## 2 Solution Method
### Numerical Method
#### 2.2.1 Governing Equations  
For the LES turbulence model, the resolved part of the field represents the \"large\" eddies that are computed numerically, while the sub grid part of the velocity represents the \"small scales\" whose effect on the resolved field is included through the subgrid-scale model. The central idea in LES turbulence, where smaller eddies are ignored in a turbulent flow simulation, is mathematically achieved by applying a filtering procedure to the full Navier-Stokes equations, whose instantaneous continuity equation, momentum equation, and energy equation for a compressible fluid can be written as Equations (1)-(3), respectively:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial\varsigma_{\rm j}} \big{[}\rho u_{j}\big{]}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial\varsigma_{\rm j }}\Big{[}\rho u_{i}u_{j}+p\varsigma_{\rm ij}-\varsigma_{\rm ij}\Big{]}=0 \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho e_{0})+\frac{\partial}{\partial x_{j}}\big{[} \rho u_{j}e_{0}+u_{j}p+q_{j}-u_{i}\varsigma_{j}\big{]}=0 \tag{3}\]  
Figure 2: Boundary conditions of the simulation scheme. For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:  
\[\tau_{ij}=2\mu S_{ij}^{*} \tag{4}\]  
where the trace-less viscous strain-rate is defined by:  
\[S_{ij}^{*}\equiv\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)-\frac{1}{3}\frac{\partial u_{k}}{ \partial x_{k}}\delta_{ij} \tag{5}\]  
The heat-flux \(q_{j}\) is given by [PERSON]'s law:  
\[q_{j}=-\lambda\frac{\partial T}{\partial x_{j}}\equiv-C_{p}\frac{\mu}{\Pr} \frac{\partial T}{\partial x_{j}} \tag{6}\]  
where the laminar Prandtl number \(\Pr\) is defined by:  
\[\Pr\equiv\frac{C_{p}\mu}{\lambda} \tag{7}\]  
To close these equations, it is also necessary to specify an equation of state. Assuming a calorically perfect gas, the following relations are valid:  
\[\gamma\equiv\frac{C_{p}}{C_{v}},p=\rho RT,e=C_{v}T,C_{p}-C_{v}=R \tag{8}\]  
where \(\gamma,C_{p},C_{v}\) and \(R\) are constant. The total energy \(e_{0}\) is defined by:  
\[e_{0}\equiv e+\frac{u_{k}u_{k}}{2} \tag{9}\]  
A closed set of partial differential equations can be formed by supplementing Equations (1)-(9) with the gas data for \(\gamma\), \(\Pr\), \(u\), and perhaps \(R\), as well as complementing with boundary conditions. As for the filtering operation, it is applied through a convolution operation with a filtering kernel. The general procedure for applying a filtering kernel to any function \(\phi\) is defined as follows:  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial\varsigma_{\rm j }}\Big{[}\rho u_{i}u_{j}+p\varsigma_{\rm ij}-\varsigma_{\rm ij}\Big{]}=0 \tag{2}\]0  
The form of this filter kernel is a box filter, meaning the value of the kernel converges to 0 below the length scale limit L and time scale limit T. This follows the same mathematical definition for a filter circuit, where a convolution operation is used to define the relationship between an unfiltered function, an impulse response function, and the final filtered response. By applying this convolution operation, the effect is to average out any phenomena that occurs on small-length scales and small time scales, thereby eliminating the influence of fast flow fluctuations (small time scales) and small-length scale flow behavior. The above average is generalized to both space and time; however, the filtering procedure could be applied only in space, or only in time, depending on the needs of the simulation. As a result, the flow field u is redefined in terms of its filtered average and the unfiltered portion of the flow occurring in the small-length scales and small time scales, called the subgrid scale portion.