## 2 Governing Equations and Subgrid-Scale Stress Modeling of LES  
### Governing Equations  
All the flow variables are decomposed into a filtered (or resolved) and unresolved (or subgrid scale, SGS) components by applying a spatial filter in LES. Therefore, a variable which can represent pressure, temperature, or velocity is divided into \(\overline{\varphi}\) (resolved) and \(\overline{\varphi}\) (subgrid) components, that is, \(\varphi=\overline{\varphi}+\overline{\varphi}\). Spatial filtering is performed on the Navier-Stokes equations by applying a low-pass filter. \[\overline{\varphi}(x,t)=\int G\big{(}x,x^{\prime}\big{)}\varphi\big{(}x^{ \prime},t\big{)}dx^{\prime} \tag{1}\]  
where \(x\) is the space coordinates, \(x^{\prime}\) denotes dummy space coordinates for each grid cell, and \(t\) is the time coordinate. \(G\) is the low-pass filter, and in this paper a top-hat filter is used. Which is defined as  
\[G\big{(}x,x^{\prime}\big{)}\ =\ \left\{ \begin{array}{c}\frac{1}{\Delta^{3}} \forall|x-x^{\prime}|\ \leq\ \frac{1}{2}\Delta\\ 0\forall|x-x^{\prime}|>\frac{1}{2}\Delta\end{array} \right\} \tag{2}\]  
where, \(\Delta=V^{\frac{1}{3}}=\big{(}\Delta_{x}\Delta_{y}\Delta_{z}\big{)}^{\frac{1} {3}}\), \(\Delta\) denotes the characteristic filter width. By applying a filter \(G\) to the incompressible Navier-Stokes equations and by neglecting the body forces, the following equations can be obtained. \[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{3}\]  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]  
\[\frac{\partial\overline{T}}{\partial t}+\frac{\partial\overline{u}_{j} \overline{T}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}[\Gamma\frac{ \partial\overline{T}}{\partial x_{j}}]-\frac{\partial\eta_{j}^{SGS}}{ \partial x_{j}} \tag{5}\]  
where the variables with an overbar represent the filtered (the locally averaged) values. Here, \(u_{i},\ p\) and \(T\) denote fluid velocity, pressure and temperature, respectively. \(\rho,\ \mu\) and \(\Gamma(=\frac{\lambda}{\mathcal{E}_{p}})\) represent density, dynamic viscosity, and the thermal diffusivity of air, respectively. \(\lambda\) and \(c_{p}\) are thermal conductivity and the specific heat of air, respectively. Using the eddy viscosity hypothesis, the subgrid stresses \(\tau_{ij}^{SGS}\) can be described as  
\[\tau_{ij}^{SGS}=\rho\big{(}\overline{u_{i}u_{j}}-\overline{u_{i}u_{j}}\big{)}= 2\mu_{t}\overline{S}_{ij}+\frac{1}{3}\tau_{kk}^{SGS}\delta_{ij} \tag{6}\]  
where, \(\tau_{kk}^{SGS}\) is the isotropic part of the subgid scale stresses and can be added to the filtered pressure or simply neglected for incompressible flow [16]. \(\delta_{ij}\) denotes the Kronecker delta. \(\overline{S}_{ij}\) represents the strain rate tensor for the resolved scale, which is defined as  
\[\overline{S}_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\overline{u}_{i}}{\partial x _{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\Bigg{)} \tag{7}\]  
In the most basic SGS model (namely the Smagorinsky model), \(\mu_{t}\) is a product of a length scale and a velocity scale, can be expressed as  
\[\mu_{t}=\rho L_{s}{}^{2}\sqrt{\overline{2S}_{ij}\overline{S}_{ij}} \tag{8}\]\[L_{s}\] is calculated using \[L_{s}=\min(kl,C_{s}\Delta)\] (9) where \[k\] is the Von Karman constant, \[l\] is the distance closest to the wall.