# Numerical Study on Airfoil Aerodynamics in Proximity to
## 4 Conclusions  
The aerodynamics and flow fields of the NACA 4412 airfoil moving over the wavy water surface with various wave amplitudes are numerically investigated in this paper. The calculations are conducted in the range of low angles of attack (\(\alpha=0^{\circ}\)-\(4^{\circ}\)), and for each angle, the wave amplitude varies from 0.05\(c\) to 0.1\(c\). Based on the result of this study, the following conclusions can be drawn. 1. The wavy water surface will cause the lift coefficient of the airfoil to fluctuate periodically. The fluctuation amplitude of the lift coefficient curves increases remarkably with successive increases in the wave amplitude and sharp peaks occur in the curves. WIG crafts flying over the ocean with high waves will generate severe oscillation. 2. During the periodical change of lift, a high wave amplitude will lead to sudden lift loss at \(\alpha=0^{\circ}\) and \(1^{\circ}\), which even causes negative lift in some cases and threats flight security for WIG crafts. With the increase of the angle of attack, the significant decrease in lift will gradually turn to the increase in lift. Therefore, WIG crafts should avoid cruising at small angles of attack. 3. At \(\alpha=2^{\circ}\) and \(4^{\circ}\), the time-averaged lift coefficient of the airfoil in proximity to the wavy water surface is larger than that of the flat ground, and it increases as the wave amplitude increases. This is consistent with the former research, which indicated that the vertical movement of wavy water surface would compress the air and lead to a significant increase. While at \(\alpha=0^{\circ}\) and \(1^{\circ}\), the time-averaged lift coefficient decreases as the wave amplitude increases, and the lift loss caused by the ground effect enhances. Figure 23: Lift coefficients of the upper and lower surfaces at \(a=0.1c\) and \(\alpha=2^{\circ}\). Figure 22: Pressure coefficient distributions on the airfoil at different moments at \(a=0.1c\) and \(\alpha=2^{\circ}\). * For a lower wave amplitude at \(\alpha=0^{\circ}\) and \(1^{\circ}\), the wavy water surface affects the flow field indirectly by driving the air to move vertically, thereby changing the shape of the actual flow channel. Under this situation, the change in the lift coefficient is mainly influenced by the contraction and expansion of the flow channel as the airfoil moves over wavy water. * For a higher wave amplitude at \(\alpha=0^{\circ}\) and \(1^{\circ}\), the pressure distributions on the airfoil are directly impacted by the vertical movement of particles on the wavy water surface. When the wave crest moves below the leading edge stagnation, the compression by the wave creates a high pressure region and prevents air from flowing under the lower surface. A low pressure region consequently occurs after the leading edge and decreases the lift coefficient significantly. This phenomenon is caused by the close distance between the leading edge stagnation and the wave crest, and disappears at higher angles of attack. * The upward movement of high amplitude waves can transmit hydrodynamic force to the airfoil by compressing the air under the airfoil. This is the reason for the significant lift increase at \(\alpha=2^{\circ}\) and \(4^{\circ}\). The distance between the leading edge and the wave crest is the critical factor which determines whether the impact will be blocked and results in the inconsistent lift change at different angles of attack. This paper studies the influence of wave amplitudes on the aerodynamics of the airfoil moving over the wavy water surface, providing a new understanding of the ground effect in the case of wavy water surface. These findings are instructive for the design and application of WIG arcerafts, especially under high grade sea conditions. Conceptualization, X.L., D.M. and H.H. ; methodology, Y.G. and H.H. ; software, X.L. ; validation, X.L. and H.H. ; formal analysis, Y.G. and H.H. ; investigation, X.L. and Y.G. ; resources, D.M. ; data curation, X.L. ; writing--original draft preparation, X.L.