UFR 3-36 Test Case

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Test Case Study

Brief Description of the Study Test Case

In the framework of the project, four different geometries with different separation behaviors were designed. Each configuration was computed with two different Reynolds numbers. The configuration described here presents an incipient separation test case where the flow is on the brink of separation but does not separate with Reynolds number ${Re_{H}=78,490}$ based on the step height ${H}$.The geometry of this UFR alongside the mesh is shown in Fig. 2. The geometry comprises three main sections: Constant-Width Forebody section ${L_{F}}$ with the largest width ${Y_{F}}$, Contoured Boat-tail section ${L_{B}}$ with the contoured width ${Y_{B}}$ and Constant-Width-Aftbody section ${L_{A}}$ with the smallest width ${Y_{A}}$. The width of the first section is fixed and the width of the last section is modified to generate the desired APG. This modification is achieved through the variation of ${\sigma }$, which is the ratio of ${Y_{A}}$ to ${Y_{F}}$. For incipient separation, $\sigma =0.62$. The width of each section is measured from a fixed arbitrary position for which $Y_{F}=0.0762$.

Figure 2: Flow Domain and grid of RANS simulations

The parametric geometry definition for the three relevant sections is given in [‌6] and is depicted in Fig. 2. The axial origin ${x=0}$ is set at the beginning of the Contoured Boat-tail section.

${\begin{aligned}y_{wall}&=Y_{F}=H/(1-\sigma )\qquad \mathrm {for} \qquad -L_{F}<x/H\leq 0\\y_{wall}&=Y_{B}=a_{1}+a_{2}x^{3}+a_{3}x^{4}+a_{4}x^{5}\qquad \mathrm {for} \qquad 0<x/H\leq L_{B}\qquad \qquad (1)\\y_{wall}&=Y_{A}=\sigma H/(1-\sigma )\qquad \mathrm {for} \qquad L_{B}<x/H\leq L_{B}+L_{A}\\\\\end{aligned}}$

with ${a_{1}=H/(1-\sigma )}$, ${a_{2}=-10H/L_{B}^{3}}$, ${a_{3}=15H/L_{B}^{4}}$ and ${a_{4}=-6H/L_{B}^{5}}$ with ${\sigma =Y_{A}/Y_{F}=0.62}$.

The different parameters with the corresponding values are listed in the table below:

Parameter	${L_{F}}$	${L_{B}}$	${L_{A}}$
Value	${11.8H}$	${5.5H}$	${14.5H}$

Table 1: Geometry parameters

CFD Methods

Reynolds-Averaged Navier-Stokes computations

SSG/LRR-$\omega$ 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 $266,112$ 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 ${y^{+}<1}$ along the body wall for all meshes and simulation scenarios. For the inflow boundary situated at ${x=-11.8H}$ 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. The turbulent kinetic energy entering the flow domain is computed according to the defined value of the turbulent intensity, which is set to $0.01$ and the turbulent viscosity is defined using the ratio of eddy to molecular dynamic viscosity $\mu _{t}/\mu _{l}$ set to $0.1$. Both values of the turbulent kinetic energy and the viscosity are required for defining reference values of the turbulent quantities used for the inflow boundary condition. For the outflow boundary at ${x=20H}$ 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 ${Z_{ref}=(-5.18H,0,6.25H)}$. The upper boundary is a permeable far-field Riemann boundary condition situated ${179H}$ from the viscous body wall and computed via the approximate Riemann method of Roe. The SSG/LRR-$\omega$ computations were conducted with a 3D solver. Hence, symmetry boundary condition is used on both side planes of the 2D domain and a 2D plugin function is switched on. Reference parameters for ${Re_{H}=78,490}$ are presented in Table 2.

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

Table 2: Boundary conditions

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

Simulations were performed using the DLR in-house software TAU [‌8] where the seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR-$\omega$ [‌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.

k-$\omega$ model

The RANS 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-$\omega$ closure model [‌28]. The steady-state numerical solutions are sought by means of a Newton’s globalization strategy named pseudo-transient continuation [‌29]. Simulations have been performed on a grid made of $19796$ quadrilateral elements with quadratic edges with a DG polynomial degree equal to $5$ (sixth order).

The flow problem is statistically two-dimensional since the turbulent flow is homogeneous in the span direction. The reference frame origin is located such that $x=0$ is the streamwise position of the step origin and $y=0$ the position in normal direction of flat plate region downstream the step. The inlet boundary is located at $x/H=-41.44$ and the freestream condition is set. The outlet boundary is placed at $x/H=28.12$ and the outlet pressure condition is imposed. Above the solid wall, no-slip adiabatic condition is set between $x/H=-13.67$ and $x/H=15.79$, while shearless condition otherwise. At the top boundary $y/H=181.58$ freestream condition is imposed.

Data provided

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

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

${\begin{aligned}\mathrm {x^{*}} &=x/H\\\mathrm {y_{wall}^{*}} &={\frac {y-Y_{A}}{H}}\\\mathrm {C_{p}} &={\frac {p-p_{ref}}{{\frac {1}{2}}\rho U_{ref}^{2}}}\\\mathrm {tau\_w} &=\mu \left.{\frac {\partial U}{\partial n}}\right|_{y=y_{wall}}\\\mathrm {\delta *} &=\int _{y_{wall}}^{\infty }\left(1-{\frac {U}{U_{edge}}}\right)dy\\\mathrm {\theta } &=\int _{y_{wall}}^{\infty }{\frac {U}{U_{edge}}}\left(1-{\frac {U}{U_{edge}}}\right)dy\end{aligned}}$

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

• profiles.dat contains the velocity and Reynolds-Stress profiles in the vertical direction: $x,y,U,V,W,u',v',w',u'v',u'w',v'w'$ for 36 different streamwise locations (${x/H}$ = -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, 12, 14 and 16).

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

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