# Numerical Study on Airfoil Aerodynamics in Proximity to
## 3 Results and Discussion
### Flow Field Analysis and Discussion  
Based on the foregoing analysis, the trend of the lift coefficient curve varying with the wave amplitude is similar for \(\alpha=0^{\circ}\) and \(1^{\circ}\). Since the lift loss phenomenon is more prominent at \(0^{\circ}\), the aerodynamics of this angle is taken as an example. We first choose a low wave amplitude case where \(a\) is \(0.05c\) for analysis. In this case, the amplitude is limited compared with the ride height. For clearer clarification of the flow mechanism, the airfoil is divided into upper and lower surfaces through the leading and trailing edge points and the lift coefficients of these two parts are defined as \(C_{L,up}\) and \(C_{L,low}\). Figure 8 shows the lift coefficient curves in two periods. It can be seen that the fluctuation amplitudes of \(C_{L,up}\) and \(C_{L,low}\) are similar in this case, and therefore the influence of the wavy water vertical movement on the lower surface is limited. Figure 7: Variation in the time–averaged lift coefficients with wave amplitudes for different angles of attack. \(a/c=0\) means flat ground. For detailed analysis, one period of the airfoil moving over the wavy water surface is divided into four time points, with the state at \(t^{*}=0\) and \(t^{*}=1\) being the same. Figure 9 shows the velocity contours at different moments and Figure 10 shows the pressure coefficient distributions on the airfoil at corresponding moments. \(V^{*}\) is defined as the ratio of the local velocity to the stream velocity. Further, the pressure coefficient \(C_{p}\) is defined as Equation (7),  
\[C_{p}=\frac{p-p_{\infty}}{1/2\rho V_{\infty}^{2}} \tag{7}\]  
where \(p\) is the local pressure, \(p_{\infty}\) and \(V_{\infty}\) are the pressure and velocity of the incoming flow far ahead respectively, and \(\rho\) is the air density. Figure 8: Lift coefficients of the upper and lower surfaces at \(a=0.05c\) and \(\alpha=0^{\circ}\). As shown in Figure 9, the leading edge of the airfoil at \(\alpha=0^{\circ}\) is close to the water surface. Therefore the cambered leading edge will lead to shrinkage at the flow channel inlet under the lower surface. For instance, the flow channel in Figure 9d is similar to a 2-D Venturi tube, which is first convergent from the inlet--Section A to the throat--Section B, and then divergent to the outlet--Section C. In this case, the incoming flow will first accelerate to a maximum velocity at the throat section and then decelerate gradually. However, under the circumstance of the wavy water surface, the vertical movement of the wave influences the flow channel. To illustrate this phenomenon, the case of the airfoil flying over the rigid wavy ground with the same wave parameters is conducted as a comparison. Figure 11 shows the velocity contours of the airfoil moving over the wavy water surface and wavy ground at \(t^{\star}=0.75\). Compared with Figure 11b, whether the high velocity region at the throat section or the low velocity regions near the leading edge and trailing edge in Figure 11a do not expand to the underlying surface. It is apparent that the actual flow channel in the wavy water case is not the exact shape formed by the lower surface of the airfoil and water surface. The reason is that the particles on the water surface move vertically, leading to the air near the water surface moving with them. The outline of the actual flow channel is drawn in Figure 11a and is compressed compared with Figure 11b. The throat--Section B moves rearward, and the width of it is narrower. Therefore the high velocity region is larger, and the maximum value of the velocity increases. During the period from \(t^{\star}=0\) to \(t^{\star}=0.25\), the wave trough moves towards the leading edge, and the distance between the leading edge of the airfoil and the water surface increases. With the cross section getting widen, the shrinkage at the inlet decreases correspondingly and weakens the acceleration effect of the airflow in the flow channel.