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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<!--{{LoremIpsum}}-->
<!--{{LoremIpsum}}-->
== Brief Description of the Study Test Case ==
== 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 with two different Reynolds numbers. The configuration described here presents a turbuluent boundary layer incipient to separation where the flow is on the brink of separation but does not separate with Reynolds number <math>{Re_H = 78,490}</math> based on the step height <math>{H}</math>. The geometry of this UFR alongside the mesh is shown in [[Lib:UFR_3-36_Test_Case#figure2|Fig. 2]]. The geometry comprises three main sections: Constant-height upstream section <math>{L_F}</math> with the largest height <math>{Y_F}</math>, curved-step section <math>{L_B}</math> with the contoured height <math>{Y_B}</math> and constant-height downstream section <math>{L_A}</math> with the smallest height <math>{Y_A}</math>. The height of the upstream section is fixed while the height of the downstream 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>. For incipient separation, <math> \sigma = 0.62 </math>. The height of each section is measured from a fixed arbitrary position for which <math> Y_F = 0.0762 </math>m.
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="figure2"></div>
<div id="figure2"></div>
{|align="center" border="0" width="700"
{|align="center" border="0" width="700"
|[[Image:Figure1_FlowDomain.png|700px]]
|[[Image: Wiki_Figure2.png|700px]]
|-
|-
!align="center"|Figure 2: 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#figure2|Fig. 2]]. The axial origin <math>{x=0}</math> is set at the beginning of the curved-step 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.  


<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 = 0.62}</math> and <math> H = Y_F - Y_A </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>.
 
The different parameters with the corresponding values are listed in the table below:
 
<div id="table1"></div>
{|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>
|-
|Value||align="center"|<math>{11.8 H}</math>||align="center"|<math>{5.5 H}</math>||align="center"|<math>{14.5 H}</math>
|}
 
<center>'''Table 1:''' Geometry parameters</center>


<span id="CFD Methods">
<span id="CFD Methods">
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=== Reynolds-Averaged Navier-Stokes computations ===  
=== Reynolds-Averaged Navier-Stokes computations ===  
==== SSG/LRR-<math>\omega</math> model ====
==== 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 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 wall for all meshes and simulation scenarios.
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 inflow boundary located 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 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 of the turbulent quantities used for the inflow boundary condition. 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 permeable far-field Riemann boundary condition located <math>{179 H}</math> from the viscous 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 [[Lib:UFR_3-36_Test_Case#table2|Table 2]].  
<div id="figure3"></div>
{|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>
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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>         
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>         


Simulations were performed using the DLR in-house software TAU [&#8204;[[Lib:UFR_3-36_References#8|8]]] where the seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR-<math>\omega</math> [&#8204;[[Lib:UFR_3-36_References#16|16]]] including the length scale correction [&#8204;[[Lib: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.
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.
 
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.


==== k-<math>\omega</math> model ====
==== k-<math>\omega</math> model ====


The RANS k-<math>\omega</math> computations have been performed 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 [&#8204;[[Lib:UFR_3-36_References#28|28]]].  
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;[[Lib:UFR_3-36_References#29|29]]][&#8204;[[Lib:UFR_3-36_References#30|30]]].  
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
The steady-state numerical solutions are sought by means of a Newton’s globalization strategy named pseudo-transient continuation, e.g, [&#8204;[[Lib: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 [[Lib:UFR_3-36_Test_Case#figure3|Fig. 3]] and [[Lib:UFR_3-36_Test_Case#figure4|Fig. 4]]) with a DG polynomial degree equal to <math>5</math> (sixth order).
Wall resolution in stream (<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" border="1" cellpadding="10"
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<math>\Delta x^{+}=\dfrac{\Delta x}{\sqrt[3]{N_{DoF}}\delta_{\nu}}</math>,
<math>\Delta x^{+}=\dfrac{\Delta x}{\sqrt[3]{N_{DoF}}\delta_{\nu}}</math>,


being <math>\delta_{\nu}</math> 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>.
<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.
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.
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. [[Lib:DNS_1-5]].  
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.  
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>.
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.
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 [[Lib:UFR_3-36_Test_Case#figure4|Fig. 4]].
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 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.
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 and computed via the exact Riemann solver.
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>
<div id="figure4"></div>
{|align="center" border="0" width="800"
{|align="center" border="0" width="900"
|[[Image:UFR3-36RoundedStep_k-omega_mesh_bottom.png|800px]]
|[[Image:UFR3-36RoundedStep_k-omega_mesh_bottom.png|900px]]
|-
|-
!align="center"|Figure 3: Detail of the near wall mesh for the RANS k-<math>\omega</math> model simulation  
!align="center"|Figure 4: Detail of the near wall mesh for the RANS k-<math>\omega</math> model simulation  
|}
|}


<div id="figure4"></div>
<div id="figure5"></div>
{|align="center" border="0" width="700"
{|align="center" border="0" width="700"
|[[Image:UFR 3-36 Rounded Step k-omega grid.png|700px]]
|[[Image:UFR 3-36 Rounded Step k-omega grid.png|700px]]
|-
|-
!align="center"|Figure 4: Detail of the computational grid around the HiFi-TURB-DLR rounded step for the RANS k-<math>\omega</math> model simulation  
!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="figure5"></div>
<div id="figure6"></div>
{|align="center" border="0" width="800"
{|align="center" border="0" width="900"
|[[Image:UFR 3-36 Rounded Step k-omega bottom boundary conditions.png|800px]]
|[[Image:UFR 3-36 Rounded Step k-omega bottom boundary conditions.png|900px]]
|-
|-
!align="center"|Figure 5: Detail of the wall region flow domain together with the imposed boundary conditions for the RANS k-<math>\omega</math> model 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  
|}
|}


=== Data provided ===  
=== Data provided ===  


Data provided here are from both SSG/LRR-<math>\omega</math> as well as k-<math>\omega</math> simulations. All quantities are non-dimensionalised using the step height <math>H</math> and the freestream velocity <math>{U_\infty}</math>:  
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>:  


* [[Media:Lib-UFR3-36 WallQuantities.dat|WallQuantities.dat]] contains the following wall quantities extracted at the curved step viscous wall:
* [[Media:Lib-UFR3-36 WallQuantities.dat|WallQuantities.dat]] contains the following wall quantities extracted at the curved step viscous wall:
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\begin{align}
\begin{align}
\mathrm{x^*} &= x/H  \\
\mathrm{x^*} &= x/H  \\
\mathrm{y_{wall}^*} &= \frac{y-Y_A}{H} \\
\mathrm{y_{wall}^*} &= y/H  \\
\mathrm{C_p} &= \frac{p-p_{ref}}{\frac{1}{2} \rho U_{ref}^2}  \\
\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{tau\_w} &= \mu \left.\frac{\partial U}{\partial n}\right|_{y=y_{wall}}  \\
Line 154: Line 148:


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.  
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.  
* [[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, W, u', v', w', u'v', u'w', v'w' </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/>
* [[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/>
<br/>
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|organisation=Deutsches Luft-und Raumfahrt Zentrum (DLR)
|organisation=Deutsches Luft-und Raumfahrt Zentrum (DLR)
}}
}}
{{UFRHeaderLib
{{UFRHeader
|area=3
|area=3
|number=36
|number=36

Latest revision as of 15:53, 17 February 2023

HiFi-TURB-DLR rounded step

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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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