## 2 CFD Simulations
### Grid Strategy
#### 2.1.3 Mesh Parameters  
The airfoil surface is the main mean vorticity and turbulence source, where the momentum and scalar transports change vigorously, and the numerical variables have high gradients. Hence, the meshing of the near wall region has a key influence on the simulation fidelity. Near the airfoil surface, there exists a thin layer called the boundary layer, which contains all the viscous effects. A successful wall-bounded turbulent flow prediction is closely related to the viscous boundary layer representation, and firstly, we should ensure the first layer cells of the boundary layer (\(y_{H}\)) are in the viscous sublayer. Generally speaking, for high accuracy demands on the wall boundary layer simulations, the \(y^{+}\) of the near wall meshes with 1 is recommended [23]. The \(y^{+}\) is calculated by the following Equation:  
\[y^{+}=\frac{\rho y_{P}u_{\tau}}{\mu} \tag{4}\]  
where \(\rho\) is the flow density, \(y_{P}\) is the height from the centroid of the wall adjacent cells to the wall, \(u_{\tau}\) is the friction velocity, \(\mu\) is the flow dynamic viscosity and \(y_{H}\) could be obtained by doubling \(y_{P}\). For the \(T_{t}\) is 81.2 K, \(\rho\) is about 0.0247 kg/m\({}^{3}\). Then according to Equation (5), the speed of propagation of acoustic disturbances in the air (speed of sound \(C_{air}\)) is 180.6 m/s. The Mach number is 10 and U\({}_{t}\) is 1806 m/s, therefore, according to Equation (6), \(\mu\) is about 4.46082 \(\times\) 10\({}^{-5}\) pasas. \[C_{air}=20.05\sqrt{T} \tag{5}\]  
\[R_{e}=\frac{\rho ULL}{\mu} \tag{6}\]And now we just have to calculate \(u_{\tau}\) to confirm \(y_{H}\). As the \(R_{e}\) is less than \(1\times 10^{9}\), the below empirical correlation formulation to estimate the skin friction coefficient (\(C_{f}\)) is used [24]:  
\[C_{f}=[2\log_{10}(R_{e})-0.65]^{-2.3} \tag{7}\]  
After \(C_{f}\) is calculated, we could obtain the wall shear stress (\(\tau_{W}\)):  
\[\tau_{W}=\frac{1}{2}\rho U^{2}C_{f} \tag{8}\]  
Now, \(u_{\tau}\) is achieved by:  
\[u_{\tau}=(\frac{\tau_{W}}{\rho})^{0.5} \tag{9}\]  
Finally, \(y_{H}\) is calculated:  
\[y_{H}=2y_{P} \tag{10}\]  
Figure 2: (a) Grid division of the sharp trailing edge; (b) Grid division of the blunt trailing edge. It is worth noting that \(C_{f}\) is an empirical formulation, so the computed \(y_{H}\) value is an estimate and would need to be updated using information from the CFD analysis. Initially, \(y^{+}\) is 1 and \(y_{H}\) is 4.65 \(\times\) 10\({}^{-5}\) m according to Equation (4) to Equation (10), while during the simulation process, the test results show that the maximum \(y^{+}\) value exceeds 1. We further lower the \(y^{+}\) value and the tests show that the value of 0.08 could guarantee the maximum \(y^{+}\) of airfoil surface within 1 and the corresponding \(y_{H}\) is 3.5 \(\times\) 10\({}^{-6}\) m. The close views of meshes near the WALL region and different trailing edge shapes are as shown in Figures 3 and 4. The mesh used in the present computations does not have such (desirable) mesh clustering around the shock or the alignment with it. This fact should be borne in mind while comparing the sharpness of the shock. Secondly, we need to perform different mesh levels to ensure grid independency, and in every refinement, the number of total grids is 1.4 times more. We take the sharp trailing edge (based on NACA4) adopting the SST k-omega as an example and the other cases present similar errors as we exemplified the errors of the meshes for the sharp trailing edge. Under x/L locations from \(-\)0.007 m to 0 m, the ratios between \(P\) and \(P_{t}\) and \(T\) and \(T_{t}\), and the error ratios compared to the reference data are shown in Tables 1-3. At 12 m far-field distance, the mean error ratios and the total mean error ratio of (P/P\({}_{t}\), T/T\({}_{t}\)) of three level meshes are (10.2%, 6.98%, 8.59%), (2.28%, 3.96%, 3.12%), and (9.88%, 4.21%, 7.05%).