## 2 Modeling and Computational Approach  
### Modeling Approach  
The motivation of the modeling approach is to cover both RANS and LES regimes via the implementation of a flow-resolving capability in RANS equations, with the understanding that the use of this model in resolving mode is equivalent to the use of LES. On the one hand, the implementation in RANS equations ensures a model complexity usually applied in RANS approaches. On the other hand, this approach avoids the typical LES problem given by the use of an artificial (i.e., non-physical) length scale (the filter width \(\Delta\)) [35]. The latter is the reason for significant LES problems: the need to use very fine grids (which implies very high computational cost) to ensure accurate results, and (very often) uncertainty regarding how well resolving a particular LES actually is. The hybridization approach applied (the set up of continuous eddy simulation (CES) methods [7; 25; 26; 27; 28; 29; 30; 31; 32]) can be used in conjunction with a variety of turbulence models, including probability density function (PDF) and Reynolds stress equation methods [7; 30; 36]. In an attempt to focus on the main advantages of this approach, it will be applied here on the level of two-equation eddy viscosity methods. The essential steps of the modeling approach will be described in the following paragraphs focusing on the equations considered, the hybridization approach, and the model functioning. We consider the compressible continuity equation and momentum equation [28],  
\[\frac{\widetilde{D}\widetilde{\rho}}{\widetilde{D}t}=-\tilde{\rho}\frac{ \partial\widetilde{U}_{k}}{\partial x_{k}},\qquad\qquad\frac{\widetilde{D} \widetilde{U}_{i}}{\widetilde{D}t}=-\frac{1}{\tilde{\rho}}\frac{\partial \widetilde{p}}{\partial x_{i}}+\frac{2}{\tilde{\rho}}\frac{\partial( \widetilde{\rho}\widetilde{v}_{ij}^{d})}{\partial x_{j}}-\frac{1}{\tilde{ \rho}}\frac{\partial(\widetilde{\rho}\widetilde{v}_{ij})}{\partial x_{j}}. \tag{1}\]  
The tilde refers to space-averaged variables, and \(\widetilde{D}/\widetilde{D}t=\partial/\partial t+\widetilde{U}_{j}\partial/ \partial x_{j}\) represents the filtered Lagrangian time derivative. Components of the velocity vector are denoted by \(\widetilde{U}_{i}\), and \(\widetilde{p},\widetilde{\rho}\), and \(\
u\), represent pressure, fluid density, and kinematic viscosity, respectively. The rate-of-strain tensor reads \(\widetilde{S}_{ij}=(\partial\widetilde{U}_{i}/\partial x_{j}+\partial\widetilde {U}_{j}/\partial x_{i})/2\), and its deviatoric part is given by \(\widetilde{S}_{ij}^{d}=\widetilde{S}_{ij}-\widetilde{S}_{mn}\delta_{ij}/3\). Throughout this paper, we use the sum notation. The right-hand side (RHS) of the momentum equation contains an unknown, the modeled stress tensor \(\tau_{ij}\). The eddy viscosity model assumes for this stress \(\tau_{ij}=2k\delta_{ij}/3-2\
u_{i}\widetilde{S}_{ij}^{d}\). Here, \(\delta_{ij}\) is the Kronecker delta, \(k\) represents the modeled kinetic energy, and \(\
u_{t}=C_{\mu}k/\omega\) is the turbulent viscosity of the flow which includes the turbulence frequency \(\omega=1/\tau\) and the model parameter \(C_{\mu}\) having a standard value \(C_{\mu}=0.09\). Here, the turbulence time scale \(\tau=k/\epsilon\) is related to \(k\) and the dissipation rate \(\epsilon\). The model viscosity \(\
u_{t}\) is closed via a two-equation turbulence model, in particular a \(k-\omega\) model that provides \(k\) and \(\omega\) for \(\
u_{t}=C_{\mu}k/\omega\). We consider the equations [37]  
\[\frac{\widetilde{D}k}{\widetilde{D}t}=P-\epsilon+D_{k},\qquad\qquad\frac{ \widetilde{D}\omega}{\widetilde{D}t}=C_{\omega_{1}}\omega^{2}\Big{(}\frac{P}{ \epsilon}-\beta\Big{)}+D_{\omega}+D_{\omega c}. \tag{2}\]  
Here, turbulence production is given by \(P=\
u_{t}S^{2}\) (which involves the strain rate \(S=(2\widetilde{S}_{mn}^{d}\widetilde{s}_{nm}^{d})^{1/2}\)), the dissipation of \(k\) is given by \(\epsilon=k\omega\), and \(\beta=C_{\omega_{2}}/(C_{k}C_{\omega_{1}})\).