## 2 Numerical Method and Validation  
In this paper, the airfoil NACA 4412 is selected. The chord length of the airfoil is 0.5 m, the incoming velocity is 30 m/s, and the corresponding Reynolds number is \(1.0\times 10^{6}\). The ride height \(h\) between the airfoil and the ground is described by the distance between the trailing edge point of the airfoil and the ground. The selection of the reference points at the trailing edge point refers to the relevant papers in this field [3; 16; 20; 21; 22], so that this study can be compared with the existing study. In this study, the ride height is set to \(h=0.1c\), where \(c\) is the chord length. The range of angles of attack is 0-18\({}^{\circ}\). The shape of the wavy ground is represented by a sine function, where the wave amplitude \(a\) is 0.05\(c\) and the wavelength \(\lambda\) is 5\(c\). The selection of the wave parameters refers to the studies mentioned in the introduction, thus this study can be compared with other research. The surface equation of wavy ground is shown in Equation (1), where \(x_{0}\) is the initial phase. \[y=a\sin(2\pi/\lambda\cdot x-x_{0}). \tag{1}\]  
The computational domain is shown in Figure 1. The sliding grid technology is adopted, and the whole calculation domain is divided into upper and lower parts. During the calculation, the upper domain remains static, the lower part of the domain moves to the right at a velocity equal to the incoming flow velocity, and the upper and lower domain exchange data through their contact surface. The inlet boundary and the upper boundary is 20\(c\) away from the airfoil, and the outlet boundary is 25\(c\). The inlet boundary condition is set as the velocity inlet, the upper boundary and the outlet boundary as the pressure outlet, the contact surface of the upper and lower domains as the interface, and the wavy ground as the non-slip wall. In order to avoid calculation divergence at the initial time, a section of flat ground is located in front of the wavy ground. The lengths of the wavy ground and the flat ground are 75\(c\) and 35\(c\), respectively. The grid of the computational domain is shown in Figure 2. The structural grid is employed, and the grid near the airfoil and the ground is refined. The numerical simulation is carried out by solving the RANS equations coupled with the turbulence model \(k\)-\(e\). The second-order implicit scheme and second-order upwind scheme are employed for time discretization and spatial discretization, respectively. The time step \(\Delta t\) is 0.00015 s, considering the stability of unsteady calculation. Equation (2) is usually satisfied. \[\frac{u\cdot\Delta t}{\Delta x}<1, \tag{2}\]  
where \(\Delta x\) is the minimum grid size and \(u\) is the incoming velocity. The convergence standard is the momentum residual and the periodicity of the aerodynamic coefficient. The residuals of the momentum, turbulent kinetic energy, and continuity are less than \(1\times 10^{-6}\). At the same time, the lift coefficient and drag coefficient change periodically. If these two conditions are satisfied simultaneously, the convergence of the numerical calculation is judged to be reached. The validation of the computational fluid dynamics (CFD) calculation is performed before numerical calculation. Many scholars have conducted a lot of studies on the NACA 4412 airfoil with  
Figure 1: Computational domain for moving over wavy ground. Figure 2: Grid for computational domain. numerical calculations or experiments. [PERSON] [28] conducted an experiment to study the aerodynamic characteristics of the NACA 4412 airfoil with the ground effect. The study here also investigates the problem of the ground effect. Therefore, the experimental data in [PERSON]'s paper are used for CFD validation. The angle of attack of the airfoil is taken as \(4^{\circ}\), the ride height of the airfoil is \(h/c=0.15\), and the Reynolds number is \(3.0\times 10^{5}\). These conditions are consistent with [PERSON]'s experimental conditions [28].