## 2 Numerical Method and Validation
The ground in the calculation domain is set as a moving wall with the velocity of airflow, and the other boundary conditions are the same as those mentioned above. Grid independence verification is carried out in order to ensure that the calculation results are independent of the grid. The number of grids increases or decreases according to the same principle. Three groups of grid numbers are set--namely, coarse grid, medium grid, and fine grid. The number of nodes in the horizontal and vertical directions is adjusted according to the consistent principle to increase or decrease the number of grids. The calculation results in Table 1 show that the calculation results are almost unchanged when the number of grids exceeds the medium grid. Hence, the medium grid is selected in this study. After convergence, the fluctuation amplitude of the lift coefficient and drag coefficient is less than \(1\times 10^{-6}\), which is far less than the lift coefficient and drag coefficient themselves. This indicates that the unphysical oscillation caused by numerical dispersion can be ignored. Figure 3 shows the pressure coefficient of the airfoil. The calculation results are in good agreement with the experimental results. This study involves unsteady calculations, so the unsteady calculation results need to be verified. Some scholars have studied the unsteady calculation of CFD [29; 30; 31]. In this paper, the unsteady numerical simulation of an NACA0015 airfoil oscillating in the pitching direction is carried out and compared with the experiment results [32]. The calculated parameters are consistent with those in the experiment. The incoming Mach number is \(0.29\), the Reynolds number is \(1.95\times 10^{6}\), the airfoil angle of attack is \(4^{\circ}\), the oscillation amplitude is \(4.2^{\circ}\), and the oscillation frequency is \(10\) Hz. The calculation results are shown in Figure 4, and the airfoil lift coefficient is in good agreement with the experimental results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mesh** & **Cell Numbers** & **Y+** & **Cl** & **Cd** \\ \hline coarse & 127,188 & 2 & 0.86890 & 0.01162 \\ medium & 209,127 & 1 & 0.86896 & 0.01143 \\ fine & 311,121 & 0.5 & 0.86892 & 0.01146 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grid independence study. Figure 3: Pressure coefficient distribution with the ground effect. There are numerical calculations of the airfoil at high angles of attack in this study. The airfoil will undergo flow separation at high angles of attack. Therefore, it is necessary to verify the accuracy of the numerical results at high angles of attack. The NASA experimental data [33] are selected for numerical verification, and the results are also compared with the numerical results of Qu [10]. The airfoil is NACA 4412, the Reynolds number is \(6\times 10^{6}\), and the free stream velocity is 88 m/s. The calculation results of the shear stress transport (SST) for two turbulence models \(k\)-\(e\) and \(k\)-\(\omega\) are shown in Figure 5.  
Since a negative angle of attack is not involved in this study, the calculation of a negative angle of attack is not considered. It can be seen from Figure 5 that the results of the two turbulence models are both in good agreement with the experimental results, as the angle of attack is less than \(12^{\circ}\). As the angle of attack is greater than \(12^{\circ}\), the calculation results of the \(k\)-\(e\) turbulence model are closer to the experimental results than the \(k\)-\(\omega\) SST and also better than Qu, which is the reason for the selection of the \(k\)-\(e\) turbulence model. However, after the stall angle of attack of \(15^{\circ}\), the calculated results do not agree well with the experimental results. At a high angle of attack, especially for angles greater than the stall angle of attack, the airfoil is faced with flow separation, and the separation region is often accompanied by a shedding vortex. Therefore, the flow field around the airfoil is more complex than that at a low angle of attack.