## 3 Results
### Hydrodynamic Analysis
#### 3.2.2 Pressure Distribution and Vortex Structure
After the first half-flapping cycle, the plate continues its motion for the second half-cycle, from \(\phi\) = 0.500 to \(\phi\) = 1.000. The evolution of three-dimensional vortex structures at the corresponding phases during the half-flapping cycle are illustrated in Figure 10 as an iso-surface of the Q-criterion (Q = 30) filled with the spanwise vorticity Y. The three vortices evolved along the side, leading edge and trailing edge of the plate, respectively. Among the three vortex structures, it is found that the vortex evolved from the trailing edge of the plate (TEV) mainly produces the low-pressure region on the plate surface. From \(\phi\) = 0.000 to \(\phi\) = 0.125, a counterclockwise TEV (red color) emerges from the trailing edge on the lower surface of the plate. Subsequently, there is a low-pressure region on the plate's lower surface as seen in Figure 9. This TEV continues to grow in size and remains attached to the trailing edge of the plate until the velocity of the trailing edge reaches maximum. At \(\phi\) = 0.250, it can be seen that the TEV has fully developed and starts to detach from the plate. This is consistent with the circular area of the low-pressure region seen at the trailing edge on the lower surface of the plate (Figure 9). After that, the TEV detaches from the plate and sheds into the downstream flow field on the lower side of the plate. At \(\phi=0.\) 375, the shedding of TEV into the wake is observed, which is well correlated with a low-pressure circular region in the downstream on the lower side of the plate as seen in Figure 9. Meanwhile, the plate holds positive- and negative-pressure regions on its lower and upper surface, respectively, due to the downward motion of the plate (Figure 9). The shedding of TEV into the downstream wake coincides with the increasing trend in thrust force. At \(\phi=0.500\), the shed counterclockwise TEV is losing its strength and starts to decay. In the corresponding pressure field (Figure 9), dissipation of the low-pressure circular region is still visible in the near wake region below the plate. This phenomenon can also be related to the trend of declination in thrust force. Specifically, the thrust force reaches maximum position just before this phase instant. The period of TEV shedding into the downstream wake coincides with the period of increasing thrust from the minimum position to the maximum position. Alternatively, a new clockwise TEV (blue color) starts to form at the trailing edge on the upper surface of the plate and the next half-cycle starts with the development of a new TEV. Figures 11a and 12 show the comparison of pressure distribution on both sides of the plate surface for all cases at a phase instant when a peak value of positive thrust appears in all cases. The corresponding vortex structures around the plate are illustrated in Figure 11b. The phase instant at which a peak of positive thrust appears in each case can be seen in Figure 7, where points a, b and c are representative phase instants corresponding to \(\phi=0.489\), \(\phi=0.561\) and \(\phi=0.611\) for case 1, case 2 and case 3, respectively. According to Figures 11a and 12, the high pressure acting on the lower side of the plate in the case of an optimal frequency \(f^{*}=1.2\) (case 2) is significantly larger than those in the other two cases. Consequently, case 2 exhibits the largest pressure difference among the three cases, which is consistent with the generation of the highest instantaneous positive thrust at the phase instant of \(\phi=0.561\) (point b in Figure 7) in case 2. It is likely that the relative deformation of the trailing edge affects the distribution of pressure around the plate surface. When examining the pressure distribution on the plate surface of all cases, it is seen that the negative-pressure regions on the upper surface of the plate appear similar as seen in Figure 12a. It would seem that the positive-pressure region below the plate surface as seen in Figure 12b is creating the increase in thrust force.