## 3 Results
### Hydrodynamic Analysis  
#### 3.2.1 Thrust Production  
The thrust force is simply a force that advances toward the flow as a component force induced by the deformation of the plate. The thrust force, \(F_{T}\), in the \(x\)-direction is computed as \(F_{T}=0.5\)\(C_{T}U^{2}\rho a\), where \(C_{T}\) is the thrust coefficient, \(U\) is the maximum translational velocity of model fin, \(\rho\) is the fluid density and \(a\) is the reference area of the plate. The mean thrust shown can be obtained by averaging the instantaneous thrust values over the entire flapping cycle. In the corresponding experiments, the thrust force was calculated from the momentum integration theorem. The plate did not move while it was vibrated in the \(z\)-direction and the velocity fields around the flapping plate were measured by PIV [25]. In the present study, the thrust force \(F_{T}\) and thrust coefficients \(C_{T}\) are calculated using a post-processing function tool available in OpenFOAM. Figure 5a shows the variation of mean thrust force with different \(f^{*}\), which can be used to reveal the effect of \(f^{*}\) on the propulsive characteristics more clearly and quantitatively. It illustrates that the thrust gradually increases with \(f^{*}\). Thrust increases with \(f^{*}\) and reaches a maximum at a certain value of \(f^{*}\). However, after reaching a certain value of \(f^{*}\), the mean thrust of the plate decreases gradually. This presents an indication that there is an optimal value of \(f^{*}\) at which the plate achieves the highest mean thrust. For the frequency range considered in our simulations, it is observed that the maximum net thrust occurs at \(f^{*}=1.2\) and the corresponding phase lag is observed to be 0.62\(\pi\) radians. The trend of variation in thrust is observed to be consistent with that in [25]. Figure 4: Comparison on the displacement of trailing edge with three different mesh resolutions. Figure 5: (**a**) Mean thrust. (**b**) The maximum relative deformation of trailing edge \(\delta_{max}\) for different non-dimensional frequencies \(f^{*}\). In this situation, the magnitude of trailing edge amplitude seems to be irrelevant to the generation of maximum thrust force since it decrease monotonically with flapping frequency as shown in Figure 3a. It would seem that the generation of maximum thrust force comes from the phase lag due to deformation, which makes a difference in the displacement of the trailing edge relative to the displacement of leading edge. Figure 5b shows the maximum deformation of the trailing edge relative to the leading edge for all cases. Note that the instantaneous displacement of leading edge \(z_{r}\) is subtracted from that of trailing edge \(z_{t}\) to obtain the instantaneous relative displacement of the trailing edge in a flapping cycle, \(z_{t}-z_{r}\), from which the maximum relative deformation of trailing edge, \(\delta_{max}\), is estimated. This means that the amplitude of the instantaneous relative displacement of trailing edge in a flapping cycle is taken as the maximum relative deformation of trailing edge \(\delta_{max}\). According to Figure 5a,b, the maximum net thrust is found to be associated with the maximum value in the relative deformation of trailing edge. It is evident that the phase lag is due to deformation of the plate, which makes a difference in the relative deformation of the trailing edge with respect to the leading edge, influences the generation of thrust force. Notably, the optimal value of \(f^{*}\) for maximum thrust generation in this plate does not coincide with the optimal value of \(f^{*}\) for the elastic structure. Figure 6 shows the instantaneous displacement of the trailing edge relative to that of the leading edge, \(z_{t}-z_{r}\), in a flapping cycle for all cases to clarify the features of maximum and minimum thrust force in a flapping cycle. In this Figure, the phase instant when the maximum and minimum thrust appears in each case is marked with red and black squares, respectively.