## 1 Introduction  
At present, with the rise of global energy prices and the advancement of the dual carbon project, the energy-saving demands of underwater vehicles are becoming increasingly prominent. Among them, underwater gliders have attracted more and more attention in the field of marine engineering by virtue of their outstanding advantages such as low navigation resistance, high energy utilization rate, and large sailing distance. However, the conventional underwater glider realizes the underwater gliding movement of a fixed track by adjusting its own buoyancy and center of gravity, which is difficult to reach a specific position to achieve multi-angle and approach-type detection functions, and lacks maneuverability. At the same time, due to the lack of the large power source of conventional underwater gliders, it is difficult to maintain a stable track in complex ocean current environments. Therefore, a high-efficiency thruster is added to the stem of a conventional underwater glider to form a hybrid-driven underwater glider [1; 2]. The Kappel tip rake propeller [3; 4] is a special propeller with special blade tip geometry, which can improve the propulsion efficiency to a certain extent, and has high application value and potential market space. The Kappel tip rake propeller is a type of special propeller that is beneficial in improving propulsion efficiency, with the blade end plate tilted towards the suction side [5]. The end plate of the propeller is similar to the winglet mounted on the wing, and the end plate can block the flow around the blade tip, thereby enlarging the pressure difference between the suction side and the pressure side, that is, becoming a positive component of thrust [6; 7; 8; 9; 10]. [PERSON] et al. [11; 12] first studied the Kappel tip rake propeller through the potential flow theory mainly based on the lift line theory. [PERSON] [13] optimized the lift line theory, considered the effect of the geometry characteristics of the blade tip rake on the induced velocity in the three-dimensional direction, and analyzed the hydrodynamic performance of the Kappel propeller. [PERSON] [14] studied the geometry characteristics of the Kappel propeller and proposed a three-dimensional coordinate conversion formula that takes into account the blade tip rake. Compared to conventional propellers, Kappel tip rake propellers have their own unique advantages. [PERSON] et al. [15; 16] showed that the Kappel tip rake propeller improved propulsion efficiency by approximately 4% compared to conventional propellers in real-scale operational tests. [PERSON] et al. [17] tested the propeller performance of the Kappel tip rake propeller in the cavitation state in a cavitation water cylinder, and the efficiency of the Kappel tip rake propeller at the design working point was 1.44% higher than that of the conventional propeller, and its end plate structure could inhibit the propeller tip vortex and block the trend of cavitation along the propeller guide edge. [PERSON] [18] focused on the cavitation performance of the Kappel tip rake propeller and found that the pitch and rake of the blade tip had a significant impact on the cavitation performance of the propeller, which provided guidance for the geometric design of the pitch and rake of the propeller. The Kappel tip rake propeller geometry has changed, and the performance is also different. [PERSON] et al. [19] obtained the propeller blades with different bending degrees of the 6-type blade tip plate by changing the tip rake and found that the propeller efficiency of the blade tip with a tip rake in the range of 0.08-0.10 was the highest, and it was difficult to improve the propeller propulsion efficiency if the blade tip rake was too large or too small. The effect of increasing the thrust coefficient caused by the increase in the bending degree of the end plate gradually weakened.