## 2 Physical Equations and Models  
The complexity of fluid motion is determined by the characteristics of the fluid itself. The inertia, compressibility and viscosity of a fluid make it difficult to describe and measure fluid motion. In addition, fluid motion can be steady or unsteady, and laminar or turbulent, and these flow characteristics also lead to the complexity of fluid motion. According to the physical characteristics of fluid motion, mathematicians and physicists have established the related equations of fluid motion in the form of mathematical expressions to describe fluid motion [24].  
1. Mass Conservation Equation \[\frac{\partial u_{i}}{\partial x_{i}}=0\] (1)
2. Momentum Conservation Equation \[f_{i}-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+v\
abla^{2}u_{i}=\frac {\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}\] (2) where \(\rho\) is fluid density, \(p\) is pressure, \(u_{i}\) is the velocity components, \(x_{i}\) is the coordinate component, \(v\) is the kinematic viscosity and \(f_{i}\) is the unit mass force.
3. Energy Conservation Equation \[\rho c_{p}\frac{\partial T}{\partial t}+\rho c_{p}u_{j}\frac{\partial T}{ \partial x_{j}}=u_{j}\frac{\partial p}{\partial x_{j}}+\frac{\partial}{ \partial x_{j}}\Bigg{(}\lambda\frac{\partial T}{\partial x_{j}}\Bigg{)}\] (3) where \(c_{p}\) is the specific heat capacity at a constant pressure, \(T\) is the temperature and \(\lambda\) is the coefficient of thermal conductivity.  
The Realizable \(k-\varepsilon\) model is selected to relate the additional term of the turbulence pulsation value to the time mean [24].  
Turbulent kinetic energy equation:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku _{i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\alpha_{k}l_{eff}\frac{\partial k }{\partial x_{j}}\Bigg{)}+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k} \tag{4}\]Epsilon turbulent dissipation rate equation:  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}(\rho \epsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\alpha_{\epsilon}\mu_{ eff}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{)}+G_{1\epsilon}\frac{\epsilon}{k}(G_{k}+G_{3 \epsilon}G_{b})-G_{2\epsilon}\rho\frac{\epsilon^{2}}{k}-R_{\epsilon}+S_{\epsilon} \tag{5}\]  
For incompressible fluids, \(G_{b}=0\) and \(Y_{M}=0\). The source term does not need to be considered in the calculation in this paper, so \(S_{k}=0\)\(S_{\epsilon}=0\). Therefore, Equations (4) and (5) become:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\alpha_{k}\mu_{eff}\frac{\partial k}{ \partial x_{j}}\Bigg{)}+G_{k}-\rho\epsilon \tag{6}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\alpha_{\epsilon} \mu_{eff}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{)}+G_{1\epsilon}\frac{ \epsilon}{k}G_{k}-G_{2\epsilon}\rho\frac{\epsilon^{2}}{k}-R_{\epsilon} \tag{7}\]  
\[\mu_{eff}=\mu+\mu_{t},\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon},\ G_{k}=- \rho\overline{u_{i}^{\prime}}l_{j}^{\alpha}\overline{\alpha}u_{j},\ R_{ \epsilon}=\frac{C_{\mu}\rho\eta^{3}(1-\eta/\eta_{0})}{1+\beta\eta^{3}}\frac{ \epsilon^{2}}{k},\]  
\[\eta=S_{\epsilon}^{k},S=\sqrt{2S_{ij}S_{ij}},\ S_{ij}=\frac{1}{2}\Big{(}\frac {\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)}.\]  
Turbulence model constant [24]:  
\(G_{1\epsilon}=1.42\), \(G_{2\epsilon}=1.68\), \(C_{\mu}=0.0845\), \(\sigma_{k}=1.0\), \(\sigma_{\epsilon}=1.3\), \(\eta_{0}=4.38\), \(\beta=0.012\).