## 4 CES-KOS vs. RANS, DES, WMLES, and WRLES
### Cost Estimates  
The comparison of computational method features will be completed by an analysis of related computational cost. We follow our previous approach for doing so [25; 40]. The simulation cost are specified by  
\[C=NN_{t}=TN/\Delta t. \tag{6}\]  
Here, \(N\) is the number of grid points applied, \(N_{t}\) is number of time steps performed, \(T=N_{t}\Delta t\) refers to the constant total physical simulation time, and \(\Delta t\) (in \(c/U_{\infty}\)) is the prescribed simulation time-step. \(N\) and \(\Delta t\) are known to vary with \(Re\) according to \(N=\alpha_{1}(Re/Re_{0})^{\beta_{1}}\), \(\Delta t=\alpha_{2}(Re/Re_{0})^{-\beta_{2}}\), where \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), and \(\beta_{2}\) are constants [40]. Here, \(Re_{0}=2.763M\), the Reynolds number of the flow considered, is used as normalization, which simplifies the comparisons of computational cost for the flow considered. There is, so far, limited information about the scaling characteristics of computational cost for complex turbulent flows. To make the best possible use of the data we have, we follow our approach presented in Ref. [25]. This means, in regard to WMLES and WRLES, that we apply scalings presented by [PERSON] et al. [58]. In regard to CES-KOS and DES, we use the same scaling assumptions as before [25]. This approach determines the \(\beta_{1}\) and \(\beta_{2}\) values shown below. The \(\alpha_{1}\) and \(\alpha_{2}\) can be determined by taking reference to simulation settings of the flow considered, which are presented in Table 7 (for the flow considered, we have \(N=\alpha_{1}\) and \(\Delta t=\alpha_{2}\)). The corresponding cost scalings are presented in Table 8, where the range of coarse/fine data in Table 7 is translated into ranges of \(\alpha_{1}\) and \(\alpha_{2}\) data in Table 8.  
Figure 20 shows the scaling of computational cost: the corresponding minima and maxima of cost per method are defined by the coarse/fine data in Table 7. For simplicity, we approximate the \(C/T\) scalings of CES-KOS, WMLES, and DES by \((Re/Re_{0})^{1.1}\) and apply the corresponding fine grid data. Then, the cost of CES to the corresponding cost of WMLES and DES are 1, 9.2, and 296, respectively, independent of \(Re\). The cost ratio of WRLES to CES-KOS is \(77.7\times 10^{3}(Re/Re_{0})^{1.62}\), showing a strong increase with \(Re\). These differences can be attributed to the CES-KOS feature of being derived through the minimization of the hybridization error. A comparison of \(\alpha_{3}\) values derived here for the finer grid,  
\[(\alpha_{3}^{KOS},a_{3}^{WRLES},\alpha_{3}^{WMLES},a_{3}^{DES})=(10.8\times 10^{ 9},8.39\times 10^{14},9.96\times 10^{10},32.0\times 10^{11}), \tag{7}\]  
with corresponding \(\alpha_{3}\) values determined regarding our previous CES simulations of the NASA wall-mounted hump flow [25] shows a reasonable agreement:  
Figure 19: CES-KOS vs. LES-type WRLES [42] and WMLES [41] models: Profiles of the normalized streamwise velocity \(\langle U\rangle/U_{ref}\), Reynolds stress \(\langle uv\rangle/U_{ref}^{2}\), pressure, and skin-friction coefficients at different locations. performance. The CES-KOS considered shows the most accurate prediction of separation bubble characteristics. Velocity, stresses, pressure and skin-friction profiles are in much better agreement with the experimental data than seen in regard to the other methods considered. This applies, in particular, to the use of the \(G_{2}\) grid. The latter is very coarse compared to grids applied in other methods. However, the use of even coarser grids can reduce the simulation performance, as illustrated in regard to the use of the \(G_{1}\) grid. In respect to computational cost features, Figure 20 and computational cost scalings presented in Table 8 speak a clear language regarding the cost differences of methods considered. Using the fine grid data in Table 8 for the flow considered here, the CES costs represent 0.0013, 10.8, 0.34% of the WRLES, WMLES, DES costs, respectively.