# Numerical Study on Airfoil Aerodynamics in Proximity to
## 2 Numerical Method and Validation
The adaptable constants have the following values: \(\beta^{*}\) = 0.09, \(\beta_{1}\) = 0.075, \(\beta_{2}\) = 0.0828, \(\sigma_{k1}\) = 0.85, \(\sigma_{k2}\) = 1.0, \(\sigma_{\omega 1}\) = 0.5, \(\sigma_{\omega 2}\) = 0.856, and \(a_{1}\) = 0.31. Figure 1: Sketch of the computational domain. Figure 2: Grids around the airfoil. A second-order implicit scheme is used for time discretization and a second-order upwind scheme is applied for space discretization. The time step \(\Delta t\) for unsteady computation is 0.0001 s to satisfy Equation (6),  
\[\frac{V_{\infty}\cdot\Delta t}{\Delta x}<1 \tag{6}\]  
where \(\Delta x\) is the minimum grid size. The SIMPLE algorithm is used for the segregated flow solver, with the under-relaxation factor set as 0.8 and 0.2 for the velocity solver and the pressure solver respectively. The momentum residual and the periodicity of the aerodynamic coefficient are selected as the convergence standard. If the residuals of the momentum, turbulent kinetic energy, and continuity are less than 1 \(\times\) 10\({}^{-6}\), and the lift coefficient changes periodically simultaneously, the convergence of the numerical calculation is considered to be reached. To verify the accuracy of the computational result, a validation for computational fluid dynamics (CFD) method is conducted. The aerodynamic characteristics of the NACA4412 airfoil under the ground effect are calculated using the CFD method, and then compared with the experimental data obtained from the existing research [3; 6; 12; 31]. The angle of attack is 4\({}^{\circ}\), the ground clearance is \(h\)/\(c\) = 0.15, and the Reynolds number is 3.0 \(\times\) 10\({}^{5}\). These conditions are consistent with those in the experiment performed by [PERSON] et al. [32]. The calculation results are presented in Figure 3, and it can be observed that the calculation results are consistent with the experiment results. At the same time, the grid independence is verified by constructing two new grids with additional refinement. Table 1 presents the calculation results under different grid numbers. The relative errors of the lift coefficient for these meshes are 1.25%, 1.25%, and 1.25% respectively, and the relative errors of the drag coefficient for these meshes are 6.61%, 4.86%, and 5.14% respectively. It can be noted that the maximum difference between the medium grid and the fine grid is 0.26%, which indicates that the medium grid can basically ensure the simulation accuracy. Therefore the medium grid is selected to perform the numerical calculations. In addition, the fluctuation amplitude of the lift coefficient and drag coefficient is less than 1 \(\times\) 10\({}^{-6}\) when the result is convergent. Hence the unphysical oscillation caused by numerical dispersion can be ignored during the simulation. According to the wave theory, the first-order linear wave is based on the inviscid hypothesis. However, the \(k\)-\(\omega\) SST turbulence model is used for CFD simulation. Therefore, it is necessary to ensure that the numerical wave is in agreement with the theoretical wave. A wave with the wavelength \(\lambda\) = 5\(c\) and the amplitude \(a\) = 0.12\(c\) is simulated in the range of the computational domain. As shown in Figure 4, the wave generated by the VOF method matches well with the theoretical wave. \[\eta=a\cos(kx-\omega t) \tag{1}\]0 Comparison of the computed and experimental pressure coefficient for NACA 4412 airfoil. In this study, the wavy ground effects were evaluated as a contrast. The NACA 0015 airfoil is chosen to verify the numerical method for predicting aerodynamics of an airfoil over the wavy ground. The simulation is conducted to compare with the wind tunnel data from [PERSON] and Tremblay-Dionne [19]. The simulation conditions are consistent with the experiment, including the chord length of the airfoil set as 25.4 mm, the Reynolds number set as 1.61 \(\times\) 10\({}^{5}\), the angle of attack set as 12\({}^{\circ}\), and the ground clearance set as _h/c_ = 0.05.