## 4 Numerical Results and Analysis
### Effect of Waves on the Foil Wake Field  
As mentioned above, the wave effect on the foil propulsive performance can be due to water particle motions disturbing the flow field of the foils. As seen in Figure 12, instantaneous dynamic pressure near the foil is given at different times in various conditions. Although the dynamic pressure changes with time and we can obtain this distribution at any time from the Fluent software, several snapshots of the pressure field of characteristic time are selected to analyze the evolution process of flow field near the foil. Based on the Bernoulli equation in hydromechanics, the dynamic pressure can also reflect the water particle velocity and vortex structures. Other parameters, such as turbulent kinetic energy, can also then be obtained by operating on the water particle velocity or post-processing the flow field profile. In addition, Figure 12c shows the flexible foil wake field of the basic case that is described in Section 4.  
Figure 12: Dynamic pressure distribution for different wave parameters in one period.  
Comparing Figure 12c,d, there is little difference in vortex intensity between the rigid foil and the flexible foil in waves, so the instantaneous thrust force is approximately equal. But the vortex distribution range of the flexible foil is more concentrated and has better continuity. In addition, the time of vortices shedding from the flexible foil is later than that of the rigid foil. Therefore, the flexible deformation can slow down the shedding of vortices and then reduce energy wasting in a flow field. As a result, the flexible foil can obtain a higher propulsive efficiency.  
On comparing Figure 12c,e, different effects of calm water and waves on the flexible foil wake fields are shown. It is obvious that the vortices from the foil in waves are stronger than those under calm water, especially at time \(T+T/2\). From a kinematics perspective, wave motion can increase foil velocity in surrounding water particles and the velocity gradient of the flow field near the foil, thus causing greater vortex intensity. From the perspective of energetics, as the foil utilizes extra wave energy, its wake field can contain more kinetic energy and slow down its vortex merging rate. Meanwhile, the output power of the foil can also be increased and the foil can generate a corresponding larger thrust force. In addition, extra input energy from waves can make the foil obtain a higher propulsive efficiency.  
Comparing Figure 12c,d the wave amplitude effect on the wake field of the flexible foil can be seen. The vortex intensity of the foil increases rapidly with larger wave amplitudes, especially at time \(T+T/2\) and \(T+3T/4\) when the orbital velocity of the foil reverses. The instantaneous maximum thrust force of the foil is then much higher and its average value is increased correspondingly. However, larger wave amplitudes make the foil wake range wider, so more energy is dissipated in the wake field. This makes the foil to not have much higher propulsive efficiency.  
Comparing Figure 12c,e, the effect of wavelength on the wake field of the flexible foil can be analyzed. As the foil velocity related to water particles with longer wavelengths is lower than that with shorter wavelengths, the foil's vortex intensity is also not higher with longer wavelengths, and the foil can only obtain a smaller thrust force. However, the vortex distribution range and the velocity gradient of the foil wake field with longer wavelengths can be smaller than those with shorter wavelengths, which means that only a small amount of energy will dissipate in the flow field and the flexible foil can achieve a relatively higher propulsive efficiency.