## 4 Numerical Results and Analysis
### Effect of Wave Encounter Frequency on Foil Performance  
In [PERSON]'s [24] opinion, when \(\omega_{0}=\omega_{f}\), higher utilization efficiency can be obtained. In this section, this viewpoint will be validated and the effect of \(\omega_{0}/\omega_{f}\) on the flexible foil propulsive performance will be analyzed in detail. In these simulations, different \(\omega_{0}/\omega_{f}\) can be obtained by changing the wavelengths and keeping \(\omega_{f}\) constant.  
With different \(\omega_{0}/\omega_{f}\), the average thrust force coefficient \(C_{T}\) and the utilization wave power coefficient \(C_{pr}\) of the rigid and flexible foil, respectively, are shown in Figure 6, in which \"-wave\" and \"-caln water\" represent corresponding parameters of the foil under the waves and the calm water, respectively; in later figures, their meanings are the same.  
Figure 5: Comparison of non-dimensional thrust in experiment and theory.  
It is obvious that the peak values of \(C_{T}\) and \(C_{pr}\) appear at \(\omega_{0}/\omega_{f}=1\); that is, the rigid and flexible foil can obtain higher thrust force and recovered wave power [30] when the foil motion frequency is equal to its wave encountering frequency, as proposed by [PERSON]. Meanwhile, as shown in Figure 6, the thrust force of the foil, whether rigid or flexible, generated under waves, is always larger than that under calm water at any \(\omega_{0}/\omega_{f}\). Therefore, we can conclude that the wave motions are beneficial to the thrust force generation, no matter what wave frequency the foil encounters. Compared with that under calm water, the thrust force of rigid and flexible foil in waves can be improved 64% and 76%, respectively, at \(\omega_{0}/\omega_{f}=1\). In addition, the thrust force and the wave power utilization coefficients of the flexible foil are 13% and 25% larger than those of rigid foil at \(\omega_{0}/\omega_{f}=1\) under waves. This phenomenon shows that flexible deformation can improve the propulsive performance of the foil, and this conclusion is in agreement with [PERSON] [17; 18].  
After obtaining \(C_{T}\) and \(C_{pr}\), the propulsive efficiency \(\eta\) and the percentage of wave energy recovery \(\eta_{r}\) can be calculated; their trends with \(\omega_{0}/\omega_{f}\) are shown in Figure 7.  
As seen in Figure 7, when \(\omega_{0}/\omega_{f}<1\), the percentage of wave energy recovery \(\eta_{r}\) of the rigid and flexible foil ascends slightly with increasing \(\omega_{0}/\omega_{f}\), and reaches peak value rapidly when \(\omega_{0}/\omega_{f}\) is close to 1, where these curve trends validate [PERSON]'s opinion. Obviously, the propulsive efficiency \(\eta\) of the flexible foil is higher than that of rigid foil in various flow field environments, while also reaching \(\eta\)  
Figure 6: Average thrust and utilization wave power coefficients for the foil versus wave encounter frequency.  
Figure 7: The percentage of wave energy recovery \(\eta_{r}\) and propulsive efficiency \(\eta\) for the foil versus wave encounter frequency.  
peak value at \(\omega_{0}/\omega_{f}=1\) in waves. This value of the flexible foil is 43% higher than that of the rigid foil. Compared with the trends of \(C_{T}\) in Figure 6, \(\eta\) changes more smoothly near \(\omega_{0}/\omega_{f}=1\). In addition, the propulsive efficiency of the flexible foil and the rigid foil in waves will become smaller than that under calm water when \(\omega_{0}/\omega_{f}=1.2\) and \(1.5\), respectively. Above all, if the motion frequency of foil does not match its encounter frequency, the foil's propulsive efficiency will be affected adversely, but the flexible deformation can reduce this adverse effect.