## 2 Governing Equations and Subgrid-Scale Stress Modeling of LES
### Governing Equations
In order to compute \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]7, a test filter denoted by a hat and of width larger than the grid filter is introduced. The dynamic constant \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]8 is calculated with a least-squares approach according to: \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]9 (10)  
Here, \(L_{ij}=\frac{\hat{u}_{i}u_{j}}-\hat{u}_{i}u_{j}\), \(M_{ij}=-2\hat{\Delta}^{2}(|\hat{\overline{S}}|\hat{\overline{S}}_{\bar{\rm ij }}-|\hat{\overline{S}}|\overline{S}_{\bar{\rm ij}}\), and \(|\overline{S}|=\sqrt{2\overline{S}_{ij}\overline{S}_{\bar{\rm ij}}}\). \(C_{d}\) is a local and instantaneous quantity and thus can vary widely in time and space. However, when \(C_{d}\) is a negative value, the simulated property may lead to numerical instabilities, and it generates instabilities in the field. Furthermore, if the total viscosity (\(\mu+\mu_{t}\)) is negative, the subgrid eddy viscosity \(\mu_{t}\) is generally clipped to zero. The test filter used in the DSLM is a symmetric discrete filter based on the trapezoidal rule:  
\[\frac{\partial\overline{T}}{\partial t}+\frac{\partial\overline{u}_{j} \overline{T}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}[\Gamma\frac{ \partial\overline{T}}{\partial x_{j}}]-\frac{\partial\eta_{j}^{SGS}}{ \partial x_{j}} \tag{5}\]0  
Overall, the DSLM which overcomes some of the drawbacks of the Smagorinsky model is a suitable alternative. However, it generates instabilities in the field, which need to be overcome. Subsequently, [PERSON] et al. [22] proposed a wall-adapting eddy viscosity model (WALE); the expression for the subgrid-scale eddy viscosity is changed so as to automatically take care of the zero value at the wall. In the WALE, the subgrid eddy viscosity \(\mu_{t}\) is defined by  
\[\frac{\partial\overline{T}}{\partial t}+\frac{\partial\overline{u}_{j} \overline{T}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}[\Gamma\frac{ \partial\overline{T}}{\partial x_{j}}]-\frac{\partial\eta_{j}^{SGS}}{ \partial x_{j}} \tag{5}\]1  
Here, \(S_{ij}^{d}=\frac{1}{2}\left(\overline{S  
Here, \(\Gamma_{t}\), \(v_{t}\), and \(Pr_{t}\) are the sub-grid scale thermal diffusivity, kinematic viscosity, and Prandtl number, respectively. In the dynamic models, the subgrid-scale turbulent Prandtl number or Schmidt number is obtained by applying the dynamic procedure originally proposed by [PERSON] et al. (2018) to the subgrid-scale flux. The system of equations and subgrid models is solved numerically by the commercial code Fluent 16.0 ([PERSON] et al., 2018).