Difference between revisions of "Best Practice Advice AC7-01"
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In the LES simulations, the volumetric ﬂowrates at the 10 terminal outlets are prescribed based | In the LES simulations, the volumetric ﬂowrates at the 10 terminal outlets are prescribed based | ||
− | on the values measured ''in vitro'' ([[ | + | on the values measured ''in vitro'' ([[Test_Data_AC7-01#table3|Table 3]]). |
These outlet conditions result | These outlet conditions result | ||
in high asymmetry in the ventilation of the two lungs: the left lung receives 29% of the | in high asymmetry in the ventilation of the two lungs: the left lung receives 29% of the | ||
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tracking. Recommended values for the parameters involved in mesh generation (initial cell | tracking. 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 | 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 Table 5. | + | the domain, number of computational cells etc.) can be found in |
+ | [[CFD_Simulations_AC7-01#table5|Table 5]]. | ||
+ | |||
==Physical Modelling== | ==Physical Modelling== | ||
===Turbulence models=== | ===Turbulence models=== | ||
In the LES simulations, the dynamic version of the Smagorinsky-Lilly subgrid scale model | In the LES simulations, the dynamic version of the Smagorinsky-Lilly subgrid scale model | ||
− | (Lilly, 1992) is adopted in order to examine the unsteady ﬂow in the realistic airway | + | ([[Best_Practice_Advice_AC7-01#lilly1992|Lilly, 1992]]) |
+ | is adopted in order to examine the unsteady ﬂow in the realistic airway | ||
geometries. Previous studies have shown that this model performs well in transitional | geometries. Previous studies have shown that this model performs well in transitional | ||
− | ﬂows in the human airways (Radhakrishnan & Kassinos, 2009; Koullapis ''et al.'', 2016). | + | ﬂows in the human airways |
+ | ([[Best_Practice_Advice_AC7-01#radhakrishnan2009|Radhakrishnan & Kassinos, 2009]]; | ||
+ | [[Best_Practice_Advice_AC7-01#koullapis2016|Koullapis ''et al.'', 2016]]). | ||
− | For the RANS simulations, the standard k-ω SST turbulence model is used due to | + | For the RANS simulations, the standard ''k-ω'' SST turbulence model is used due to |
− | its good prediction of such | + | its good prediction of such wall-bounded ﬂow (i.e. using a blending between wall and |
free-stream region) and low computational cost. Other turbulence models have not been | free-stream region) and low computational cost. Other turbulence models have not been | ||
considered but based on prior experince they are exprected to perform worse. | considered but based on prior experince they are exprected to perform worse. | ||
Line 96: | Line 101: | ||
are important when considering the transport of micron—sized particles in the human | are important when considering the transport of micron—sized particles in the human | ||
airways. This is mainly because the particle density is much greater than the density of | airways. This is mainly because the particle density is much greater than the density of | ||
− | the air | + | the air |
+ | <math>{(\rho_p/\rho_f\ge 1000)}</math>. | ||
Furthermore, in numerical simulations the particles are assumed | Furthermore, in numerical simulations the particles are assumed | ||
spherical and, as a result, the important forces that need to be taken into account in lung | spherical and, as a result, the important forces that need to be taken into account in lung | ||
deposition studies are drag, gravity and Brownian motion force. However, Lift force due | deposition studies are drag, gravity and Brownian motion force. However, Lift force due | ||
to shear can also be important as particle size increases. This is evident in ﬁgure 21, that | to shear can also be important as particle size increases. This is evident in ﬁgure 21, that | ||
− | plots deposition fraction in the | + | plots deposition fraction in the mouth-throat / trachea (segments 1&2) of the benchmark |
− | geometry for three particle sizes (1, 4.3 and 10 | + | geometry for three particle sizes (1, 4.3 and 10<math>{\mu m}</math>) at a ﬂowrate of |
− | In this case, the expression for the shear lift is obtained by | + | 60 L / min (LES results). |
− | and Mei (1992). It is observed that the | + | In this case, the expression for the shear lift is obtained by |
− | deposition of 10 | + | [[Best_Practice_Advice_AC7-01#saffman1965|Saffman (1965)]], |
+ | [[Best_Practice_Advice_AC7-01#saffman1968|Saffman (1968)]] | ||
+ | and [[Best_Practice_Advice_AC7-01#mei1992|Mei (1992)]]. | ||
+ | It is observed that the Saffman lift force results in an increase in the | ||
+ | deposition of 10<math>{\mu m}</math> particles by approximately 5%. | ||
Line 113: | Line 123: | ||
|align="center"|[[Image:AC7-01_fig21.png|650px]] | |align="center"|[[Image:AC7-01_fig21.png|650px]] | ||
|- | |- | ||
− | |align="left"|'''Figure 21:''' LES predictions of Deposition fraction in the | + | |align="left" width=650|'''Figure 21:''' LES predictions of Deposition fraction in the mouth-throat / trachea (segments 1&2) of the benchmark geometry for three particle sizes (1, 4.3 and 100<math>{\mu}</math>m) at a ﬂowrate of 60 L/min. |
|} | |} | ||
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In our LES the grid size is larger than the Kolmogorov scale and thus this | In our LES the grid size is larger than the Kolmogorov scale and thus this | ||
condition is satisﬁed. Speciﬁcally, the minimum ratio of resolved scales (mesh size near | condition is satisﬁed. Speciﬁcally, the minimum ratio of resolved scales (mesh size near | ||
− | the wall) to the particle size is 40 for 1 | + | the wall) to the particle size is 40 for 1<math>{\mu m}</math> particles and |
+ | 4 for 10<math>{\mu m}</math>. | ||
In point-particle LES, the resolved velocities are used in the calculation of the forces | In point-particle LES, the resolved velocities are used in the calculation of the forces | ||
Line 137: | Line 148: | ||
deposition in the airways has not been investigated. | deposition in the airways has not been investigated. | ||
− | Armenio ''et al.'' (1999) examined the effects of small-scale velocity ﬂuctuations on the | + | [[Best_Practice_Advice_AC7-01#armenio1999|Armenio ''et al.'' (1999)]] |
+ | examined the effects of small-scale velocity ﬂuctuations on the | ||
motion of tracer and inertial particles in a turbulent channel ﬂow at | motion of tracer and inertial particles in a turbulent channel ﬂow at | ||
− | + | <math>{Re_\tau=175}</math>. | |
They concluded that well-resolved LES with an adequate LES model can provide fairly accurate | They concluded that well-resolved LES with an adequate LES model can provide fairly accurate | ||
particle statistics for moderate Reynolds number ﬂows. They also observed that errors in | particle statistics for moderate Reynolds number ﬂows. They also observed that errors in | ||
Line 145: | Line 157: | ||
errors. Speciﬁcally, good agreement in dispersion statistics was found between DNS and | errors. Speciﬁcally, good agreement in dispersion statistics was found between DNS and | ||
well-resolved LES with the dynamic model (differences less than 8%) whereas rather higher | well-resolved LES with the dynamic model (differences less than 8%) whereas rather higher | ||
− | errors were recorded for the coarser meshes. The study of Armenio ''et al.'' (1999) provides | + | errors were recorded for the coarser meshes. The study of |
+ | [[Best_Practice_Advice_AC7-01#armenio1999|Armenio ''et al.'' (1999)]] provides | ||
conservative estimates of the accuracy of LES in the prediction of particle-laden ﬂow since | conservative estimates of the accuracy of LES in the prediction of particle-laden ﬂow since | ||
tracer particles were used, which are the most sensitive to the small-scale ﬂuctuations, In | tracer particles were used, which are the most sensitive to the small-scale ﬂuctuations, In | ||
Line 160: | Line 173: | ||
In the RANS simulations, the minimum ratio of resolved scales (mesh size near the | In the RANS simulations, the minimum ratio of resolved scales (mesh size near the | ||
− | wall) to the particle size is 30 for 1 | + | wall) to the particle size is 30 for 1<math>{\mu m}</math> particles and 3 for |
+ | 10<math>{\mu m}</math> and therefore the particles | ||
can be considered as point-particles and Lagrangian simulations can be performed. For | can be considered as point-particles and Lagrangian simulations can be performed. For | ||
particle tracking in RANS simulations, the generation of the ﬂuid ﬂuctuating velocity | particle tracking in RANS simulations, the generation of the ﬂuid ﬂuctuating velocity | ||
Line 193: | Line 207: | ||
‘Simulation and pharmaceutical technologies for advanced patient-tailored inhaled | ‘Simulation and pharmaceutical technologies for advanced patient-tailored inhaled | ||
medicines’, supported by COST (European Cooperation in Science and | medicines’, supported by COST (European Cooperation in Science and | ||
− | Technology — www.cost.eu). | + | Technology — [https://www.cost.eu www.cost.eu]). |
+ | |||
==References== | ==References== | ||
− | Armenio, V., Piomelli, U. & Fiorotto, V. 1999 Effect of the subgrid scales on particle | + | <div id="armenio1999"></div> |
− | motion. ''Physics of Fluids'' '''11''' (10), 3030 – 3042. | + | Armenio, V., Piomelli, U. & Fiorotto, V. 1999 |
+ | :Effect of the subgrid scales on particle motion. ''Physics of Fluids'' '''11''' (10), 3030 – 3042. | ||
Line 218: | Line 234: | ||
:''The Mechanics of Inhaled Pharmaceutical Aerosols''. Academic Press, New York. | :''The Mechanics of Inhaled Pharmaceutical Aerosols''. Academic Press, New York. | ||
− | Jasak, H. 1996 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. | + | <div id="jasak1996"></div> |
+ | Jasak, H. 1996 | ||
+ | :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. | ||
Line 286: | Line 304: | ||
:The lift on a small sphere in a slow shear ﬂow. ''Journal of Fluid Mechanics'' '''22''' (2), 385 – 400. | :The lift on a small sphere in a slow shear ﬂow. ''Journal of Fluid Mechanics'' '''22''' (2), 385 – 400. | ||
− | Saffman, P. G. 1968 The lift on a small sphere in a slow shear ﬂow | + | |
− | of Fluid Mechanics 31 (3), | + | <div id="saffman1968"></div> |
+ | Saffman, P. G. 1968 | ||
+ | :The lift on a small sphere in a slow shear ﬂow – corrigendum. ''Journal of Fluid Mechanics'' '''31''' (3), 624 – 624. | ||
Line 297: | Line 317: | ||
<div id="schmidt2004"></div> | <div id="schmidt2004"></div> | ||
Schmidt, Andreas, Zidowitz, Stephan, Kriete, Andres, Denhard, Thorsten, Krass, Stefan | Schmidt, Andreas, Zidowitz, Stephan, Kriete, Andres, Denhard, Thorsten, Krass, Stefan | ||
− | & Peitgen, | + | & Peitgen, Heinz-Otto 2004 |
:A digital reference model of the human bronchial tree. ''Computerized Medical Imaging and Graphics'' '''28''' (4), 203 – 211. | :A digital reference model of the human bronchial tree. ''Computerized Medical Imaging and Graphics'' '''28''' (4), 203 – 211. | ||
Revision as of 11:11, 10 October 2019
Contents
- 1 Aerosol deposition in the human upper airways
- 2 Best Practice Advice
Aerosol deposition in the human upper airways
Application Challenge AC7-01 © copyright ERCOFTAC 2019
Best Practice Advice
Key Fluid Physics and Deposition Mechanisms
Airﬂow in the human upper airways transitions to turbulence due to geometric effects, such as the bent in the oropharyngeal region and the constriction at the glottis. The bent in the oropharynx causes substantial ﬁltering of inhaled aerosols due to inertial impaction on the airway walls. Filtering in the extrathoracic airways increases as the particle size and inhalation ﬂowrate increase.
As we move in the tracheobronchial airways, the Reynolds number is reduced because the air travels through a larger total cross-sectional area. As a result, airﬂow relaminarizes in the ﬁrst generations. At the ﬂowrate examined in the present AC, the main deposition mechanism in this region is inertial impaction, with signiﬁcant deposition at the bents and the bifurcations. At lower ﬂowrates, deposition can also be inﬂuenced by gravitational sedimentation because the residence times of the particles in the bronchial airways is longer.
Application Uncertainties
The differences between measurements and simulations can result from several uncertainties involved in the tests. A ﬁrst source of uncertainty are the in vitro inlet conditions, which might be different from the velocity and particle proﬁles assumed in the CFD simulations. In the experimental setup, various devices were placed upstream of the mouth inlet (see figure 5) and these devices are expected to alter the inlet ﬂow and particle conditions from what is prescribed in the simulations.
Another source of uncertainty between the experiment and the simulations is the size of the particles. Monodisperse particles have been assumed in the simulations whereas the aerosols generated in the experiments had a standard geometric deviation of size smaller than .
Computational Domain and Boundary Conditions
The geometry of the extrathoracic airways must be included because turbulence is generated in this region and alters transport and deposition of particles in the distal airways. In addition, signiﬁcant ﬁltering occurs in the mouth and throat, which affects the amount of inhaled aerosols that will eventually reach the desirable lung generations. In the present AC, in both LES and RANS tests the inlet at the mouth of the model was extruded in order to generate turbulent velocity conditions. This strategy was adopted due to the absence of a more realistic inlet velocity proﬁle.
Concerning the boundary conditions, the inlet velocity proﬁle and the particle distribution are important determinants of particle deposition and thus realistic inlet conditions should be used. At the outlets, it is important to apply correct pressures such that the ventilation of the airway tree is realistic. Otherwise, both the air and particle distribution in the trachea will not be predicted accurately.
In the LES simulations, the volumetric ﬂowrates at the 10 terminal outlets are prescribed based on the values measured in vitro (Table 3). These outlet conditions result in high asymmetry in the ventilation of the two lungs: the left lung receives 29% of the inhaled air whereas the right lung receives 71%.
In the RANS calculations, a simpliﬁed boundary condition setup was applied. Instead of applying prescribed ﬂowrates at the outlets of the system similarly to the experiments, a simpler strategy of applying the ﬂowrate at the inlet and zero pressure at the outlets was used. Using these boundary conditions, an overall good agreement with the experimental data was observed with small differences on the ventilation distribution after the third branching level. This approach may be used to obtain preliminary results, however, the correct application of the ﬂow ﬁeld for all the outlets is recommended in future works to better predict the ﬂow in the further downstream located sections of the system.
Discretisation and Grid Resolution
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. 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. Despite the application of interpolated properties for the particle positions, a better agreement was observed when a reﬁnement was applied to the wall layers. Hence, a ﬁner grid in the vicinity of the wall is recommended for allowing more accurate particle tracking. 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 Table 5.
Physical Modelling
Turbulence models
In the LES simulations, the dynamic version of the Smagorinsky-Lilly subgrid scale model (Lilly, 1992) is adopted in order to examine the unsteady ﬂow in the realistic airway geometries. Previous studies have shown that this model performs well in transitional ﬂows in the human airways (Radhakrishnan & Kassinos, 2009; Koullapis et al., 2016).
For the RANS simulations, the standard k-ω SST turbulence model is used due to its good prediction of such wall-bounded ﬂow (i.e. using a blending between wall and free-stream region) and low computational cost. Other turbulence models have not been considered but based on prior experince they are exprected to perform worse.
In order to better validate the numerical predictions of LES and RANS, a second application challenge will follow that will focus on airﬂow characteristics in the same geometry. Moreover, in this second AC, numerical predictions will be compared against PIV measurements.
Lagrangian particle tracking
Lagrangian particle tracking has been adopted in the present application. Although there is a number of forces acting on the particles (Drag, buoyancy, Basset (or history), pressure gradient force, lift due to shear and rotation and Brownian forces), only few of them are important when considering the transport of micron—sized particles in the human airways. This is mainly because the particle density is much greater than the density of the air . Furthermore, in numerical simulations the particles are assumed spherical and, as a result, the important forces that need to be taken into account in lung deposition studies are drag, gravity and Brownian motion force. However, Lift force due to shear can also be important as particle size increases. This is evident in ﬁgure 21, that plots deposition fraction in the mouth-throat / trachea (segments 1&2) of the benchmark geometry for three particle sizes (1, 4.3 and 10) at a ﬂowrate of 60 L / min (LES results). In this case, the expression for the shear lift is obtained by Saffman (1965), Saffman (1968) and Mei (1992). It is observed that the Saffman lift force results in an increase in the deposition of 10 particles by approximately 5%.
Figure 21: LES predictions of Deposition fraction in the mouth-throat / trachea (segments 1&2) of the benchmark geometry for three particle sizes (1, 4.3 and 100m) at a ﬂowrate of 60 L/min. |
As described in the discrete phase modelling, the time step of the particle tracking
calculation should automatically and independently be adapted along the trajectories by
considering all relevant time scales, which are also changing throughout the ﬂow ﬁeld.
This allows for numerically eﬂicient particle tracking. If such an approach is not possible,
a veriﬁcation of the relevant time scales should be calculated by using averaged values in
order to apply a correct time step for the simulations.
Modelling of unresolved ﬂow velocities
In Lagrangian point-particle methods the particle size must be smaller that the Kolmogorov scale. In our LES the grid size is larger than the Kolmogorov scale and thus this condition is satisﬁed. Speciﬁcally, the minimum ratio of resolved scales (mesh size near the wall) to the particle size is 40 for 1 particles and 4 for 10.
In point-particle LES, the resolved velocities are used in the calculation of the forces acting on the particles whereas the unresolved part of the air velocities is lost due to the ﬁltering procedure. Thus, the effect of the small-scale (unresolved) motion on particle dispersion and deposition must be either modelled separately, or neglected. In the present LES calculations, the effect of the unresolved scales of the continuous phase on particle deposition in the airways has not been investigated.
Armenio et al. (1999) examined the effects of small-scale velocity ﬂuctuations on the motion of tracer and inertial particles in a turbulent channel ﬂow at . They concluded that well-resolved LES with an adequate LES model can provide fairly accurate particle statistics for moderate Reynolds number ﬂows. They also observed that errors in LES are mainly attributed to the LES ﬁltering operation rather than to sub-grid modelling errors. Speciﬁcally, good agreement in dispersion statistics was found between DNS and well-resolved LES with the dynamic model (differences less than 8%) whereas rather higher errors were recorded for the coarser meshes. The study of Armenio et al. (1999) provides conservative estimates of the accuracy of LES in the prediction of particle-laden ﬂow since tracer particles were used, which are the most sensitive to the small-scale ﬂuctuations, In the case of particles with inertia, the errors are expected to be smaller.
In the present application challenge, airﬂow Reynolds numbers are low to moderate in the extrathoracic airways and the motion of inertial particles is simulated. Therefore, it is expected that if sufﬁcient mesh resolution is employed then a model for accounting the effect of the unresolved scales on particle motion is not necessary. However, there are cases where the inhalation ﬂowrate and thus the Reynolds number is higher, such as the inhalation from low-resistance inhalers, coughing etc. Therefore, further studies are needed to determine the conditions under which a model accounting for the effect of the unresolved scales on particle motion is required.
In the RANS simulations, the minimum ratio of resolved scales (mesh size near the wall) to the particle size is 30 for 1 particles and 3 for 10 and therefore the particles can be considered as point-particles and Lagrangian simulations can be performed. For particle tracking in RANS simulations, the generation of the ﬂuid ﬂuctuating velocity acting on the particle is an essential step, which in this study has been done assuming isotropic turbulence. Moreover, the relevant turbulent time and length scales have been determined based on this assumption. Nevertheless, a spurious drift may occur driving ﬁne particles to the wall and unrealistically enhancing deposition. This spurious drift has to be corrected appropriately.
Recommendations for Future Work
The present application challenge focuses on aerosol deposition and the airﬂow in the geometry has not been studied. In order to better validate the numerical predictions of LES and RANS, a second application challenge will follow that will focus on airﬂow characteristics in the same geometry. Moreover, in this second AC, numerical predictions will be compared against PIV measurements.
Concerning the dispersed phase, it is known that forces due to collision between particles become important in the case of dense aerosol suspensions. In other words, as the volume fraction of the particle phase increases, collisions between particles inﬂuence the simulation results. In addition, the ﬂow is affected by the presence of the particles and thus particle force source terms must be included in the momentum equations of the ﬂuid phase. This case is known as four-way coupling. If the particle volume fraction is sufﬁciently low, the particles do not collide with each other and also the ﬂow remains unaffected by their presence. This is known as one-way coupling. In an intermediate state, named two-way coupling, the ﬂow is affected by the particles, but particle collisions do not have signiﬁcant impact and therefore collision effects can be discarded in the simulation. The limits of validity of the coupling regimes are reported by Elghobashi (1994). In the present study, we have assumed one-way coupling, however, it is important to examine how deposition in the human airways is inﬂuenced when two- and four-way coupling effects are present.
Acknowledgements
The present application challenge is based upon work from COST Action MP1404 SimInhale ‘Simulation and pharmaceutical technologies for advanced patient-tailored inhaled medicines’, supported by COST (European Cooperation in Science and Technology — www.cost.eu).
References
Armenio, V., Piomelli, U. & Fiorotto, V. 1999
- Effect of the subgrid scales on particle motion. Physics of Fluids 11 (10), 3030 – 3042.
Baker, T. J. 2003
- Interpolation from a cloud of points. pp. 55 – 63. Santa Fe, New Mexico.
Chan, Tai & Lippmann, Morton 1980
- Experimental measurements and empirical modelling of the regional deposition of inhaled particles in humans. American Industrial Hygiene Association Journal 41 (6), 399 – 409.
Crowe, C. T., Schwarzkopf, J. D., Sommerfeld, M. & Tsuji, Y. 2012
- Multiphase ﬂows with droplets and particles. CRC Press.
Finlay, W. H. 2001
- The Mechanics of Inhaled Pharmaceutical Aerosols. Academic Press, New York.
Jasak, H. 1996
- 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.
Von Karman, T. & Horwarth, L., ed. 1938
- On the statistical theory of isotropic turbulence, vol. A164. Proc. Royal Society London.
Koullapis, P., Kassinos, S. C., Muela, J., Perez-Segarra, C., Rigola, J., Lehmkuhl, 0., Cui, Y., Sommerfeld, M., Elcner, J., Jicha, M., Saveljic, I., Filipovic, N., Lizal, F. & Nicolaou, L. 2018
- 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 – 94.
Koullapis, P. G., Kassinos, S.C., Bivolarova, M. P. & Melikov, A. K. 2016
- 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 – 2212.
Legg, B. J. & Raupach, M.R,. 1982
- Markov-chain simulations of particle dispersion in inhomogeneous ﬂows: The mean drift velocity induced by a gradient eulerian velocity variance. Boundary-Layer Meteorology 24 (3 – 13).
Li, A. & Ahmadi, G. 1992
- Dispersion and deposition of spherical particles from point sources in a turbulent channel ﬂow. Aerosol Science and Technology 16, 2097226.
Lilly, D. K. 1992
- A proposed modiﬁcation of the Germano subgrid-scale closure method. Physics of Fluids A 4 (3), 6337635.
Lizal, Frantisek, Belka, Miloslav, Adam, Jan, Jedelsky, Jan & Jicha, Miroslav 2015
- 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 – 757.
Lizal, Frantisek, Elcner, Jakub, Hopke, Philip K, Jedelsky, Jan & Jicha, Miroslav 2012
- Development of a realistic human airway model. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 226 (3), 197 – 207.
Macpherson, G. B., Nordin, N. & Weller, G. 2009
- Particle tracking in unstructured, arbitrary polyhedral meshes for use in CFD and molecular dynamics. Communications in Numerical Methods in Engineering 25 (3), 263 – 273.
Matida, E. A., Finlay, W. H., Lange, C. F. & Grgic, B. 2004
- Improved numerical simulation of aerosol deposition in an idealized mouth-throat. Aerosol Science 35, 1 – 19.
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- An approximate expression for the shear lift force on a spherical particle at finite reynolds number. International Journal of Multiphase Flow 18 (1), 145 – 147.
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Saffman, P. G. 1965
- The lift on a small sphere in a slow shear ﬂow. Journal of Fluid Mechanics 22 (2), 385 – 400.
Saffman, P. G. 1968
- The lift on a small sphere in a slow shear ﬂow – corrigendum. Journal of Fluid Mechanics 31 (3), 624 – 624.
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- Ver. Deutsch Ing.
Schmidt, Andreas, Zidowitz, Stephan, Kriete, Andres, Denhard, Thorsten, Krass, Stefan & Peitgen, Heinz-Otto 2004
- A digital reference model of the human bronchial tree. Computerized Medical Imaging and Graphics 28 (4), 203 – 211.
SOMMERFELD, M., KOHNEN, G. & RUGER, M., ed. 1993
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Contributed by: *** — ***