## 3 CES-KOS Resolution Features  
### Resolution Features  
Characteristic resolution features of the CES-KOS model will be illustrated first. Figure 5 and Table 2 show velocity fluctuations (\(u\) and \(v\)) for different \(Re\) and resolution conditions, in particular the \(Re=\) 5.5 M (\(G_{2},G_{1}\)) and \(Re=\) 2.763 M (\(G_{2},G_{1}\)) cases. These cases cover a range of flow resolution conditions, from nearly fully resolved (\(Re=\) 2.763 M, \(G_{2}\)) to predominantly RANS-type (\(Re=\) 5.5 M, \(G_{1}\)) conditions. The \(Re=\) 2.763 M(\(G_{2}\)) case shows a diverse range of small-scale vortices within the recirculation region, indicating a high level of flow resolution. On the other hand, the \(Re=\) 5 M (\(G_{1}\)) case exhibits an unsteady RANS-type regime, in which fluctuations persist, but small-scale turbulence structures are hardly resolved anymore. The transitional cases (\(Re=\) 2.763 M (\(G_{1}\)) and  
Figure 4: Illustration of inflow simulation.  
\(Re=5.5\) M (\(G_{3}\))) show less fine-scale structures than the (\(Re=2.763\) M, \(G_{2}\)) case due to the grid coarsening and increased \(Re\), respectively. Specific observations are the following ones: (i) Grid coarsening and higher \(Re\) cause continuous changes of fluctuations, without sudden shifts. The most noticeable change is the spatial clustering of fluctuations, while the overall strength of fluctuations remains largely unaffected. (ii) In particular, under unstationary RANS conditions, fluctuations are still present (they do not disappear). It is interesting to see that the strength of fluctuations is comparable to the strength of fluctuations under resolving conditions.  
Figure 6 shows local resolution characteristics of the CES-KOS model, specifically \(L_{+}=L/L_{tot}\), \(k_{+}=k/k_{tot}\), and \(\epsilon_{+}=\epsilon/\epsilon_{tot}\) for different grids at different positions. Apart from looking at specifics, the \(k_{+}\) profile, for example, shows significant changes in the flow resolution ranging from almost RANS regimes close to the walls to (depending on the grid) resolving regimes in the core of the flow. It is essential to note that these resolution variations take place smoothly; there are no oscillations in the resolution indicators. The latter is a consequence of the model set-up assuming, for example, spatially invariant variational contributions \(\delta k/k\). The grid effect is as required: the use of a finer grid implies more resolution, which decreases \(L_{+}=L/L_{tot}\) and \(k_{+}=k/k_{tot}\). As observed also otherwise [25], \(\epsilon_{+}\) values were found close to unity, but grid effects are clearly present.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Fluc.** & \(\mathbf{y/c=0.48}\) & \(\mathbf{y/c=0.6}\) & \(\mathbf{y/c=0.8}\) \\ \hline \(u\) & \(-0.002\leq u\leq 0.002\) & \(-0.003\leq u\leq 0.003\) & \(0.004\leq u\leq 0.004\) \\ \(v\) & \(-0.003\leq v\leq 0.003\) & \(-0.004\leq v\leq 0.004\) & \(0.004\leq v\leq 0.004\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Range of velocity fluctuations for the cases described in Figure 5.  
Figure 5: Velocity fluctuations \(u,v\) (top to bottom) within and off the recirculation region in \(xz\) planes at \(y=(0.48,0.6,0.8)c\) (left to right) for (**a**) \(Re=5.5\) M (\(G2\)), (**b**) \(Re=5.5\) M (\(G1\)), (**c**) \(Re=2.763\) M (\(G2\)), (**d**) \(Re=2.763\) M (\(G1\)) cases. The color scale is given in Table 2. Positive (negative) fluctuations are represented by red (blue) regions.