## 2 CFD Simulations
### Numerical Methods  
#### 2.2.1 Turbulence Models  
At middle to high _Re_, the unsteady random motion could be observed, which is called turbulence. By the NS equations, turbulence could be described mathematically. For the enormous demand for the computational resources of the direct numerical simulation (DNS) in practice, the Reynolds average (RANS) method is widely used [25]. RANS introduces additional unknown terms into equations, which need to be provided by suitable turbulence models. Therefore, the adopted turbulence model would determine the simulation quality. Typical RANS models include k-epsilon and k-omega, and SA. k-epsilon are two-equation models, which are designed for traditional industrial CFD and have disadvantages of insensitivity to the boundary layer separation and adverse pressure gradients [26]. For the external aerodynamics studied in this paper, the k-epsilon is not suitable. The standard k-omega models, however, are better at predicting adverse pressure gradient boundary layer flows and have advantages relative to the k-epsilon. Further, SST k-omega overcomes the deficiency of standard k-omega models in that the solution of which has a strong sensitivity to the freestream values. So, the SST k-omega turbulence model is suitable for external aerodynamic flows [27]. It should be noted that excessive turbulence energy generation is a draw-back of the SST k-omega and the production limiter is suggested. The SA model is specifically designed for aeronautics and aerospace applications involving wall-bounded flows [28]. Therefore, the SST k-omega and SA turbulence models are selected.  
SST k-omega is a two-equation turbulence model and the turbulent viscosity \(\mu_{t}\) is computed as a function of the turbulence kinetic energy k and the specific dissipation rate \(\omega\). K and \(\omega\) are obtained from the following transport equations:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\Bigg{)}+G_{k}-Y_{k}+S_{k}+G_{b} \tag{11}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{i}}(\rho \omega u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\Bigg{)}+G_{\omega}-Y_{\omega}+S_{\omega}+G_{ \omega b}+D_{\omega} \tag{12}\]  
SA is a one-equation RANS turbulence model and the transport equation for the turbulent kinematic viscosity \(\widetilde{\
u}\) is:  
\[\frac{\partial}{\partial t}(\rho\widetilde{\
u})+\frac{\partial}{\partial x_{ i}}(\rho\widetilde{\
u}u_{i})=\frac{1}{\sigma_{\widetilde{\
u}}}\Bigg{\{} \frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\rho\widetilde{\
u})\frac{ \partial\widetilde{\
u}}{\partial x_{j}}\Bigg{]}+C_{b2}\rho\Bigg{(}\frac{ \partial\widetilde{\
u}}{\partial x_{j}}\Bigg{)}^{2}\Bigg{\}}+G_{v}-Y_{v}+S_{ \widetilde{v}} \tag{13}\]  
Attention should be paid to the modified turbulent viscosity order selection of the SA turbulence model. Taking the sharp trailing edge (based on NACA4), adopting SA at three far-field distances as examples and the comparison of the results between two order types are shown in Table 7. Unlike under the incompressible environment [3], the mean error ratios of \(\mathrm{P/P_{t}}\) and \(\mathrm{T/T_{t}}\) of the first-order under 12 m far-field distance are (9.18%, 4.96%), and those of the second-order are (4.42%, 2.16%). Similarly, the results under 16 m and 20 m far-fields distances are (9.38%, 7.71%), (5.33%, 5.24%), (7.42%, 7.12%), and (4.69%, 4.28%) in turn and the error ratios of the second-order are preferred, hence, the second-order upwind Modified turbulent viscosity is chosen.