# Numerical Study on Airfoil Aerodynamics in Proximity to
## 1 Introduction
It was also found that lower flight height leads to higher aerodynamic coefficient amplitudes and strengthens the flow separation at high angles of attack. [PERSON] et al. [18] examined a transonic airfoil over the wavy ground and observed different shock behaviors. Results of different Mach numbers and ground clearances showed that the lower surface shock can either remain attached to the ground or detach from the ground surface, which depends on the flow conditions. In addition to numerical simulations, other authors also conducted experiments. [PERSON] and [PERSON] [19] used surface pressure and particle image velocimetry measurements to obtain the aerodynamics and the flow field of a NACA 0015 airfoil over the wavy ground. It revealed that the peak values of aerodynamic coefficients were larger than those in the flat ground case especially when the ground distance is small, and the variation trend of different aerodynamic coefficients was not consistent. In 2019, they further studied the effect of the trailing-edge flap under the same experimental conditions and reported that the fluctuation of aerodynamic coefficients is weakened due to the deflected flap [20]. It should be noted that the heaving motion exists in water waves and the wavy behavior is unsteady compared with the rigid wavy ground. There were only a few researchers focusing on the numerical studies on the ground effect over the wavy water surface. [PERSON] et al. established the nonlinear lifting theory with the discrete vortex method for two-dimensional airfoils [21] and three-dimensional wings [22] near the wavy water surface. The method was further extended to the research on the heaving airfoils over progressive water waves [23] in 2014. [PERSON] [24] numerically surveyed the aerodynamics of a heaving airfoil over an air-water interface and highlighted the impact of the water surface on the lift force. [PERSON] et al. [25] compared the ground effect over the wavy wall with the wavy water surface and reported that the wavy water disturbance velocity results in the differences in aerodynamics, especially in a small ground clearance. They demonstrated that the flow properties change due to the interaction between the water wave and the airflow under the airfoil but did not expound the mechanism of such interaction. [PERSON] et al. [26] discovered that the fluctuation of the wing aerodynamic coefficients in the wavy water surface case is much more extensive than that in the wavy ground case. Analysis of the flow field clarified that the air compression caused by the vertical movement of the water surface is the critical factor for the significant lift increase. There are also some studies focusing on the design of new vertical taking off and landing WIG crafts, such as the work conducted by [PERSON] [27], which studied the ducted fan performance over the ground, static water and dynamic waves respectively. He demonstrated that the wave with a large wave height affected the aerodynamic performance greatly and the wave surface was impacted by the jet flow. Despite that a great deal of work has been carried out for the exploration of the ground effect, the research on the airfoils over the wavy water surface is very limited. Some researchers have attempted to clarify the interaction mechanism between the airfoil and the wavy water surface, but the impact of the wave amplitude on the aerodynamics of the airfoil has not been discussed adequately. Consequently, it is essential to investigate the aerodynamics of the airfoil moving over the wavy water surface with various wave amplitudes. The aim of the paper is to present guidance for the design of WIG crafts which may face different sea conditions. The remainder of this paper is organized as follows: Firstly, the numerical calculation model is introduced, and the accuracy is verified. Then, the aerodynamic characteristics of the NACA 4412 airfoil in proximity to the wavy water surface are numerically calculated for different wave amplitudes.