## 4 CES-KOS vs. RANS, DES, WMLES, and WRLES
### CES-KOS vs. WMLES and WRLES  
Given the notable shortcomings of DES-type models particularly in the attached boundary layer region, as shown above, let us consider next the corresponding comparisons of CES-KOS with WMLES and WRLES. Table 6 shows characteristic differences of the methods considered in regard to separation and reattachment locations and bubble lengths. The reported WMLES misidentifies the reattachment location and overestimates the separation bubble size by up to 17.5%. While WRLES captures the separation point accurately, its predicted bubble length size has a 20% error. In contrast, the CES-KOS model provides the highest accuracy regarding the estimation of the separation length, with an error of only \(\approx\)4%.  
Figure 19 shows in its first row streamwise velocity profiles obtained by CES-KOS, WMLES [41], and WRLES [42]. It may be seen that the CES-KOS model predicts the streamwise velocity more accurately than WMLES and WRLES. In regard to turbulent shear stress profiles shown in the second row, we see a reasonable agreement of WMLES, WRLES, and CES-KOS with experimental data. It is of interest to note that CES-KOS and WRLES provide very similar results. In attached flow regions, WMLES over-predicts the turbulent shear stress (in the same amount as IDDES, as seen above). Due to its delayed reattachment point, WMLES predicts a faster separated shear layer growth and higher maximum Reynolds stresses compared to CES-KOS. These trends have been noted in past numerical studies using alternative models [56; 57].  
Figure 19 also shows pressure coefficient distributions obtained by CES-KOS, WMLES, and WRLES. The figure helps to illustrate and validate the accuracy of CES-KOS predictions. The CES-KOS and WRLES models accurately predict pressure coefficient profiles due to their sufficient flow resolution ability. In contrast, WMLES predicts a linearly increasing pressure distribution within \(x/c=(0.7,1.1)\), which fails to accurately capture the separation zone. Furthermore, both CES-KOS and WRLES show reasonable predictions of the shock location and post-shock pressure recovery. The WRLES results agree slightly better with the experimental data downstream of the bump (between \(x/c=1.1\) and \(1.3\)) compared to the CES-KOS model. Figure 19 also shows skin-friction coefficient distributions obtained by CES-KOS, WMLES, and WRLES. Evidently, WMLES significantly underestimates the skin-friction coefficient in the separation region and fails to accurately represent the post-separation flow physics. The predictions of CES-KOS and  
\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 \\ WRLES [42] & 24B & 0.69(0.68) & 1.17(1.16) & 0.48 & 20 \\ WMLES [34] & 24.09M & 0.7 & 1.17 & 0.47 & 17.5 \\ \hline \hline \end{tabular}
\end{table}
Table 6: LES-type models: separation and reattachment characteristics.  
Figure 18: CES-KOS vs. DES-type WA-DES, SA-DES [50], and IDDES [34] models: Normalized Reynolds stress \(\left<uv\right>/U_{ref}^{2}\), pressure and skin-friction coefficient distributions.  
with the exception that CES-KOS better agrees with the experimental data in the \(C_{f}\) plateau region upstream of separation. Overall, CES-KOS provides the most accurate predictions.