## 3 Numerical Simulation Method  
### Mathematical Model and Numerical Method  
The Reynolds-averaged Navier-Stokes method is adopted for numerical simulation of flow around airfoils. The basic governing equations of compressible fluid flow under study are as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\big{[}\mu(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{\partial x_{ k}})]+\frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline{u_{i}^{\prime}u_{j}^{ \prime}}\Big{)} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho H)+\frac{\partial}{\partial x_{j}}\Big{(} \rho u_{j}H\Big{)}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}k_{x}+\frac {c_{p}\mu_{t}}{\Pr_{t}}\bigg{)}\frac{\partial T}{\partial x_{j}}\bigg{]}-p \frac{\partial u_{j}}{\partial x_{j}}+\frac{\partial u_{i}}{\partial x_{j}} \mu_{eff}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{\partial x_{j}} \bigg{)} \tag{3}\]  
where \(u_{i}\) is [PERSON] average velocity component with average sign omitted, \(u_{i}^{\prime}\) is fluctuating velocity, \(\delta_{ij}\) is component of Kronecker tensor, \(\Pr_{t}\) is turbulent Prandtl number, \(\mu_{eff}\) is the sum of turbulent viscosity and molecular viscosity, which is called effective viscosity, and \(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) can be calculated by the Boussinesq hypothesis:  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\Bigg{(}\frac{\partial u_ {i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}-\frac{2}{3 }\bigg{(}\rho k+\mu_{t}\frac{\partial u_{t}}{\partial x_{t}}\bigg{)}\delta_{ij} \tag{4}\]  
The turbulence model chooses the SST k-\(\omega\) model [62], and its control equation is:  
\[\frac{\partial(\rho k)}{d_{t}}+\frac{\partial}{\partial x_{j}}\big{(}\rho uj k\big{)}=P_{k}-\rho\theta^{*}\omega k+\frac{\partial}{ \partial x_{j}}\Bigg{[}(\mu+\sigma_{k1}\mu_{t})\frac{\partial k}{\partial x_{ j}}\Bigg{]} \tag{5}\]  
\[\frac{\partial(\rho\omega)}{d_{t}}+\frac{\partial}{\partial x_{j}}\big{(}\rho uj \omega\big{)}=P_{\omega}-\rho\beta\omega^{2}+\frac{\partial}{\partial x_{j}} \Bigg{[}(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{j}} \Bigg{]}+2\rho(1-F_{1})\sigma_{\omega 2}\frac{1}{\omega}\frac{\partial k}{ \partial x_{i}}\frac{\partial\omega}{\partial x_{j}} \tag{6}\]  
The formula of eddy viscosity coefficient in the above formula is:  
\[\mu_{t}=\min\bigg{(}\frac{\rho k}{\omega},\frac{\partial_{1}\rho k}{\Omega F_{ 2}}\bigg{)} \tag{7}\]  
where \(\partial_{1}=0.31\).  
The simulations adopt an implicit algorithm, coupled solver, and choose the second-order upwind scheme as the discretization scheme. The boundary condition of the airfoil wall is a non-slip boundary condition, and the inlet and outlet of the far-field boundary, respectively, adopt pressure far-field and pressure outlet boundary conditions.  
Figure 1: RAE 2822 airfoil variable camber and chord-length diagram.