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Semi-confined flows

Underlying Flow Regime 3-36

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Key Physics

The key flow physics to be accurately captured in this UFR are turbulent boundary layers subjected to an adverse pressure gradient over a curved surface with and without separation and reattachment.

Numerical Modelling

With the FV-Code TAU a second order discretization for convective fluxes was used including the turbulence model. For grid details please refer to the previous section ("CFD Methods"). Otherwise, standard rules of grid generation apply. For the RANS computations, the inflow boundary is a reservoir-pressure inflow boundary condition with prescribed total pressure and total density. The inflow direction is by default perpendicular to the boundary face, uniform inflow was assumed at the inlet. For the outflow boundary an exit-pressure outflow boundary condition is used. The exit pressure is adapted during the simulation to match the reference pressure at the reference locations, i.e. at ${Z_{ref}=(-5.18H,0,6.25H)}$ . In spanwise direction symmetry boundary conditions are used, while the body wall is set to a viscous wall boundary. The upper wall is set to a far-field boundary condition located at a distance of ${179H}$ opposed to the viscous wall upstream of the APG region. A study was performed to find the optimal position for the far-field boundary condition, revealing that a distance to the viscous wall less than ${179H}$ shows an influence on the computed pressure distributions. With a distance larger than ${179H}$ the influence vanishes. Results of this study are given in Fig. 7.

Figure 7: Pressure distributions for varying distances of the far-field boundary condition

Physical Modelling

The test case for this numerical experiment needs to be well-defined and all boundary conditions as well as the flow conditions at suitable reference positions need to be specified. The reason behind this is to produce a reliable basis for subsequent DNS computations of the test case. Therefore, a study of the grid space was conducted. Three key features were investigated: the far-field position and the inlet position and laminar-turbulent transition. Each of these features was varied to ensure they do not affect the flow behavior.

Furthermore, five RANS turbulence models already released in TAU and one additional model modification were used for the simulations:

Four variations of the one-equation Spalart-Allmaras RANS turbulence model:
- SA-neg [‌9]
- Non-linear SA-neg model version with rotation and curvature correction (RC) and quadratic constitutive relation (QCR) [‌10][‌11]: SA-RC-QCR
- SA model version with Low-Re-modifications [‌12][‌13]: SA-LRe
- Non-linear SA-neg model version with rotation and curvature correction (RC) and quadratic constitutive relation (QCR) as well as Low-Re-modifications: SA-RC-QCR-LRe
Two variations of the two-equation Menter SST RANS turbulence model:
- SST2003 [‌14]
- Non-linear model version SST2003-RC-QCR [‌15][‌11]
The seven-equation omega-based Differential Reynolds stress turbulence model SSG/LRR- $\omega$ [‌16] including the length scale correction [‌17]

Figure 8: skin friction and pressure coefficient distributions with different RANS turbulence models

Application Uncertainties

The Reynolds numbers used for the test case were determined, on the one hand, by affordable cost of the DNS computations and on the other hand, by avoiding re-laminarization in the accelerated part of the boundary layer. Two different criteria for possible re-laminarization were applied, the acceleration parameter

$K_{acc}=-{\frac {\nu }{\rho U_{\infty }^{3}}}{\frac {\partial p}{\partial s}}$

as well as the pressure-gradient parameter

$\Delta _{p}=-{\frac {\nu }{\rho U_{\tau }^{3}}}{\frac {\partial p}{\partial s}}$

and compared to limiting values for re-laminarization according to the literature [‌24], [‌25] and [‌26]. Here, $U_{\tau }$ is the friction velocity. Fig. 8 shows the distribution of $K_{acc}$ and $\Delta _{p}$ for the Reynolds number $Re_{H}=78,490$ together with the threshold values for both parameters, which this Reynolds number does not exceed.