# Numerical Study on Airfoil Aerodynamics in Proximity to
## 1 Introduction  
When the aircraft flies in close proximity to an underlying surface, its lift-to-drag ratio will increase due to the ground effect. Wing-in-ground (WIG) crafts are designed to utilize the favorable influence of the ground effect for higher operating efficiency and have been developed by many countries [1]. Compared to the conventional aircraft, the WIG craft has some unique features in the application environment and aerodynamic characteristics. The flow physics is also more complicated. To fully exploit the benefit of the ground effect on the WIG craft, a considerable number of studies have been conducted such as the research on two-dimensional airfoils in the static ground effect [2; 3; 4; 5], airfoils in the dynamic ground effect [6; 7] and three-dimensional aircrafts or wings in proximity to the ground [8; 9; 10; 11]. Although extensive researches on the ground effect have been carried out, most of them considered the underlying surface as a rigid flat ground. However, considering that the WIG craft is designed to fly at a fairly low altitude, the ocean is a more proper environment for application without the danger of encountering hills, trees, or other obstacles on the land. Therefore, WIG crafts usually cruise on the water surface for transportation. The water surface is deformable and free, therefore it cannot be substituted absolutely by ground for research. When the WIG craft flies over the ocean surface, waves often occur and sometimes have high amplitudes. Therefore, it is necessary to explore theaerodynamics of WIG crafts in proximity to the wavy water surface, especially for various wave amplitudes. To eliminate the inconsistency between the actual flight situation and the theoretical research, numerous studies have attempted to explore the impact of the free surface on the ground effect. Among them, [PERSON] [12] investigated the deformation of the water surface under WIG crafts and showed that the deformation was negligible in the two-dimensional situation. However, three-dimensional CFD results suggested that the water surface deformation behind the wing could not be neglected, which was probably caused by wingtip vortices instead of the pressure distribution beneath the wing. [PERSON] [13] and [PERSON] [14] took use of the lifting line theory to reveal lifting mechanisms of two-dimensional and three-dimensional WIG crafts. The results showed that the free surface deformation significantly affects the WIG craft aerodynamics at a low Froude number, especially for the low altitude. But when the Froude number is high enough, the water free-surface can be regarded as a rigid wall. [PERSON] [15] proposed a modified iterative boundary element numerical method for two-dimensional airfoils and three-dimensional wings moving steadily over the water free-surface. It was demonstrated that the free surface generated Kelvin wave deformation due to the ground effect, which might cause substantial effects on the WIG craft performance. Therefore, it is necessary to consider the free surface effects during the design of the WIG craft. Since waves over the ocean surface complicate the flow field around WIG crafts, more recent attention has been focused on the ground effect over the wavy ground. [PERSON] and [PERSON] [16] investigated aerodynamics of a NACA 6409 airfoil over the wavy ground by solving the Euler equations. Numerical simulations were conducted under various ride heights, wave amplitudes and wavelengths, and the results showed that the wavelength affects the slope of aerodynamic phase curves. By adopting the sliding mesh technology, [PERSON] et al. [17] conducted a series of numerical simulations for a three-dimensional WIG craft flying over the wavy ground. Their work argued that although the aerodynamic coefficients are periodic under the wavy ground circumstance, the average values of the coefficients vary in the same manner as the flat ground case when the flight height varies.