## 3 CES-KOS Resolution Features
### Flow Simulations: Grid Effects  
Figure 7 shows the spatial distribution of fluctuations, velocity magnitude, and pressure. A \(\lambda\)-structure-like shock pattern is seen over the bump region in particular in pressure and velocity contour plots. The pressure contour shows a diminishing pressure within the shock region compared to the downstream distribution. The interplay between the impinging shock wave and boundary layer triggers the downstream boundary layer propagation and leads to an unsteady separation bubble formation near the trailing edge. Figure 8 shows corresponding modeled kinetic energy \(k\), Mach number, and density variation for the two grids applied. The shock structure can be clearly seen in the Mach number and density distributions. Both grids predict a Mach number up to 1.4 prior to flow separation. The modeled kinetic energy \(k\) is particularly large in the separation and post-separation regions. The fine grid producing more flow resolution provides slightly smaller values of the modeled kinetic energy \(k\) and correspondingly more pronounced fine-scale structures. In Figure 9, we compare CES-KOS predictions of mean velocities with experimental data at six locations: the equilibrium region (at \(x/c=-0.25\)), near separation (\(x/c=0.688\)), within the separation region (\(x/c=0.813\)), close to reattachment (\(x/c=1.00\), \(1.125\)), and in the recovery region (\(x/c=1.25\)). Overall, there is an excellent agreement between CES-KOS predictions on both grids applied and experimental data: minor improvements can be seen if the finer \(G_{2}\) grid is implied. An almost perfect agreement of simulations and  
Figure 6: CES-KOS flow resolution features: \(L+\), \(k+\), and \(e_{+}\) (first, second, and third row) are shown at the given \(x/c\). Figure 7: CES-KOS simulations on \(G_{2}\): contour plots of fluctuations, velocity magnitude, and pressure. measurements is found at \(x/c=-0.25\). At \(x/c=0.688\), which is in the region after the shock location but prior to detachment, there are noticeable but small deviations in the \(G_{2}\) grid results close to the boundary layer margins and away from the wall. However, at positions \(x/c=(0.813,1.0,1.125)\), which fall within the separation zone, the predictions align well with the measurements. Beyond the separation zone at \(x/c=1.25\), following the flow reattachment, there is again an excellent agreement. Overall, the predictions are in very good agreement with measurements, comparable to results in [PERSON]'s work [47], which applies low-diffusion AUFSR schemes. The Reynolds-stress profiles presented in Figure 10 show minor discrepancies between the simulated and measured stresses. Close to the reattachment location at \(x/c=1.00\), the simulated stresses are lower than the measured stresses near the wall, with a maximum difference of 31% near the peak of the bump. At \(x/c=1.25\) in the post-reattachment region, the simulated Reynolds stress profile closely matches the experimental profile in the far field. It is worth noting that the \(G_{1}\) grid results deviate somewhat more from the measurements compared to the \(G_{2}\) grid profile, indicating the relevance of grid refinements. Figure 8: CES-KOS simulations: contour plots of modeled kinetic energy \(k\), Mach number, and density on \(G_{1}\) (**first row**) and \(G_{2}\) (**second row**). Figure 10: CES-KOS simulations: profiles of the normalized Reynolds stress \(\langle uv\rangle/U_{ref}^{2}\) at different axial locations for the two grids applied. Figure 9: CES-KOS simulations: profiles of the normalized streamwise velocity \(\langle U\rangle/U_{ref}\) at different axial locations for the two grids applied. Figure 11 shows flow streamlines colored by mean velocity magnitude for the two grids applied. The shock-wave line is clearly visible. The plot shows a noticeable difference between the streamlines of grid \(G_{1}\) and \(G_{2}\): the \(G_{2}\) result shows a considerably larger separation bubble.