# Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow
## 3 Results and Discussion  
### Flow around an Isolated Flexible Plate  
Figure 5 shows the force and displacement _versus_ aspect ratio \(AR\), including (a) mean drag coefficient \(\overline{C_{D}}\) and maximum lift coefficient \(C_{L,\text{max}}\) and (b) flapping amplitude \(A_{f}\) and Strouhal number \(St\). In this study, the aspect ratio is changed from 0.5 to 2.0. As shown in Figure 5a, both the mean drag coefficient and the maximum lift coefficient increase almost linearly with the increasing aspect ratio. As regards the increasing slope, the maximum lift coefficient is clearly higher than that of the mean drag coefficient. This indicates that the maximum lift coefficient is much more sensitive to the aspect ratio. Despite both of the force coefficients monotonically increasing with the aspect ratio, the slopes are less than 1.0. This means that the forces on the plate are not able to increase with the increasing forced area by an equivalent percentage. According to Equation (6), it seems that the force coefficients depend on the plate's forced area, but actually, the influence of the plate's geometry shape also has to be considered. As regards the value, the maximum lift coefficient is obviously higher than the mean drag coefficient, and the difference between them also increases with the increasing aspect ratio. This indicates that incoming flow has a stronger influence on the plate in the transverse direction than in the streamwise direction. Figure 4: Vortical structure of the flow over an isolated flexible plate (Supplementary Video SI is available online). As shown in Figure 5b, both the flapping amplitude and Strouhal number increase monotonically with the increasing aspect ratio. Moreover, their increasing slope tends to decrease, which indicates that the effect of aspect ratio on the plate's flapping amplitude gets weaker as it increases. It is obvious that the two curves are not wholly similar in the increasing trends. Concerning the definition of the Strouhal number, \(St=fA_{f}/U_{\infty}\), the flapping amplitude \(A_{f}\) and flapping frequency \(f\) are both important influencing factors. Regarding the value, the Strouhal number falls in the range from 0.16 to 0.27 in the present study. Despite the difference in the Reynolds number, the similar values of the Strouhal number were obtained in the simulations of flexible filaments. It is much more interesting that even for an active flapping rigid hydrofoil propelled in a tank, the higher thrust efficiency was found to fall in the range of \(0.2\sim 0.4\), which is close to the present study [48]. Furthermore, most fish and birds have been shown to swim or fly near an optimal Stvalue at 0.3 due to the high propulsion efficiency [49]. The similarity in Strouhal number value in animals' propulsion has been found to be strongly related to the formation of the optimal vortex [50]. These comparisons indicate that wake instability existing in the passive flapping plate is similar to that of swimming fish or flying birds in nature. The hydrodynamics and flapping motion have close associations with the fluid characteristics. Even if the flow is moving, the relation of fluid-fluid and fluid-structure can be identified to some extent. Figure 6 shows the snapshot of tracing particles from the top and side view, including (a) \(AR=0.5\) and (b) \(AR=2.0\). The particles are released from the sources at the upstream of the plate with a time interval of 10. It should be noted that the particles' motion is following the local fluid velocity, but does not affect the fluid motion. As shown in the figure, each pathline of the tracing particle shows a spiral trajectory in the streamwise direction, which is formed in the process of the plate's flapping. Due to the flexible plate deforming at most instants, the free flapping part of the structure always resists the incoming flow to move downstream in the streamwise direction. Hence, the particle driven by the flow is inclined to separate from the side edge when it encounters the resistance of the trailing edge.