# 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 ${\displaystyle {Re_{H}=78,490}}$ based on the step height ${\displaystyle {H}}$. The geometry of this UFR is shown in Fig. 2. The geometry comprises three main sections: An upstream section with ${\displaystyle {11.84H}}$, the curved-step section with length ${\displaystyle {L=5.53H}}$ and the downstream section with ${\displaystyle {14.47H}}$. The length ${\displaystyle {L}}$ of the step is fixed for all geometries, while the height ${\displaystyle {H}}$ is adapted to generate the desired APG. For the incipient separation testcase the step height equals ${\displaystyle {H=0.028956m}}$.

The parametric geometry definition for the curved-step sections is given in [‌6] and is depicted in Fig. 2. The origin ${\displaystyle {x=0}}$ is set at the beginning of the curved-step section.

{\displaystyle {\begin{aligned}&y_{wall}&&=H&&\mathrm {for} \,-11.84H

with ${\displaystyle {a_{1}=H}}$, ${\displaystyle {a_{2}=-10H/L^{3}}}$, ${\displaystyle {a_{3}=15H/L^{4}}}$ and ${\displaystyle {a_{4}=-6H/L^{5}}}$.

## CFD Methods

### Reynolds-Averaged Navier-Stokes computations

#### SSG/LRR-${\displaystyle \omega }$ 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 ${\displaystyle 266,112}$ points. Along the bottom contour ${\displaystyle 448}$ points are used in streamwise direction with a smaller spacing in the curved-step region. In the wall-normal direction ${\displaystyle 298}$ points are used, ${\displaystyle 98}$ of which are concentrated near the wall. The wall-normal growth ratio is approximatively ${\displaystyle 1,077}$ and the dimensionless distance from the wall is ${\displaystyle {y^{+}<1}}$ along the bottom wall for all meshes and simulation scenarios. The computational mesh close to the step is shown in Figure Fig. 3.

For the inflow boundary located at ${\displaystyle {x=-11.84H}}$ 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 ${\displaystyle Ma}$. The turbulent kinetic energy entering the flow domain is computed according to the defined value of the turbulent intensity, which is set to ${\displaystyle 0.01}$ and the turbulent viscosity is defined using the ratio of eddy to molecular dynamic viscosity ${\displaystyle \mu _{t}/\mu _{l}}$ set to ${\displaystyle 1}$. 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 ${\displaystyle {x=20H}}$ 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 ${\displaystyle {Z_{ref}=(-5.18H,0,6.22H)}}$ in agreement with the original Disotell case. The upper boundary is a permeable far-field Riemann boundary condition located ${\displaystyle {179H}}$ from the bottom wall and computed via the approximate Riemann method of Roe. The SSG/LRR-${\displaystyle \omega }$ 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 ${\displaystyle {Re_{H}=78,490}}$ are presented in Table 2.

 Parameter ${\displaystyle {Ma}}$ ${\displaystyle {P_{exit}}}$ ${\displaystyle {P_{t,inflow}}}$ ${\displaystyle {\rho _{t,inflow}}}$ ${\displaystyle {T_{t,inflow}}}$ ${\displaystyle {P_{s,ref}}}$ ${\displaystyle {\rho _{s,ref}}}$ ${\displaystyle {T_{s,ref}}}$ ${\displaystyle {P_{(Z{ref})}}}$ Value ${\displaystyle {0.13455}}$ ${\displaystyle {89769Pa}}$ ${\displaystyle {93074Pa}}$ ${\displaystyle {1.0943kg/m^{3}}}$ ${\displaystyle {296.36K}}$ ${\displaystyle {89593.581Pa}}$ ${\displaystyle {1.065kg/m^{3}}}$ ${\displaystyle {293.15K}}$ ${\displaystyle {89300Pa}}$
Table 2: Boundary conditions

with the Mach number ${\displaystyle Ma}$, the exit pressure ${\displaystyle {P_{exit}}}$, total inflow pressure ${\displaystyle {P_{t,inflow}}}$, total inflow density ${\displaystyle {\rho _{t,inflow}}}$, total inflow temperature ${\displaystyle {T_{t,inflow}}}$, static pressure ${\displaystyle {P_{s,ref}}}$ at the reference position ${\displaystyle Z_{ref}}$, static density ${\displaystyle {\rho _{s,ref}}}$ at the reference position ${\displaystyle Z_{ref}}$, static temperature ${\displaystyle {T_{s,ref}}}$ at the reference position ${\displaystyle Z_{ref}}$ and the static pressure ${\displaystyle {P_{(Z{ref})}}}$ at the reference position ${\displaystyle Z_{ref}}$

Simulations were performed using the DLR in-house software TAU [‌8] where the seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR-${\displaystyle \omega }$ [‌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-${\displaystyle \omega }$ model results, hence, at a pre-defined check-point at ${\displaystyle x_{ref}=-3.5H}$ boundary-layer quantities were extracted to be matched by the DNS.

#### k-${\displaystyle \omega }$ model

The RANS k-${\displaystyle \omega }$ 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-${\displaystyle \omega }$ 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 ${\displaystyle 19,796}$ quadrilateral elements with quadratic edges (see Fig. 4 and Fig. 5) with a DG polynomial degree equal to ${\displaystyle 5}$ (sixth order). Wall resolution in streamwise (${\displaystyle {\Delta x^{+}}}$) and wall-normal (${\displaystyle {\Delta y^{+}}}$) directions at different streamwise locations is reported in the following table

 ${\displaystyle {x/H}}$ ${\displaystyle {\Delta x^{+}}}$ ${\displaystyle {\Delta y_{1}^{+}}}$ ${\displaystyle {-3.5}}$ ${\displaystyle {258.35}}$ ${\displaystyle {0.91}}$ ${\displaystyle {3}}$ ${\displaystyle {15.19}}$ ${\displaystyle {0.31}}$ ${\displaystyle {4}}$ ${\displaystyle {12.10}}$ ${\displaystyle {0.28}}$ ${\displaystyle {5}}$ ${\displaystyle {41.83}}$ ${\displaystyle {0.50}}$ ${\displaystyle {6}}$ ${\displaystyle {61.62}}$ ${\displaystyle {0.67}}$
Table 3: Wall resolution

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

${\displaystyle \Delta x^{+}={\dfrac {\Delta x}{{\sqrt[{3}]{N_{DoF}}}\delta _{\nu }}}}$,

${\displaystyle \delta _{\nu }}$  being the viscous length scale and ${\displaystyle N_{DoF}}$ the number of Degrees of Freedom per equation within the mesh element, here ${\displaystyle N_{DoF}=56}$.


The flow problem is statistically two-dimensional since the turbulent flow is homogeneous in the span direction. As the k-${\displaystyle \omega }$ 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-${\displaystyle \omega }$ simulations is designed to match the value of the boundary layer thickness, ${\displaystyle \delta _{99}^{uDNS}}$, 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 ${\displaystyle L}$ was carried out. To mimic a free-stream approaching the plate, the domain was extended ${\displaystyle 2L}$ 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 ${\displaystyle x_{ref}/H=-3.5}$ was set equal to the distance from the leading edge of the precursor flat plate to the streamwise coordinate where ${\displaystyle \delta _{99}^{RANS}=\delta _{99}^{uDNS}}$, i.e., ${\displaystyle \Delta x/H=10.17}$. Accordingly, the leading edge of the no-slip solid wall is located at ${\displaystyle x/H=-13.67}$. To mitigate spurious perturbations possibly originating at the outlet boundary, a “sacrificial buffer” is created downstream of the rounded step, i.e., at ${\displaystyle x/H>15.79}$. 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 ${\displaystyle x/H=-41.44}$, the total pressure and temperature are set to the values ${\displaystyle {P_{t,inflow}}}$ and ${\displaystyle {T_{t,inflow}}}$, respectively. The turbulent intensity at the domain entrance is ${\displaystyle Tu=0.001}$ and the ratio of eddy to molecular dynamic viscosity is equal to ${\displaystyle \mu _{t}/\mu _{l}=0.01}$. At the outlet boundary, placed at ${\displaystyle x/H=28.12}$, the static pressure ${\displaystyle {P_{s,ref}}}$ is imposed. The upper boundary is a permeable far-field Riemann boundary condition and is located ${\displaystyle 179H}$ from the solid wall upstream of the step and computed via the exact Riemann solver.

### Data provided

Data provided here are the wall quantities and profiles with results from both, SSG/LRR-${\displaystyle \omega }$ as well as k-${\displaystyle \omega }$, simulations and Reynolds Stress profiles from SSG/LRR-${\displaystyle \omega }$ simulations. All quantities are non-dimensionalised using the step height ${\displaystyle H}$ and the freestream velocity ${\displaystyle {U_{\infty }}}$:

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

{\displaystyle {\begin{aligned}\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{aligned}}}

where ${\displaystyle \rho }$ and ${\displaystyle \mu }$ are local values, ${\displaystyle U_{ref}}$ and ${\displaystyle p_{ref}}$ are reference values at the reference position ${\displaystyle Z_{ref}}$ and ${\displaystyle U_{edge}}$ is the magnitude of wall-tangential velocity at the boundary edge.

• profiles.dat contains the velocity and Reynolds-Stress profiles in the vertical direction:
${\displaystyle x,y,U,V,{\overline {u'u'}},{\overline {v'v'}},{\overline {w'w'}},{\overline {u'v'}}}$
for 37 different streamwise locations (${\displaystyle {x/H}}$ = -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)