## 2 Modeling and Computational Approach
### Modeling Approach
The diffusion terms involved read  
\[D_{k}=\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\
u+\
u_{t}\Big{)}\frac{ \partial k}{\partial x_{j}}\Bigg{]},\quad D_{\omega}=\frac{\partial}{ \partial x_{j}}\Bigg{[}\Big{(}\
u+\frac{\
u_{t}}{\sigma_{\omega}}\Big{)}\frac{ \partial\omega}{\partial x_{j}}\Bigg{]},\quad D_{\omega c}=\frac{C_{\omega}}{ k}(\
u+\
u_{t})\frac{\partial k}{\partial x_{j}}\frac{\partial\omega}{ \partial x_{j}} \tag{3}\]  
The model parameters involved have the values  
\[C_{\omega 1}=0.49,\ \ C_{\omega 2}=0.072,\ \ C_{\omega}=1.1,\ \ C_{k}=0.09,\ \ \sigma_{\omega}=1.8.\]  
The cross-diffusion term \(D_{\omega c}\) is added to \(D_{w}\), although its relevance is subject to questions (it is supposed to reduce the inaccurate behavior of \(k-\omega\) models for free shear layer flow simulations) [38]. Numerical errors induced by the integration of the \(\omega\) equation across the viscous sub-layer can compromise the accurate simulation of the velocity profile [39]. To circumvent this problem, \(\omega=2\
u/d^{2}\) is applied to specifically set the value of \(\omega\) at the first cells above the wall, while \(d\) accounts for the distance between the wall and the center of the first cell. We note that the expression \(\omega=2\
u/d^{2}\) is derived under the assumption that the first cell center lies within the viscous sub-layer, alternative settings of which are available in [38]. The hybridization can be accomplished in several ways [27]. By following our recent approach [25], we address the hybridization by a modification of the dissipation in the scale equation by the introduction of an unknown function \(\beta^{*}\), which replaces \(\beta\) in Equation (2),  
\[\frac{\widetilde{D}k}{\widetilde{D}t}=P-\epsilon+D_{k},\qquad\qquad\frac{ \widetilde{D}\omega}{\widetilde{D}t}=C_{\omega_{1}}\omega^{2}\Big{(}\frac{P} {\epsilon}-\beta^{*}\Big{)}+D_{\omega}+D_{\omega c}. \tag{4}\]  
We refer to the model described in the following as CES-KOS (or simply KOS), where KO refers to the use of an \(k-\omega\) model and KOS refers to a hybridization of the scale equation. Essential steps of the hybridization described elsewhere [27] are the following ones. The production term in the \(\omega\) equation is replaced by using the \(k\) equation and the first variation in the resulting \(\omega\) equation is considered. The hybridization error becomes minimal for a zero first-order variation. The latter provides a relationship between variations of model parameters (\(\beta^{*}\)) and model variables (like \(k\) and \(\omega\)). The integration of this relationship from a complete modeling state to a state of partial modeling implies  
\[\beta^{*}=1+R(\beta-1). \tag{5}\]  
Here, \(R\) (which characterizes the flow resolution degree) will be discussed in the next paragraph. In order to obtain \(\beta^{*}=1+R(\beta-1)\), we neglected for simplicity \(D_{\omega c}\) and the substantial derivatives \(\widetilde{D}k/\widetilde{D}t\) and \(\widetilde{D}\epsilon/\widetilde{D}t\). It is possible to involve the latter terms at the cost of more complex equations [29]. Relevant assumptions also made to obtain \(\beta^{*}=1+R(\beta-1)\) are normalized first variations \(\delta k/k\) and \(\delta\omega/\omega\) which are invariant in space and time. It is worth noting that the latter assumptions do not involve any restriction of the modeling approach, as these assumptions stabilize the flow resolution in space and time. These equations were applied in conjunction with the elliptic blending approach used in our previous work to account for wall damping [40] (in the RANS-mode for all the flow fields). The model functioning is essentially related to the variable \(R\). The hybridization approach provides the relation \(R=L_{+}^{2}\). Here, \(L=k^{3/2}/\epsilon\) refers to the length scale of modeled motions, and \(L_{+}=L/L_{tot}\) is the turbulence length scale resolution ratio incorporating both modeled (\(L\)) and total contributions (\(L_{tot}\)) [7].