## 4 Conclusions  
This paper investigated an existing flexible hydrofoil device that imitates the swimming characteristics of fish with a view to addressing the hydrodynamic insufficiency of ecological water bodies in the plains of southern China. The hydrodynamic characteristics of the NACA0012 two-dimensional flexible symmetric hydrofoil model with a chord length of \(c=1\) m under different hardnesses and oscillation modes were numerically simulated and analyzed by bidirectional fluid-structure interaction. The results demonstrated the superior hydrodynamic performance of the flexible hydrofoil under specific hardness and oscillation modes, and the primary conclusions are as follows:  
Figure 19: Comparison of the vortex nephograms of the flow field under different oscillation modes. (**a**) Outward oscillation. (**b**) Inward oscillation. (**c**) Pitch-only oscillation. * Among the five groups of hardness selected in this paper, the hydrofoils with a Shore hardness of D50 (e.g., LDPE) performed optimally in terms of propulsion efficiency, pumping efficiency, and drag-to-lift ratio. This level of hardness ensured good shape stability of the hydrofoil while allowing for proper chord-wise deformation, which optimized the effective angle of attack and drag-to-lift ratio and improved propulsion efficiency. Excessively high hydrofoil hardness could impede trailing edge vortex shedding, while excessively low hardness could cause additional energy loss and vortex dissipation, both of which led to reduced hydrodynamic performance. Hydrofoils with very low hardness experienced reduced force stability during pitch-only and outward oscillation motions, generating additional negative thrust, and further diminishing their overall hydrodynamic performance. The findings from the simulations clearly showed that a balanced Shore hardness (D50) provided optimal performance by enhancing the hydrofoil's ability to maintain shape stability and improve thrust generation. * Among the three selected common motion ways, the inward oscillation motion way has the best performance among all the tested oscillation modes and had significant hydrodynamic improvement effects for both the rigid and flexible hydrofoils. For the inward oscillation motion way, the chord-wise deformation of the hydrofoil reduced the surface resistance between the shedding vortex and the hydrofoil surface and increased the effective angle of attack of the hydrofoil, which was conducive to the enhancement of the trailing edge vortex strength. The hydrodynamic performance of the flexible hydrofoil was better than that of the rigid hydrofoil in both outward oscillation and inward oscillation modes, while the hydrodynamic performance of the rigid hydrofoil was better than that of the flexible hydrofoil in the pitch-only oscillation mode. * Combined with the comparative analysis of the vortex structure, velocity distribution of the flow field, and the observation of the instantaneous thrust curve, we found that the vortex structure formed by the inward oscillation motion way was more conducive to inducing a more distant and stable jet. For the inward oscillation motion way, the trailing edge vortex was elongated along the direction perpendicular to the incoming velocity, which enhanced the central vorticity of the trailing edge vortex, increased the kinetic energy along the direction of the incoming velocity when the trailing edge vortex was dislodged, and reduced the dissipation rate of the vortex substantially. In contrast, the outward oscillation and pitch-only oscillation motions led to rapid vortex dissipation due to the different direction of the force on the trailing edge vortex shedding, which affected the formation of the reversed Karman vortex street and reduced the distance and stability of the jet, which in turn affected the overall hydrodynamic performance of the hydrofoil. This study provided a systematic framework for optimizing the hydrodynamic performance of flexible hydrofoils, highlighting their significant advantages over rigid designs.