Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations

Abstract

Modeling of wall-bounded turbulent flows, in particular the hybridization of the Reynolds-averaged Navier-Stokes (RANS) and large eddy simulation (LES) methods, has faced serious questions for decades. Specifically, there is continuous research of how usually applied methods such as detached eddy simulation (DES) and wall-modeled LES (WMLES) can be made more successful in regard to complex, high-Reynolds-number ($Re$) flow simulations. The simple question is how it is possible to enable reliable and cost-efficient predictions of high-$Re$ wall-bounded turbulent flows in particular under conditions where data for validation are unavailable. This paper presents a strict analysis of strategies for the design of seamlessly resolving turbulent flow simulations for a wide class of turbulence models. The essential conclusions obtained are the following ones: First, by construction, usually applied methods like DES are incapable of systematically spanning the range from modeled to resolved flow simulations, which implies significant disadvantages. Second, a strict solution for this problem is given by novel continuous eddy simulation (CES) methods, which perform very well. Third, the design of a computational simplification of CES that still outperforms DES appears to be very promising.

1. Introduction

The introduction of two-equation Reynolds-averaged Navier-Stokes (RANS) turbulence models by Kolmogorov [1] has marked a milestone in turbulence research. Such models involve two basic ingredients: an equation for the kinetic energy of turbulence k (which reflects the intensity of turbulent velocity fluctuations) and an equation for the scale of turbulence (an equation for the dissipation rate $ϵ$ of turbulent kinetic energy, or the turbulence dissipation time scale $\tau =k/ϵ$, or the turbulence frequency $\omega =1/\tau$, or the turbulence length scale $L=k^{3/2}/ϵ$). The structure of the transport equation for k is well established, but there is still significant uncertainty about the most appropriate scale variable that should be considered in conjunction with k and the appropriate structure of the scale variable equation [2,3,4,5]. The most common structure of scale equations currently applied follows simple return-to-equilibrium concepts [6,7] similar to the Bhatnagar–Gross–Krook (BGK) approximation [8] of the Boltzmann equation.

Interestingly, Kolmogorov [1] did not include a production term in his $\omega$-equation. Rotta [9,10] made an attempt to provide a solution to these questions from a fundamental view point; he developed a $kL$ model. This approach addresses a relevant question: the fact that forming a source term equilibrium using usually applied $k-ϵ$ or $k-\omega$ models does not allow for the determination of a length L (or related) scale [5]. Specifically, this source term equilibrium enables it to determine one turbulence variable ($\omega$) but no second turbulence variable. On the other hand, Rotta’s $kL$ equation requires an extra term near the wall to be consistent with the log law [5,11]. Menter and Egorov [5] addressed these questions based on a thorough analysis of Rotta’s approach, leading to a revision of Rotta’s model. However, there are different lines to argue regarding the concrete model formulation: a linear dependency on the second velocity derivative was proposed first [12], whereas a quadratic formulation has been used later [5]. Such structural uncertainty characterizes all currently applied turbulence models. For example, the Spalart–Allmaras (SA) model, one of the most often applied models, is purely based on empirical arguments [13]. The use of large eddy simulation (LES) [14,15,16,17,18,19] which focuses on an almost complete flow resolution is no way to ignore these RANS problems. As is well known, LES requires unaffordable computational costs in regard to high-Reynolds-number ($Re$), complex turbulent flow simulations [20,21,22,23]. A major concern with LES is also the missing involvement of a reliable measure of its resolution ability [24,25].

The no-alternative solution is the hybridization of the RANS and LES methods. Many different ways were suggested in this regard. One very popular way is the wall-modeled LES (WMLES) [15,18,19,20,21,26,27,28,29,30,31,32,33,34,35,36], where RANS components are involved close to solid walls. Variations of this approach given by the development of the Reynolds-stress-constrained LES (RSC-LES) [37,38,39,40,41,42,43,44,45,46,47,48] are discussed elsewhere [6]. Another very popular way is given by detached eddy simulation (DES) [49,50,51,52,53,54,55,56,57,58,59,60], where the performance of RANS models is improved by switching from the RANS turbulence length scale applied close to the wall to a much smaller LES-type length scale away from the wall. A variety of alternative methods were suggested [6,61], including unified RANS-LES (UNI-LES) [62,63,64,65,66,67], partially averaged Navier–Stokes (PANS) [68,69,70,71,72,73,74,75,76,77,78,79], partially integrated transport modeling (PITM) [80,81,82,83,84,85,86,87,88,89,90], and scale adaptive simulation (SAS) methods [5,12,60,91,92,93,94]. However, such hybrid RANS-LES methods suffer from a variety of problems. Being mostly based on RANS, these methods fully reflect the structural uncertainty of equations and the choice of model variables. On top of that, such equations suffer from the uncertainty of how RANS and LES equation elements should be combined and how simulation settings should be chosen. Such issues apply to aerospace and wind energy problems but also to a variety of other problems, such as, for example, mesoscale and microscale modeling in regard to atmospheric simulations and many technical applications [95,96,97]. It is also worth mentioning the following: The use of machine learning (ML) methods becomes increasingly popular. Such developments are promising, but there is currently no indication that the use of such ML methods in regard to the hybridization of LES relates to essential methodological improvements; see the recent review in Ref. [98].

The difficulty of dealing with these issues is the lack of a direct mathematical approach to derive such turbulence equations. There are mathematical approaches to analyze turbulence models like renormalization group theory (RNG) [99,100,101,102], consistency with thermodynamics [103,104], and realizability [105,106], but these approaches are based on a given equation structure. However, a promising approach in this regard is the following: A rudimentary expectation is a turbulence model’s ability to cover different scaling regimes, e.g., to seamlessly transition between non-resolving and resolving regimes. This may be seen to pose requirements on a turbulence scale equation. Such an approach has been presented recently by the development of continuous eddy simulation (CES) methods [6,98,107,108,109,110,111,112,113,114,115,116,117,118,119,120]. The motivation of this paper is to significantly extend this approach to a wide class of turbulence models and various scaling regimes. On this basis, new insight can be obtained in regard to both the appropriate structure of turbulence equations and the appropriate transitioning between several scaling regimes. This paper is organized in the following way: A generic two-equation turbulence model is introduced in Section 2 followed by an analysis of hybridization frameworks in Section 3. Section 4 deals with an analysis of hybridization strategies, and applications will be discussed in Section 5. Conclusions are presented in Section 6. An illustration of this paper’s structure is given in Figure 1.

2. Generic Two-Equation Turbulence Model

As a basis for the following discussion, we introduce first the $k-ϵ$ two-equation turbulence model [2,121]. The model considered is given by the incompressible continuity equation $\partial {\tilde{U}}_i/\partial x_i=0$ and momentum equation

$${\displaystyle \frac{D{\tilde{U}}_i}{Dt}}=-{\displaystyle \frac{\partial (\tilde{p}/\rho +2k/3)}{\partial x_i}}+2{\displaystyle \frac{\partial (\nu +{\nu }_t){\tilde{S}}_{ik}}{\partial x_k}}.$$

(1)

Here, $D/Dt=\partial /\partial t+{\tilde{U}}_k\partial /\partial x_k$ denotes the filtered Lagrangian time derivative, and the sum convention is used throughout this paper. ${\tilde{U}}_i$ refers to the $i^{th}$ component of the spatially filtered velocity. We have here the filtered pressure $\tilde{p}$, $\rho$ is the constant mass density, k is the modeled energy, $\nu$ is the constant kinematic viscosity, and ${\tilde{S}}_{ij}=(\partial {\tilde{U}}_i/\partial x_j+\partial {\tilde{U}}_j/\partial x_i)/2$ is the rate-of-strain tensor. The modeled viscosity is given by ${\nu }_t=C_{\mu }{k}^{1/2}L$. Here, $C_{\mu }$ is a model parameter with a standard value $C_{\mu }=0.09$, and L is a characteristic length scale. The latter can be related to the dissipation rate $ϵ=k/\tau$ of modeled kinetic energy, or the dissipation time scale $\tau$, or turbulence frequency $\omega =1/\tau$ by $L=k^{3/2}/ϵ=k^{1/2}\tau =k^{1/2}/\omega$, respectively. We note that RANS and LES approaches differ by different settings of characteristic length scales L in ${\nu }_t=C_{\mu }{k}^{1/2}L$ and the different grids applied. For k and $ϵ$, we consider the transport equations

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,\mathrm{where}{D}_k={\displaystyle \frac{\partial }{\partial x_j}}{\displaystyle \frac{{\nu }_t}{{\sigma }_k}}{\displaystyle \frac{\partial ϵ}{\partial x_j}},\end{array}$$

(2a)

$$\begin{array}{c}\hfill {\displaystyle \frac{Dϵ}{Dt}}=C_{{ϵ}_1}{\displaystyle \frac{{ϵ}^2}{k}}\left({\displaystyle \frac{P}{ϵ}}-\alpha \right)+D_{ϵ},\mathrm{where}{D}_{ϵ}={\displaystyle \frac{\partial }{\partial x_j}}{\displaystyle \frac{{\nu }_t}{{\sigma }_{ϵ}}}{\displaystyle \frac{\partial ϵ}{\partial x_j}}.\end{array}$$

(2b)

Here, $P={\nu }_t{S}^2$ is the production of k, where $S={\left(2{\tilde{S}}_{mn}{\tilde{S}}_{nm}\right)}^{1/2}$ is the characteristic shear rate. $C_{{ϵ}_1}$ is a constant with standard value $C_{{ϵ}_1}=1.44$, and ${\sigma }_{ϵ}=1.3$. In addition we have $\alpha =C_{{ϵ}_2}/C_{{ϵ}_1}$, where $C_{{ϵ}_2}=1.92$ [2] implies $\alpha =1.33$. For simplicity we show the diffusion term for high $Re$ flows, i.e., the contribution of $\nu$ to the diffusivity is neglected. For clarity purposes we do account for ${\sigma }_k$, which is usually set equal to unity.

It is relevant to note that the methods presented are applicable to stratified flows if appropriate extensions are considered [116]. There is a need for the addition of a potential temperature equation and the addition of buoyancy and Coriolis terms in the mean flow equations. In regard to the further analysis presented in this paper, it is relevant to note that the analysis of turbulence models is unchanged with the understanding that the buoyancy production has to be added to the shear production P.

There is not only the $k-ϵ$ model [2,121] that is applied to determine the turbulence scale; there is also a large variety of models that are applied instead of the $k-ϵ$ model: $k-\omega$ models [2,122], $k-L$ models [123,124,125], $k-kL$ models [5,9,10,126], $k-{\nu }_t$ models [5,12], and stand-alone ${\nu }_t$ models without the k-equation [13,127,128]. To enable the discussion of a large variety of usually applied turbulence models, we generalize the scale equation considered (the ϵ Equation (2b)). Specifically, we follow Umlauf and Burchard [129] by introducing the generic scale variable $G=C_G{k}^m{L}^n$. Here, m and $n\ne 0$ are any numbers, and $C_G$ is a function of $C_{\mu }$ that depends on the model: e.g., we have

$$C_G=(1;1;1;C_{\mu })\mathrm{for}G=\left(\omega ={\displaystyle \frac{k^{1/2}}{L}};L;ϵ={\displaystyle \frac{k^{3/2}}{L}};{\nu }_t=C_{\mu }{k}^{1/2}L\right).$$

(3)

The use of the $k-ϵ$ model in conjunction with the definition $G=C_G{k}^m{L}^n$ enables the derivation of the following generic turbulence model:

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,\end{array}$$

(4a)

$$\begin{array}{c}\hfill {\displaystyle \frac{DG}{Dt}}={\displaystyle \frac{G}{k}}[(m+{\displaystyle \frac{3n}{2}}-n{C}_{ϵ1})P-(m+{\displaystyle \frac{3n}{2}}-n{C}_{ϵ2}+{\displaystyle \frac{n{C}_{\mu }}{{\sigma }_{ϵ}}}{F}_G)ϵ+nq{D}_k(1-{\displaystyle \frac{{\sigma }_k}{{\sigma }_{ϵ}}})]+D_{Gϵ}.\end{array}$$

(4b)

Here, $q=m/n+3/2$, $ϵ=C_G^{1/n}{k}^q/G^{1/n}$, and the following abbreviations are applied:

$$D_{Gϵ}={\displaystyle \frac{\partial }{\partial x_j}}{\displaystyle \frac{{\nu }_t}{{\sigma }_{ϵ}}}{\displaystyle \frac{\partial G}{\partial x_j}},F_G=L^2\left[\left({\displaystyle \frac{1}{n}}+{\displaystyle \frac{1}{n^2}}\right){\displaystyle \frac{1}{G^2}}{\displaystyle \frac{\partial G}{\partial x_j}}{\displaystyle \frac{\partial G}{\partial x_j}}-{\displaystyle \frac{2q}{nGk}}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial G}{\partial x_j}}+{\displaystyle \frac{q^2-q}{k^2}}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial k}{\partial x_j}}\right].$$

(5)

Three examples for scale equations implied in this way are given in

Table 1. Equations for $\omega =ϵ/k$, $L=k^{3/2}/ϵ$, and ${\nu }_t=C_{\mu }{k}^2/ϵ$ derived from the $k-ϵ$ model.

${\displaystyle \frac{D\omega }{Dt}=\frac{\omega }{k}\left[(C_{ϵ1}-1)P-\left(C_{ϵ2}-1-F_{\omega }\frac{C_{\mu }}{{\sigma }_{ϵ}}\right)k\omega \right]+D_k({\sigma }_k/{\sigma }_{ϵ}-1)\frac{\omega }{k}+\frac{\partial }{\partial x_j}\frac{{\nu }_t}{{\sigma }_{ϵ}}\frac{\partial \omega }{\partial x_j}}$,

${\displaystyle \frac{DL}{Dt}=\frac{L}{k}\left[(3/2-C_{ϵ1})P-\left(3/2-C_{ϵ2}+F_L\frac{C_{\mu }}{{\sigma }_{ϵ}}\right)\frac{k^{3/2}}{L}\right]+D_k(1-{\sigma }_k/{\sigma }_{ϵ})\frac{3}{2}\frac{L}{k}+\frac{\partial }{\partial x_j}\frac{{\nu }_t}{{\sigma }_{ϵ}}\frac{\partial L}{\partial x_j}}$,

${\displaystyle \frac{D{\nu }_t}{Dt}=\frac{{\nu }_t}{k}\left[(2-C_{ϵ1})P-\left(2-C_{ϵ2}+F_{\nu }\frac{C_{\mu }}{{\sigma }_{ϵ}}\right)\frac{C_{\mu }{k}^2}{{\nu }_t}\right]+2D_k(1-{\sigma }_k/{\sigma }_{ϵ})\frac{{\nu }_t}{k}+\frac{\partial }{\partial x_j}\frac{{\nu }_t}{{\sigma }_{ϵ}}\frac{\partial {\nu }_t}{\partial x_j}}$,

, where the corresponding $D_{Gϵ}$ and $F_G$ are given in terms of Equation (5). Two relevant observations are the following ones: First, the source terms should not include diffusive turbulent transport terms, leading to the requirement that ${\sigma }_k={\sigma }_{ϵ}$. We will assume from now on ${\sigma }_k=1$ as is usually assumed. Nevertheless, to stress the generality of further developments, we will keep ${\sigma }_{ϵ}$ as an adjustable parameter (keeping in mind that turbulence models do not always strictly follow requirements). Second, the appearance of cross-diffusion terms in the dissipation term given by $F_G$ is not in line with turbulence modeling principles. (i) There is no reason to assume that the $k-ϵ$ model has a higher degree of reliability than other two-equation models. By using another turbulence model than the $k-ϵ$ model, we would obtain different cross-diffusion terms $F_G$ in regard to every two-equation turbulence model considered. Thus, these $F_G$ terms simply represent spurious source terms, which should be disregarded. (ii) In addition, the nondimensional $F_G$ is unbounded, and it can be positive or negative. Thus, the inclusion of $F_G$ in the dissipation term may imply random effects. (iii) The strongest argument against the inclusion of $F_G$ is that it prevents the hybridization of the generic model presented below. As argued in the introduction, this fact speaks against physical principles.

Based on these arguments we will consider the following generic turbulence model:

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,\end{array}$$

(6a)

$$\begin{array}{c}\hfill {\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[P-\alpha ϵ\right]+D_G=C_1{\displaystyle \frac{G}{\tau }}\left[{\displaystyle \frac{P}{ϵ}}-\alpha \right]+D_G.\end{array}$$

(6b)

Here, $C_1$ and $C_2$ are constants, and $D_G=\partial [({\nu }_t/{\sigma }_G)\partial G/\partial x_j]/\partial x_j$. The last expression in Equation (6b) indicates the spirit of this model, the trend toward a production-to-dissipation equilibrium. As pointed out in Appendix A, there are three conditions imposed on this model. In addition to $C_{\mu }=0.09$, the log law implies

$$\begin{array}{c}\hfill n^2{\kappa }^2=C_{\mu }^{1/2}{\sigma }_G(C_2-C_1).\end{array}$$

(7)

On top of this there is the constraint $C_1=m$. The latter condition is usually not strictly satisfied, but most turbulence models apply settings in line with this requirement.

3. Hybridization Frameworks

The generic turbulence model provides the basis for hybridizations, but there are actually three ways to address this question. A first way is to hybridize dissipation terms in the generic model. A second way is to hybridize the modeled viscosity in the generic turbulence model equation. A third way is the reduction of the generic two-equation model to a one-equation model with corresponding hybridization. Details of these three techniques will be presented in the following section. These results are summarized in

Table 2. Summary of hybrid models obtained, where $\alpha =C_2/C_1$ and $f_{\Delta }=({\Delta }_C/L_{tot})/{[1+{({\Delta }_C/L_{tot})}^3]}^{1/3}$. Characteristic-related settings for the reference length scale $L_0$ are given on the right-hand side. Based on $G=C_G{k}^m{L}^n$, the log-law requirements for CES-K*S, CES-K*K, and CES-K*V are $n^2{\kappa }^2=C_{\mu }^{1/2}{\sigma }_G(C_2-C_1)$. The last two rows (CES-SAS and CES-DES) describe specific choices of CES-K*S. For the $k-{\nu }_t$ model we have ${\kappa }^2=C_{\mu }^{1/2}{\sigma }_{\nu }(C_{\nu 2}-C_{\nu 1})$, and for the ${\nu }_t$ model we have ${\kappa }^2={\sigma }_{\nu }(c_{dL}{\kappa }^2/C_{\mu }^{3/2}-c_p)$ and ${\kappa }^2={\sigma }_{\nu }(c_{d\nu }{\kappa }^2/C_{\mu }^{3/2}-c_p)$, respectively.

${\displaystyle \mathrm{CES}-\mathrm{K}*\mathrm{S}:\frac{Dk}{Dt}=P-ϵ+D_k,\frac{DG}{Dt}=C_1\frac{G}{k}\left[P-{\alpha }^{*}ϵ\right]+D_G,{\alpha }^{*}=1+({\alpha }_0-1)L_{*}^2}$	$L_0=L_{tot}$
${\displaystyle \mathrm{CES}-\mathrm{K}*\mathrm{K}:\frac{Dk}{Dt}=P-{\beta }^{*}ϵ+D_k,\frac{DG}{Dt}=C_1\frac{G}{k}\left[P-\alpha ϵ\right]+D_G,{\beta }^{*}={\alpha }_0-({\alpha }_0-1)L_{*}^2}$	$L_0=L_{tot}$
${\displaystyle \mathrm{CES}-\mathrm{K}*\mathrm{V}:\frac{Dk}{Dt}={\gamma }^{*}P-ϵ+{\gamma }^{*}{D}_k,\frac{DG}{Dt}=C_1\frac{G}{k}\left[{\gamma }^{*}P-\alpha ϵ\right]+{\gamma }^{*}{D}_G,{\gamma }^{*}={\gamma }_0{\left(\frac{L_0}{L}\right)}^2}$	$L_0=f_{\Delta }{L}_{tot}$
${\displaystyle [{\nu }_t^{*}={\gamma }^{*}{\nu }_t]}$	$[\approx min({\Delta }_C,L_{tot})]$
${\displaystyle \mathrm{CES}-\mathrm{SAS}:\frac{Dk}{Dt}=P-ϵ+D_k,\frac{D{\nu }_t}{Dt}=C_{{\nu }_1}\frac{{\nu }_t}{k}\left[P-{\alpha }_{\nu }^{*}ϵ\right]+D_{\nu },{\alpha }_{\nu }^{*}=1+({\alpha }_0-1)L_{*}^2}$	$L_0=min({\Delta }_C,y)$
${\displaystyle \mathrm{CES}-\mathrm{DES}:\frac{D{\nu }_t}{Dt}={\nu }_{t}S\left[c_p-c_{dL}{L}_{*}^2\right]+D_{\nu }=c_p{\nu }_{t}S-c_{d\nu }\frac{{\nu }_t^2}{C_{\mu }^{3/2}{L}_0^2}+D_{\nu }}$	$L_0={\kappa }^{-1}{L}_{vK}$

.

3.1. Hybridization of the Generic Two-Equation Model

According to Equation (6), the turbulence model considered is expressed as

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,\end{array}$$

(8a)

$$\begin{array}{c}\hfill {\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[P-{\alpha }^{*}ϵ\right]+D_G.\end{array}$$

(8b)

In the RANS equations, we have a constant ${\alpha }^{*}=\alpha =C_2/C_1$. In contrast to that, ${\alpha }^{*}$ is considered to be an adjustable parameter here. As pointed out in the introduction, a rudimentary expectation is a model’s ability to seamlessly cover resolving and modeling flow regimes. However, it requires care with the understanding of the meaning of resolving and modeling flow regimes. Used on appropriate grids, Equation (8) can be used in the resolving mode; the turbulence variables such as L and k become much smaller than in RANS mode in this case. However, a transition between different regimes cannot trivially be accomplished by using Equation (8) on appropriate (fine or coarser) grids in conjunction with ${\alpha }^{*}=\alpha =C_2/C_1$. In this case, Equation (8) would operate as an unsteady RANS with a random, uncontrolled inclusion of some resolved motion (which is known to be an inappropriate concept; see LES performed on coarse grids). Consequently, this transition has to be driven by the model itself. The latter can be accomplished by variational analysis that requires the model to ensure a transition between different scaling regimes. As shown in Appendix B, such variational analysis implies

$${\alpha }^{*}=1+L_{*}^2({\alpha }_0-1),$$

(9)

where $L_{*}=L/L_0$ refers to the modeled reference length scale ratio (no assumption regarding the reference length scale $L_0$ will be made right now). An assumption made in this analysis is the neglect of substantial derivatives $Dk/Dt$ and $DG/Dt$ only in regard to the calculation of ${\alpha }^{*}$. This assumption simplifies equations; it was found to be very well justified in previous applications [109,110,119].

The hybridization of the dissipation in the scale equation is not the only option; it is also possible to apply a corresponding hybridization of the dissipation in the k-equation,

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-{\beta }^{*}ϵ+D_k,\end{array}$$

(10a)

$$\begin{array}{c}\hfill {\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[P-\alpha ϵ\right]+D_G,\end{array}$$

(10b)

where the adjustable hybridization parameter ${\beta }^{*}$ is introduced. By using the same analysis as before (see Appendix B) we obtain

$${\beta }^{*}={\alpha }_0-L_{*}^2({\alpha }_0-1).$$

(11)

It is remarkable to see that $L_{*}^2$ drives the hybridization for the large family of turbulence models considered and for different types of hybridizations. As expected and desired by Rotta [9,10], we find that a length scale ($L_0$) enters the turbulence production–dissipation terms in addition to the time scale $S^{-1}$.

3.2. Viscosity Hybridization of the Generic Two-Equation Model

Hybridizations of the generic model by modifying dissipation terms in the scale or k equation are not the only options: there is another option focusing on the hybridization of the modeled viscosity ${\nu }_t$. The generic turbulence model equations considered for this case are given by

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}={\gamma }^{*}P-ϵ+{\gamma }^{*}{D}_k,\end{array}$$

(12a)

$$\begin{array}{c}\hfill {\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[{\gamma }^{*}P-\alpha ϵ\right]+{\gamma }^{*}{D}_G.\end{array}$$

(12b)

Here, $\alpha =C_2/C_1$ is a constant as considered before, and the adjustable ${\gamma }^{*}$ is introduced. The latter stands for a modification of the modeled viscosity, with the understanding that ${\gamma }^{*}$ is used inside the gradient term of turbulent transport terms. As shown in Appendix C by using a corresponding variational analysis as before, we obtain for this case

$${\gamma }^{*}={\gamma }_0{\left({\displaystyle \frac{L_0}{L}}\right)}^2,\mathrm{which}\mathrm{implies}{\nu }_t^{*}={\gamma }^{*}{\nu }_t.$$

(13)

Here, $L_0$ and ${\gamma }_0$ refer to any reference state. A significant difference to the hybridizations in terms of ${\alpha }^{*}$ and ${\beta }^{*}$ considered before is that fact that ${\gamma }^{*}$ scales with $L_0/L$ instead of the scaling with $L/L_0$ obtained in regard to ${\alpha }^{*}$ and ${\beta }^{*}$.

3.3. The One-Equation Hybrid Generic Model

The reduction of the generic model to a one-equation model can be obtained as follows (more specifically, we derive first a two-equation model that enables the transition to an one-equation model). This derivation is only meaningful by considering a modeled viscosity equation: $G={\nu }_t$; no other assumption is made,

$$\begin{array}{c}\hfill {\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,\end{array}$$

(14a)

$$\begin{array}{c}\hfill {\displaystyle \frac{D{\nu }_t}{Dt}}=C_{{\nu }_1}{\displaystyle \frac{{\nu }_t}{k}}\left[P-{\alpha }_{\nu }^{*}ϵ\right]+D_{\nu }.\end{array}$$

(14b)

Here, ${\alpha }_{\nu }^{*}$ refers to an adjustable hybridization parameter. According to Equation (A13), ${\alpha }_{\nu }^{*}$ is given by

$${\alpha }_{\nu }^{*}=1+({\alpha }_0-1)L^2/L_0^2.$$

(15)

Here, $L_0$ is used again as a non-specified reference scale.

The transition to a one-equation model is described in Appendix D. We obtain in this way

$$\begin{array}{c}\hfill {\displaystyle \frac{D{\nu }_t}{Dt}}={\nu }_{t}S\left[c_p-c_{dL}{\displaystyle \frac{L^2}{L_0^2}}\right]+D_{\nu },\end{array}$$

(16)

see Equation (A23). An alternative writing is given by Equation (A24),

$$\begin{array}{c}\hfill {\displaystyle \frac{D{\nu }_t}{Dt}}=c_p{\nu }_{t}S-c_{d\nu }{\displaystyle \frac{{\nu }_t^2}{C_{\mu }^{3/2}{L}_0^2}}+D_{\nu },\end{array}$$

(17)

In these equations, $c_p$, $c_{dL}$, and $c_{d\nu }$ are model parameters. An essential fact is the following: The latter parameters are obtained by model variables including $r=C_{\mu }^{1/2}k/\left[{\nu }_{t}S\right]$, which has a log-law value of unity. No attempt is made to satisfy log-law requirements via setting $r=1$. Instead, the log-law requirements are imposed as conditions on model parameters given by ${\kappa }^2={\sigma }_{\nu }(c_{dL}{\kappa }^2/C_{\mu }^{3/2}-c_p)$ and ${\kappa }^2={\sigma }_{\nu }(c_{d\nu }{\kappa }^2/C_{\mu }^{3/2}-c_p)$, respectively; see Appendix D.

4. Analysis of Hybridization Strategies

A summary of available modeling frameworks as presented in Section 3 is given in Table 2. The actual hybridization requires a specification of the reference state, in particular the setting of $L_0$. This is very essential: the range covered by L to $L_0$ variations determines the hybrid range covered by the simulation method. Several possibilities for $L_0$ settings (which depend on the specific modeling framework considered) will be discussed in the following section. An overview of these options is provided in Figure 2. For clarity purposes we only discuss here the implications of $L_0$ settings in regard to Table 2 methods; no further assumptions as usually applied will be made.

4.1. Simple Empirical Strategies

The first most simple strategy is to follow the DES concept, characterized by a switch of RANS and LES length scales. DES can be applied in a variety of versions. For the following discussion we focus on a comparison with a one-equation modeled viscosity equation (see Equations (16) and (17))

$${\displaystyle \frac{D{\nu }_t}{Dt}}={\nu }_{t}S\left[c_p-c_{dL}{L}_{*}^2\right]+D_{\nu }=c_p{\nu }_{t}S-c_{d\nu }{\displaystyle \frac{{\nu }_t^2}{C_{\mu }^{3/2}{L}_0^2}}+D_{\nu }.$$

(18)

The DES concept is to apply $L_0=min({\Delta }_C,y)$ [49]. Here, $\Delta$ is the filter width, ${\Delta }_C=C_{+}\Delta$, and y is the distance from the nearest wall. The constant $C_{+}$ is given by $C_{+}={(3C_K/2)}^{3/2}/\pi$, where $C_k$ refers to the Kolmogorov constant. Values $C_k=(1.3,1.43,1.5)$, e.g., imply $C_{+}=(0.876,1.0,1.074)$. In DES methods, the typical notation applied is ${\Delta }_C=C_{DES}\Delta$, where the characteristic DES constant $C_{DES}$ is introduced. The idea of this approach is to enable the simulation of instationary flow on appropriate grids. In this mode, the model length scale L is much smaller than in RANS mode; L can be of the order of the filter width $\Delta$. Then, there is little variation of $L/L_0$; see the illustration in Figure 2. This means, $L_0$ as given by DES does not allow it to span regimes ranging from total modeling to a total flow resolution: the DES concept is therefore an inappropriate concept to set up seamlessly operating hybrid simulation methods. This conclusion is fully in line with the experience obtained by DES simulations. There is research for about 30 years of how to stabilize the uncontrolled interaction of modeled and resolved flow without any convergence accomplished so far [130].

A second simple strategy (applicable in the CES-K*S framework) is the use of the von Kármán length scale $L_{vK}=\kappa |S/S^{\prime }|$ as the reference length scale as applied in SAS methods, where $S^{\prime }={[{\partial }^2{\tilde{U}}_i/\partial x_k^2\times {\partial }^2{\tilde{U}}_i/\partial x_j^2]}^{1/2}$. In the logarithmic flow region we find $C_{\mu }^{3/4}L=L_{vK}=\kappa y$. A simple way to illustrate the SAS approach is to take reference to Equation (14),

$${\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,{\displaystyle \frac{D{\nu }_t}{Dt}}=C_{{\nu }_1}{\displaystyle \frac{{\nu }_t}{k}}\left[P-{\alpha }_{\nu }^{*}ϵ\right]+D_{\nu }.$$

(19)

By using ${\alpha }_{\nu }^{*}=1+({\alpha }_0-1)L^2/L_0^2$ and $P={\nu }_t{S}^2$ and $ϵ=C_{\mu }{k}^2/{\nu }_t$, the ${\nu }_t$ equation can be written as

$${\displaystyle \frac{D{\nu }_t}{Dt}}=C_{{\nu }_1}{\displaystyle \frac{{\nu }_t^2{S}^2}{k}}-{\alpha }_{\nu }^{*}{C}_{{\nu }_1}{C}_{\mu }k+D_{\nu }=C_{{\nu }_1}{\displaystyle \frac{{\nu }_t^2{S}^2}{k}}-C_{{\nu }_1}{C}_{\mu }k-({\alpha }_0-1)C_{{\nu }_1}{C}_{\mu }k{\displaystyle \frac{L^2}{L_0^2}}+D_{\nu }.$$

(20)

The typical SAS setting applied is then $L_0={\kappa }^{-1}{L}_{vK}$. Based on the ability of $L_{vK}$ to pick up flow instabilities, reference to a resolving flow regime is included in this way [5]. This approach is successful [5,12,91,92,93,94], but concerns arise from both a theoretical viewpoint [131,132] and with respect to the simulation performance [133]. These issues can be traced back to the concept of not spanning a complete variability range of the model in between fully resolved and fully modeled regimes (see the illustration in Figure 2). Something of interest is the following: there is ambiguity in the SAS approach of how to include the effect of $L_{vK}$: a linear dependency on the second velocity derivative was proposed first [12], whereas a quadratic formulation has been used later [5]. The analysis presented here confirms the need for a quadratic formulation.

4.2. Unbounded Extended Range

A systematic strategy (applicable in the CES-K*V framework) for introducing $L_0$ such that a variety of resolution regimes is spanned is to match LES scaling with modeling. This concept may be seen to safeguard an LES simulation performed on relatively coarse grids (grids that cannot ensure an appropriate LES resolution). By using $L_0=L_{\Delta }$, the latter can be accomplished based on [120]

$$\begin{array}{c}\hfill L_{\Delta }={\displaystyle \frac{{\Delta }_C}{{\left[1+{({\Delta }_C/L_{tot})}^3\right]}^{1/3}}}=f_{\Delta }{L}_{tot},\end{array}$$

(21)

where ${\Delta }_C=C_{+}\Delta$, and $f_{\Delta }=({\Delta }_C/L_{tot})/{[1+{({\Delta }_C/L_{tot})}^3]}^{1/3}$. For a sufficiently small $\Delta$, Equation (21) implies $L={\Delta }_C$. The relation $L={\Delta }_C$ is simply a consequence of calculating L by integration over the Kolmogorov spectrum [81]. Equation (21) can be simplified by replacing $L_{\Delta }$ by ${\Delta }_C$ combined with an appropriate bounding. This leads to

$$L_{\Delta }=\left\{\begin{array}{cc}{\Delta }_C& \mathrm{if}{\Delta }_C/L_{tot}\le 1\hfill \\ L_{tot}& \mathrm{if}{\Delta }_C/L_{tot}\ge 1\hfill \end{array}\right\}.$$

(22)

This bounding is relatively well justified; see Figure 3, where $L_{\Delta +}=L_{\Delta }/L_{tot}$ is shown.

It is of interest to note that this strategy is only meaningful within the CES-K*V concept, which involves scaling with $L_0/L$ in contrast to the $L/L_0$ in the other modeling approaches. Hence, we consider

$${\displaystyle \frac{Dk}{Dt}}={\gamma }^{*}P-ϵ+{\gamma }^{*}{D}_k,{\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[{\gamma }^{*}P-\alpha ϵ\right]+{\gamma }^{*}{D}_G.$$

(23)

The use of ${\gamma }^{*}={\gamma }_0{(L_0/L)}^2$ combined with $L_0=L_{\Delta }$ according to Equation (22) and (for simplicity) ${\gamma }_0=1$ results in

$${\gamma }^{*}=\left\{\begin{array}{cc}{({\Delta }_C/L)}^2& \mathrm{if}{\Delta }_C/L_{tot}\le 1\hfill \\ {(L_{tot}/L)}^2& \mathrm{if}{\Delta }_C/L_{tot}\ge 1\hfill \end{array}\right\}.$$

(24)

4.3. Bounded Full Range

The only way to enable a transitioning between fully resolved and fully modeled regimes (applicable in the CES-K*S/K framework) is the identification $L_0=L_{tot}$, where $L_{tot}$ is the total length scale involving both modeled and resolved contributions (see below). This approach can be implemented in CES-K*S,

$${\displaystyle \frac{Dk}{Dt}}=P-ϵ+D_k,{\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[P-{\alpha }^{*}ϵ\right]+D_G.$$

(25)

and CES-K*K models,

$${\displaystyle \frac{Dk}{Dt}}=P-{\beta }^{*}ϵ+D_k,{\displaystyle \frac{DG}{Dt}}=C_1{\displaystyle \frac{G}{k}}\left[P-\alpha ϵ\right]+D_G.$$

(26)

According to Equation (9), we have here ${\alpha }^{*}=1+L_{+}^2(\alpha -1)$. In contrast to the notation applied before, we apply here $L_{*}=L_{+}$, where $L_{+}=L/L_{tot}$. Correspondingly, we switch from ${\alpha }_0$ to $\alpha$, where $\alpha$ is the RANS value of ${\alpha }^{*}$. By definition we have $0\le L_{+}\le 1$; $L_{+}=0$ refers to complete flow resolution, whereas $L_{+}=1$ refers to complete modeling. The mechanism of variations between the RANS state and LES can be seen by considering Equation (26). $L_{+}=1$ implies that ${\beta }^{*}=1$. An increase in ${\beta }^{*}$ via $L_{+}$ variations increases the dissipation of k and it decreases k. For $L_{+}=0$ we have ${\beta }^{*}=\alpha$. Then, the production–dissipation terms of both the k and G equations are driven by $P-\alpha ϵ$, and the source terms finally disappear, leading to a zero modeled viscosity (the DNS limit). The computational methods obtained in this way are referred to as CES.

The difference between CES as presented here to usually applied RANS methods is the appearance of the resolution indicator $L_{+}=L/L_{tot}$. The modeled contribution is calculated by $L={〈k〉}^{3/2}/〈ϵ〉$ (the brackets refer to averaging in time). The total length scale is calculated correspondingly by $L_{tot}=k_{tot}^{3/2}/{ϵ}_{tot}$. Here, $k_{tot}=〈k〉+k_{res}$ is the sum of modeled and resolved contributions, where the resolved contribution is calculated by

$$k_{res}={\displaystyle \frac{1}{2}}\left(〈{\tilde{U}}_i{\tilde{U}}_i〉-〈{\tilde{U}}_i〉〈{\tilde{U}}_i〉\right).$$

(27)

Correspondingly, ${ϵ}_{tot}$ is the sum of modeled and resolved contributions, ${ϵ}_{tot}=〈ϵ〉+{ϵ}_{res}$, where

$${ϵ}_{res}=\nu \left[〈{\displaystyle \frac{\partial {\tilde{U}}_i}{\partial x_j}}{\displaystyle \frac{\partial {\tilde{U}}_i}{\partial x_j}}〉-〈{\displaystyle \frac{\partial {\tilde{U}}_i}{\partial x_j}}〉〈{\displaystyle \frac{\partial {\tilde{U}}_i}{\partial x_j}}〉\right].$$

(28)

Both $k_{res}$ and ${ϵ}_{res}$ are calculated on the fly (during the simulation) by using Equations (27) and (28) to process the resolved fluctuations produced by the model. It is of interest to note that $L_{+}$ is related to the corresponding kinetic energy ratio $k_{+}=k/k_{tot}$ by $L_{+}=k_{+}^{3/2}/{ϵ}_{+}$, where ${ϵ}_{+}=ϵ/{ϵ}_{tot}$. Away from walls, ${ϵ}_{+}=1$ represents a good approximation which implies $L_{+}=k_{+}^{3/2}$.

5. Performance of Hybridization Strategies

To further scrutinize the suitability of hybridization strategies, we turn now to illustrations of their flow simulation performance. This discussion will focus on two complex high-$Re$ wall-bounded turbulent flows that include flow separation: flow over periodic hills [134,135] and flow over a NASA wall-mounted hump [136]. For both of these flows, applications of the simulation methods under consideration are available.

An illustration of the periodic hill flow considered is given in Figure 4 [119]. This flow is a channel flow involving periodic restrictions. The flow, which is used a lot for the evaluation of turbulence models [6], involves features such as separation, recirculation, and natural reattachment [134,135]. Specifically, we consider this flow at the highest $Re=37$ K for which experimental data for model evaluation are still available [119,134,135]. The NASA wall-mounted hump flow is illustrated in Figure 5. Seifert and Pack [136] developed the wall-mounted hump model to investigate unsteady flow separation, reattachment, and flow control at a high Reynolds number $Re=c{\rho }_{ref}{U}_{ref}/\mu \approx 936$ K based on the chord length c and freestream velocity $U_{ref}$. Here, $\mu$ is the dynamic viscosity, and the subscript abbreviation $ref$ indicates the reference freestream conditions, which are determined at the axial point $x/c=-2.14$. The model reflects the upper surface of a 20-thick Glauert-Goldschmied airfoil that was originally designed for flow control purposes in the early twentieth century. As a benchmark for comparison, we used the experiment conducted by Greenblatt et al. [137] without flow control. This benchmark case has been extensively documented on the NASA Langley Research Center’s Turbulence Modeling Resource webpage and has been widely used for evaluating different turbulence modeling techniques, as discussed in the 2004 CFD Validation Workshop. We see in Figure 5 a strongly convex region just before the trailing edge, which induces flow separation.

The computational methods considered in this section are presented in

Table 3. Computational methods involved in performance analysis. Here, SST refers to the shear–stress transport (SST) $k-\omega$ turbulence model [140], used as the RANS baseline model.

Methods	Remarks
GAS-SST [133]	The GAS version of CES-K*V, SST is hybridized according to GAS.
SAS-SST [133]	SST hybridized according to scale adaptive simulation (SAS) [5,12].
DDES-SST [133]	SST hybridized according to delayed detached eddy simulation (DDES) [55].
RANS-SST [133]	SST RANS baseline model.
CES-KOS [111,119]	CES-K*S applied to $k-\omega$ turbulence model.
CES-KOKU [119]	CES-K*K version (equivalent to CES-KOS), $k-\omega$ model, hybridization via time scale.

. For reasons explained in Section 4, the main focus will be on methods having the potential to systematically cover wide hybridization regimes (methods described in Section 4.2 and Section 4.3). In particular, CES-K*V has not been strictly applied so far. However, there is a version of it (referred to as the grid-adaptive simulation (GAS) method), which is very similar to CES-K*V. The GAS approach has been applied in Refs. [133,138,139] with two significant differences to CES-K*V: instead of ${({\Delta }_C/L)}^2$ in Equation (24), these authors applied ${({\Delta }_C/L)}^{2/3}$, and the bounding $L_{tot}$ has been replaced by L,

$${\gamma }^{*}=\left\{\begin{array}{cc}{({\Delta }_C/L)}^{2/3}& \mathrm{if}{\Delta }_C/L\le 1\hfill \\ 1& \mathrm{if}{\Delta }_C/L\ge 1\hfill \end{array}\right\}.$$

(29)

The latter [i.e., ${({\Delta }_C/L)}^{2/3}$] is a representation of the modeled-to-total k ratio in difference to the squared modeled-to-total L ratio derived here by variational analysis. As shown in Figure 3, such a modification certainly can affect the performance of simulations. An attractive feature of this approach is its computational simplicity; simulations can be performed without any need to calculate additional variables (like $L_{tot}$). The disadvantage is the following: modeling variables such as k and L do not represent RANS variables in partially resolving simulations, i.e., the concept considered cannot provide a transition between fully resolved and fully modeled regimes.

5.1. Simple Empirical Strategies

DES and SAS face significant conceptual shortcomings; see the discussion in Section 4.1. Therefore, the DES and SAS methods will only be briefly discussed as part of the discussion of GAS methods in the next subsection. It is worth noting that the DES and SAS results discussed here were obtained by the same simulating settings as used in regard to GAS methods [133].

With regard to the periodic hill flow, DES and SAS results for reattachment point predictions are reported in

Table 4. Reattachment point locations and errors to experiments for several methods. Left: periodic hill flow at $Re=37$ K. In experiments, the reattachment point is reported as ${(x/h)}_r=3.76$ [134]. Right: NASA hump flow at $Re=936$ K. The reattachment point is reported as ${(x/h)}_r=1.10$ [137].

Methods	Cells	${(\mathit{x}/\mathit{h})}_{\mathit{r}}$	Error	Methods	Cells	${(\mathit{x}/\mathit{h})}_{\mathit{r}}$	Error
GAS-SST [133]	0.216M	3.70	$-1.6\%$	GAS-SST [133]	0.77M	1.18	$+6.3\%$
	0.392M	3.98	$+5.9\%$		1.53M	1.18	$+6.3\%$
	0.768M	4.00	$+6.4\%$
SAS-SST [133]	0.216M	7.49	$+99.2\%$	SAS-SST [133]	0.77M	1.29	$+16.2\%$
	0.768M	4.45	$+18.4\%$
DDES-SST [133]	0.216M	7.40	$+96.8\%$	DDES-SST [133]	0.77M	1.54	$+38.7\%$
	0.768M	4.17	$+10.9\%$
RANS-SST [133]	0.216M	7.43	$+97.6\%$	RANS-SST [133]	0.77M	1.27	$+14.4\%$
CES-KOKU [119]	0.5M	3.78	$+0.5\%$	CES-KOS [119]	1.7M	1.11	$+0.3\%$
	0.12M	3.70	$-1.6\%$		3.9M	1.10	$+0.0\%$
WRLES [141]	33.6M	4.00	$+6.4\%$	WRLES [142]	210M	1.095	$-1.4\%$
				WMLES [143]	4.4M	1.045	$-5.9\%$
					11.6M	1.105	$-0.5\%$

. Such reattachment point predictions represent a valuable criterion of how well simulation methods can deal with separation zone characteristics. It may be seen that both SAS-SST and DDES-SST provide rather inaccurate predictions. Used on the same grid as RANS-SST, they behave basically like RANS (there is no advantage of using a hybrid method).

With regard to the NASA hump flow, the corresponding DES and SAS results for reattachment point predictions are also reported in Table 4. Although the results are less drastic, the message is pretty much the same as obtained for the hill flow case. In particular, there is no substantial advantage of SAS-SST and DDES-SST compared to RANS-SST. More specifically, the performance of DDES-SST is much worse than the corresponding RANS-SST prediction. The SAS-SST and DDES-SST pressure coefficient ($C_p$) and skin friction coefficients ($C_p$) distributions are shown in Figure 6; the SAS-SST and DDES-SST mean velocity predictions are shown in Figure 7. The unsatisfactory performance of SAS-SST and DDES-SST with respect to these predictions, in particular DDES-SST, is clearly visible. The corresponding SAS-SST and DDES-SST predictions of the total Reynolds stress contour plots (not shown here) reveal that SAS-SST and DDES-SST incorrectly characterize the overall flow structure [133].

5.2. Unbounded Extended Range

With regard to the periodic hill flow, the GAS-SST predictions show a very different picture. The reattachment point predictions are reported in Table 4. Overall, the results are good. Nevertheless, in contrast to expectations, we observe that a grid refinement implies a systematic increase in the error. Similarly, a grid refinement leads to slightly less accurate predictions of the pressure coefficient distribution (not shown here). Velocity and Reynolds stress predictions on the coarsest grid show a very good agreement with measurements (not shown here).

With regard to the NASA hump flow, GAS-SST reattachment point predictions are also reported in Table 4. The error is relatively small and unaffected by grid refinement. GAS-SST skin friction ($C_f$) and pressure coefficient ($C_p$) distributions are shown in Figure 6. The results show significant improvements compared to SAS-SST, DDES-SST, and RANS-SST. Nevertheless, the double-peak structure of the $C_f$ distribution is not well reflected and is no different to RANS-SST. The corresponding mean velocity profiles are shown in Figure 7. Overall, there is a reasonable agreement with experimental results, although the profiles at $x/c=(1.1,1.2,1.3)$ cannot be seen to be accurate. The grid effect on these results is not reported. The Reynolds stresses are presented reasonably well. Contour plots of total Reynolds stress (not shown here) also show that GAS-SSTS describes the overall flow structure really well compared to experimental results and are much improved compared to SAS-SST and DDES-SST [133].

5.3. Bounded Full Range

In regard to periodic hill flows, the CES-K*K version applied (CES-KOKU, see Table 3) shows an almost perfect prediction of the reattachment point, even the use of a very coarse (0.12M) grid provides excellent reattachment point predictions (see Table 4). A thorough evaluation of the model performance for this flow can be found elsewhere [119]. A significant range of $Re$ ($Re=[37,250,500]$ K) and a variety of coarse and finer grids ($G_{500}$, $G_{250}$, and $G_{120}$ involving 500 K, 250 K, and 120 K grid points, respectively) were considered. In particular, these simulations cover the whole range of simulations under almost resolving and almost completely modeled regimes. These studies reveal an excellent ability of this model to represent the characteristics of the recirculation zone, the structure of the velocity field, and the Reynolds stresses.

In regard to the NASA hump flow, a significant range of $Re$ ($Re=[0.935,5,10]$M) and a variety of coarse and finer grids were considered, including a $G_3$ grid with 1.7M grid points and a $G_4$ grid with 3.9M grid points. The reattachment point predictions of the CES-K*S version applied (CES-KOS, see Table 3) are also shown in Table 4. The results are excellent; there are errors of 0.3% and 0% on the $G_3$ and $G_4$ grids, respectively, which is even much below the error of GAS-SST. An example of CES advantages is given in Figure 8 in regard to comparisons with WRLES and WMLES [111]. We see that all methods involved show a reasonable agreement with the experimental pressure coefficient profiles. Figure 8 also shows the mean skin friction coefficient obtained by CES-KOS, WMLES, and WRLES simulations, demonstrating their agreement with experimental values. In the separation zone, from $0\le x/c\le 0.65$, WRLES underpredicts the skin friction coefficient, while WMLES overestimates the actual peak. In regard to post-reattachment, however, the $C_f$ profiles of WRLES and CES-KOS match relatively well, despite using different frameworks, mesh sizes, and grid resolutions. In regard to GAS-SST, we see that the CES-K*S version applied represents the $C_f$ peak structure better than GAS-SST. Figure 9 shows corresponding mean velocity profiles. The conclusion is that the CES-K*S version applied provides predictions that are equivalent to or better than WRLES, better than WMLES, and better than GAS-SST.

6. Conclusions

This paper presents an analysis of strategies for the design of seamlessly resolving turbulent flow simulations. The basis for this study is a generic two-equation turbulence model which covers all usually applied two-equation turbulence models like $k-ϵ$ or $k-\omega$ models. Technically, there are two new ingredients in this approach. First, three basic hybridization frameworks are introduced for this wide class of turbulence models, which are referred to as CES-K*S (hybridization of dissipation in scale equation), CES-K*K (hybridization of dissipation in k equation), and CES-K*V (hybridization of modeled viscosity). The results obtained are based on exact variational analysis. The second technical novelty is the consideration of a flexible hybridization (related to the specification of the reference length scale $L_0$), which can be adjusted to several objectives. A first strategy (the simple empirical strategy) is the specification of $L_0$ according to DES-type and SAS-type models. A second strategy (the unbounded extended range strategy) is to take explicit reference to LES scaling in terms of the filter width $\Delta$. A third strategy (the bounded, full-range strategy) is to take reference to the total (RANS type) variables. As illustrated in Figure 2, these three strategies differ significantly with respect to computational requirements and their ability to cover various simulation regimes.

Relevant conclusions presented here can be summarized as follows:

• Turbulence modeling. An essential ingredient of turbulence models is a scale equation, e.g., an $ϵ$ or $\omega$ equation. There is no theoretical basis for such an equation; these equations are designed by taking reference to empirical arguments. There exists a huge variety of such equations: the $k-ϵ$ model [2,121], $k-\omega$ models [2,122], $k-L$ models [123,124,125], $k-kL$ models [5,9,10,126], $k-{\nu }_t$ models [5,12], and stand-alone ${\nu }_t$ models without a k-equation [13,127,128]. On top of that, a variety of equation structures (including or excluding cross-diffusion terms) is applied. The analysis presented here cannot provide strict guideline for further developments, but it leads to very valuable conclusions. First, the requirement for turbulence models to be applicable to various scaling regimes excludes the inclusion of cross-diffusion terms, which prohibits the hybridization of equations. Second, the diffusivities of the k and scale equations (e.g., the ${\sigma }_k$ and ${\sigma }_{ϵ}$ applied) should be the same. Third, the analysis presented clarifies the structure of dissipation terms; it provides corresponding evidence that is missing in SAS and DES approaches.
• Usually applied hybrid RANS-LES. Based on the fundamental conceptual shortcomings of RANS and LES (to often provide unreliable predictions of separated turbulent flows or to be inapplicable to very high-$Re$ flows seen in reality because resolution requirements cannot be met), the use of hybrid RANS-LES is the no-alternative approach to deal with these problems. Most hybrid RANS-LES applications are performed with DES-type or WMLES-type models. WMLES is known to be relatively inaccurate [6,110,111]; explicit evidence for this view is provided by the discussions in Section 5. In addition, WMLES faces significant uncertainty of predictions caused by different ways to combine RANS and LES elements in WMLES equations. The situation is different in regard to DES. There is research over decades hoping to improve DES predictions by empirical model improvements combined with variations of how DES is used. For the first time, this paper provides explicit mathematical evidence that DES cannot systematically span a range of modeled-to-resolved flow regimes because of the scaling applied (the setting of $L_0$). As specified in Figure 2, DES only triggers uncontrolled instabilities. This is often helpful, but is no guarantee for systematic improvements of simulation results compared to RANS. As shown here, the same issue applies to SAS. The flow simulation results presented in Section 5 fully confirm this view.
• Optimal solution: core CES: A way to overcome these problems is the use of the full-range strategy, which is equivalent to CES methods that apply $L_{+}$ to explicitly drive the hybrid model in between fully modeled (RANS) and fully resolved (LES) regimes. Applications of this approach [110,111,119] demonstrate an impressive ability of this approach to provide very good flow predictions at relatively low computational cost. A specific feature of CES is its well balanced predictive ability, in contrast to the DES and WMLES methods that can provide some flow characteristics well at the cost of other flow characteristics. Comparisons with WRLES and WMLES results (see, e.g., the comparisons presented in Section 5) show that CES performs clearly better than WMLES and at least as good or better than WRLES at a small fraction of the WRLES cost. It is essential to note that CES methods include via $L_{+}$ their own flow resolution indicator in contrast to LES methods. This matters in regard to the known difficulty of assessing the resolution ability of LES [24,25]. It is also worth mentioning that the methods presented here (based on the generic model consideration) enable the use of LES in conjunction with a variety of turbulence models that may be considered according to specific requirements.
• Interesting option: WMLES-type CES. In contrast to the usually applied hybrid methods, the CES core methods require the explicit calculation of $L_{+}$ by processing statistics of resolved motion. Currently available experience shows that this adds only a little fraction to computational costs, but it requires corresponding computational code modifications. An interesting alternative pointed out here is the use of the unbounded extended range strategy, which takes explicitly reference to LES scaling in terms of the filter width $\Delta$. In this way, the involvement of additional simulation ingredients (like $L_{+}$) can be avoided. This concept (which represents a version of consistently formulated WMLES) may be seen to safeguard an LES simulation performed on relatively coarse grids (grids that cannot ensure an appropriate LES resolution). Nevertheless, because of the reference scaling applied, there is no guarantee that this concept works well in regimes on coarse grids, well away from the LES regime. A variant of using this approach has been discussed here by taking reference to GAS simulation results; see Section 5. The results are much better than the corresponding RANS, DES, and SAS results, but not as good as the CES results. It is worth noting that the GAS concept differs from the corresponding result obtained here by variational analysis by applying, for example, an inaccurate scaling with $\Delta$ (see the discussion related to Equation (29)).