## 4 CES-KOS vs. RANS, DES, WMLES, and WRLES
### Cost Estimates
In regard to our previous CES simulations of the NASA wall-mounted hump flow [25], we found for the flow considered corresponding CES cost ratios of 0.012, 0.51, 31.9% of WRLES, WMLES, DES costs, respectively (see Equation (8). This comparison shows that significant variations in cost ratios can be found, but the CES costs are always much below the costs of other methods. These CES advantages enable the reliable simulations of high \(Re\) flows that cannot be properly studied by other methods. This was demonstrated by the simulation of the flow considered at the much higher \(Re=5.5\) m. The conclusion was that there are very minor \(Re\) effects, i.e., the flow considered is basically in the asymptotic \(Re\) regime. The consideration of cost depending on \(Re\) is only one way of looking at the question considered. The overall message is that CES opens a new dimension for studying the influence of significant grid refinements to improve predictions via involving a higher fraction of resolved motion. In respect to functionality features, we see the following. WRLES is constrained by the need to use sufficiently fine grids. Usually, it is not evident at all whether the grids applied enable fully resolved simulations. Hybrid RANS-LES like WMLES methods are known to be seriously affected by simulation settings, the use of different (equilibrium or non-equilibrium) wall models, definitions of regions where different models and grids are applied, different mesh distributions, and set-up options to manage the information exchange between such different flow regions. A similar dependence on simulation settings is found for DES methods: the results depend on the concrete model applied and the definition of differently treated simulation zones. For the same number of grid points, DES and WMLES are known to depend on the mesh organization. An appropriate setting of simulation options requires validation data which are often unavailable. CES methods are independent of such functionality requirements, and the model can be used as is, which is a significant advantage. The CES features enable reliable predictions under conditions where validation data are unavailable. The present paper on Bachalo-Johnson axisymmetric transonic bump flow simulations continues our previous simulations of periodic hill flows at high \(Re\)[31] and the NASA wall-mounted hump flow [25]. A natural continuation of this work would be the simulation of even more complex flows, as given by Gaussian bump flows [60]. The latter is outside the scope of this paper. However, corresponding CES simulations appear to be highly promising due to the inherent nature of CES, the incorporation of actual resolved motion, and related stable model response. Such simulations are expected to significantly improve RANS and hybrid RANS-LES as WMLES and DES results. As demonstrated here, it is even plausible to expect improvements of WRLES at a fraction of the computational cost. Conceptualization, S.H. ; Methodology, S.H. ; Software, A.F. ; Validation, A.F. and S.H. ; Formal analysis, A.F. ; Investigation, A.F. ; Resources, S.H. ; Data curation, A.F. ; Writing--original draft, A.F. ; Writing--review & editing, S.H. ; Supervision, S.H. ; Project administration, S.H. ; Funding acquisition, S.H. All authors have read and agreed to the published version of the manuscript. This research received no external funding. This research received no external funding. Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data are contained within the article. We are very thankful for computational resources provided by the Advanced Research Computing Center and the Wyoming-NCAR Alliance at the University of Wyoming. Many thanks to [PERSON] for technical support. Financial support of [PERSON] through the University of Wyoming (College of Engineering and Physical Sciences (CEPS), Dean's Office) is gratefully acknowledged. [PERSON] would like to acknowledge support from the National Science Foundation (AGS, Grant No.