## 3 Results
### Hydrodynamic Analysis
#### 3.2.2 Pressure Distribution and Vortex Structure
Figure 10: Three-dimensional vortex structures around the plate during a half-flapping cycle for the case of \(f^{*}=0.93\) (case 1). In the case of \(f^{*}=0.93\) (case 1), as shown in Figure 11a, the leading edge is at the phase instant of \(\phi=0.489\) and approaching the end of its acceleration phase with downward motion to its center. Meanwhile, the trailing edge is also on the way of a return stroke with downwards motion to its center in an acceleration phase. Since both the leading edge and  
Figure 11: (**a**) Pressure distributions on the spanwise symmetry plane of the plate and (**b**) three-dimensional vortex structures at the phase instant when a peak value of thrust appears in each case. Figure 12: Pressure distributions on the; (**a**) top and (**b**) bottom surfaces of the plate at a phase instant when a peak value of positive thrust appears in each case (The symmetry plane and leading edge are represented by a black line and red line, respectively). trailing edge of the plate are in the accelerating phase with downward motion, the positive pressure is observed to be widely distributed along most of the lower surface of the plate. From the first diagram of Figure 12b, it is seen that the positive-pressure region is enlarged from the leading edge to the trailing edge of the plate surface. Due to the relative velocity of the trailing edge and leading edge, asymmetric pressure is created on the lower surface of the plate. The positive-pressure field is more concentrated near the trailing edge region due to increasing displacement of the trailing edge. The pressure difference between the two surfaces is observed to be lowest among the three cases, which is consistent with the generation of smallest positive thrust in case 1. The lowest pressure difference in this case is attributed to the smallest relative displacement of trailing edge with that of leading edge as shown in Figure 6 (see point a). In the cases of \(f^{*}=1.2\) and \(f^{*}=1.47\), the trailing edge exhibits out-of-phase motion and a shorter time of in-phase motion with the leading edge throughout most of a flapping stroke due to larger phase lag. As the \(f^{*}\) increases, the high-pressure region on the lower surface of the plate shifts from the region near the leading edge to the region near the trailing edge and develops more around the trailing edge region as shown in Figure 11a. As illustrated in Figure 11a, the leading edge of case 2 (\(f^{*}=1.2\)) is starting its second half-cycle with downward motion from its center position while the trailing edge is reversing its direction of motion to its center. Consequently, the leading edge is in a decelerating phase while the trailing edge is in an accelerating phase. Due to deceleration of the leading edge during its second half-cycle, it was observed that the positive pressure around half of the leading edge region dissipates while acceleration of the trailing edge leads to a notable development of positive pressure on most of the trailing edge region. The pressure difference between the two surfaces of the plate is observed to be the largest among the three cases, which is related to the generation of highest positive thrust at the phase instant of \(\phi=0.561\) (point b in Figure 7) in case 2. The reason for the largest pressure difference at this phase instant is consistent with the largest displacement of trailing edge relative to the leading edge as illustrated in Figure 6 (see point b). As the \(f^{*}\) increases, the out-of-phase motion of the trailing edge becomes more pronounced due to larger phase lag of the trailing edge. In Figure 11a, the leading edge of case 3 (\(f^{*}=1.47\)) is already on the way of its second half-cycle with downward motion from its center position while the trailing edge is at the beginning of its reverse motion to the center. Consequently, the leading edge is in a decelerating phase while the trailing edge is at the start of an accelerating phase. Deceleration of the leading edge causes most of its surrounding pressure to disappear.