# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 1 Introduction  
Exploring the ocean requires a variety of marine observation equipment. In particular, the marine mobile vehicle provides new means for the ocean's development and utilization, which significantly expands the human ability to explore, understand, and exploit the ocean. Underwater propulsion technology is one of the critical technologies of the marine mobile vehicle, which directly affects the maneuverability, operability, and endurance of vehicles. Compared with the traditional propeller, the propulsion system based on flapping foil has high stability, high efficiency, excellent maneuverability, and environmental friendliness [1; 2; 3]. As a result of so many advantages, it is necessary to understand the hydrodynamic performance of the propulsion system of flapping foil fully and deeply. Analyzing the impact of different parameters on the flapping foil's propulsion performance and trying to improve its propulsion performance is of great significance for revealing the propulsion mechanism and popularizing applications. The fluid-structure interaction between the flapping foil and flow field is complicated and unsteady [4; 5]. In recent decades, many researchers have explored the propulsion mechanism of flapping foil and the effects of multiple factors on the propulsion performance through natural observation, numerical simulation, and experimental study [6; 7; 8]. Previous studies have found that factors such as the motion amplitude and reduced frequency [9], motion trajectory [10], St (Strouhal number) [11], pivot location [12], phase difference between heaving and pitching motions [13], shape and flexibility of the foil [14; 15] can directly affect the hydrodynamic performance of the flapping foil. The reduced frequency and the amplitude of heaving and pitching motion are the main factors affecting the flapping flow field structure [9]. [PERSON] and [PERSON] [16] study the hydrodynamic performance and wake vortex of the pure pitching flapping foil at a small amplitude through the tank experiment. The results show that the wake vortex structure can be effectively controlled by controlling the reduced frequency, pitching amplitude, and motion trajectory, which can significantly affect the flapping foil's thrust. [PERSON] [17] and [PERSON] [18] analyze the effect of the reduced frequency on the propulsion performance of full-active flapping foil. They find that there is a critical value for the reduced frequency. When the reduced frequency is less than the critical value, the mean thrust coefficient increases monotonously. However, when the reduced frequency is higher than the critical value, the propulsion performance of flapping foil decreases significantly. The influencing mechanism of heaving and pitching motion amplitude is similar to the reduced frequency [7; 17]. The thickness, chord length, curvature, and flexibility also affect the foil's hydrodynamic performance [11; 19; 20]. Besides, some researchers study the influence of non-sinusoidal motion trajectories on the propulsion performance of flapping foil and intend to improve it [21; 22; 23; 24; 25]. According to the above research, most researchers assume the freestream as uniform and steady when studying the hydrodynamic performance of flapping foil. In the marine environment, especially when approaching the surface and floor of the sea, the flow field is usually non-uniform and time-varying due to the seabed topography or the wind and waves. The conclusions based on the above assumption may not reflect the hydrodynamic performance of flapping foil under real conditions. The numerical simulation results obtained by simplifying the flow to a uniform flow may not be accurate enough. Therefore, it is of great practical significance to analyze the influence of non-uniform and time-varying freestream velocity on the propulsion performance of the flapping foil. We noticed a few studies on the influence of non-uniform and time-varying flow velocity on the propulsion performance of the flapping foil.