## 1 Introduction  
In recent years, many countries have paid more attention to ocean scientific research and resource exploration [1; 2; 3]. As a result, ocean delivery platforms with better performance have been the focus of research, in which unmanned underwater vehicles (UUVs) play an important role. The UUV has been widely used in various missions of ocean development [4] owing to its excellent environmental adaptation and scientific-observation abilities. However, a conventional UUV usually uses a screw propeller to provide thrust force, and this design has certain shortcomings; such as the propulsive system occupying a large space, consuming more energy, and a higher noise level. However, a bionic propulsor, which imitates the flapping caudal fin of fish, has several advantages; including higher efficiency, better maneuverability, lower noise, coupling of propulsion and control systems, etc. [5; 6]. In addition, fish can utilize the surrounding unstable flow field energy using their caudal fin to improve swimming performance [7; 8; 9; 10]. Therefore, studies on the propulsive performance and flow field evolution mechanism of bionic foil, flapping near the wave surface, are meaningful for the UUV in many aspects; such as utilizing wave favorable impact on foils propulsion, improving UUV navigation performance near the sea surface, increasing its endurance, and reducing ocean pollution. When operating at sea, a UUV, propelled by a bionic propulsor, needs to navigate near the surface for a long time in various situations (such as data exchange). During this time, wave motions strongly affect the surrounding flow field of the UUV and its hydrodynamic performance. Another key issue is that most UUVs use gas or battery as their power source, which means that they have limited endurance and cause pollution easily. However, a new idea was proposed recently that suggested that UUVs extract energy from surrounding flows as their driving power or auxiliary propulsion [11], as fish do. Similar to this idea, researchers have conducted studies on the application of flapping foils near the wave surface as an auxiliary propulsor for ships [12; 13; 14], but fewer studies have been done for UUVs. Researchers have analyzed the hydrodynamic performance of the flapping foil in an infinite flow field, which means the whole flow field is filled with water and there is no free surface. They usually focus on its mechanical parameters and flow field distribution; such as thrust force, resistance, input power, usable power, propulsive efficiency, and instantaneous vortex shape near the foil. [PERSON] [15] investigated the effect of caudal length on the hydrodynamic performance of flapping foils. In detail, [PERSON] considered that the hydrodynamic performance of a flapping foil is optimum at lower and higher Strouhal numbers (\(St<0.2\) and \(St>0.6\)). Additionally, [PERSON] demonstrated the possibility of improving foil propulsive efficiency at moderate Strouhal numbers (\(0.2<St<0.6\)) by manipulation of caudal length. However, comparing with rigid foil in above studies, a flexible foil is more similar to the caudal fin of fish. Therefore, [PERSON] [16] simply treated the flexible foil as an inextensible filament. Then, [PERSON] used a solver to investigate the hydrodynamic effects of passive flexibility in chord direction on a self-propelled plunging foil in two-dimensions--incompressible and laminar flows. The solver coupled the immersed boundary method for the flow and the finite difference method for the structure. Considering the flexible deformation in a span-wise direction of bionic caudal fin, [PERSON] [17; 18] found that its performance could be improved by adjusting the motion and flexibility parameters. The span-wise flexibility of caudal fin could increase thrust force with high propulsive efficiency.