## 2 Methods
### Solution Strategy and Validation
#### 2.3.3 Validation of the Fluid-Solid Coupling Method  
The bidirectional fluid-structure interaction method offered significant advantages over traditional dynamic meshing methods, enabling real-time feedback between the fluid forces and structural deformations to capture more accurate and realistic interactions in dynamic environments. This approach avoided the common limitations of dynamic meshing techniques, such as the inability to effectively deal with passive deformation in fluid mechanics. In fluid-solid coupling calculations, the fluid domain is coupled with the solid domain through a System Coupling module, a process that is inevitably accompanied by an increase in the potential for error and uncertainty. [PERSON], [PERSON] et al. [17] employed a water hole experiment to measure the change response of the thrust coefficient with heaving amplitude for three groups of airfoils, subsequently verifying the reliability of the measured data. To guarantee the dependability of the bidirectional fluid-structure interaction method presented in this paper, a model of the water hole experiment was constructed for the purpose of verifying the accuracy of the calculations presented in this paper. This is illustrated in Figure 8.  
Figure 7: Grid-independent verification: (**a**) Fluid domain mesh validation; (**b**) Solid domain mesh validation.  
To evaluate the efficacy of the methodology presented in this paper, two sets of experiments on rigid and flexible hydrofoils, as documented in the literature, were selected for comparison with the calculated values obtained through the proposed method. The comparison results are illustrated in Figure 9. In the calculation, the hydrofoil was a pure having single degree of freedom motion; the equation of motion is shown in Equation (3), with the heaving amplitude \(h_{max}\) = 0.175 m, the heaving phase difference \(\varphi_{h}\) = 0, and the frequency \(f\) = 2 Hz. The hydrofoil consisted of rectangular stainless-steel reinforcement with E = 200 G-Pa, \(\rho\) = 7800 kg/m\({}^{3}\), and DPMS rubber with E = 2.5 \(\times\) 10\({}^{-4}\) G-Pa, \(\rho\) = 970 kg/m\({}^{3}\), the length of the flow channel at 1.8 m, and the height and width at 1 m.  
A comparison of the above results revealed that the numerical calculations of flexible hydrofoils were consistent with the experimental trend, with an insignificant error remaining after the neglect of time phase error had been eliminated. In contrast, the numerical calculations of the rigid hydrofoils exhibited relatively larger errors. This discrepancy may be attributed to the substantial vibration of the rigid hydrofoil during its movement within the water hole test, which resulted in considerable signal interference. The comparison results demonstrated that the trend of the thrust coefficient's change with time is  
Figure 8: Modeling of the validation methods.  
Figure 9: Comparison of test results with calculated results.  
essentially consistent. However, the numerical calculation of the rigid hydrofoil effectively circumvented the signal interference caused by the transmission shaft or its own frequency resonance. This aligned more closely with the requisite calculation accuracy for this study, particularly in comparison to the experimental measurements. Considering the analyses, it can be concluded that the numerical calculation method proposed in this paper met the requisite standards of accuracy and effectiveness.