''226''' (3), 197&amp;nbsp;&amp;ndash;&amp;nbsp;207.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Matida2004&quot;&gt;&lt;/div&gt;<br /> Matida, E. A., Finlay, W. H., Lange, C. F. &amp; Grgic, B. 2004 <br /> :Improved numerical simulation of aerosol deposition in na idealized mouth-throat. ''Aerosol Science'' '''35''', 1&amp;nbsp;&amp;ndash;&amp;nbsp;19.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Mead-Hunter2013&quot;&gt;&lt;/div&gt;<br /> Mead-Hunter, Ryan, King, Andrew J.C., Larcombe, Alexander N. &amp; Mullins, Benjamin J. 2013 <br /> :The influence of moving walls on respiratory aerosol deposition modelling. ''Journal of Aerosol Science'' '''64''', 48&amp;nbsp;&amp;ndash;&amp;nbsp;59.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Muela2019&quot;&gt;&lt;/div&gt;<br /> Muela, J., Rigoala, J., Oliet, C., Perez-Segarra, C.D. &amp; Oliva, A. 2019 <br /> :Assessment of numerical aspects using LES in particle separation devices. ''In 8th European Conference for Aeronautics and Aerospace Sciences (EUCASS)'', p. 678.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Nicoud1999&quot;&gt;&lt;/div&gt;<br /> Nicoud, Franck &amp; Ducros, Frederic 1999 <br /> :Subgrid-scale stress modelling based on the square of the velocity gradient tensor. ''Flow, turbulence and Combustion'' '''62''' (3), 183&amp;nbsp;&amp;ndash;&amp;nbsp;200.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;OpenFOAM2013a&quot;&gt;&lt;/div&gt;<br /> OpenFOAM Foundation 2013a <br /> :OpenFOAM Programmer’s Guide, version 2.2.1 edn. London, UK.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;OpenFOAM2013b&quot;&gt;&lt;/div&gt;<br /> OpenFOAM Foundation 2013b<br /> :OpenFOAM User Guide, version 2.2.1 edn. London, UK.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Radhakrishnan2009&quot;&gt;&lt;/div&gt;<br /> Radhakrishnan, H. &amp; Kassinos, S. 2009 <br /> :CFD modeling of turbulent flow and particle deposition in human lungs. ''31st Annual International Conference of the IEEE EMBS'', Mineapolis, Minnesota, USA pp. 2867&amp;nbsp;&amp;ndash;&amp;nbsp;2870.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Sagaut2006&quot;&gt;&lt;/div&gt;<br /> Sagaut, P. 2006 <br /> :Large Eddy Simulation for Incompressible Flows: An Introduction. ''Springer''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Schmidt2004&quot;&gt;&lt;/div&gt;<br /> Schmidt, Andreas, Zidowitz, Stephan, Kriete, Andres, Denhard, Thorsten, Krass, Stefan &amp; Peitgen, Heinz-Otto 2004 <br /> :A digital reference model of the human bronchial tree. ''Computerized Medical Imaging and Graphics'' '''28''' (4), 203&amp;nbsp;&amp;ndash;&amp;nbsp;211.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Sciacchitano2016&quot;&gt;&lt;/div&gt;<br /> Sciacchitano, A. and Wieneke, B. 2016<br /> :PIV uncertainty propagation. ''Meas. Sci. Technol.'' '''27''' (084006), 20pp.<br /> &lt;br/&gt;<br /> &lt;div id=&quot;Smagorinsky1963&quot;&gt;&lt;/div&gt;<br /> Smagorinsky, Joseph 1963 <br /> :General circulation experiments with the primitive equations: I. The basic experiment. ''Monthly weather review'' '''91''' (3), 99&amp;nbsp;&amp;ndash;&amp;nbsp;164.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Tabor2004&quot;&gt;&lt;/div&gt;<br /> Tabor, G. R., Baba-Ahmadi, M. H., de Villiers, E. &amp; Weller, H. G. 2004 <br /> :Construction of inlet conditions for LES of turbulent channel flow. ''European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS)''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Tawhai2011&quot;&gt;&lt;/div&gt;<br /> Tawhai, M.H. &amp; Lin, C.-L. 2011 <br /> :Airway gas flow. ''Comprehensive Physiology'' '''1''', 1135&amp;nbsp;&amp;ndash;&amp;nbsp;1157. <br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Trias2011&quot;&gt;&lt;/div&gt;<br /> Trias, F. X. &amp; Lehmkuhl, O. 2011<br /> :A self-adaptive strategy for the time integration of Navier-Stokes equations. ''Numerical Heat Transfer. Part B'' '''60''' (2), 116&amp;nbsp;&amp;ndash;&amp;nbsp;134.<br /> &lt;br/&gt;<br /> &lt;div id=&quot;Vazquez2016&quot;&gt;&lt;/div&gt;<br /> Vazquez, A. M., Houzeaux, G., Koric, S., Artigues, A., Aguado-Sierra, J., Aris, R., Mira, D., Calmet, H., Cucchietti, F., Owen, H., Casoni, E., Taha, A., Burness, E. D., Cela, J. M. &amp; Valero, M. 2016 <br /> :Alya: Multiphysics engineering simulation towards exascale. ''Journal Computational Sciences'' '''14''', 15&amp;nbsp;&amp;ndash;&amp;nbsp;27.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2011&quot;&gt;&lt;/div&gt;<br /> Verstappen, Roel 2011 <br /> :When does eddy viscosity damp subfilter scales sufficiently? ''Journal of Scientific Computing'' '''49''' (1), 94.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2010&quot;&gt;&lt;/div&gt;<br /> Verstappen, R.W.C.P., Bose, S.T., Lee, J., Choi, H. &amp; Moin, P.P 2010 <br /> :A dynamic eddy-viscosity model based on the invariants of the rate-of-strain. ''In Proceedings of the summer program'', pp. 183&amp;nbsp;&amp;ndash;&amp;nbsp;192. Center for Turbulence Research, Stanford University Stanford.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2003&quot;&gt;&lt;/div&gt;<br /> Verstappen, R.W.C.P. &amp; Veldman, A.E.P. 2003 <br /> :Symmetry-preserving discretization of turbulent flow. ''Journal of Computational Physics'' '''187''', 343&amp;nbsp;&amp;ndash;&amp;nbsp;368.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Weibel1963&quot;&gt;&lt;/div&gt;<br /> Weibel, E. R. 1963 <br /> :Morphometry of the human lung. Springer-Verlag, Berlin.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Wu2014&quot;&gt;&lt;/div&gt;<br /> Wu, Dan, Tawhai, Merryn H, Hoffman, Eric A &amp; Lin, Ching-Long 2014 <br /> :A numerical study of heat and water vapor transfer in MDCT-based human airway models. ''Ann Biomed Eng'' '''42''' (10), 2117&amp;nbsp;&amp;ndash;&amp;nbsp;2131.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Xi2018&quot;&gt;&lt;/div&gt;<br /> Xi, Jinxiang, April Si, Xiuhua, Dong, Haibo &amp; Zhong, Hualiang 2018 <br /> :Effects of glottis motion on airflow and energy expenditure in a human upper airway model. ''European Journal of Mechanics - B/Fluids'' '''72''', 23&amp;nbsp;&amp;ndash;&amp;nbsp;37.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Yin2010&quot;&gt;&lt;/div&gt;<br /> Yin, Youbing, Choi, Jiwoong, Hoffman, Eric A., Tawhai, Merryn H. &amp; Lin, Ching-Long 2010 <br /> :Simulation of pulmonary air flow with a subject-specific boundary condition. ''Journal of Biomechanics'' '''43''' (11), 2159&amp;nbsp;&amp;ndash;&amp;nbsp;2163.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zang1993&quot;&gt;&lt;/div&gt;<br /> Zang, Yan, Street, Robert L. &amp; Koseff, Jeffrey R. 1993 <br /> :A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. ''Physics of Fluids A: Fluid Dynamics'' '''5''' (12), 3186&amp;nbsp;&amp;ndash;&amp;nbsp;3196.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zhang2004&quot;&gt;&lt;/div&gt;<br /> Zhang, Z. &amp; Kleinstreuer, C. 2004 <br /> :Airflow structures and nano-particle deposition in a human upper airway model. ''Journal of Computational Physics'' '''198''', 178&amp;nbsp;&amp;ndash;&amp;nbsp;210.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zhou2005&quot;&gt;&lt;/div&gt;<br /> Zhou, Yue &amp; Cheng, Yung-Sung 2005 <br /> :Particle deposition in a cast of human tracheobronchial airways. ''Aerosol Science and Technology'' '''39''' (6), 492&amp;nbsp;&amp;ndash;&amp;nbsp;500.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Best_Practice_Advice_AC7-02&diff=38842 Best Practice Advice AC7-02 2020-06-15T12:17:48Z <p>Kassinos: /* Turbulence Models */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> =Best Practice Advice=<br /> ==Key Fluid Physics==<br /> &lt;!--<br /> Briefly describe the key fluid physics/flow regimes which exert an influence on the DOAPs. Ideally this should draw together into a coherent picture the associated UFR descriptions together with any important interactions which are AC specific. Mention the UFRs associated with this AC that you have considered in drafting your best practice advice. ''Access the Knowledge Base to find the UFRs associated with your AC''.<br /> --&gt;<br /> <br /> In the present AC, experiments and simulations were conducted at a flowrate of 60 L/min through an upper airway geometry. <br /> At this flow conditions, the Reynolds number for air in the trachea is 4920, which is well within the turbulent regime. <br /> Geometric effects, such as the bent in the oropharyngeal region and the constriction at the laryngeal glottis (just upstream of the trachea, see [[#Figure25|figure 25]]<br /> ) enhance turbulence levels as the air moves from the inlet to the region of the trachea. <br /> Turbulent kinetic energy levels reach a peak in the shear layer formed between the high speed laryngeal jet and the surrounding (low speed) air (see [[#Figure25|figure 25]]). <br /> The characteristics of the laryngeal jet formation bear a resemblance to the flow through a constricted pipe, which can be classified as a free shear flow where the wall serves to confine the spreading of the jet rather than producing turbulence ([[Lib:Best Practice Advice AC7-02#Tawhai2011|Tawhai &amp; Lin, 2011]]). <br /> High turbulence levels persist in the region of the first bifurcation (stations H1-H2 &amp; J1-J2 in [[Lib:CFD Simulations AC7-02#Figure12|figure 12]](b)).<br /> <br /> ==Application Uncertainties==<br /> &lt;!--<br /> List any uncertainties which make a high fidelity CFD model difficult to assemble. Typical examples might include:<br /> *a gas leakage between two components which is difficult to resolve on practical meshes, andeven if it is resolved, the leakage flow conditions may not be known.<br /> *flow conditions at inlet to the AC (or indeed other boundaries) which may be complex and not precisely known.<br /> *fine details of the geometry are imprecise.<br /> Briefly discuss the sensitivity of the DOAP predictions to these uncertainties and their impact on the BPA. In particular, can clear, unequivocal BPA be given or is it necessary to introduce appropriate caveats.<br /> --&gt;<br /> <br /> The differences between measurements and simulations can result from several uncertainties involved in the tests. <br /> A first source of uncertainty are the inlet conditions, which are not perfectly matched between the measurements and the computations. <br /> In the experiments, the lung model was placed in an open liquid tank with a piston diaphragm pump attached to a linear actuator to achieve a quasi-stationary inspiratory flow. <br /> The stroke of the piston followed a cyclic triangular function with an adjustable falling constant slope and thus constant velocity to match different flow rates during inspiration. <br /> The measured mean velocity at the inlet of the model, shown in [[Lib:Test Data AC7-02#Figure7|figure 7]], is asymmetric, probably due to the action of the piston diaphragm pump. <br /> In the computations, instead of reproducing the measured inlet conditions, either uniform or turbulent inlet velocity profiles were prescribed. <br /> Due to a leakage flow between the upper and lower part of the model in the experiments, the achieved flowrate within the main bifurcation and bronchi region was about 10% lower than in the upper part of the model. <br /> As a result, a maximum flowrate of &lt;math&gt; Q_{w/g} &lt;/math&gt; = 28.56 L/min could be achieved in the measurements. <br /> This value is slightly lower than &lt;math&gt; Q_{w/g} &lt;/math&gt; = 31.75 L/min, which is the target value for an equivalent air flowrate of 60L/min through the model. <br /> Although the flow is well within the turbulent regime, the theoretical maximum Reynolds number decreases from 4921 to 4286.<br /> <br /> ==Computational Domain and Boundary Conditions==<br /> <br /> The geometry of the extrathoracic airways must be included because turbulence is generated in this region that propagates in the first airway generations. <br /> Concerning the boundary conditions, the inlet velocity profile is important and thus realistic inlet conditions should be used. <br /> At the outlets, it is important to apply correct pressures such that the ventilation of the airway tree is physiologically realistic ([[Lib:Best Practice Advice AC7-02#Yin2010|Yin et al., 2010]]). <br /> In the present AC, in order to simplify the experimental setup and be able to perform the flow measurements, uniform pressures were prescribed at all outlets.<br /> <br /> Concerning the inlet conditions for the turbulent variables in RANS calculations, the application of a turbulence intensity of 5% for the k-ω SST model at the extended inlet (10xDinlet) yielded higher turbulent kinetic energy values close to the inlet of the model compared to the mapped inlet condition (Inlet 1). <br /> The k-ε models were found to provide overall higher turbulence levels than the k-ω SST model, especially at the near-wall regions.<br /> <br /> ==Discretisation and Grid Resolution==<br /> <br /> Since it is not possible to generate a structured hexahedral grid for the present geometry due to its complexity, a higher refinement ratio should be applied to avoid numerical diffusion. <br /> In addition to that, layers of prismatic elements should be added near the wall boundaries for a better prediction of this region, not only with regard to flow properties itself, but the flow conditions seen by the particles, i.e. mean velocity and turbulence properties. <br /> Airflow through the glottis constriction at the larynx, illustrated in [[#Figure25|figure 25]], bears a resemblance to flow through a constricted pipe. <br /> This type of flow may be classified as a free shear flow in which the wall serves to confine the spreading of the jet rather than producing turbulence. <br /> In this case, turbulence is most active at the interface between two free streams (high and low speed) and along the jet core and therefore, fine mesh resolution should be placed accordingly to capture strong turbulence activities. <br /> Use of a strict y+ = 1 condition in the generation of the near-wall mesh but extremely coarse mesh in the core region of the airway model is conceptually wrong ([[Lib:Best Practice Advice AC7-02#Tawhai2011|Tawhai &amp; Lin, 2011]]).<br /> Recommended values for the parameters involved in mesh generation (initial cell height, average expansion ratio, number of near-wall prism layers, average cell volume in the domain, number of computational cells etc.) can be found in Tables 4,5 (LES) and 6 (RANS).<br /> <br /> LES were found to give similar results independent of the discretisation method used (Finite Volume or Finite Element).<br /> <br /> &lt;div id=&quot;Figure25&quot;&gt;&lt;/div&gt;<br /> [[File:jet.png|center|thumb|500px|'''Figure 25''': Laryngeal jet formed at the glottis constriction. (a) Isosurfaces of &lt;math&gt; |&lt;u&gt;| &lt;/math&gt; = 6.7m/s ; (b) Mean velocity contours; (c) Turbulent kinetic energy contours.]]<br /> <br /> ==Turbulence Models==<br /> <br /> <br /> The ability of four LES subgrid-scale models was assessed by comparing their predictions to the PIV data. <br /> These models were the WALE, the Smagorinsky, the variational multiscale (VMS) WALE and the QR eddy-viscosity model from Verstappen. <br /> All the models were found to provide very similar results. <br /> It is concluded that the influence of the subgrid scales on the airflow in the human upper airways is small, and the choice of subgrid-scale turbulence model is not as important as in cases with higher Reynolds numbers.<br /> <br /> In general the RANS prediction for the velocity magnitude follow trends similar to those of the measured data. <br /> RANS however show poor performance at locations with shear layers, recirculation and flow separation. <br /> Moreover, RANS simulations consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This is a direct consequence of discarding many of the elements of the underlying turbulence physics when solving only for the mean flow (see discussion in section [[Lib:Evaluation AC7-02|Evaluation]). <br /> The k-ε models provides overall higher turbulence levels than the k-ω SST turbulence model.<br /> <br /> Results presented in AC7-01 showed that the deposition fractions obtained with RANS models achieved a quite good agreement with LES and measurements. <br /> This might seem surprising given the tendency of RANS to underpredict turbulence intensities at several stations downstream of the glottis constriction. <br /> This apparent paradox probably relates to the fact that the dominant deposition mechanism in the upper airways is inertial impaction. <br /> While inertial impaction tends to be dominated by mean flow effects, turbulent dispersion still plays an important role, especially in regions where there is significant large-scale anisotropy in the turbulence. <br /> Hence, it is known that using only the time-averaged air velocity field for deposition studies (this is the case in RANS) leads to deposition overpredictions ([[Lib:Best Practice Advice AC7-02#Matida2004|Matida et al., 2004]]).<br /> Still, RANS deposition predictions can be improved when they are used together with a turbulent dispersion model, as was done in AC7-01. <br /> When using a turbulent dispersion model, individual particles are allowed to interact successively with discrete eddies, each eddy having length, velocity and lifetime characteristic scales obtained from the primary flow calculation results. <br /> It is therefore important to correctly select the turbulent dispersion model and it’s parameters for accurate deposition predictions. <br /> LES on the other hand, since they resolve the large scale eddies, do not need a model to account for these in particle-laden flows. <br /> The study of [[#armenio1999|Armenio et al. (1999)]] has shown that the motion of inertial particles in low to moderate Reynolds number flows, is not influenced from the unresolved sub-grid scales in LES.<br /> <br /> In conclusion, LES are more capable than RANS in predictions of airflow in the human upper airways, since they can account better for the physics of the turbulent flow without the need to adjust model parameters.<br /> <br /> ==Recommendations for Future Work==<br /> <br /> The present application challenge focuses on the airflow that develops in the human upper airways. <br /> The airway geometry has been considered rigid. <br /> In reality, the lung expands and contracts and the airway walls deform during inhalation and exhalation ([[Lib:Best Practice Advice AC7-02#Mead-Hunter2013|Mead-Hunter et al., 2013]]). <br /> In addition, there is periodic movement of the glottal aperture during tidal breathing that regulates the respiratory airflow dynamics ([[Lib:Best Practice Advice AC7-02#Xi2018|Xi et al., 2018]]).<br /> It is therefore important to assess the effect of wall deformation on the developed flow features inside the airways.<br /> <br /> The higher temperature and humidity of the human body compared to the inhaled ambient air results in heat and water vapor transfer as the air is transported in the airways <br /> ([[Lib:Best Practice Advice AC7-02#Wu2014|Wu et al., 2014]]).<br /> During inhalation, air is heated and humidified by the airway walls until it reaches the body temperature and approximately 100% relative humidity. <br /> Such effects can play a role in air and inhaled aerosols transport inside the airways and should be further examined in the future studies.<br /> <br /> ==Acknowledgements==<br /> <br /> The present application challenge is based upon work from COST Action MP1404 SimInhale &quot;Simulation and pharmaceutical technologies for advanced patient-tailored inhaled medicines&quot;, supported by COST (European Cooperation in Science and Technology - www.cost.eu).<br /> <br /> ==References==<br /> &lt;div id=&quot;Adrian2011&quot;&gt;&lt;/div&gt;<br /> Adrian, R.J. and Westerweel, J. 2011 <br /> :Particle Image Velocimetry. ''Cambridge University Press, Cambridge.''<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;armenio1999&quot;&gt;&lt;/div&gt;<br /> Armenio, V., Piomelli, U. &amp; Fiorotto, V. 1999<br /> :Effect of the subgrid scales on particle motion. ''Physics of Fluids'' '''11''' (10), 3030&amp;nbsp;&amp;ndash;&amp;nbsp;3042.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Charnyi2017&quot;&gt;&lt;/div&gt;<br /> Charnyi, Sergey, Heister, Timo, Olshanskii, Maxim A. &amp; Rebholz, Leo G. 2017 <br /> :On conservation laws of Navier-Stokes Galerkin discretizations. ''Journal of Computational Physics'' '''337''', 289&amp;nbsp;&amp;ndash;&amp;nbsp;308.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Codina2001&quot;&gt;&lt;/div&gt;<br /> Codina, R. 2001 <br /> :Pressure stability in fractional step finite element methods for incompressible flows. ''Journal of Computational Physics'' '''170''', 112&amp;nbsp;&amp;ndash;&amp;nbsp;140.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Hughes2000&quot;&gt;&lt;/div&gt;<br /> Hughes, Thomas J.R., Mazzei, Luca &amp; Jansen, Kenneth E. 2000 <br /> :Large eddy simulation and the variational multiscale method. ''Computing and Visualization in Science'' '''3''' (1), 47&amp;nbsp;&amp;ndash;&amp;nbsp;59.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Janke2019&quot;&gt;&lt;/div&gt;<br /> Janke, T., Koullapis, P., Kassinos, S.C. &amp; Bauer, K. 2019 <br /> :PIV measurements of the Siminhale benchmark case. ''European Journal of Pharmaceutical Sciences'' '''133''', 183&amp;nbsp;&amp;ndash;&amp;nbsp;189.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Jasak1996&quot;&gt;&lt;/div&gt;<br /> Jasak, H. 1996 <br /> :Error analysis and estimation for the finite volume method with applications to fluid flows. ''PhD thesis, Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, UK''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Jofre2014&quot;&gt;&lt;/div&gt;<br /> Jofre, L., Lehmkuhl, O., Ventosa, J., Trias, F.X. &amp; Oliva, A. 2014 <br /> :Conservation properties of unstructured finite-volume mesh schemes for the Navier-Stokes equations. ''Numerical Heat Transfer, Part B: Fundamentals'' '''54''' (1), 53&amp;nbsp;&amp;ndash;&amp;nbsp;79.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Kim1985&quot;&gt;&lt;/div&gt;<br /> Kim, J. &amp; Moin, P. 1985 <br /> :Application of a fractional-step method to incompressible Navier-Stokes equations. ''Journal of Computational Physics'' '''59''' (2), 308&amp;nbsp;&amp;ndash;&amp;nbsp;323.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Koullapis2018&quot;&gt;&lt;/div&gt;<br /> Koullapis, P., Kassinos, S. C., Muela, J., Perez-Segarra, C., Rigola, J., Lehmkuhl, O., Cui, Y., Sommerfeld, M., Elcner, J., Jicha, M., Saveljic, I., Filipovic, N., Lizal, F. &amp; Nicolaou, L. 2018 <br /> :Regional aerosol deposition in the human airways: The SimInhale benchmark case and a critical assessment of ''in silico'' methods. ''European Journal of Pharmaceutical Sciences'' '''113''', 77&amp;nbsp;&amp;ndash;&amp;nbsp;94.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Koullapis2016&quot;&gt;&lt;/div&gt;<br /> Koullapis, P. G., Kassinos, S.C., Bivolarova, M. P. &amp; Melikov, A. K. 2016 <br /> :Particle deposition in a realistic geometry of the human conducting airways: Effects of inlet velocity profile, inhalation flowrate and electrostatic charge. ''Journal of Biomechanics'' '''49''', 2201&amp;nbsp;&amp;ndash;&amp;nbsp;2212.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lehmkuhl2019&quot;&gt;&lt;/div&gt;<br /> Lehmkuhl, O., Houzeaux, G., Owen, H., Chrysokentis, G. &amp; Rodriguez, I. 2019 <br /> :A low-dissipation finite element scheme for scale resolving simulations of turbulent flows. ''Journal of Computational Physics'' '''390''', 51&amp;nbsp;&amp;ndash;&amp;nbsp;65.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lehmkuhl2007&quot;&gt;&lt;/div&gt;<br /> Lehmkuhl, O., Perez Segarra, C.D., Borrell, R., Soria, M. &amp; Oliva, A. 2007 <br /> :Termofluids: A new Parallel unstructured CFD code for the simulation of turbulent industrial problems on low cost PC cluster. ''Proceedings of the Parallel CFD Conference'' pp. 1&amp;nbsp;&amp;ndash;&amp;nbsp;8.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lilly1992&quot;&gt;&lt;/div&gt;<br /> Lilly, D. K. 1992 <br /> :A proposed modification of the Germano subgrid-scale closure method. ''Physics of Fluids A'' '''4''' (3), 633&amp;nbsp;&amp;ndash;&amp;nbsp;635.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lin2007&quot;&gt;&lt;/div&gt;<br /> Lin, C.-L., Tawhai, M.H., Mclennan, G. &amp; Hoffman, E.A. 2007 <br /> :Characteristics of the turbulent laryngeal jet and its effect on airflow in the human intra-thoracic airways. ''Respiratory Physiology and Neurobiology'' '''157''', 295&amp;nbsp;&amp;ndash;&amp;nbsp;309.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lizal2015&quot;&gt;&lt;/div&gt;<br /> Lizal, Frantisek, Belka, Miloslav, Adam, Jan, Jedelsky, Jan &amp; Jicha, Miroslav 2015 <br /> :A method for ''in vitro'' regional aerosol deposition measurement in a model of the human tracheobronchial tree by the positron emission tomography. ''Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine'' '''229''' (10), 750&amp;nbsp;&amp;ndash;&amp;nbsp;757.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Lizal2012&quot;&gt;&lt;/div&gt;<br /> Lizal, Frantisek, Elcner, Jakub, Hopke, Philip K, Jedelsky, Jan &amp; Jicha, Miroslav 2012 <br /> :Development of a realistic human airway model. ''Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine'' '''226''' (3), 197&amp;nbsp;&amp;ndash;&amp;nbsp;207.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Matida2004&quot;&gt;&lt;/div&gt;<br /> Matida, E. A., Finlay, W. H., Lange, C. F. &amp; Grgic, B. 2004 <br /> :Improved numerical simulation of aerosol deposition in na idealized mouth-throat. ''Aerosol Science'' '''35''', 1&amp;nbsp;&amp;ndash;&amp;nbsp;19.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Mead-Hunter2013&quot;&gt;&lt;/div&gt;<br /> Mead-Hunter, Ryan, King, Andrew J.C., Larcombe, Alexander N. &amp; Mullins, Benjamin J. 2013 <br /> :The influence of moving walls on respiratory aerosol deposition modelling. ''Journal of Aerosol Science'' '''64''', 48&amp;nbsp;&amp;ndash;&amp;nbsp;59.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Muela2019&quot;&gt;&lt;/div&gt;<br /> Muela, J., Rigoala, J., Oliet, C., Perez-Segarra, C.D. &amp; Oliva, A. 2019 <br /> :Assessment of numerical aspects using LES in particle separation devices. ''In 8th European Conference for Aeronautics and Aerospace Sciences (EUCASS)'', p. 678.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Nicoud1999&quot;&gt;&lt;/div&gt;<br /> Nicoud, Franck &amp; Ducros, Frederic 1999 <br /> :Subgrid-scale stress modelling based on the square of the velocity gradient tensor. ''Flow, turbulence and Combustion'' '''62''' (3), 183&amp;nbsp;&amp;ndash;&amp;nbsp;200.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;OpenFOAM2013a&quot;&gt;&lt;/div&gt;<br /> OpenFOAM Foundation 2013a <br /> :OpenFOAM Programmer’s Guide, version 2.2.1 edn. London, UK.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;OpenFOAM2013b&quot;&gt;&lt;/div&gt;<br /> OpenFOAM Foundation 2013b<br /> :OpenFOAM User Guide, version 2.2.1 edn. London, UK.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Radhakrishnan2009&quot;&gt;&lt;/div&gt;<br /> Radhakrishnan, H. &amp; Kassinos, S. 2009 <br /> :CFD modeling of turbulent flow and particle deposition in human lungs. ''31st Annual International Conference of the IEEE EMBS'', Mineapolis, Minnesota, USA pp. 2867&amp;nbsp;&amp;ndash;&amp;nbsp;2870.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Sagaut2006&quot;&gt;&lt;/div&gt;<br /> Sagaut, P. 2006 <br /> :Large Eddy Simulation for Incompressible Flows: An Introduction. ''Springer''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Schmidt2004&quot;&gt;&lt;/div&gt;<br /> Schmidt, Andreas, Zidowitz, Stephan, Kriete, Andres, Denhard, Thorsten, Krass, Stefan &amp; Peitgen, Heinz-Otto 2004 <br /> :A digital reference model of the human bronchial tree. ''Computerized Medical Imaging and Graphics'' '''28''' (4), 203&amp;nbsp;&amp;ndash;&amp;nbsp;211.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Sciacchitano2016&quot;&gt;&lt;/div&gt;<br /> Sciacchitano, A. and Wieneke, B. 2016<br /> :PIV uncertainty propagation. ''Meas. Sci. Technol.'' '''27''' (084006), 20pp.<br /> &lt;br/&gt;<br /> &lt;div id=&quot;Smagorinsky1963&quot;&gt;&lt;/div&gt;<br /> Smagorinsky, Joseph 1963 <br /> :General circulation experiments with the primitive equations: I. The basic experiment. ''Monthly weather review'' '''91''' (3), 99&amp;nbsp;&amp;ndash;&amp;nbsp;164.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Tabor2004&quot;&gt;&lt;/div&gt;<br /> Tabor, G. R., Baba-Ahmadi, M. H., de Villiers, E. &amp; Weller, H. G. 2004 <br /> :Construction of inlet conditions for LES of turbulent channel flow. ''European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS)''.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Tawhai2011&quot;&gt;&lt;/div&gt;<br /> Tawhai, M.H. &amp; Lin, C.-L. 2011 <br /> :Airway gas flow. ''Comprehensive Physiology'' '''1''', 1135&amp;nbsp;&amp;ndash;&amp;nbsp;1157. <br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Trias2011&quot;&gt;&lt;/div&gt;<br /> Trias, F. X. &amp; Lehmkuhl, O. 2011<br /> :A self-adaptive strategy for the time integration of Navier-Stokes equations. ''Numerical Heat Transfer. Part B'' '''60''' (2), 116&amp;nbsp;&amp;ndash;&amp;nbsp;134.<br /> &lt;br/&gt;<br /> &lt;div id=&quot;Vazquez2016&quot;&gt;&lt;/div&gt;<br /> Vazquez, A. M., Houzeaux, G., Koric, S., Artigues, A., Aguado-Sierra, J., Aris, R., Mira, D., Calmet, H., Cucchietti, F., Owen, H., Casoni, E., Taha, A., Burness, E. D., Cela, J. M. &amp; Valero, M. 2016 <br /> :Alya: Multiphysics engineering simulation towards exascale. ''Journal Computational Sciences'' '''14''', 15&amp;nbsp;&amp;ndash;&amp;nbsp;27.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2011&quot;&gt;&lt;/div&gt;<br /> Verstappen, Roel 2011 <br /> :When does eddy viscosity damp subfilter scales sufficiently? ''Journal of Scientific Computing'' '''49''' (1), 94.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2010&quot;&gt;&lt;/div&gt;<br /> Verstappen, R.W.C.P., Bose, S.T., Lee, J., Choi, H. &amp; Moin, P.P 2010 <br /> :A dynamic eddy-viscosity model based on the invariants of the rate-of-strain. ''In Proceedings of the summer program'', pp. 183&amp;nbsp;&amp;ndash;&amp;nbsp;192. Center for Turbulence Research, Stanford University Stanford.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Verstappen2003&quot;&gt;&lt;/div&gt;<br /> Verstappen, R.W.C.P. &amp; Veldman, A.E.P. 2003 <br /> :Symmetry-preserving discretization of turbulent flow. ''Journal of Computational Physics'' '''187''', 343&amp;nbsp;&amp;ndash;&amp;nbsp;368.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Weibel1963&quot;&gt;&lt;/div&gt;<br /> Weibel, E. R. 1963 <br /> :Morphometry of the human lung. Springer-Verlag, Berlin.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Wu2014&quot;&gt;&lt;/div&gt;<br /> Wu, Dan, Tawhai, Merryn H, Hoffman, Eric A &amp; Lin, Ching-Long 2014 <br /> :A numerical study of heat and water vapor transfer in MDCT-based human airway models. ''Ann Biomed Eng'' '''42''' (10), 2117&amp;nbsp;&amp;ndash;&amp;nbsp;2131.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Xi2018&quot;&gt;&lt;/div&gt;<br /> Xi, Jinxiang, April Si, Xiuhua, Dong, Haibo &amp; Zhong, Hualiang 2018 <br /> :Effects of glottis motion on airflow and energy expenditure in a human upper airway model. ''European Journal of Mechanics - B/Fluids'' '''72''', 23&amp;nbsp;&amp;ndash;&amp;nbsp;37.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Yin2010&quot;&gt;&lt;/div&gt;<br /> Yin, Youbing, Choi, Jiwoong, Hoffman, Eric A., Tawhai, Merryn H. &amp; Lin, Ching-Long 2010 <br /> :Simulation of pulmonary air flow with a subject-specific boundary condition. ''Journal of Biomechanics'' '''43''' (11), 2159&amp;nbsp;&amp;ndash;&amp;nbsp;2163.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zang1993&quot;&gt;&lt;/div&gt;<br /> Zang, Yan, Street, Robert L. &amp; Koseff, Jeffrey R. 1993 <br /> :A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. ''Physics of Fluids A: Fluid Dynamics'' '''5''' (12), 3186&amp;nbsp;&amp;ndash;&amp;nbsp;3196.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zhang2004&quot;&gt;&lt;/div&gt;<br /> Zhang, Z. &amp; Kleinstreuer, C. 2004 <br /> :Airflow structures and nano-particle deposition in a human upper airway model. ''Journal of Computational Physics'' '''198''', 178&amp;nbsp;&amp;ndash;&amp;nbsp;210.<br /> &lt;br/&gt;<br /> <br /> &lt;div id=&quot;Zhou2005&quot;&gt;&lt;/div&gt;<br /> Zhou, Yue &amp; Cheng, Yung-Sung 2005 <br /> :Particle deposition in a cast of human tracheobronchial airways. ''Aerosol Science and Technology'' '''39''' (6), 492&amp;nbsp;&amp;ndash;&amp;nbsp;500.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38841 Evaluation AC7-02 2020-06-15T12:11:44Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section <br /> [[Lib:CFD Simulations AC7-02#RANS Simulations#Numerical Accuracy|Numerical Accuracy]]). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38840 Evaluation AC7-02 2020-06-15T12:09:50Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section [[Lib:CFD Simulations AC7-02# RANS Simulations#Numerical Accuracy]]). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=Evaluation_AC7-02&diff=38839 Evaluation AC7-02 2020-06-15T12:07:56Z <p>Kassinos: /* Evaluation */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==Evaluation==<br /> <br /> [[#Figure23|Figure 23]] shows comparisons of normalised in-plane mean velocity contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES1-3 and RANS computation using the RNG k-ε model. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth-throat and trachea, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_x^2+u_y^2} \qquad (8).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> |u| = \sqrt{u_y^2+u_z^2} \qquad (9).<br /> &lt;/math&gt;<br /> <br /> The velocity to normalise the simulation results is the bulk speed of air in the trachea at a flowrate of 60L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> Deviations from the PIV-measured data are evident in the calculations near the inlet of the model, at station B1-B2. <br /> These are more pronounced in LES3, in which a uniform velocity profile was prescribed at the inlet, and persist further downstream at station C1-C2. <br /> The differences near the inlet (station B1- B2) between the measurements and the results from the rest of the computations (LES1&amp;2 and RANS) are probably associated to the differences in the inlet velocity conditions between the experiment and the simulations. <br /> Notable deviations are evident in RANS predicted velocity profile at stations D, H and J (see discussion in section [[Lib:CFD Simulations AC7-02# RANS#Numerical Accuracy]]). <br /> At most of the stations, LES predictions for the mean velocity are in reasonable accordance independently of the method used.<br /> <br /> [[#Figure24|Figure 24]] shows comparisons of normalised turbulent kinetic energy contours and profiles at the locations of the PIV measurement planes IV-VI, between the measurements, LES and RANS. <br /> In PIV, only the in-plane velocity components were measured. <br /> Therefore, at planes IV &amp; V (mouth and trachea), the in-plane TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_x' u_x'&gt; + &lt;u_y' u_y'&gt; ) \qquad (10).<br /> &lt;/math&gt;<br /> <br /> <br /> At planes VI (main bifurcation and bronchi, see [[Lib:Test Data AC7-02#Figure6|figure 6]](b)), the in-plane mean velocity is calculated as:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{4} (&lt;u_y' u_y'&gt; + &lt;u_z' u_z'&gt; ) \qquad (11).<br /> &lt;/math&gt;<br /> <br /> Compared to RANS, LES perform better in capturing the locations of TKE peaks as well as the TKE levels. <br /> Deviations between LES results and measurements are more evident at stations B and C. <br /> The differences between LES and the measurements can be attributed to factors such as the different turbulent sampling, inlet conditions and of course errors in both the numerical and experimental parts. <br /> RANS consistently underestimate turbulent kinetic energy levels compared to PIV and LES data. <br /> This can be explained by the fact that in RANS, the turbulence model must represent a very wide range of scales. <br /> In contrast, in LES only the smallest scales are modeled. <br /> While the small scales depend only on viscosity, the large ones are affected strongly on boundary conditions. <br /> As a result, RANS cannot capture the effect of the large scales of turbulence as good as the LES and tend to underpredict turbulent kinetic energy levels.<br /> <br /> In summary, LES show better agreement to experimental data than RANS in both the mean and turbulent flowfields. <br /> On the other hand, although RANS predictions for the mean flow are in fair agreement to the measured values, RANS systematically under predict TKE values.<br /> <br /> &lt;div id=&quot;Figure23&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure23.jpeg|center|thumb|500px|'''Figure 23''': Comparison of normalised mean velocity contours and profiles at selected stations - shown in (a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;div id=&quot;Figure24&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure24.jpeg|center|thumb|500px|'''Figure 24''': Comparison of normalised turbulent kinetic energy contours and profiles at selected stations - shown in fig. 22(a) - between PIV, LES and RANS data.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38838 CFD Simulations AC7-02 2020-06-15T12:05:47Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[#Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38837 CFD Simulations AC7-02 2020-06-15T12:03:31Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section [[Numerical Accuracy|Numerical Accuracy]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38836 CFD Simulations AC7-02 2020-06-15T11:43:00Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation]], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38835 CFD Simulations AC7-02 2020-06-15T11:41:14Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[#Lib:Evaluation AC7-02|Evaluation], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38834 CFD Simulations AC7-02 2020-06-15T11:40:12Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section [[Lib:Evaluation AC7-02|Evaluation], the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38833 CFD Simulations AC7-02 2020-06-15T11:35:52Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[#Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[#Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38832 CFD Simulations AC7-02 2020-06-15T11:35:10Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section [[Numerical Accuracy|Numerical Accuracy]], a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section [[Comparison of different LES models|Comparison of different LES subgrid-scale models]] and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38831 CFD Simulations AC7-02 2020-06-15T11:32:29Z <p>Kassinos: /* Solution strategy and boundary conditions &amp;mdash; Airflow */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section [[Lib:Evaluation AC7-02|Evaluation]].<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38830 CFD Simulations AC7-02 2020-06-15T11:31:48Z <p>Kassinos: /* Comparison of different LES subgrid-scale models */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section [[Lib:Evaluation AC7-02|Evaluation]] and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38829 CFD Simulations AC7-02 2020-06-15T11:28:02Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38828 CFD Simulations AC7-02 2020-06-15T11:27:18Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38827 CFD Simulations AC7-02 2020-06-15T11:24:01Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38826 CFD Simulations AC7-02 2020-06-15T11:23:19Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Airflow in the human upper airways#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38825 CFD Simulations AC7-02 2020-06-15T11:22:28Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Lib:CFD Simulations AC7-02#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38824 CFD Simulations AC7-02 2020-06-15T11:21:07Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38823 CFD Simulations AC7-02 2020-06-15T11:20:38Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Large Eddy Simulations - Case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38822 CFD Simulations AC7-02 2020-06-15T11:19:39Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section [[#Large Eddy Simulations - case LES1#Computational domain and meshes|Computational domain and meshes]].<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38821 CFD Simulations AC7-02 2020-06-15T11:18:16Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38820 CFD Simulations AC7-02 2020-06-15T11:17:53Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[Large Eddy Simulations - case LES1#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38819 CFD Simulations AC7-02 2020-06-15T11:17:04Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[Lib:CFD Simulations AC7-02#Large Eddy Simulations - case LES1#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38818 CFD Simulations AC7-02 2020-06-15T11:14:18Z <p>Kassinos: /* Computational domain and meshes */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section [[Lib:CFD simulations AC7-02#Large Eddy Simulations - case LES1#Numerical accuracy|Numerical accuracy]]. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38817 CFD Simulations AC7-02 2020-06-15T11:04:04Z <p>Kassinos: /* Overview of CFD Simulations */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section 3.2.3. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38816 CFD Simulations AC7-02 2020-06-15T11:00:27Z <p>Kassinos: /* Overview of CFD Simulations */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section 2.4. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section 3.2.3. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos https://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC7-02&diff=38815 CFD Simulations AC7-02 2020-06-15T11:00:10Z <p>Kassinos: /* Overview of CFD Simulations */</p> <hr /> <div>{{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> __TOC__<br /> =Airflow in the human upper airways=<br /> '''Application Challenge AC7-02'''&amp;nbsp;&amp;nbsp;&amp;nbsp;© copyright ERCOFTAC 2020<br /> ==CFD Simulations==<br /> ==Overview of CFD Simulations==<br /> <br /> Three LES (LES1-3) and one RANS simulation were carried out in the benchmark geometry at an air flowrate of 60 L/min. <br /> This flowrate results in a Reynolds number of 4921 in the model’s trachea. <br /> This is slightly higher than the Reynolds number in the experiments, due to the reasons discussed in section [[Lib:Test Data AC7-02#Measurement errors|Measurement errors]]. <br /> The details of the numerical tests are given in the following paragraphs. <br /> In summary, the main differences between the simulations are:<br /> &lt;br/&gt;<br /> 1. '''Computational meshes''': LES1&amp;2 employ the same comp. meshes, whereas LES3 and RANS use different meshes.<br /> &lt;br/&gt;<br /> 2. '''Turbulence modeling''': LES1 uses the dynamic version of the Smagorinsky-Lilly subgrid scale model, whereas LES2&amp;3 employ the Wall-adapting eddy viscosity (WALE) SGS model. <br /> In RANS simulations, the k-ω SST, standard k-ε and RNG k-ε turbulence model have been tested.<br /> &lt;br/&gt;<br /> 3. '''Discretisation method''': LES1&amp;2 and RANS use the Finite Volume approach whereas LES3 employs the Finite Element Method.<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES1==<br /> ===Computational domain and meshes===<br /> The geometry used in the calculations is the same as the one used in the experiments developed by the group at Brno University of Technology (BUT). <br /> The computational domain, shown in [[#Figure8|figure 8]], has one inlet and ten different outlets, for which appropriate boundary conditions must be specified in the simulations.<br /> <br /> <br /> &lt;div id=&quot;Figure8&quot;&gt;&lt;/div&gt;<br /> [[File:RANS_geom.png|center|thumb|500px|'''Figure 8''': Computational domain viewed from different angles.]]<br /> <br /> <br /> The digital model of the physical geometry was used to generate a proper computational mesh in order to perform the simulations. <br /> For LES1, two meshes were generated to allow us to examine the sensitivity of the results to the mesh resolution.<br /> The coarser mesh includes 10 million computational cells and the finer approximately 50 million cells. <br /> In these meshes, the near-wall region was resolved with prismatic elements, while the core of the domain was meshed with tetrahedral elements. <br /> Cross-sectional views of these meshes at seven stations are shown in [[#Figure9|figure 9]]. <br /> A grid convergence analysis was carried out in order to determine the appropriate resolution for the simulations. <br /> This analysis is presented in section 3.2.3. <br /> [[#Table3|Table 3]] reports grid characteristics, such as the height of the wall-adjacent cells (&lt;math&gt; \Delta r_{min}&lt;/math&gt;), the number of prism layers near the walls, the average expansion ratio of the prism layers (&lt;math&gt; \lambda &lt;/math&gt;), the total number of computational cells, the average cell volume (V) and the average and maximum y+ values. <br /> The higher y+ values (above 1) are found near the glottis constriction and the bifurcation carinas, which are characterised by high wall shear stresses.<br /> <br /> &lt;div id=&quot;Figure9&quot;&gt;&lt;/div&gt;<br /> [[File:LES_meshes4.png|center|thumb|500px|'''Figure 9''': Cross-sectional views of the three generated meshes employed in LES1&amp;2 at seven stations (A-G).]]<br /> <br /> &lt;div id=&quot;Table3&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table3.png|center|thumb|500px|'''Table 3''': Characteristics of Meshes 1-3. &lt;math&gt; \Delta r_{min}&lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> LES were performed using the dynamic version of the Smagorinsky-Lilly subgrid scale model with localized filtering ([[Lib:Best Practice Advice AC7-02#Lilly1992|Lilly, 1992]]; [[Lib:Best Practice Advice AC7-02#Zang1993|Zang et al., 1993]]) in order to examine the unsteady flow in the realistic airway geometries. <br /> Previous studies have shown that this model performs well in transitional flows in the human airways ([[Lib:Best Practice Advice AC7-02#Radhakrishnan2009|Radhakrishnan &amp; Kassinos, 2009]]; [[Lib:Best Practice Advice AC7-02#Koullapis2016|Koullapis et al., 2016]]). <br /> The airflow is described by the filtered set of incompressible Navier-Stokes equations,<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_j}{\partial x_j} = 0 \qquad (2)<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{\partial \overline u_i}{\partial t} + \overline u_j \frac{\partial \overline u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \overline p}{\partial x_i} + \frac{\partial}{\partial x_j} \bigg[ (\nu + \nu_{sgs}) \frac{\partial \overline u_i}{\partial x_j} \bigg] \qquad (3),<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \overline u_i &lt;/math&gt;, p, &lt;math&gt; \rho = 1.2kg/m^3 &lt;/math&gt;, &lt;math&gt; \nu=1.59 \times 10^{-5} m^2/s &lt;/math&gt;<br /> and &lt;math&gt; \nu_{sgs} &lt;/math&gt; are the filtered velocity component in the i-direction, the filtered pressure, the density and kinematic viscosity of air, and the subgrid-scale (SGS) turbulent eddy viscosity, respectively. <br /> The overbar denotes resolved quantities.<br /> <br /> The governing equations are discretized using a finite volume method and solved using OpenFOAM, an open-source CFD code ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013a|OpenFOAM Foundation, 2013a,b]]).<br /> In this framework, unstructured boundary fitted meshes are used with a collocated cell-centred variable arrangement. <br /> The finite volume method in OpenFOAM is in general 2nd-order accurate in space, depending on the convection differencing scheme (CDS) used. <br /> Whenever possible (usually in the cases with lower Reynolds numbers), the 2nd-order linear CDS is used. <br /> The order of accuracy had to be decreased in some cases in order to stabilize the simulation. <br /> In these cases, the clippedLinear scheme was used, which provides a good compromise between the accuracy of the (2nd order) linear scheme and the stability of the (1st order) mid-point scheme ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> The temporal derivative is discretized using backward differencing, which is is also second order accurate in time and implicit. <br /> To ensure numerical stability the time step used is &lt;math&gt; 2.5 \times 10^{-6}s &lt;/math&gt; at inhalation flowrate of 60 L/min. <br /> The non-linearity in the momentum equation is lagged in OpenFOAM (linearization of equation before discretisation). <br /> The system of partial differential equations is treated in a segregated way, with each equation being solved separately with explicit coupling between the results. <br /> In turbulent flow applications, where the time step is kept small enough to capture the smaller turbulent time scales, the pressure-implicit split-operator algorithm, or PISO-algorithm, is used.<br /> <br /> At the inhalation flowrate of 60 L/min, the Reynolds number at the inlet, based on the inflow bulk velocity and inlet tube diameter, is <br /> &lt;math&gt; Re_{in} = \frac{u_{in} D_{in}}{\nu} = 3745&lt;/math&gt;, which lies in the turbulent regime. <br /> In order to generate turbulent inflow conditions, a mapped inlet, or recycling, boundary condition was used ([[Lib:Best Practice Advice AC7-02#Tabor2004|Tabor et al., 2004]]).<br /> To apply this boundary condition, the pipe at the inlet was extended by a length equal to ten times its diameter, as shown in [[#Figure10|figure 10]]. <br /> The pipe section was initially fed with an instantaneous turbulent velocity field generated in a precursor pipe flow LES. <br /> During the simulation of the airway geometry, the velocity field from the mid-plane of the pipe domain was mapped to the extended pipe’s inlet boundary ([[#Figure10|figure 10]]<br /> ). Scaling of the velocities was applied to enforce the specified bulk flow rate. <br /> In this manner, turbulent flow is sustained in the extended pipe section, and a turbulent velocity profile enters the mouth inlet. <br /> To match the experimental conditions, uniform pressure is prescribed in the simulations at the 10 terminal outlets. <br /> A no-slip velocity condition is imposed on the airway walls.<br /> <br /> &lt;div id=&quot;Figure10&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure10.png|center|thumb|500px|'''Figure 10''': Explanatory figure for the mapped inlet, or recycling, boundary condition applied at the inlet of the model.]]<br /> <br /> ===Numerical accuracy===<br /> <br /> The sensitivity of the calculations to the mesh resolution was tested and the results are presented in this section. <br /> [[#Figure11|Figure 11]] displays contours and profiles of the mean velocity magnitude at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. <br /> Good agreement is observed for the mean velocities between the coarse and fine meshes. <br /> Slight deviations are found at station D1-D2, located just downstream of the glottis. <br /> Airflow recirculation is evident at this station that is not well captured on the coarse mesh.<br /> <br /> [[#Figure12|Figure 12]] displays contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat/trachea, the main bifurcation and the left/right bronchi for the two different meshes examined. TKE is calculated as:<br /> <br /> &lt;math&gt;<br /> TKE = \frac{1}{2} &lt;u_i' u_i'&gt; \qquad (4).<br /> &lt;/math&gt;<br /> <br /> <br /> Higher levels of TKE are recorded on the finer mesh. These are mainly generated in the shear layers. As observed from the 2D profiles, TKE peaks are underpredicted on the coarse mesh. <br /> In conclusion, although the coarse mesh yields results that are in reasonable agreement with the finer mesh predictions, in the following comparisons between CFD and PIV measurements, the finer LES predictions are considered.<br /> <br /> &lt;div id=&quot;Figure11&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure11.jpeg|center|thumb|500px|'''Figure 11''': LES1: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure12&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure12.jpeg|center|thumb|500px|'''Figure 12''': LES1: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES2==<br /> ===Computational domain and meshes===<br /> <br /> <br /> The computational domain and the meshes employed in these simulations are the ones described in Section 3.2.1.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> In LES2, an eddy-viscosity-type model following the Boussinesq hypothesis ([[Lib:Best Practice Advice AC7-02#Sagaut2006|Sagaut, 2006]]) is used to close the system of filtered Navier-Stokes equations, (2&amp;3).<br /> The Wall-adapting eddy viscosity model (WALE) SGS model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) has been employed. <br /> This model is based on the square of the velocity gradient tensor. The SGS viscosity obtained with this model takes into account the strain and the rotation rate of the smallest resolved turbulent fluctuations. <br /> Some features of this model are its capability of switching off in two-dimensional flows and in laminar flows. <br /> It also has a cubic behaviour near walls with respect to the normal direction of the wall.<br /> <br /> The CFD simulations presented in this section have been carried out using the in-house CFD software TermoFluids ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2007|Lehmkuhl et al., 2007]]), based on the finite volume method (FVM). <br /> This CFD code is parallel, highly scalable and designed to work in both structured and unstructured meshes. <br /> The convective term in the momentum equation (3) is discretized using a second order Symmetry-Preserving (SP) scheme ([[Lib:Best Practice Advice AC7-02#Verstappen2003|Verstappen &amp; Veldman, 2003]]).<br /> This discretization scheme constructs an anti-symmetric discrete convective operator. <br /> This operator does not introduce artificial dissipation in the momentum equation, and therefore the kinetic-energy is preserved. <br /> The diffusive operator is discretized by means of a second-order Central Differencing Scheme (CDS). <br /> Both convective and diffusive operators are integrated explicitly by means of a one-leg second-order time-integration scheme ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This scheme includes a free-parameter allowing to adapt its stability region. <br /> The free-parameter is dynamically selected maximizing the stability region in function of the eigenvalues of the discrete operators. <br /> Moreover, this time-integration strategy also selects the optimal time-step at each iteration. <br /> The pressure-velocity coupling is solved by means of the Fractional Step projection method ([[Lib:Best Practice Advice AC7-02#Kim1985|Kim &amp; Moin, 1985]]).<br /> The idea behind this technique is to split the momentum within two steps: a first explicit step where an intermediate velocity is obtained, followed by a second step where the pressure is solved implicitly and the intermediate velocity is corrected obtaining the physical divergence-free velocity field. <br /> The Poisson equation is solved by means of the iterative Conjugate-Gradient (CG) method with Jacobi diagonal scaling.<br /> <br /> The presented case has an inlet flowrate Q=60 L/min, which falls within the turbulent regime. <br /> Therefore, in order to generate the turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the streamwise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time. <br /> At the respiratory airways walls, the boundary condition is defined as no-slip. <br /> Regarding the outlets, two different conditions were employed. <br /> For the cases presented in section 3.3.3.1, a constant flowrate was fixed en each one of the outlets. <br /> On the other hand, for the cases solved in section 3.3.3.2 and in Section 4, the pressure is fixed at the ten outlets with a value equal to atmospheric pressure.<br /> <br /> ===Numerical accuracy===<br /> <br /> Two of the main numerical aspects that will determine the accuracy of a CFD simulation employing a LES approach are the mesh and the turbulence model employed for the subgrid scales. <br /> Therefore, in the present section these two parameters have been analyzed in detail.<br /> <br /> <br /> ====Mesh convergence analysis====<br /> The effects of the mesh on the simulation results are studied comparing the same experiment and geometry using two different meshes: <br /> a fine one with 50 million control volumes (CVs) and a coarse one with 10 million CVs (see 3.2.1). <br /> The results for both mean velocities and TKE (&lt;math&gt; = 1/2 &lt; u_i' u_i' &gt; &lt;/math&gt;) obtained with the two meshes are depicted in [[#Figure13|figure 13]] and [[#Figure14|14]], respectively.<br /> <br /> As can be seen in [[#Figure13|figure 13]], the agreement for the mean velocities is very good between both meshes and the results are practically identical. <br /> On the other hand, some differences can be appreciated for TKE between the two studied meshes. <br /> As observed in the 2D profiles ([[#Figure14|figure 14]]), the fine mesh yields higher levels of TKE than the coarse one. <br /> However, both meshes are able to capture the same trend, obtaining the highest levels of TKE in the same positions, which belongs to the shear layer regions. <br /> Hence, it is clear that, although first order quantities are well captured by both meshes, special care must be taken if second order quantities must be reproduced accurately, since the latter are very sensitive to mesh resolution. <br /> In conclusion, sufficiently fine meshes must be employed in order to correctly capture second order quantities, although first order ones can be obtained accurately with coarser meshes. <br /> Obviously, the recommended grid-spacing in this kind of simulations will be the result of a compromise between accuracy and computational cost.<br /> <br /> &lt;div id=&quot;Figure13&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure13.jpeg|center|thumb|500px|'''Figure 13''': LES2: Comparison of mean velocity contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> &lt;div id=&quot;Figure14&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure14.jpeg|center|thumb|500px|'''Figure 14''': LES2: Comparison of turbulent kinetic energy contours and profiles at selected stations between coarse (Mesh 1) and fine (Mesh 2) meshes.]]<br /> <br /> ====Comparison of different LES subgrid-scale models====<br /> <br /> In order to assess the influence of the subgrid-scale turbulence model in LES of airflow in the human upper airways, four different turbulence models were compared. <br /> Besides the WALE model ([[Lib:Best Practice Advice AC7-02#Nicoud1999|Nicoud &amp; Ducros, 1999]]) previously introduced, three additional turbulence models have been studied. <br /> The first one is the model proposed by [[Lib:Best Practice Advice AC7-02#Smagorinsky1963|Smagorinsky (1963)]], based on the Prandtl mixing length applied to subgrid-scale modelling. <br /> In this model the turbulent viscosity is proportional to the strain. <br /> The second model is the variational multiscale (VMS) approach applied in WALE. <br /> This method was originally formulated for the Smagorinsky model by [[Lib:Best Practice Advice AC7-02#Hughes2000|Hughes et al. (2000)]].<br /> Using this framework the modelling is confined to the effect of a small-scale Reynolds stress, in contrast to classical LES in which the entire SGS stress is modelled. <br /> The latter is the model proposed by Verstappen known as the QR eddy-viscosity model ([[Lib:Best Practice Advice AC7-02#Verstappen2010|Verstappen et al., 2010]]; [[Lib:Best Practice Advice AC7-02#Verstappen2011|Verstappen et al., 2011]]).<br /> This model is based on two invariants of the rate-of-strain tensor.<br /> The method is computationally very efficient, switches off in the case of back-scattering, laminar and two-dimensional flows, and the subgrid turbulent viscosity is proportional to the cube with respect to the normal direction of the wall.<br /> <br /> The results for the in-plane mean velocities and TKE (see section 4 and equations 8-11) are depicted in [[#Figure15|figure 15]] and [[#Figure16|16]], respectively. <br /> As can be seen, the four models predict very similar results, except in shear layer regions for TKE levels (just downstream glottis constriction - station D), where some differences can be seen between the different models. <br /> However, these deviations are very small and not relevant. Therefore, it can be stated that all LES subgrid-scale models are able to provide results that are in reasonable agreement with the measurements.<br /> <br /> Nonetheless, in previous works where different turbulence models in confined flows were compared, it was found that some turbulence models were not able to correctly predict the flow pattern. <br /> In the work of [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]], the Smagorinsky model was not able to correctly model the subgrid dissipation in the simulated confined flow, being too dissipative. The main difference to the cases examined in these works is the Reynolds number. <br /> In the present study, the Reynolds number in the trachea at 60 L/min is around Re = 4921, while the one of the case presented in [[Lib:Best Practice Advice AC7-02#Muela2019|Muela et al. (2019)]] is two orders of magnitude higher (&lt;math&gt; Re = 3.47 \cdot 10^5&lt;/math&gt;).<br /> <br /> In simulations of the airflow in the human upper airways, the Reynolds number is in most cases below Re &lt; 10000 and therefore the flow is either laminar or with low turbulence. <br /> Due to that, the influence of the subgrid scales in this kind of flows is small, and the turbulence model is not as relevant as in more turbulent cases. <br /> Therefore, the turbulence model in simulations of the airflow in the human upper airways using LES modeling does not play a key role.<br /> <br /> &lt;div id=&quot;Figure15&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure15.jpeg|center|thumb|500px|'''Figure 15''': Comparison of normalised mean velocity profiles at selected stations -shown in (a)- between PIV and different LES subgrid-scale models.]]<br /> <br /> &lt;div id=&quot;Figure16&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure16.jpeg|center|thumb|500px|'''Figure 16''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different LES models.]]<br /> <br /> ==Large Eddy Simulations &amp;mdash; Case LES3==<br /> ===Computational domain and meshes===<br /> <br /> Although the geometry used in the LES3 calculations is the same as the one used in the LES1&amp;2 (shown in [[#Figure8|figure 8]]), the mesh employed in the LES3 simulations is different. <br /> It consists of 7 million linear finite elements (1.7 million degrees of freedom) including 3 layers of prismatic elements near the airway geometry walls. <br /> The adequacy of the employed mesh resolution is discussed in section 3.4.3.<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> Alya ([[Lib:Best Practice Advice AC7-02#Vazquez2016|Vazquez et al., 2016]]), which is a parallel multi-physics/multi-scale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments is used in LES3. <br /> The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed ([[Lib:Best Practice Advice AC7-02#Charnyi2017|Charnyi et al., 2017]]), which conserves linear and angular momentum, and kinetic energy at the discrete level. <br /> Both second- and third-order spatial discretizations are used. <br /> Neither upwinding nor any equivalent momentum stabilization is employed. <br /> In order to use equal-order elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina, 2001), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes ([[Lib:Best Practice Advice AC7-02#Jofre2014|Jofre et al., 2014]]). <br /> The set of equations is integrated in time using a third-order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator ([[Lib:Best Practice Advice AC7-02#Trias2011|Trias &amp; Lehmkuhl, 2011]]). <br /> This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach ([[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al., 2019]]). <br /> WALE SGS closure is used as LES model.<br /> <br /> Atmospheric pressure and a laminar uniform velocity profile (zero turbulence) are applied at the mouth inlet of the model. <br /> At the 10 outlets of the model, zero-gradient pressure condition is imposed whereas the velocities are extrapolated from the boundary-adjacent cells using specified flow rates at each of the outlet. <br /> Although the outlet boundary condition in LES3 is different compared to the PIV setup, since the comparisons concern the proximal region of the model (mouth, trachea and main bifurcation) this mismatch is not expected to affect our findings.<br /> This is confirmed by the comparisons shown in section 4.<br /> <br /> ===Numerical accuracy===<br /> In order to assess the independence of LES3 results from grid resolution, a mesh convergence analysis was carried out. <br /> In the results presented in the following of this section, the computational domain was truncated at the end of the trachea and simulations were performed in the mouth-trachea region. <br /> Four computational meshes were generated, as shown in [[#Figure17|figure 17]]. <br /> Details of these meshes are reported in [[#Table4|Table 4]]. <br /> Three meshes have identical resolution in the bulk region and different degrees of refinement of the near-wall prismatic elements (M1a-c). <br /> Mesh M2 has the finest resolution in both the bulk and near-wall regions. <br /> For the purpose of the mesh convergence analysis, the inlet conditions were different to the ones used in the simulation of the entire geometry for the LES3 case (i.e uniform velocity). <br /> Specifically, in order to generate a turbulent velocity profile for the inlet section, an auxiliary simulation of a pipe with the same diameter as the inlet and periodic boundary condition in the stream-wise direction have been employed. <br /> The velocity field generated at the mid-plane of the pipe is saved and later imposed at the inlet section of the simulation during running time (similar to LES2 inlet conditions).<br /> <br /> [[#Figure18|Figure 18]] displays contours and profiles of the mean velocity magnitude and [[#Figure19|figure 19]] shows contours and profiles of the turbulent kinetic energy (TKE) at cross sections in the mouth-throat and trachea (stations A-E). <br /> Remarkable agreement is observed between the mesh resolutions examined, suggesting mesh convergent results even on the coarse resolution (M1b). <br /> Mesh M1a has overall good agreement with the rest of the meshes, however has difficulties in predicting the low speed region at the start of the trachea (just downstream glottis constriction - station D), suggesting that at least 5 elements inside the boundary layer are needed to properly solve that region of the geometry. <br /> However, it is remarkable how the coarse mesh M1b gives results with reasonable agreement to those of the fine mesh (M2), showing the capability of low dissipation FEM to remain second order accurate even with very complex geometries. <br /> Here the key is the capability of such schemes to preserve the kinetic energy of the convective operator without the need of reducing the accuracy of the scheme like in unstructured FVM (for more information see [[Lib:Best Practice Advice AC7-02#Lehmkuhl2019|Lehmkuhl et al. (2019)]]).<br /> <br /> &lt;div id=&quot;Table4&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table4.jpeg|center|thumb|500px|'''Table 4''': Characteristics of meshes used for the LES3 mesh convergence analysis. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> &lt;div id=&quot;Figure17&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure17.jpeg|center|thumb|500px|'''Figure 17''': Cross-sectional views of the meshes near the inlet of the model (station A in fig. 10(b)) used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure18&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure18.jpeg|center|thumb|500px|'''Figure 18''': Comparison of mean velocity contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> &lt;div id=&quot;Figure19&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure19.jpeg|center|thumb|500px|'''Figure 19''': Comparison of turbulent kinetic energy contours and profiles at selected stations (shown in (a)), between meshes used for the LES3 mesh convergence analysis.]]<br /> <br /> ==RANS Simulations==<br /> <br /> <br /> The gas flowfield calculations in the airway system were conducted for the steady-state by solving the Reynolds-Averaged Navier-Stokes (RANS) equations in connection with an appropriate turbulence model using the open-source platform OpenFOAM 4.1 ([[Lib:Best Practice Advice AC7-02#OpenFOAM2013b|OpenFOAM Foundation, 2013b]]). <br /> For coupling the velocity and pressure fields, the SIMPLE (Semi-Implicit-Method-Of-Pressure-Linked-Equations) algorithm is applied. <br /> Hence, the module simpleFOAM is used in order to solve the conservation and momentum equations for a steady-state, incompressible and turbulent flow. <br /> For comparison three turbulence models were considered, the standard k-ω SST, the standard k-ε and the RNG k-ε turbulence model.<br /> <br /> ===Computational domain and meshes===<br /> <br /> The geometry used in the RANS calculations is essentially the same geometry as the one used in the LES case (shown in [[#Figure8|figure 8]]).<br /> <br /> The meshing process is one of the most important parts in solving the airways system. <br /> A mesh with insufficient quality of the computational cells (high mesh non-orthogonality or skewness) can result in numerical diffusion and consequently prediction of the gas phase with significant errors ([[Lib:Best Practice Advice AC7-02#Jasak1996|Jasak, 1996]]).<br /> The geometry of the airways system is extremely complex with several changes of sections, branches, constrictions and expansions. <br /> Therefore, it is not possible to generate a hexahedral and structured mesh. <br /> The solution found for this problem lies in the use of tetrahedral elements, a configuration being more adaptable to the complexity of the mesh. <br /> However, this kind of mesh structure can lead to numerical errors mainly in the boundary layers, e.g. near-wall region, making it necessary to create a layer of prismatic elements close to the wall. <br /> Two meshes were generated with different near-wall resolutions. <br /> Cross-sectional views of these meshes at five stations are shown in [[#Figure20|figure 20]]. <br /> In the coarser mesh (Mesh 1), only three layers of prismatic elements were used close to the wall; however the results obtained with such mesh resolution were not satisfactory. <br /> This problem was solved by refining the mesh close to the wall. <br /> Specifically, the near-wall element was divided three times having the two parts closer to the wall at 25% of the original element thickness and the third one having a thickness of 50%, as can be observed in [[#Figure20|figure 20]](b). <br /> [[#Table5|Table 5]] reports grid characteristics for meshes 1 and 2. <br /> Mesh 2 was successfully applied to particle deposition studies ([[Lib:Best Practice Advice AC7-02#Koullapis2018|Koullapis et al., 2018]]) and therefore is also used here for the RANS simulations without further analysis of grid convergence. <br /> The focus of this study relates to the influence of turbulence model as well as the applied inlet and outlet boundary conditions.<br /> <br /> &lt;div id=&quot;Figure20&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure20.jpeg|center|thumb|500px|'''Figure 20''': (a) Cross-sectional views of the two generated meshes at five stations (A-E). (b) Cross sectional views at the entrance pipe, showing coarser mesh with three near-wall prism layers and finer mesh with the near-wall element divided by three.]]<br /> <br /> &lt;div id=&quot;Table5&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table5.jpeg|center|thumb|500px|'''Table 5''': Characteristics of meshes 1-2. &lt;math&gt; \Delta r_{min} &lt;/math&gt; is the height of the wall-adjacent cells, &lt;math&gt; \lambda &lt;/math&gt; the average expansion ratio of the prism layers and &lt;math&gt; V_{cell,avg.} &lt;/math&gt; the average cell volume in the domain.]]<br /> <br /> ===Solution strategy and boundary conditions &amp;mdash; Airflow===<br /> <br /> The present RANS computations were conducted only with Mesh 2 and for a flow rate of 60 L/min. <br /> The gas density and the dynamic viscosity are set to 1.184 kg/m3 and &lt;math&gt; 19.1\cdot 10^{-6} kg/(m \cdot s) &lt;/math&gt;, respectively. <br /> In the following the applied inlet/outlet and boundary conditions are described. <br /> The outlet boundary conditions for the present case which should closely mimic the experimental situation are based on a zero-gradient strategy (see [[#Table6|Table 6]]). <br /> No-slip boundary conditions are used at the pipe walls and a zero pressure is assigned to all outlet boundaries. <br /> Wall functions are applied in order to solve the turbulence properties in the near wall region, therefore not demanding extra refinements near the wall for a proper solution. <br /> As inlet boundary conditions two cases are considered: <br /> <br /> '''Inlet 1''': A mapped boundary condition is applied, where the inlet pipe in front of the oral cavity has a length of 30mm. <br /> The mapping of the inlet profile was done over a quite short distance of 20mm. <br /> With such an approach a developed inlet flow is obtained without the need of simulating a longer pipe at the entrance of the lung model. <br /> This procedure is performed by initially setting an averaged value for the desired property (e.g. velocity, turbulent kinetic energy, etc.) at the inlet and defining a reference plane at a certain downstream distance (i.e. in this case 20 mm from the inlet). <br /> Following, the new mapped inlet BC takes the value of the desired property from this offset plane and uses it for the next iteration step.<br /> <br /> '''Inlet 2''': For this case the short inlet pipe was extended by a straight pipe with a length of 10 times the inlet diameter. <br /> The extended grid had the same cross-sectional structure as the short inlet. <br /> At the new inlet boundary, a plug flow inlet with constant values of all properties (e.g. velocity, turbulent kinetic energy, specific dissipation rate and dissipation rate) was specified. <br /> Hence, the flow developed over the length of the inlet pipe and then entered the lung model with a more or less developed but symmetric profile. The inlet turbulent kinetic energy was fixed and constant based on 5% turbulence intensity:<br /> <br /> &lt;math&gt;<br /> k = \frac{3}{2} (U \cdot I)^2, \quad (I=0.05) \qquad (5).<br /> &lt;/math&gt;<br /> <br /> The dissipation rate (ε) and specific dissipation rate (ω) at the inlet were determined from using the mixing length l and the inlet diameter &lt;math&gt; D_{inlet} &lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> \epsilon = C_{\mu}^{0.75} \frac{k^{1.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (6),<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \omega = C_{\mu}^{-0.25} \frac{k^{0.5}}{l}, \quad (l=0.07 \cdot D_{inlet}) \qquad (7).<br /> &lt;/math&gt;<br /> <br /> More information about the boundary conditions used in the calculations, as well as all the wall functions and properties needed in the calculations are extensively detailed in the OpenFOAM user guide. <br /> A summary of the operational conditions and the boundary condition setup is shown in [[#Table6|Table 6]].<br /> <br /> &lt;div id=&quot;Table6&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Table6.jpeg|center|thumb|500px|'''Table 6''': Configuration of the boundary conditions for the gas-phase calculations considering a gas flow of 60 L/min for Inlet 1 (mapped) and Inlet 2 (pipe extension).]]<br /> <br /> ===Numerical accuracy===<br /> <br /> [[#Figure21|Figure 21]] shows comparisons of normalised mean velocity profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS computations with different turbulence models and inlet conditions. <br /> The velocity to normalise the simulation results is the bulk velocity of air in the trachea at a flowrate of 60 L/min, &lt;math&gt; u_T = 4.8m/s &lt;/math&gt;. <br /> In general the RANS prediction for the velocity magnitude follow similar trends to the measured data. <br /> However notable deviations exist at stations D, H and J which are located at regions with shear layers, recirculation and flow separation (D is located just downstream the glottis constiction whereas H&amp;J are downstream the first bifurcation in the left and right main bronchi, respectively). <br /> Among the different RANS computations, there are no significant differences. <br /> Only the Inlet 1 condition used with the k-ω SST turbulence model gives slightly lower velocities in the upper region of the oral cavity ([[#Figure21|figure 21]], profile B). <br /> At stations D and H, the simulations with the standard k-ε model yield larger differences in comparison to PIV than those obtained with the RNG k-ε and k-ω SST models. <br /> In conclusion, it is not possible based on the RANS predicted mean velocity profiles to give a preference to a certain turbulence model or the selection of the inlet boundary conditions.<br /> <br /> [[#Figure22|Figure 22]] shows comparisons of normalised turbulent kinetic energy profiles at the locations of the PIV measurement planes IV-VI, between the measurements and RANS. <br /> All the RANS computations yield much lower turbulent kinetic energy throughout the lung model compared to the measured values. <br /> The mapped inlet (Inlet 1) results in extremely low k-values at the first profile B, which is very unrealistic. <br /> With the inlet pipe of 10xDinlet (Inlet 2) and an inlet turbulence intensity of 5% the k-ω SST turbulence model yields higher turbulent kinetic energy values. <br /> At the rest of stations in the lung model, the k-ω SST turbulence model produces the same turbulence levels no matter which inlet condition was applied. <br /> The k-ε models provide overall higher turbulence levels than the k-ω SST model, especially in the near wall regions of the more distal stations (G, H and J). <br /> The TKE peaks are associated with near-wall maxima which are also found in some of the measured turbulent kinetic energy profiles. <br /> However, the agreement to the measured turbulent kinetic energy levels is still not satisfactory.<br /> <br /> &lt;div id=&quot;Figure21&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure21.jpeg|center|thumb|500px|'''Figure 21''': Comparison of normalised mean velocity profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;div id=&quot;Figure22&quot;&gt;&lt;/div&gt;<br /> [[File:Brno_PIV_Figure22.jpeg|center|thumb|500px|'''Figure 22''': Comparison of normalised turbulent kinetic energy profiles at selected stations -shown in [[#Figure15|figure 15]](a)- between PIV and different RANS models and inlet conditions.]]<br /> <br /> &lt;br/&gt;<br /> ----<br /> <br /> {{ACContribs<br /> |authors=P. Koullapis&lt;sup&gt;a&lt;/sup&gt;, J. Muela&lt;sup&gt;b&lt;/sup&gt;, O. Lehmkuhl&lt;sup&gt;c&lt;/sup&gt;, F. Lizal&lt;sup&gt;d&lt;/sup&gt;, J. Jedelsky&lt;sup&gt;d&lt;/sup&gt;, M. Jicha&lt;sup&gt;d&lt;/sup&gt;, T. Janke&lt;sup&gt;e&lt;/sup&gt;, K. Bauer&lt;sup&gt;e&lt;/sup&gt;, M. Sommerfeld&lt;sup&gt;f&lt;/sup&gt;, S. C. Kassinos&lt;sup&gt;a&lt;/sup&gt;<br /> |organisation=&lt;br&gt;&lt;sup&gt;a&lt;/sup&gt;Department of Mechanical and Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus&lt;br&gt;<br /> &lt;sup&gt;b&lt;/sup&gt;Heat and Mass Transfer Technological Centre, Universitat Politècnica de Catalunya, Terrassa, Spain&lt;br&gt;<br /> &lt;sup&gt;c&lt;/sup&gt;Barcelona Supercomputing center, Barcelona, Spain &lt;br&gt;<br /> &lt;sup&gt;d&lt;/sup&gt;Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic&lt;br&gt;<br /> &lt;sup&gt;e&lt;/sup&gt;Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, Germany&lt;br&gt;<br /> &lt;sup&gt;f&lt;/sup&gt;Institute Process Engineering, Otto von Guericke University, Halle (Saale), Germany &lt;br&gt;<br /> }}<br /> {{ACHeader<br /> |area=7<br /> |number=02<br /> }}<br /> <br /> © copyright ERCOFTAC 2020</div> Kassinos