## 4 CES-KOS vs. RANS, DES, WMLES, and WRLES
### CES-KOS vs. RANS  
The CES-KOS results were compared with four RANS models, in particular the Wray-Agarwal (WA) model [50], the Spalart-Allmaras (SA) model [50], the shear-stress transport (SST) \(k-\omega\) model [which will be simply referred to as SST model] [51], and the \(k-kL\) model [51]. Table 4 reports separation and reattachment locations, bubble length, and % error results from different RANS-type turbulence models. The SST, SA, and \(k-kL\) models predict a larger bubble size than given by the experimental data. The WA model had an error of \(-23.5\%\), the smallest bubble size, due to late separation. The CES-KOS prediction error is less than \(4\%\). Most RANS models properly predicted the location of separation around \(x/c=0.70\) (with the exception of the WA model). The SST and SA models struggle with an accurate prediction of the reattachment point; they indicate delayed reattachment. The WA and \(k-kL\) model predictions are closer to the experiment measurement at 1.10. The best result follows from the CES-KOS simulation with an error of \(1.7\%\). Model comparisons were performed at five axial locations: \(x/c=-0.25\) (initial region), \(x/c=0.688\) (the region after the shock location but prior to detachment), \(x/c=(0.813,1.125)\) (within the separation zone), and \(x/c=1.375\) (in the post-reattachment region). Figure 15 shows mean velocity predictions of CES-KOS compared with the predictions of RANS models mentioned above and the experimental results [33]. At \(x/c=-0.25\), all models provide similar predictions of the average velocity profile. Most of the RANS models predict the velocity trend over the bump fairly well except at some locations, in particular within the separation region (\(x/c=0.688\)) where the traditional \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Cases** & **Grid** & **Separation** & **Reattachment** & **Bubble Length** & **Error in Bubble** \\  & **Points** & **Location (\(x/c\))** & **Location (\(x/c\))** & **Ax/c** & **Length Prediction (\%)** \\ \hline Expt. [33] & - & 0.70 & 1.10 & 0.4 & - \\ CES-KOS & 5.4 M & 0.703 & 1.1188 & 0.4158 & 3.95 \\ SA [16; 50; 52] & - & 0.688 & 1.1600 & 0.472 & 18 \\ SST-k-w [51; 53] & 0.21 M & 0.665 & 1.1600 & 0.495 & 23.7 \\ k-kL [51; 54; 55] & 0.21 M & 0.669 & 1.1200 & 0.450 & 12.75 \\ WA [16; 50; 52] & - & 0.817 & 1.1230 & 0.306 & \(-23.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RANS-type models: Separation and reattachment characteristics. Figure 14: CES-KOS simulations: \(Re\) effect on profiles of the normalized streamwise velocity \(\left\langle U\right\rangle/U_{ref}\) and normalized Reynolds shear stress \(\left\langle uv\right\rangle/U_{ref}^{2}\) at different axial locations. SA model clearly overestimates the actual mean velocity profile. CES-KOS matches the experimental data and improves the prediction of SA, WA, SST, and k-kL models in the near shock-wave and post-separation regions. In the post-reattachment region, all predictions are relatively similar, in particular far away from the wall. Close to the wall, SST, SA and CES-KOS models slightly underpredict the measured velocity. Figure 15 also shows a corresponding comparison of Reynolds shear stresses in the post-separation region. The predicted profiles exhibit a qualitative similarity with the experimental data, although the Reynolds stresses are under-predicted. Compared to SST and k-kL turbulence models, the CES-KOS prediction is in much better agreement with the experimental results. In particular, SST and \(k-kL\) turbulence models underpredict the shear stress maximum (at about \(y/c=0.018\)) by up to 45% for the case considered, whereas the CES-KOS model shows its capability of predicting a shear stress very similar to the experimental results. Figure 16 shows the time-averaged distributions of surface wall pressure and skin-friction coefficients from RANS models and CES-KOS simulations.