# Numerical Study on Airfoil Aerodynamics in Proximity to
## 3 Results and Discussion  
### Lift Behavior  
The aerodynamics of the airfoil flying over the wavy water surface is periodic. For clearer analysis, a relative time \(t^{\star}\) is defined to describe the relative position between the \begin{table}
\begin{tabular}{c c c c c} \hline
**Mesh** & **Number of Cells** & \(y^{\star}\) & \(C_{L}\) & \(C_{D}\) \\ \hline Coarse & 0.13 million & 2 & 0.86890 & 0.01162 \\ Medium & 0.21 million & 1 & 0.86896 & 0.01143 \\ Fine & 0.31 million & 0.5 & 0.86892 & 0.01146 \\ Experiment & – & – & 0.8582 & 0.0109 \\ \hline \end{tabular}
\end{table} 
Table 1: Grid independence study. Figure 4: Comparison of the computed and theoretical wave elevation at \(t\) = 1.2 s.  
Figure 5: Comparison of experimental and computed pressure coefficient distributions for NACA 0015 airfoil. EXP = experimental. airfoil and wave. Supposing that \(T\) represents a wave period which is the time of the airfoil moving past one wavelength, the relative time is defined as \(t^{*}=t/T\). At the initial moment of one period, the trailing edge of the airfoil is directly above the wave crest. For each angle of attack, the lift coefficients of various amplitudes are shown in Figure 6. For comparison, the lift coefficients under the case of free stream and flat ground effect at the corresponding angle are calculated as well. As can be seen from Figure 6, the shape of the lift coefficient curves at low wave amplitudes is similar to sinusoidal waves. With the wave amplitude increasing, the fluctuation amplitude of the lift coefficient curves increases remarkably and the shape of the curve changes. It is also revealed that the trend of aerodynamics varying with the wave amplitude is inconsistent at different angles. When \(\alpha=0^{\circ}\), the most significant change occurs at the valley of the lift coefficient curve, with the minimum value decreasing considerably as the wave amplitude increases. When the wave amplitude is larger than 0.0875c, the minimum value of the lift coefficient is even less than zero, and this will threaten flight safety seriously. While the fluctuation amplitude at the peak increases as well, the maximum value of the lift coefficient still decreases. For the case of \(\alpha=1^{\circ}\), the amplitudes of both the wave crest and the trough increase, but the minimum value decreases more. As for the case of \(\alpha=2^{\circ}\) and \(4^{\circ}\), the maximum value of the lift coefficient increases notably, and the wave trough value decreases rarely especially for \(\alpha=4^{\circ}\). It can be concluded that as the angle of attack increases, the enhancing effect caused by the amplitude increase transfers from the valley of the lift curve to the crest, in the range of low attack angles. The time-averaged lift coefficients of different angles are then calculated and presented in Figure 7, and the result matches with the aforementioned analysis. For \(\alpha=0^{\circ}\) and \(1^{\circ}\), the time-averaged lift coefficients decreases as the wave amplitude increases and those of \(\alpha=2^{\circ}\) and \(4^{\circ}\) change inversely. The significant increase in the lift coefficient at \(\alpha=4^{\circ}\) when the airfoil moves in proximity to the wavy water surface has been discovered in our previous research and explained in detail [26]. The flow mechanism is that the air under the airfoil is compressed by the particles on the water surface due to their vertical movement, and the hydrodynamic force is then transmitted to the airfoil. According to this theory, the vertical disturbance of the wavy water enhances as the amplitude increases, and hence the time-averaged lift coefficient rises. However, the theory cannot directly explain the lift loss at \(\alpha=0^{\circ}\) and \(1^{\circ}\). Compared the dashed line with the dotted line in Figure 6a, the lift coefficient in the ground effect case is lower than that in the free stream case at \(\alpha=0^{\circ}\).