# Numerical Simulation of Non-Isothermal Mixing Flow Characteristics with ELES Method
## 2 Turbulence Model and CFD Methodology  
### Governing Equations of LES/WMLES  
The ELES method splits the domain into RANS and LES portions. The velocity field \(U\) (\(x\), \(t\)) is decomposed into a filtered component \(U\) (\(x\), \(t\)) and a subgrid-scale component \(u^{\prime}\)(\(x\), \(t\)) in the LES portion as below:  
\[U(x,t)=\overline{U}(x,t)+u^{\prime}(x,t) \tag{1}\]  
Unlike the mean (time-averaged) component \(U\) (\(x\), \(t\)) in the Reynolds decomposition, here, \(U\) (\(x\), \(t\)) represents an instantaneous field. The filtered momentum equation can then be derived once the velocity decomposition is brought into the Navier-Stokes equation [23]:  
\[\frac{\partial\overline{U}_{j}}{\partial t}+\frac{\partial\overline{U_{i}U_{j }}}{\partial x_{i}}=v\frac{\partial^{2}\overline{U}_{j}}{\partial x_{i} \partial x_{i}}-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{j}} \tag{2}\]  
where \(p\) (\(x\), \(t\)) is the filtered pressure field, which contains the body force influence, and the residual stress tensor arising from the subgrid-scale motion is defined as Equation (3):  
\[\tau_{ij}=\rho(\overline{U_{i}U_{j}}-\overline{U_{i}}\overline{U_{j}}) \tag{3}\]  
Then,  
\[\frac{\partial\overline{U}_{j}}{\partial t}+\frac{\partial(\overline{U}_{i} \overline{U}_{j})}{\partial x_{i}}=v\frac{\partial^{2}\overline{U}_{j}}{ \partial x_{i}\partial x_{i}}-\frac{\partial\tau_{ij}}{\partial x_{i}}-\frac{ 1}{\rho}\frac{\partial\overline{p}}{\partial x_{j}} \tag{4}\]  
Closure needs to be achieved by additional modeling of \(\tau_{ij}\).  
In addition, the filtered continuity equation is defined as Equation (5):  
\[\overline{(\frac{\partial U_{i}}{\partial x_{i}})}=\frac{\partial\overline{U} _{i}}{\partial x_{i}}=0 \tag{5}\]  
The energy governing equation is defined as Equation (6):  
\[\frac{\partial(\rho\overline{h})}{\partial t}+\overline{U}_{i}\frac{\partial (\rho\overline{h})}{\partial x_{i}}=\frac{\partial}{\partial x_{i}}\left[ \left(k+\frac{\mu_{t}c_{p}}{\mathrm{Pr}_{t}}\right)\frac{\partial\overline{T }}{\partial x_{j}}\right] \tag{6}\]  
where \(h\) represents the filtered enthalpy, \(k\) is the thermal conductivity, \(c_{p}\) is the constant pressure-specific heat of the fluid, \(Pr_{t}\) is a turbulent Prandtl number, and \(\mu_{t}\) is the eddy viscosity.  
Back to the closure modeling of the residual stress tensor \(\tau_{ij}\), it can be computed based on the Boussinesq hypothesis, and the expression is as Equation (7) [24]:  
\[\tau_{ij}=-2\mu_{t}\overline{S}_{ij}+\frac{1}{3}\tau_{ik}\delta_{ij} \tag{7}\]  
where \(\mu_{t}\) and \(S_{ij}\) are the eddy viscosity and the filtered rate-of-strain tensor, respectively. \(S_{ij}\) is defined as Equation (8):  
\[\overline{S}_{ij}=\frac{1}{2}(\frac{\partial\overline{U}_{i}}{\partial x_{j}} +\frac{\partial\overline{U}_{j}}{\partial x_{i}}) \tag{8}\]The Wall-Modeled LES (WMLES) method is adopted for the entire LES domain and reduces the stringent and \(Re\) number-dependent grid resolution requirements of classical wall-resolved LES [19]. The original Algebraic WMLES formulation was put forward by [PERSON] et.al. [25], which incorporates several models, such as a modified Smagorinsky model [26], the wall-damping function of Piomelli [27], and a mixing length model. The eddy viscosity \(\mu_{t}\) in the [PERSON] et al. model is defined as Equation (9).  
\[\mu_{t}=\rho\text{min}\Big{[}\left(\kappa d_{w}\right)^{2},\left(C_{Smag} \Delta\right)^{2}\Big{]}\bullet 5\bullet\Big{\{}1-\exp\Big{[}-\big{(}y^{+}/25 \big{)}^{3}\Big{]}\Big{\}} \tag{9}\]  
where \(d_{w}\) is the wall distance, \(\kappa=0.41\) is the Von Karman constant, and \(C_{Smag}=0.2\). \(S\) is the strain rate, and \(y^{+}\) is the normal wall inner scaling. \(\Delta\) is defined as Equation (10).  
\[\Delta=\text{min}(\max(C_{w}\bullet d_{w};C_{w}\bullet)l_{\max},h_{wn});h_{\max}) \tag{10}\]  
Here, \(h_{max}\) is the maximum edge length for a rectilinear hexahedral cell, \(h_{wn}\) is the wall-normal grid spacing, and \(C_{w}\) is a constant 0.15.