HiFi-TURB-DLR rounded step

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 and Boundary Conditions

With the DLR-TAU Code 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.

To ensure that the upper boundary will not influence the flow over the curved wall, a study was performed to find the optimal position for the far-field boundary. It was found that a distance to the curved wall less than ${179H}$ shows an influence on the computed pressure distributions. For distances larger than ${179H}$ the influence vanishes. These results were generated during the initial design stage of the testcase when a cheaper Spalart-Allmaras turbulence model ( the SA-neg variant) was applied. Results of this study are given in Fig. 7. Minor deviations (not visible) only vanished for a distance larger than ${179H}$ .

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

Besides studying the far-field position, three simulations were performed to study the sensitivity of the inlet position: the upstream section of the curved step was extended, a Dirichlet boundary condition was applied and laminar-turbulent transition was considered applying a RANS-based transition transport model. The results showed no significant effect and did not change the incipient character of the flow, however, an influence of the predicted skin friction coefficient distribution of the SSG/LRR- $\omega$ model in the recovery region was observed. As the RANS computations were performed to design the testcase, no DNS data were initially available to evaluate this influence.

Physical Modelling

In the early design phase of the testcase, five different RANS turbulence models already released in the DLR-TAU code and one additional model modification were used for assessment of the flow.

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]

For the detailed analysis of the flow only the SSG/LRR- $\omega$ model was evaluated, results for the other RANS turbulence models are not given in detail. An indication of the performance of these models is displayed in Fig. 8.

Figure 8: Skin friction (left) and pressure coefficient (right) distributions for different RANS turbulence models

Considering the skin friction coefficient distribution, the SA-neg turbulence model shows very good agreement with the DNS results, while the corrections RC + QCR outlined above move the flow closer to separation. Adding the LowRe modifiction yields results in between the original SA-neg and the SA-neg-RC-QCR model. Predictions of the Menter-SST turbulence model are close to separation, adding the corrections RC + QCR yields separated flow. The SSG/LRR- $\omega$ model shows a deviation in the recovery region downstream of the step which might be improved by extending the extent of the upstream section of the curved step.

In the ("Evaluation") section the results of the DNS are compared in detail to the results of the SSG/LRR- $\omega$ model of DLR and the Wilcox k- $\omega$ model of UniBG. From the pressure coefficient distribution it is visible that the Wilcox k- $\omega$ model overpredicts the suction and pressure peak compared to the DNS while the SSG/LRR- $\omega$ model shows reverse behaviour but is closer to the DNS. The skin friction coefficient reveals a stronger tendency of the SSG/LRR- $\omega$ model to separation compared with the DNS and the Wilcox k- $\omega$ model. Downstream of the step both models deviate in a similar manner from the DNS, however, at x/H = 5 the SSG/LRR- $\omega$ model is in better agreement with the DNS data. The streamwise velocity profiles reflect the results of the skin friction coefficient: the SSG/LRR- $\omega$ model predicts a smaller velocity in the APG region between 0 < x/H < 5 compared to the DNS and here the Wilcox k- $\omega$ model is closer to DNS, but the former model is in better agreement with DNS at 5 < x/H < 9. For the streamwise normal stresses the SSG/LRR- $\omega$ model is not able to capture the characteristics visible in the DNS data at $0.001\leq y_{W}/H<0.01$ upstream of the APG region, while these stresses are overpredicted close to the boundary-layer edge downstream of the APG region. For the shear-stresses, the SSG/LRR- $\omega$ model better captures the DNS data compared to the Wilcox k- $\omega$ model. Finally, for the turbulent kinetic energy, both RANS models show very similar distributions compared to the DNS.

Based on these results there is no clear indication as to which RANS turbulence model performs better for the testcase. Both models do not capture the flow in the APG-region correctly compared to the DNS data. From a design point of view, the results of the SSG/LRR- $\omega$ model are more conservative as the predicted flow is closer to separation compared to the DNS data. Again, it needs to be emphasized that the SSG/LRR- $\omega$ model was used to design the testcase and the results were generated in a "blind" manner i.e. the target DNS results were not known. For future computations of this testcase it is recommended to increase the extent of the upstream section of the flow domain when applying the SSG/LRR- $\omega$ model.

Application uncertainties

The Reynolds numbers for the testcase were determined, on the one hand, low enough to be affordable for DNS computations and on the other hand, high enough to avoid re-laminarization in the accelerated region of the turbulent boundary layer upstream of the APG region. The latter was ensured by applying two different criteria for possible re-laminarization, 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. 9 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, not exceeded for this Reynolds number.

Figure 9: Assessment of possible re-laminarization

Recommendations for Future Work

This testcase was designed to be part of a family of testcases with varying pressure gradient and Reynolds number to provide a comprehensive data base for data-driven turbulence modeling. As soon as DNS data for the other configurations are available, these cases should be added to the test case description and RANS computations for these testcases should be added. Additionally, RANS turbulence models augmented by a Machine Learning method that were generated and tested on this testcase during the HiFi-TURB project should be described, their computational results added and evaluated in comparison to the traditional models.

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