## 4 CES-KOS vs. RANS, DES, WMLES, and WRLES
### Ces-Kos _vs. Des_  
The inadequacy of RANS to accurately represent unsteady flow characteristics resulted in the development of DES methods. In this subsection, we will compare CES-KOS predictions with three DES methods: the Wray-Agarwal detached eddy simulation (WA-DES) method [50], the Spalart-Allmaras-based DES (SA-DES) [41] method, and the zonal improved delayed DES (IDDES) [34] method. The zonal IDDES technique uses a synthetic turbulence generator at an adaptive RANS-LES interface located at \(x/c=-0.5\) (\(0.5c\) upstream of the bump). Due to the different flow regions considered, it is considered as a zonal approach. Table 5 shows characteristic differences in the methods considered in regard to separation and reattachment locations and bubble lengths. IDDES shows early reattachment around \(x/c=1.08\). WA-DES predicts a longer bubble and delayed reattachment at \(x/c=1.166\). The CES-KOS model provides the most accurate predictions.  
Figure 17 shows a comparison between velocity profiles provided by DES-based models and the CES-KOS model. The velocity distribution predicted by SA-DES deviates significantly from the empirical data, especially at \(x/c=0.938\) and \(x/c=1.125\). Profiles at other locations also show considerable differences between the model and experiments, indicating SA-DES issues with resolving the thin shear layer. WA-DES results also provide inaccurate results; see the profiles at \(x/c=0.688\) and \(x/c=1.375\). Overall, the CES-KOS model provides the most accurate results.  
Figure 18 shows Reynolds-stress profiles of DES-type models and CES-KOS predictions. We see significant differences, IDDES shows the largest difference of up to 37% compared to experimental data. The Reynolds stress within the separation zone is notably low in IDDES, making it challenging to capture velocity fluctuations. SA-DES and CES-KOS overestimate the stresses off the boundary layer region. Corresponding comparisons of pressure coefficient distributions are shown in the middle of Figure 18. The IDDES results indicate reduced pressure levels in the post-separation region (\(0.85\leq x/c\leq 1.2\)) compared to the experimental data. Notably, IDDES substantially underestimates skin-friction coefficients for \(0.12<x/c<0.65\). Changes in slope in the skin-friction distribution  
Figure 17: CES-KOS vs. DES-type WA-DES, SA-DES [50], and IDDES [34] models: Profiles of the normalized streamwise velocity \(\langle U\rangle/U_{ref}\) at different locations.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Cases** & **Grid** & **Separation** & **Reattachment** & **Bubble Length** & **Error in Bubble** \\  & **Points** & **Location (\(x/c\))** & **Location (\(x/c\))** & **\(\Delta x/c\)** & **Length Prediction (\%)** \\ \hline Expt. [33] & - & 0.70 & 1.10 & 0.4000 & - \\ CES-KOS & 5.4M & 0.703 & 1.1188 & 0.4158 & 3.95 \\ IDDES [34] & 1.6B & 0.7 & 1.0800 & 0.3800 & -5.00 \\ WA-DES [16; 50; 52] & - & 0.696 & 1.1660 & 0.4700 & 17.50 \\ \hline \hline \end{tabular}
\end{table}
Table 5: DES-type models: Separation and reattachment characteristics.  
correlate with longitudinal curvature reversal of the bump's surface at \(x/c\approx 0.12\) and \(x/c\approx 1.1\). The low pressure values from IDDES correlate with the skin-friction coefficient discrepancies. Specifically, the shear stress reversal predicted by IDDES occurs excessively downstream, consistent with the misplaced shock-wave position.