## 1 Introduction  
In the low-lying plains and river network areas of the southern China, insufficient hydraulic dynamics in ecological water bodies pose a widespread challenge [1]. Previous studies have shown that low-head axial flow pumps perform inadequately in practical applications [2]. [PERSON] et al. [3] demonstrated that hydrofoils, which mimic the movement of fish, offer several advantages, including low resistance, high maneuverability, low noise, and a suitability for ultra-low head conditions. These properties make hydrofoils particularly significant for enhancing the hydraulic dynamics of ecological water bodies under ultra-low head conditions in plain areas. Despite these benefits, hydrofoils have not been widely adopted as a pumping strategy, primarily due to the lack of comprehensive theoretical research on their hydrodynamic behavior. The earliest research on hydrofoils was mainly based on the aerodynamic transformation of airfoils. Scholars combined the aerodynamic advantages of airfoils with the swimming mechanism of fish and obtained the hydrofoil device through the active oscillatory movement of the airfoil wing shape in the water. The pumping mechanism of a hydrofoil is the formation of reversed Karman vortex streets by the shedding of trailing edge vortex pairs, which in turn generates a jet. To further optimize the hydrodynamic performance of the hydrofoil, scholars have conducted extensive research on the hydrofoil parameters. [PERSON] et al. [4] employed numerical methods to simulate and analyze the water propulsion effects of four different hydrofoil motion modes, thereby identifying the optimal motion modes of the hydrofoil. [PERSON] et al. [5] integrated the underwater motion model of the hydrofoil with the method of thrust efficiency estimation, analyzed the numerical results of the propulsive performance under various motion conditions, and determined the optimal thrust. The optimal efficiency points for the lifting-and-pitching amplitude and frequency of the rigid hydrofoil were determined. An overlapping grid strategy was employed by [PERSON] et al. [6] to numerically simulate a hydrofoil, with consideration given to wall effects. Based on the analysis of the flow mechanism, the optimal wall spacing for achieving maximum propulsive efficiency was determined to be 0.8c (where c is the hydrofoil chord length). In research on stiffer hydrofoils, scholars have assumed the hydrofoils to be rigid models. This assumption has had the effect of neglecting the deformation caused by forces during the study of their hydrodynamic characteristics. As a result, the complexity of hydrofoil research is simplified. However, when hydrofoils are in motion, they inevitably deform due to the reactive forces from the fluid. This deformation is not only dependent on the hardness of the hydrofoil but is also influenced by factors such as oscillation frequency and the Reynolds number. By ignoring this aspect, the rigid model introduces a certain degree of computational error. This inaccuracy is magnified when investigating flexible hydrofoils with large deformations and therefore cannot be simplified to a rigid model. Flexible hydrofoils demonstrate markedly enhanced hydrodynamic performance under specific conditions when compared to rigid hydrofoils. [PERSON] et al. [7] employed Digital Particle Image Velocity (DPIV) to quantify the aerodynamic forces, torque, and flow vector fields of both rigid and flexible foils equipped with leading-edge veins. Their findings revealed that minimal foil surface deformation not only reduces the distance between vortices and the foil surface but also delays vortex dispersion, enhances foil flow intensity, and thus generates a higher initial peak lift. This suggests that within a specificrange of chord-wise flexibility, flexible foils exhibit enhanced efficiency. Furthermore, [PERSON] et al.