## 1 Introduction
[8] demonstrated through numerical simulations that the flexible deformation of hydrofoils contributes to increased vortex strength and reduced energy dissipation, thereby enhancing propulsion performance[PERSON] Additionally, the role of flexibility in enabling energy-efficient movement has drawn increasing attention, inspired by natural swimmers such as fish and birds, which utilize intermittent movements to maximize thrust efficiency and minimize energy consumption. [PERSON] et al. [9] investigated the hydrodynamic performance of a bionic tail fin and found that spreading flexibility can increase thrust and maintain high propulsive efficiency[PERSON] As a result, flexible hydrofoils represent an important area of research, but many research deficiencies and gaps remain. Some recent studies have referred to the flexible deformation of hovering airfoils and proposed the use of active flexibility to replace passive deformation in hydrofoils [10]. However, this approach does not consider the coupling force exerted by the fluid during hydrofoil motion and has some limitations. In contrast, the bidirectional fluid-structure interaction method employed in this study enables real-time feedback between fluid forces and structural deformation. This approach provided a more comprehensive and accurate simulation of the complex mechanical interactions of flexible hydrofoils in dynamic fluid environments, overcoming the limitations of unidirectional coupling models and improving computational precision. The research conducted by [PERSON] et al. [11] indicated that flexible hydrofoils can generate superior thrust compared to their rigid counterparts at a low frequency of 1 Hz. However, alterations in the hardness of the hydrofoil will unavoidably result in fluctuations in deformation, and the hydrodynamic performance of hydrofoils constructed from disparate materials will also diverge. Furthermore, during actual motion, flexible hydrofoils will deform in varying ways contingent on the motion way employed. Therefore, this study focused on analyzing the effects of Shore hardness and motion modes, providing a basis for optimal design choices in practical applications. Furthermore, about the matter of model dimensions, [PERSON] et al. [12] demonstrated that hydrofoils exhibit three-dimensional instability under a specific pitching amplitude through Floquet stability analysis. To circumvent the three-dimensional instability factor in the study of the impact of hydrofoil chord-wise flexibility on hydrodynamic performance, this paper employed a chord-wise two-dimensional model of the hydrofoil. The central inquiry of this study pertained to the influence of Shore hardness and oscillation mode on the hydrodynamic performance of flexible hydrofoils under ultra-low head conditions. The objective of this study was to enhance the comprehension of the passive deformation characteristics of flexible hydrofoils and their application in the context of water flow dynamics in ecological water bodies. To this end, a two-dimensional model of flexible hydrofoils was constructed, and the chord-wise deformation behavior of hydrofoils was simulated under different conditions. This was achieved by applying a coupling force based on the bidirectional fluid-structure interaction technique. The differences in hydrodynamic characteristics between flexible hydrofoils and rigid hydrofoils were analyzed, and the hydrodynamic characteristics of flexible hydrofoils with varying chord-wise deformations were compared with those of flexible hydrofoils with different low-frequency motions. The grid and time-step independence of the model were verified, thereby improving the accuracy and reliability of the model. The viability of the simulation approach was validated by comparing the results with the experimental data from existing studies. Subsequently, the optimal hardness of the flexible hydrofoil was determined by considering the pumping efficiency and propulsion efficiency as evaluation criteria.