## 3 Results
### Hydrodynamic Analysis
#### 3.2.2 Pressure Distribution and Vortex Structure  
In this section, the pressure distributions around the plate surfaces and the relative deformation of trailing edge are examined to investigate why the thrusts produced by the plate differ with \(f^{*}\). The pressure difference between the upper and lower surface of the plate is the source of propulsion. The non-dimensional frequency of \(f^{*}\) = 0.93, 1.2 and 1.47 are termed as case 1, case 2 and case 3, respectively. Figure 9 shows the pressure distributions on the spanwise symmetry plane of the plate in a half-flapping cycle for case 1 at the phase instants of \(\phi\) = 0, 0.125, 0.25, 0.375 and 0.5. In the first half-cycle, from \(\phi\) = 0 to \(\phi\) = 0.5, the leading edge of the plate shifts from its center position with the upward motion and then returns to its center position with the downward motion. Figure 8: Variation of mean thrust coefficient with different \(f^{*}\). Figure 9: Pressure distributions on the spanwise symmetry plane of the plate during a half-flapping cycle for the case of \(f^{*}\) = 0.93 (case 1). At \(\phi\) = 0.000, the plate is at the start of the first half-cycle with the maximum velocity of the leading edge and its leading edge beginning to move from its center position with upward motion. As the plate is gaining speed, a very large pressure field is observed in both positive and negative around both surface of the plate. Consequently, a high pressure difference between the two surfaces is created, which results in the generation of thrust in the positive x direction at this phase instant. At \(\phi\) = 0.125, the leading edge continues its upward motion and the velocity of the leading edge decreases. During the decelerating phase of the leading edge, the positive- and negative-pressure regions around the leading edge seem to be decayed and distributed mostly on the region near the trailing edge of the plate as shown in the second diagram of Figure 9. With the decreasing trend of thrust in Figure 7, the pressure difference between the two surfaces of the plate is also decreasing. This demonstrates that the plate produces a smaller positive thrust than the phase instant described above, which is well correlated with the declining of thrust at this phase instant. At \(\phi\) = 0.250, the leading edge reaches the end of its upward motion and the reversal phase with zero velocity. It is observed that a very low-pressure region is attached to the trailing edge of the plate, which results in a significant decrease in thrust force at this phase instant. A region of very low-pressure difference is also observed to be generated near the region of the leading edge, and the plate cannot force water backward adequately to produce forward motion. This is consistent with the generation of the negative value of thrust at this phase instant. At \(\phi\) = 0.375, the leading edge is en route back to its center with the downward motion with velocity of the leading edge increasing. As the leading edge is moving downwards in the accelerating phase, the plate pushes against the water with the lower surface creating the positive-pressure region around it. Meanwhile, the plate pulls the fluid on the upper surface of the plate, which creates a negative-pressure region on the upper surface of the plate. As a result, the plate creates a larger pressure difference between the two surfaces, which leads to a larger positive thrust at this phase instant. At \(\phi\) = 0.500, the leading edge reaches the end of the accelerating phase with downward motion to its center. As the leading edge is at the maximum velocity, the positive and negative pressures develop highly below and above the entire surface of the plate. As a result, the thrust forces are greatly enhanced at this time instant. In particular, the thrust force of the plate has already reached the maximum value before this phase instant. These phenomena are well correlated with the trend of thrust in a half-flapping cycle.