## 3 Results  
### Hardness Effects on Hydrofoil Hydrodynamic Characteristics  
The deformation of a flexible hydrofoil in a water flow not only affects the thrust and lift of the hydrofoil but also has an impact on its pumping efficiency and propulsion efficiency. To gain a deeper understanding of these effects, the hydrodynamic performance of flexible hydrofoils under different strain conditions was simulated based on the fluid-solid coupling model. In this paper, we classified the hydrofoil materials into D10, D30, D50, D60, and D70 according to the Shore hardness, because for the materials, in the context of bidirectional fluid-structure interaction, the Shore hardness does not provide a direct definition, necessitating a classification system. The corresponding material data were selected for parameter mapping in this study based on the Shore hardness data from the Springer Material Data Handbook [18], as illustrated in Table 1. By comparing the hydrodynamic properties of hydrofoils with varying Shore hardnesses, a more rational hydrofoil material selection could be evaluated. As postulated by [PERSON] et al. [19], a minor chord-wise strain can effectively enhance the aerodynamic attributes of hydrofoils. In accordance with this, the chord-wise strain has the capacity to enhance the drag-to-lift ratio of hydrofoils, thereby exerting a beneficial influence on their hydrodynamic characteristics. To analyze the chord-wise strain of the hydrofoils of different materials and its relationship with hydrodynamic characteristics, this paper compared the average chord-wise strain (\(D_{d}\)) of the inward oscillation motion of hydrofoils under steady flow fields with different hydrofoil materials, as illustrated in Figure 10.  
The results showed that the chord-wise strain of hydrofoils was basically inversely proportional to the hardness in a linear relationship. When the Shore hardness of the hydrofoil was D10, the chord-wise strain was the largest, 0.0349 m. When the Shore hardness was D70, the chord-wise strain was the smallest, 0.0089 m. To analyze the effect of the size of the chord-wise strain on the instantaneous thrust of the hydrofoils more clearly, this paper compared the instantaneous thrust change curves of the hydrofoils of five hardness materials in a single oscillation cycle in a steady flow field, as shown in Figure 11. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Hardness** & **D10** & **D30** & **D50** & **D60** & **D70** \\ \hline Mapping material & SiR & TPU & LDPE & UHMWPE & PP \\ Elastic Modulus (G\(\cdot\)Pa) & 0.005 & 0.015 & 0.4 & 0.7 & 1.3 \\ Densities (g/cm\({}^{3}\)) & 1.1 & 1.1 & 0.92 & 0.93 & 0.9 \\ \hline \end{tabular}
\end{table} 
Table 1: Parametric mapping of hydrofoil materials. Silicone rubber (SiR), thermoplastic polyurethane elastomer (TPU), low-density Polyethylene (LDPE), ultrahigh-molecular-weight polyethylene (UHMWPE), and polypropylene (PP). According to Figure 11, it can be seen that the hydrofoil obtained the largest peak instantaneous thrust at a Shore hardness of D50, which was 108.1 N; at a Shore hardness of D10, the peak instantaneous thrust of the hydrofoil was the smallest, which plummeted to 54.05 N; and at Shore hardness of D30, D60, and D70, the peak instantaneous thrusts were close to each other, which were 93.1, 95.6, and 91.9 N, respectively. It was obvious that the proper chord-wise deformation of the hydrofoil could have a favorable effect on the peak instantaneous thrust of the hydrofoil, but too much chord-wise deformation led to a decrease in the instantaneous thrust of the hydrofoil. This result is in some agreement with the study results of [PERSON] et al. [19]  
In a recent study, [PERSON] et al. [20] conducted an analysis of the structure of vorticity and turbulence fields in the vicinity of a hydrofoil's surface utilizing direct numerical simulation (DNS).