## 2 Basic Equations and Parameter Settings  
### Control Equations and Turbulence Models  
In computational fluid dynamics, according to the different forms of control equations, it is divided into conservation equation or non-conservation equation. The conservation control equations are conducive to algorithm design and reduce the difficulty of solving. The numerical simulation of the flow field is a two-dimensional flow field, so the control equation is a two-dimensional conservation Navier-Stokes equation in the Cartesian coordinate system. The continuity equation, momentum equation, and energy equation can be expressed by the same expression, that is:  
\[\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{ \partial y}\ =\ J \tag{1}\]  
We consider \(U,F,G,J\) as column vectors. The Formula (1) represents the entire conservation type equation. \(U,F,G,J\) are given below. \[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]  
\[F\ =\ [\rho u,\rho u^{2}+p-\tau_{xx},\rho vu-\tau_{xy},\rho(e+\frac{V^{2}}{2})u +pu-k\frac{\partial T}{\partial x}-u\tau_{xx}-v\tau_{xy}] \tag{3}\]  
\[G\ =\ [\rho v,\rho uv-\tau_{yx},\rho v^{2}+p-\tau_{yy},\rho(e+\frac{V^{2}}{2})v +pv-k\frac{\partial T}{\partial y}-u\tau_{yx}-v\tau_{yy}] \tag{4}\]  
\[J\ =\ \Big{[}0,\rho f_{x},\rho f_{y},\rho(uf_{x}+vf_{y})+\rho q\Big{]} \tag{5}\]  
\[e\ =\ C_{\mathrm{v}}T \tag{6}\]  
\[\tau_{xy}\ =\ \tau_{yx}\ =\ \mu(\frac{\partial v}{\partial x}+\frac{\partial u }{\partial y}) \tag{7}\]  
\[\tau_{\mathrm{xx}}\ =\ \lambda(\mathrm{VV})+2\mu\frac{\partial u}{\partial x} \tag{8}\]  
\[\tau_{yy}\ =\ \lambda(\mathrm{VV})+2\mu\frac{\partial v}{\partial y} \tag{9}\]  
\[q_{x}\ =\ -k\frac{\partial T}{\partial x} \tag{10}\]\[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]0  
Among the equations, \(\rho\), \(u\), \(v\), and \(p\) are the density of the flow field, velocity, and pressure in the x and y directions, \(e\), \(\tau\), and \(q\) are the internal energy, shear stress, and volume heating rate per unit mass, respectively. \(k\), and \(C_{p}\) are the thermal conductivity and the constant heat capacity. \(\mu\) is a molecular viscosity coefficient, and \(\lambda\) is a second viscosity coefficient. [PERSON] puts forward the hypothesis that,  
\[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]1  
In order to close the control equation, we also need to introduce the ideal gas equation, where \(R\) is the gas constant. \[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]2  
The \(k\)-\(\varepsilon\) model can quickly and accurately predict the statistical characteristics of turbulence and contains more turbulent pulsation structure information and has a wide range of applications. The k-\(\varepsilon\) model is based on the generation and dissipation phase equilibrium of turbulent flow energy to solve the Reynolds time-average equation numerically. The specific way is to link the vortex viscosity coefficient with the turbulent energy and dissipation rate. \[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]3  
where \(\
u_{t}\) is the vortex viscosity coefficient, \(C_{\mu}\) is the dimensionless coefficient, \(k\) is the turbulent flow energy, and \(\varepsilon\) is the kinetic energy dissipation rate. The \(k\) and \(\varepsilon\) equations solved by the k-\(\varepsilon\) model are as follows [21]:  
\[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]4  
\[U\ =\ [\rho,\rho u,\rho v,\rho(e+\frac{V^{2}}{2})]^{\mathrm{T}} \tag{2}\]5  
In Formulas (15) and (16):  
\[