## 3 Results
### Motion Way Effects on Hydrofoil Hydrodynamic Characteristics  
To investigate the effect of the motion way on the hydrodynamic characteristics of the flexible hydrofoil, the material of the flexible hydrofoil in this paper was defined as LDPE with a Shore hardness of D50, based on the results in section A. The pumping and propulsion efficiencies of the flexible hydrofoil and the rigid hydrofoil were compared under steady flow conditions, considering pitch-only motion, and lifting-and-pitching-coupled inward and outward motions, as shown in Figure 14. The results showed that among the three modes of motion of the flexible hydrofoil, the inward oscillation mode had the most favorable hydrodynamic characteristics, followed by the outward oscillation mode. In contrast, the pitch-only oscillation motion mode had the least favorable hydrodynamic characteristics. The pitch-only oscillation motion resulted in a significant reduction in the propulsion and pumping efficiency of the flexible hydrofoil. On the contrary, the pitch-only oscillation motion of the rigid hydrofoil was superior to the outward oscillation motion under the same conditions. This result was consistent with the previous findings of [PERSON] et al. [21]. Further, this paper discussed and analyzed the differences between the motion modes of flexible hydrofoils and their hydrodynamic performance compared to rigid hydrofoils. Figure 14: Comparison of pumping efficiency and propulsion efficiency between rigid and flexible hydrofoils with different oscillation modes. To gain a more intuitive understanding of the hydrodynamic performance of the hydrofoils in three modes of oscillation, this paper presented a comparative analysis of the vortex structures and velocity distributions of the flexible hydrofoils in different modes of oscillation in a steady flow field. Figure 15 illustrates the vortex nephograms of the three modes in the steady flow field. From the vortex nephograms, it was observed that the trailing edge vortex center strength was higher in the inward oscillating motion mode, the stability of the reverse Karman vortex street was higher, and the maximum dissipation distance was farther. This was followed by the outward oscillating motion mode, in which the reverse Karman vortex street dissipated more quickly, and the vortex strength was relatively low. A comparison of the structures of the vortex street in the three modes of motion revealed that in the inward oscillation motion, the distance between the positive vortex and the negative vortex was relatively large in the Y-direction. This made the vortex street more effective in inducing a relatively smooth jet in the X-direction. In contrast, the relatively compact reverse Karman vortex consumed a greater quantity of kinetic energy in the Y-direction in the pitch-only oscillation motion mode. This was due to the negative angle-of-attack condition of the hydrofoil during the outward pendulum motion, which resulted in the trailing edge vortex being pulled obliquely by the reverse force during shedding. This in turn weakened the strength of the center vortex and the shedding vortex's traversing velocity, leading to a concentration of the vortex near the hydrofoils, which reduced the average flow rate. To show the change process of the surface vortex pattern during hydrofoil motion, this paper compared the changes in surface vortex between the rigid hydrofoil and flexible hydrofoil with the inboard oscillation motion and outer oscillation motion ways for a stable flow field in a single oscillation cycle. Figure 16 shows the comparison of the vortex structure between the rigid hydrofoil and flexible hydrofoil under the outer oscillation motion in a single cycle.