UFR 3-36 Test Case: Difference between revisions

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= HiFi-TURB-DLR rounded step =
= HiFi-TURB-DLR rounded step =
{{UFRHeaderLib
{{UFRHeader
|area=3
|area=3
|number=36
|number=36
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== Brief Description of the Study Test Case ==
== Brief Description of the Study Test Case ==
In the framework of the project, four different geometries with four different separation behaviors were  The geometry of this UFR alongside the mesh is shown in [[Lib:UFR_3-36_Test_Case#figure1|Fig. 1]]. The geometry comprises three main sections: Constant-Width Forebody section <math>{L_F}</math> with the largest width <math>{Y_F}</math>, Contoured Boat-tail section <math>{L_B}</math> with the contoured width <math>{Y_B}</math> and Constant-Width-Aftbody section <math>{L_A}</math> with the smallest width <math>{Y_A}</math>. The width of the last section is modified to generate the desired APG. This modification is achieved through the variation of <math>{\sigma}</math>, which is the ratio of <math>{Y_A}</math> to <math>{Y_F}</math>.  
In the framework of the HiFi-TURB project, four different geometries with different boundary-layer properties were designed. Each configuration was computed for two different Reynolds numbers. The configuration concerns a turbuluent boundary layer incipient to separation where the flow is on the brink of separation but does not separate at the Reynolds number <math>{Re_H = 78,490}</math> based on the step height <math>{H}</math>. The geometry of this UFR is shown in [[UFR_3-36_Test_Case#figure2|Fig. 2]]. The geometry comprises three main sections: An upstream section with <math>{11.84 H}</math>, the curved-step section with length <math>{L = 5.53 H}</math> and the downstream section with <math>{14.47 H}</math>. The length <math>{L}</math> of the step is fixed for all geometries, while the height <math>{H}</math> is adapted to generate the desired APG. For the incipient separation testcase the step height equals <math>{H = 0.028956 m}</math>.


<div id="figure1"></div>
<div id="figure2"></div>
{|align="center" border="0" width="550"
{|align="center" border="0" width="700"
|[[Image:Figure1_FlowDomain.png]]
|[[Image: Wiki_Figure2.png|700px]]
|-
|-
!align="center"|Figure 1: Flow Domain and grid of RANS simulations
!align="center"|Figure 2: Flow Domain
|}
|}


The parametric geometry definition for the three relevant sections is given in [&#8204;[[Lib:UFR_3-36_References#6|6]]] and is depicted in [[Lib:UFR_3-36_Test_Case#figure1|Fig. 1]]. The axial origin <math>{x=0}</math> is set at the beginning of the Contoured Boat-tail section.  
The parametric geometry definition for the curved-step sections is given in [&#8204;[[UFR_3-36_References#6|6]]] and is depicted in [[UFR_3-36_Test_Case#figure2|Fig. 2]]. The origin <math>{x=0}</math> is set at the beginning of the curved-step section.  


<!--
{{DisplayEquation|eqn=y_{wall} = (1 - R_1) + \sqrt{R_1^2 - x_2}                                  \qquad \qquad \qquad \mathrm{for} \qquad 0 < x/H \leq 2.3}}
{{DisplayEquation|no=1|eqn=y_{wall} = y_2 - \sqrt{\frac{1}{4}R_1^2 - ( x_2 - x)^2}                \qquad \mathrm{for} \qquad 2.3 < x/H \leq 2.835}}
{{DisplayEquation|eqn=y_{wall} = R_2 - \sqrt{R_2^2 - (3 - x)^2} \qquad \qquad \qquad \mathrm{for} \qquad 2.835 < x/H \leq 2.937}}
-->
<div id="eqn1"></div>
<div id="eqn1"></div>
<center><math>{
<center><math>{
\begin{align}
\begin{align}
y_{wall} &= y_F = H/(1-\sigma) \qquad \mathrm{for} \qquad -L_F < x/H \leq 0 \\
&y_{wall} &&= H &&\mathrm{for}\, -11.84 H < x \leq 0 \\
y_{wall} &= y_B = a_1 + a_2 x^3 + a_3 x^4 + a_4 x^5 \qquad \mathrm{for} \qquad 0 < x/H \leq L_B
&y_{wall} &&= a_1 + a_2 x^3 + a_3 x^4 + a_4 x^5 &&\mathrm{for}\, 0 < x \leq 5.53 H \\
\qquad\qquad (1) \\
&y_{wall} &&= 0 &&\mathrm{for}\, 5.53 H < x \leq 20 H \\
y_{wall} &= Y_A = \sigma H/(1 - \sigma) \qquad \mathrm{for} \qquad L_B < x/H \leq L_B + L_A \\
\\
\\
\end{align}
\end{align}
}</math></center>
}</math></center>


with <math>{a_1 = H/(1-\sigma)}</math>, <math>{a_2 = - 10 H/L_B^3}</math>, <math>{a_3 = 15 H/L_B^4}</math> and <math>{a_4 = - 6 H/L_B^5}</math> with <math>{\sigma = Y_A/Y_F}</math>.
with <math>{a_1 = H}</math>, <math>{a_2 = - 10 H/L^3}</math>, <math>{a_3 = 15 H/L^4}</math> and <math>{a_4 = - 6 H/L^5}</math>.
 
<span id="CFD Methods">


== CFD Methods ==
== CFD Methods ==
</span>
=== Reynolds-Averaged Navier-Stokes computations ===  
=== Reynolds-Averaged Navier-Stokes computations ===  
==== SSG/LRR-<math>\omega</math> model ====
For the entire computational domain, a structured 2D mesh was created using Pointwise V18.2. Sensitivity studies were carried out on various meshes and the final mesh used in this UFR contains a total of <math>266,112</math> points. Along the bottom contour <math> 448 </math> points are used in streamwise direction with a smaller spacing in the curved-step region. In the wall-normal direction  <math> 298 </math> points are used, <math> 98 </math> of which are concentrated near the wall. The wall-normal growth ratio is approximatively <math> 1,077 </math> and the dimensionless distance from the wall is <math>{y^+<1}</math> along the bottom wall for all meshes and simulation scenarios. The computational mesh close to the step is shown in Figure [[UFR_3-36_Test_Case#figure2|Fig. 3]].


For the entire computational domain, a structured 2D mesh was created using Pointwise V18.2. Sensitivity studies were carried out on various meshes and the final mesh used in this UFR contains a total of 266.112 points. Along the body contour 448 points are used in streamwise direction with a smaller spacing in the focus region. In the normal direction to the body wall 298 points are used, 98 of which are concentrated near the body wall region. The body wall-normal growth ratio is approximatively 1.077 and the dimensionless distance from the wall is <math>{y^+<1}</math> along the body wall for all  meshes and simulation scenarios.
<div id="figure3"></div>
For the inflow boundary situated at <math>{x = -11.8 H}</math> a reservoir-pressure inflow boundary condition is used. This boundary condition prescribes total pressure and total density. The inflow direction is by default perpendicular to the boundary face. For the outflow boundary at <math>{x = 20 H}</math> an exit-pressure outflow boundary condition is used. The exit pressure is adapted during the simulation to match the reference pressure at the coordinate point <math>{Z_{ref}  = (-5.18 H , 0 , 6.25 H)}</math>. The upper boundary is a far-field boundary condition situated <math>{179 H}</math> from the viscous body wall. Symmetry boundary condition is used on both side planes of the 2D domain. Different Reynolds numbers were simulated, two of them are presented in [[Lib:UFR_3-36_Test_Case#table2|Table 2]] with the corresponding reference parameters. Here we use a definition of the Reynolds number based on <math>{H}</math>  as Reynolds length.
{|align="center" border="0" width="500"
|[[Image: Wiki_Mesh_Figure3.png|500px]]
|-
!align="center"|Figure 3: Computational Mesh for RANS SSG/LRR-<math>\omega</math> simulation
|}
 
For the inflow boundary located at <math>{x = -11.84 H}</math> a reservoir-pressure inflow boundary condition is used. This boundary condition prescribes total pressure and total density. The inflow direction is by default perpendicular to the boundary face and a constant velocity is prescribed defined by the Mach number <math>Ma</math>. The turbulent kinetic energy entering the flow domain is computed according to the defined value of the turbulent intensity, which is set to <math>0.01</math> and the turbulent viscosity is defined using the ratio of eddy to molecular dynamic viscosity <math> \mu_{t}/\mu_{l}</math> set to <math>1</math>. Both values of the turbulent kinetic energy and the turbulent viscosity are required for defining reference values for the turbulent quantities (Reynolds stresses and specific turbulent dissipation rate) used for the inflow boundary condition and are selected based on best-practise. For the outflow boundary at <math>{x = 20 H}</math> an exit-pressure outflow boundary condition is used. The exit pressure is adapted during the simulation to match the reference pressure at the coordinate point <math>{Z_{ref}  = (-5.18 H , 0 , 6.22 H)}</math> in agreement with the original Disotell case. The upper boundary is a permeable far-field Riemann boundary condition located <math>{179 H}</math> from the bottom wall and computed via the approximate Riemann method of Roe. The SSG/LRR-<math>\omega</math> computations were conducted with a 3D solver. Hence, a symmetry boundary condition is used on both side planes of the 2D domain and the solver is operated in a 2D manner (only one cell in spanwise direction). Reference parameters for <math>{Re_H = 78,490}</math>  are presented in [[UFR_3-36_Test_Case#table2|Table 2]].  


<div id="table2"></div>
<div id="table2"></div>
{|align="center" border="1" cellpadding="10"
{|align="center" border="1" cellpadding="10"
|-
|-
|Parameter||align="center"|<math>{L_F}</math>||align="center"|<math>{L_B}</math>||align="center"|<math>{L_A}</math>
|Parameter||align="center"|<math>{Ma}</math>||align="center"|<math>{P_{exit}}</math>||align="center"|<math>{P_{t,inflow}}</math>||align="center"|<math>{\rho_{t,inflow}}</math>||align="center"|<math>{T_{t,inflow}}</math>||align="center"|<math>{P_{s,ref}}</math>||align="center"|<math>{\rho_{s,ref}}</math>||align="center"|<math>{T_{s,ref}}</math>||align="center"|<math>{P_{(Z{ref})}}</math>
|-
|-
|Value||align="center"|<math>{11.8 H}</math>||align="center"|<math>{5.5 H}</math>||align="center"|<math>{14.5 H}</math>
|Value||align="center"|<math>{0.13455}</math>||align="center"|<math>{89769 Pa}</math>||align="center"|<math>{93074 Pa}</math>||align="center"|<math>{1.0943 kg/m^3}</math>||align="center"|<math>{296.36 K}</math>||align="center"|<math>{89593.581 Pa}</math>||align="center"|<math>{1.065 kg/m^3}</math>||align="center"|<math>{293.15 K}</math>||align="center"|<math>{89300 Pa}</math>
|}
|}


<center>'''Table 2:''' Geometry parameters</center>
<center>'''Table 2:''' Boundary conditions </center>


=== DNS Computations ===
with the Mach number <math> Ma </math>, the exit pressure <math>{P_{exit}}</math>, total inflow pressure <math>{P_{t,inflow}}</math>, total inflow density <math>{\rho_{t,inflow}}</math>, total inflow temperature <math>{T_{t,inflow}}</math>, static pressure <math>{P_{s,ref}}</math> at the reference position <math> Z_{ref} </math>, static density <math>{\rho_{s,ref}}</math> at the reference position <math> Z_{ref} </math>, static temperature <math>{T_{s,ref}}</math> at the reference position <math> Z_{ref} </math> and the static pressure <math>{P_{(Z{ref})}}</math> at the reference position <math> Z_{ref} </math>       


The test case is designed as a numerical experiment with the aim of comparing RANS results to DNS data. For the set-up of the DNS, the inflow boundary conditions are different, e.g. a recycling method can be used to generate the turbulent input or synthetic turbulence can be injected. It is also possible to numerically trip the boundary layer from laminar to turbulent to generate the desired turbulent boundary layer. Hence, to ensure a comparison to the results achieved with RANS turbulence models, a reference position upstream of the APG-area is defined where boundary-layer properties need to match between RANS and DNS computations to permit the comparison downstream in the region of interest. The reference position is located at <math>{x/H = -3.51}</math>. Depending on the generation of turbulence at the inlet, the computational domain needs to be adapted to ensure the correct boundary-layer properties at the reference position. If numerical tripping is performed, the laminar and turbulent distances need to be determined upstream of the reference position by precursor simulation as displayed in [[Lib:UFR_3-36_Test_Case#figure3|Fig. 3]].
Simulations were performed using the DLR in-house software TAU [&#8204;[[UFR_3-36_References#8|8]]] where the seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR-<math>\omega</math> [&#8204;[[UFR_3-36_References#16|16]]] including the length scale correction [&#8204;[[UFR_3-36_References#17|17]]] is already implemented. TAU is a Finite-Volume-based unstructured cell-centered on dual-grids code of second-order accuracy. For the computations performed here, the mean-flow and turbulence convective terms are discretized using second-order central schemes together with Matrix Dissipation. Low-Mach number preconditioning was applied and steady computations using a LU-SGS scheme were performed. All results presented in this report are based on fully converged simulations.


<div id="figure3"></div>
This UFR was designed based on the SSG/LRR-<math>\omega</math> model results, hence, at a pre-defined check-point at <math>x_{ref} = -3.5 H</math> boundary-layer quantities were extracted to be matched by the DNS.
{|align="center" border="0" width="750"
 
|[[Image:UFR3-36_fig3.png|740px]]
==== k-<math>\omega</math> model ====
 
The RANS k-<math>\omega</math> computations have been performed by the University of Bergamo using the CFD code MIGALE. The solver uses the Discontinuous Galerkin (DG) method for the spatial discretization of the governing equations, here the compressible Reynolds-averaged Navier-Stokes equations coupled with the k-<math>\omega</math> closure model of Wilcox [&#8204;[[UFR_3-36_References#32|32]]][&#8204;[[UFR_3-36_References#28|28]]].
The DG method implemented in code MIGALE is able to guarantee high-accuracy (here up to sixth-order) on meshes made of elements of arbitrary shape [&#8204;[[UFR_3-36_References#29|29]]][&#8204;[[UFR_3-36_References#30|30]]]. The steady-state numerical solutions are sought by means of a Newton’s globalization strategy named pseudo-transient continuation, e.g, [&#8204;[[UFR_3-36_References#31|31]]]. Simulations have been performed on a 2D grid made of <math>19,796</math> quadrilateral elements with quadratic edges (see [[UFR_3-36_Test_Case#figure3|Fig. 4]] and [[UFR_3-36_Test_Case#figure4|Fig. 5]]) with a DG polynomial degree equal to <math>5</math> (sixth order).
Wall resolution in streamwise (<math>{\Delta x^{+}}</math>) and wall-normal (<math>{\Delta y^{+}}</math>) directions at different streamwise locations is reported in the following table
 
{|align="center" border="1" cellpadding="10"
|-
|align="center"|'''<math>{x/H}</math>'''||align="center"|'''<math>{\Delta x^{+}}</math>'''||align="center"|'''<math>{\Delta y_{1}^{+}}</math>'''
|-
|align="center"|'''<math>{-3.5}</math>'''||align="center"|'''<math>{258.35}</math>'''||align="center"|'''<math>{0.91}</math>'''
|-
|align="center"|'''<math>{3}</math>'''||align="center"|'''<math>{15.19}</math>'''||align="center"|'''<math>{0.31}</math>'''
|-
|align="center"|'''<math>{4}</math>'''||align="center"|'''<math>{12.10}</math>'''||align="center"|'''<math>{0.28}</math>'''
|-
|align="center"|'''<math>{5}</math>'''||align="center"|'''<math>{41.83}</math>'''||align="center"|'''<math>{0.50}</math>'''
|-
|align="center"|'''<math>{6}</math>'''||align="center"|'''<math>{61.62}</math>'''||align="center"|'''<math>{0.67}</math>'''
|}
<center>'''Table 3:''' Wall resolution</center>
 
The wall resolution takes into account of the degree of the DG polynomial approximation. In particular, it is defined as
 
<math>\Delta x^{+}=\dfrac{\Delta x}{\sqrt[3]{N_{DoF}}\delta_{\nu}}</math>,
 
<math>\delta_{\nu}</math>  being the viscous length scale and <math>N_{DoF}</math> the number of Degrees of Freedom per equation within the mesh element, here <math>N_{DoF}=56</math>.
 
The flow problem is statistically two-dimensional since the turbulent flow is homogeneous in the span direction.
As the k-<math>\omega</math> computations were perfomed with a 3D solver, a symmetry boundary condition is used on the side planes of the two-dimensional domain and only one cell in the spanwise direction is considered.
 
The computational domain of the RANS k-<math>\omega</math> simulations is designed to match the value of the boundary layer thickness, <math>\delta_{99}^{uDNS}</math>, predicted by the uDNS at the checkpoint, cf. [[DNS_1-5]].
To this purpose, a precursor computation on a zero-pressure-gradient flat plate of length <math>L</math> was carried out. To mimic a free-stream approaching the plate, the domain was extended <math>2L</math> upstream of the solid wall and a symmetry condition was set on the lower part of this extension.
For the HiFi-TURB-DLR rounded step, the length of the no-slip wall upstream of the checkpoint <math>x_{ref}/H=-3.5</math> was set equal to the distance from the leading edge of the precursor flat plate to the streamwise coordinate where <math>\delta_{99}^{RANS}=\delta_{99}^{uDNS}</math>, i.e., <math>\Delta x/H = 10.17</math>. Accordingly, the leading edge of the no-slip solid wall is located at <math>x/H = -13.67</math>.
To mitigate spurious perturbations possibly originating at the outlet boundary, a “sacrificial buffer” is created downstream of the rounded step, i.e., at <math>x/H > 15.79</math>. In this region, a symmetry condition is set on the horizontal boundary and mesh coarsening is applied to reduce the solution gradients.
A near-wall detail of the domain together with the imposed boundary conditions is shown in [[UFR_3-36_Test_Case#figure5|Fig. 6]].
 
At the inlet boundary, located at <math> x/H=-41.44 </math>, the total pressure and temperature are set to the values <math>{P_{t,inflow}}</math> and <math>{T_{t,inflow}}</math>, respectively. The turbulent intensity at the domain entrance is <math>Tu = 0.001</math> and the ratio of eddy to molecular dynamic viscosity is equal to <math>\mu_{t}/\mu_{l} = 0.01</math>.
At the outlet boundary, placed at <math> x/H=28.12 </math>, the static pressure <math>{P_{s,ref}}</math> is imposed.
The upper boundary is a permeable far-field Riemann boundary condition and is located <math>179 H</math> from the solid wall upstream of the step and computed via the exact Riemann solver.
 
<div id="figure4"></div>
{|align="center" border="0" width="900"
|[[Image:UFR3-36RoundedStep_k-omega_mesh_bottom.png|900px]]
|-
!align="center"|Figure 4: Detail of the near wall mesh for the RANS k-<math>\omega</math> model simulation
|}
 
<div id="figure5"></div>
{|align="center" border="0" width="700"
|[[Image:UFR 3-36 Rounded Step k-omega grid.png|700px]]
|-
!align="center"|Figure 5: Detail of the computational grid region around the rounded step for the RANS k-<math>\omega</math> model simulation
|}
 
<div id="figure6"></div>
{|align="center" border="0" width="900"
|[[Image:UFR 3-36 Rounded Step k-omega bottom boundary conditions.png|900px]]
|-
|-
!align="center"|Figure 3: Set-up for DNS Simulation
!align="center"|Figure 6: Detail of the wall region flow domain together with the imposed boundary conditions for the RANS k-<math>\omega</math> model simulation
|}
|}


At the reference position, the properties of the turbulent boundary layer are determined by the Reynolds number based on the momentum thickness <math>{Re_{\theta} = 1496}</math> and the Reynolds number based on the friction velocity <math>{Re_{\tau} = 609}</math> computed with SA-neg model with RC and QCR extensions and with Low-Reynolds modification [&#8204;[[Lib:UFR_3-36_References#13|13]]].
=== Data provided ===
 
Data provided here are the wall quantities and profiles with results from both, SSG/LRR-<math>\omega</math> as well as k-<math>\omega</math>, simulations and Reynolds Stress profiles from SSG/LRR-<math>\omega</math> simulations. All quantities are non-dimensionalised using the step height <math>H</math> and the freestream velocity <math>{U_\infty}</math>:  


Between the different flow condition proposed, UNIBG focused the effort on the incipient separation configuration case with <math>{Re_H = 78,490}</math>. A precursory computational campaign for the turbulent flow over a flat plate has been performed with the purpose of (i) assessing the effectiveness of the Synthetic Inlet Turbulence (SIT) injection strategy inspired by the work of Housseini et al. [&#8204;[[Lib:UFR_3-36_References#18|18]]] and Schlatter and Örlü [&#8204;[[Lib:UFR_3-36_References#19|19]]]; (ii) investigating the influence of the mesh density on the solution; (iii) defining the inlet boundary position and flow condition that ensure the target boundary layer integral parameters at the reference location <math>{x_{ref}}</math> (see Figure 3). According to the outcomes of this campaign, the inlet boundary is set at <math>{x/H = -12.71}</math>, where the Blasius laminar velocity profile computed at <math>{Re_x=650,000}</math>, the uniform static pressure and the uniform total temperature (see Table 2) are imposed. At location <math>{x/H = -12.1}</math>, within the laminar boundary layer region, the tripping source term is activated to promote the transition to turbulence [&#8204;[[Lib:UFR_3-36_References#18|18]]][&#8204;[[Lib:UFR_3-36_References#19|19]]]. The outlet boundary is positioned at <math>{x/H = 23.95}</math>, with a pressure outflow condition having the same value of the inlet static pressure and at the end of a streamwise coarsened mesh region starting at <math>{x/H = 13.82}</math>. Following the setup of the RANS computations, the no-slip adiabatic boundary condition is set while at the upper boundary, situated at <math>{179 H}</math> from the wall upstream the smooth bump, the far-field boundary condition is imposed. Side planes, instead, are considered as periodic with a distance from each other of <math>{\Delta z = 3 H}</math>.
* [[Media:Lib-UFR3-36 WallQuantities.dat|WallQuantities.dat]] contains the following wall quantities extracted at the curved step viscous wall:
<br/><center><math>{
\begin{align}
\mathrm{x^*} &= x/H  \\
\mathrm{y_{wall}^*} &= y/H \\
\mathrm{C_p} &= \frac{p-p_{ref}}{\frac{1}{2} \rho U_{ref}^2}  \\
\mathrm{tau\_w} &= \mu \left.\frac{\partial U}{\partial n}\right|_{y=y_{wall}}  \\
\mathrm{\delta*}&= \int_{y_{wall}}^{\infty} \left( 1 - \frac{U}{U_{edge}} \right) dy \\
\mathrm{\theta}&= \int_{y_{wall}}^{\infty} \frac{U}{U_{edge}} \left( 1 - \frac{U}{U_{edge}} \right) dy
\end{align}
}</math></center><br/>  


The DNS have been performed by using the UNIBG in-house software MIGALE [&#8204;[[Lib:UFR_3-36_References#20|20]]]. MIGALE is an implicit high-order discontinuous Galerkin solver for the compressible and incompressible Navier-Stokes equations. Godunov fluxes are treated with the exact solution of the local Riemann problems while viscous fluxes are handled by means of the BR2 scheme [&#8204;[[Lib:UFR_3-36_References#21|21]]]. The time integration is performed with linearly implicit Rosenbrock type Runge-Kutta schemes with optimal stability properties up to order five.
where <math> \rho </math> and <math> \mu </math> are local values, <math> U_{ref} </math> and <math> p_{ref} </math> are reference values at the reference position <math> Z_{ref} </math> and <math> U_{edge} </math> is the magnitude of wall-tangential velocity at the boundary edge.
The computational mesh is made of 15,016,384 hexahedral elements with quadratic edges concentrated near the wall region. The wall-normal growth ratio is approximatively 1.2 with a first cell height of <math>{y^+ \approx 1}</math>. Time integration is performed with the fifth order – eight stages Rosenbrock scheme ROD5_1 [&#8204;[[Lib:UFR_3-36_References#22|22]]] using a global time step adaptation strategy [&#8204;[[Lib:UFR_3-36_References#23|23]]]. The corresponding average step size is <math>{\Delta t = 19/14000}</math> CTU, where the convective time unit (CTU) is defined as the ratio between <math>{H}</math> and the freestream velocity. Turbulence statistics have been collected for 26 CTU.
* [[Media:Lib-UFR3-36 profiles.dat|profiles.dat]] contains the  velocity and Reynolds-Stress profiles in the vertical direction:<center> <math> x, y, U, V, \overline{u'u'},  \overline{v'v'}, \overline{w'w'}, \overline{u'v'} </math></center> for 37 different streamwise locations (<math>{x/H}</math> = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 4, 4.5, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14).</li><br/>


<br/>
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|organisation=Deutsches Luft-und Raumfahrt Zentrum (DLR)
}}
}}
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{{UFRHeader
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© copyright ERCOFTAC {{CURRENTYEAR}}
© copyright ERCOFTAC {{CURRENTYEAR}}
-

Latest revision as of 15:53, 17 February 2023

HiFi-TURB-DLR rounded step

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

Semi-confined flows

Underlying Flow Regime 3-36

Test Case Study

Brief Description of the Study Test Case

In the framework of the HiFi-TURB project, four different geometries with different boundary-layer properties were designed. Each configuration was computed for two different Reynolds numbers. The configuration concerns a turbuluent boundary layer incipient to separation where the flow is on the brink of separation but does not separate at the Reynolds number based on the step height . The geometry of this UFR is shown in Fig. 2. The geometry comprises three main sections: An upstream section with , the curved-step section with length and the downstream section with . The length of the step is fixed for all geometries, while the height is adapted to generate the desired APG. For the incipient separation testcase the step height equals .

Wiki Figure2.png
Figure 2: Flow Domain

The parametric geometry definition for the curved-step sections is given in [‌6] and is depicted in Fig. 2. The origin is set at the beginning of the curved-step section.

with , , and .

CFD Methods

Reynolds-Averaged Navier-Stokes computations

SSG/LRR- model

For the entire computational domain, a structured 2D mesh was created using Pointwise V18.2. Sensitivity studies were carried out on various meshes and the final mesh used in this UFR contains a total of points. Along the bottom contour points are used in streamwise direction with a smaller spacing in the curved-step region. In the wall-normal direction points are used, of which are concentrated near the wall. The wall-normal growth ratio is approximatively and the dimensionless distance from the wall is along the bottom wall for all meshes and simulation scenarios. The computational mesh close to the step is shown in Figure Fig. 3.

Wiki Mesh Figure3.png
Figure 3: Computational Mesh for RANS SSG/LRR- simulation

For the inflow boundary located at a reservoir-pressure inflow boundary condition is used. This boundary condition prescribes total pressure and total density. The inflow direction is by default perpendicular to the boundary face and a constant velocity is prescribed defined by the Mach number . The turbulent kinetic energy entering the flow domain is computed according to the defined value of the turbulent intensity, which is set to and the turbulent viscosity is defined using the ratio of eddy to molecular dynamic viscosity set to . Both values of the turbulent kinetic energy and the turbulent viscosity are required for defining reference values for the turbulent quantities (Reynolds stresses and specific turbulent dissipation rate) used for the inflow boundary condition and are selected based on best-practise. For the outflow boundary at an exit-pressure outflow boundary condition is used. The exit pressure is adapted during the simulation to match the reference pressure at the coordinate point in agreement with the original Disotell case. The upper boundary is a permeable far-field Riemann boundary condition located from the bottom wall and computed via the approximate Riemann method of Roe. The SSG/LRR- computations were conducted with a 3D solver. Hence, a symmetry boundary condition is used on both side planes of the 2D domain and the solver is operated in a 2D manner (only one cell in spanwise direction). Reference parameters for are presented in Table 2.

Parameter
Value
Table 2: Boundary conditions

with the Mach number , the exit pressure , total inflow pressure , total inflow density , total inflow temperature , static pressure at the reference position , static density at the reference position , static temperature at the reference position and the static pressure at the reference position

Simulations were performed using the DLR in-house software TAU [‌8] where the seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR- [‌16] including the length scale correction [‌17] is already implemented. TAU is a Finite-Volume-based unstructured cell-centered on dual-grids code of second-order accuracy. For the computations performed here, the mean-flow and turbulence convective terms are discretized using second-order central schemes together with Matrix Dissipation. Low-Mach number preconditioning was applied and steady computations using a LU-SGS scheme were performed. All results presented in this report are based on fully converged simulations.

This UFR was designed based on the SSG/LRR- model results, hence, at a pre-defined check-point at boundary-layer quantities were extracted to be matched by the DNS.

k- model

The RANS k- computations have been performed by the University of Bergamo using the CFD code MIGALE. The solver uses the Discontinuous Galerkin (DG) method for the spatial discretization of the governing equations, here the compressible Reynolds-averaged Navier-Stokes equations coupled with the k- closure model of Wilcox [‌32][‌28]. The DG method implemented in code MIGALE is able to guarantee high-accuracy (here up to sixth-order) on meshes made of elements of arbitrary shape [‌29][‌30]. The steady-state numerical solutions are sought by means of a Newton’s globalization strategy named pseudo-transient continuation, e.g, [‌31]. Simulations have been performed on a 2D grid made of quadrilateral elements with quadratic edges (see Fig. 4 and Fig. 5) with a DG polynomial degree equal to (sixth order). Wall resolution in streamwise () and wall-normal () directions at different streamwise locations is reported in the following table

Table 3: Wall resolution

The wall resolution takes into account of the degree of the DG polynomial approximation. In particular, it is defined as

,

  being the viscous length scale and  the number of Degrees of Freedom per equation within the mesh element, here .

The flow problem is statistically two-dimensional since the turbulent flow is homogeneous in the span direction. As the k- computations were perfomed with a 3D solver, a symmetry boundary condition is used on the side planes of the two-dimensional domain and only one cell in the spanwise direction is considered.

The computational domain of the RANS k- simulations is designed to match the value of the boundary layer thickness, , predicted by the uDNS at the checkpoint, cf. DNS_1-5. To this purpose, a precursor computation on a zero-pressure-gradient flat plate of length was carried out. To mimic a free-stream approaching the plate, the domain was extended upstream of the solid wall and a symmetry condition was set on the lower part of this extension. For the HiFi-TURB-DLR rounded step, the length of the no-slip wall upstream of the checkpoint was set equal to the distance from the leading edge of the precursor flat plate to the streamwise coordinate where , i.e., . Accordingly, the leading edge of the no-slip solid wall is located at . To mitigate spurious perturbations possibly originating at the outlet boundary, a “sacrificial buffer” is created downstream of the rounded step, i.e., at . In this region, a symmetry condition is set on the horizontal boundary and mesh coarsening is applied to reduce the solution gradients. A near-wall detail of the domain together with the imposed boundary conditions is shown in Fig. 6.

At the inlet boundary, located at , the total pressure and temperature are set to the values and , respectively. The turbulent intensity at the domain entrance is and the ratio of eddy to molecular dynamic viscosity is equal to . At the outlet boundary, placed at , the static pressure is imposed. The upper boundary is a permeable far-field Riemann boundary condition and is located from the solid wall upstream of the step and computed via the exact Riemann solver.

UFR3-36RoundedStep k-omega mesh bottom.png
Figure 4: Detail of the near wall mesh for the RANS k- model simulation
UFR 3-36 Rounded Step k-omega grid.png
Figure 5: Detail of the computational grid region around the rounded step for the RANS k- model simulation
UFR 3-36 Rounded Step k-omega bottom boundary conditions.png
Figure 6: Detail of the wall region flow domain together with the imposed boundary conditions for the RANS k- model simulation

Data provided

Data provided here are the wall quantities and profiles with results from both, SSG/LRR- as well as k-, simulations and Reynolds Stress profiles from SSG/LRR- simulations. All quantities are non-dimensionalised using the step height and the freestream velocity :

  • WallQuantities.dat contains the following wall quantities extracted at the curved step viscous wall:



where and are local values, and are reference values at the reference position and is the magnitude of wall-tangential velocity at the boundary edge.

  • profiles.dat contains the velocity and Reynolds-Stress profiles in the vertical direction:
    for 37 different streamwise locations ( = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 4, 4.5, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14).




Contributed by: Erij Alaya and Cornelia Grabe — Deutsches Luft-und Raumfahrt Zentrum (DLR)

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