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Usually applied simulation methods for turbulent flows as large eddy simulation (LES), wall-modeled LES (WMLES), and detached eddy simulation (DES) face significant challenges: they are characterized by improper resolution variations and essential practical simulation problems given by huge computational cost, imbalanced resolution transitions, and resolution mismatch. Alternative simulation methods are described here. By using an extremal entropy analysis, it is shown how minimal error simulation methods can be designed. It is shown that these methods can overcome the typical shortcomings of usually applied simulation methods. A crucial ingredient of this analysis is the identification of a mathematically implied general hybridization mechanism, which is missing in existing methods. Applications to several complex high Reynolds number flow simulations reveal essential performance, functionality, and computational cost advantages of minimal error simulation methods. 

Reynolds-averaged Navier–Stokes (RANS), large eddy simulation (LES), and hybrid RANS-LES, first of all wall-modeled LES (WMLES) and detached eddy simulation (DES) methods, are regularly applied for wall-bounded turbulent flow simulations. Their characteristic advantages and disadvantages are well known: significant challenges arise from simulation performance, computational cost, and functionality issues. This paper describes the application of a new simulation approach: continuous eddy simulation (CES). CES is based on exact mathematics, and it is a minimal error method. Its functionality is different from currently applied simulation concepts. Knowledge of the actual amount of flow resolution enables the model to properly adjust to simulations by increasing or decreasing its contribution. The flow considered is a high Reynolds number complex flow, the Bachalo–Johnson axisymmetric transonic bump flow, which is often applied to evaluate the performance of turbulence models. A thorough analysis of simulation performance, computational cost, and functionality features of the CES model applied is presented in comparison with corresponding features of RANS, DES, WMLES, and wall-resolved LES (WRLES). We conclude that CES performs better than RANS, DES, WMLES, and even WRLES at a little fraction of computational cost applied for the latter methods. CES is independent of usual functionality requirements of other methods, which offers relevant additional advantages. 

The usual concept of simulation methods for turbulent flows is to impose a certain (partial) flow resolution. This concept becomes problematic away from limit regimes of no or an almost complete flow resolution: discrepancies between the imposed and actual flow resolution may imply an unreliable model behavior and high computational cost to compensate for simulation deficiencies. An exact mathematical approach based on variational analysis provides a solution to these problems. Minimal error continuous eddy simulation (CES) designed in this way enables simulations in which the model actively responds to variations in flow resolution by increasing or decreasing its contribution to the simulation as required. This paper presents the first application of CES methods to a moderately complex, relatively high Reynolds number turbulent flow simulation: the NASA wall-mounted hump flow. It is shown that CES performs equally well or better than almost resolving simulation methods at a little fraction of computational cost. Significant computational cost and performance advantages are reported in comparison to popular partially resolving simulation methods including detached eddy simulation and wall-modeled large eddy simulation. Characteristic features of the asymptotic flow structure are identified on the basis of CES simulations. 

Hybrid RANS-LES methods are supposed to provide major contributions to future turbulent flow simulations, in particular for reliable flow predictions under conditions where validation data are unavailable. However, existing hybrid RANS-LES methods suffer from essential problems. A solution to these problems is presented as a generalization of previously introduced continuous eddy simulation (CES) methods. These methods, obtained by relatively minor extensions of standard two-equation turbulence models, represent minimal error simulation methods. An essential observation presented here is that minimal error methods for incompressible flows can be extended to stratified and compressible flows, which opens the way to addressing relevant atmospheric science problems (mesoscale to microscale coupling) and aerospace problems (supersonic or hypersonic flow predictions). It is also reported that minimal error methods can provide valuable contributions to the design of consistent turbulence models under conditions of significant modeling uncertainties. 

A significant extension of previously introduced continuous eddy simulation methods is presented by introducing minimal error partially and fully resolving simulation methods for turbulent flows. This approach represents a machine learning strategy for the hybridization of modeling-focused and resolution-focused simulation methods. It can be applied to well-known equation structures (Spalart–Allmaras type equations, usually applied two-equation models), and it can be used for different hybridization types and in different computational versions. Physically, minimal error methods implement a mode interplay, which ensures that the resolution imposed by a model equals the actual flow resolution. Differently formulated simulation methods reveal two typical errors, and they cannot be expected to provide reliable predictions under conditions where validation data are unavailable. These problems can be avoided by minimal error formulations of model structures considered.