## 3 CES-KOS Resolution Features
### Flow Simulations: Reynolds Number Effects  
The study of \(Re\) effects is certainly of interest, and the CES concept provides a much improved basis to address such questions than given by other modeling approaches (see introduction). Corresponding results of CES-KOS simulations will be reported here by expanding simulations of the \(Re=2.763\) M case to \(Re=5.5\) M simulations. A qualitative impression is provided by Figure 13, which shows streamline structures for the low- and high-\(Re\) regimes. Correspondingly, as \(Re\) is increased from \(Re=2.763\) to \(5.5\) M, a larger separation bubble is observed. Table 3 presents a summary of the variations of separation and reattachment points at different \(Re\) along with the experimental data. It may be seen that increasing \(Re\) causes an increasing size of the separation bubble. The difference of separation bubbles between \(Re=2.673\) M and \(Re=5.5\) M is less than 10%.  
Overall, Figure 14 shows that higher \(Re\) values have a minimal influence on the velocity profile of the flow [48]. In particular at \(x/c=1.125\) and \(1.375\), the boundary layer thickness near the lower wall seems to expand with increasing \(Re\). Similar observations are made in regard to the Reynolds shear stress profile in the mid-separation region: there are only minor \(Re\) effects. Figure 14 also shows corresponding profiles of the pressure coefficient (\(C_{p}\)) and skin-friction coefficient (\(C_{f}\)). It may be seen that the \(C_{p}\) profiles are hardly affected by \(Re\) effects. Although the separation locations for the two cases are not very different, the lower \(Re\) case indicates an earlier reattachment at \(x/c\approx 1.1188\), while the reattachment occurs at approximately \(x/c\approx 1.1652\) for \(Re=5.5\) M (see Table 3). Prior to flow separation, the \(C_{f}\) peak is significantly higher in the \(Re=5\) M case than for the \(Re=2.763\) M case, indicating that the turbulence is highly intermittent in this zone. This finding aligns with the common understanding that higher Reynolds numbers correspond to increased skin friction, identified as the dominant drag in numerous practical applications. [PERSON] and [PERSON] [49] attributed this effect to the growing influence of the large-scale turbulence structures, strong and accelerated flow over the bump and an implied impact on the wall-shear stress. These factors contribute to an amplified growth rate of the separated shear layer and consequently impact the reattachment location reported in Table 3.  
Figure 13: Time-averaged streamline across the flow domain colored by mean velocity magnitude: \(Re=2.763\) M case left, \(Re=5.5\) M case right.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Cases** & **Re** & **Separation** & **Reattachment** & **Bubble Length** & **Error in Bubble** \\  & & **Location (\(x/c\))** & **Location (\(x/c\))** & **\(\Delta x/c\)** & **Length Prediction (\%)** \\ \hline Expt. [33] & 2.763 M & 0.70 & 1.10 & 0.4 & - \\ G\({}_{1}\) & 2.763 M & 0.8338 & 1.1479 & 0.3141 & 21.5 \\ G\({}_{2}\) & 2.763 M & 0.703 & 1.1188 & 0.4158 & 3.95 \\ \hline \(G_{2}\) & 5.5 M & 0.7105 & 1.1652 & 0.4547 & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: CES-KOS separation and reattachment characteristics.