# Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow
## 3 Results and Discussion
### Flow around Two Flexible Plates in Side-by-Side Arrangement
#### 3.2.1 Coupling Motions and Vortical Structures
In the single-plate mode, the two plates are flapping in phase with just a row of vortex existing in the wake. Figure 9 shows the time evolutions of the transverse displacements of points \(B_{1}\) and \(B_{2}\) at \(G_{z}=0.1\). As shown in the figure, the two plates flap synchronously with the same phase. Meanwhile, the two plates' flapping motion share the equivalent frequency and amplitude. Despite the two displacement curves not overlapping, the peak of \(Z_{B,up}\) is equal to the absolute value of minimum \(Z_{B,un}\). Therefore, the motion of each plate is asymmetry about its local \(x_{b}\)-axis, but the two plates' flapping amplitude is symmetrical about the flow domain's transverse center-plane. Figure 10 shows the envelops of the two flapping flexible plates at \(G_{z}=0.1\), including (a) envelops in 3D and (b) envelops of the plates' mid-lines. In Figure 10a, the upper and lower plate are displayed in a semi-apparent mesh and a solid surface, respectively. As shown in Figure 10a, the deformation of the upper plate is always similar to the lower one. Additionally, the deformation in the spanwise direction still exists in this situation and keeps symmetrical about the flow domain's center-plane in the \(Y\)-axis direction. As shown in Figure 10b, each of the trailing edge's mid-point shows an \"8\" shape trajectory from the side view. The point \(N_{1}\) denotes the intersection node of the trajectory of point \(B_{1}\), so as with point \(N_{2}\) to point \(B_{2}\). It is observed that \(N_{1}\) and \(N_{2}\) are symmetrical about the transverse mid-plane; this result is in accord with Figure 9.  
Figure 10: Envelops of the two flapping flexible plates at \(G_{z}=0.1\): (**a**) envelops in 3D; (**b**) envelops of the plates’ mid-lines. Figure 9: Time evolutions of the transverse displacements of points \(B_{1}\) and \(B_{2}\) at \(G_{z}=0.1\). Figure 11 shows the time evolutions of the vortical structures of two side-by-side flapping plates at \(G_{z}=0.1\). The instants \(t_{0}\sim t_{3}\) in the figure are equal to those in Figure 9. As the plates move towards the positive \(Z\)-axis direction, a clockwise vortex gradually forms on the upper surface of the upper plate. As the plates start to move towards the negative \(Z\)-axis direction, this fully-developed vortex V2 starts to shed from the trailing edge. Concerning the fluid-induced flapping, the strength of vorticity in both sides of the plates is stronger than in the gap flow; this causes the plate's asymmetrical flapping motion to some extent. Although there are two flexible plates in the flow domain, just one vortex sheds in half of the flapping cycle. The vorticity pattern in this situation is similar to an isolated plate in the same flow condition, so it is called single-plate mode. In the symmetrical flapping mode, the two plates are flapping out of phase, and two rows of vortices with opposite rotating directions exist in the wake. Figure 12 shows the time evolutions of the transverse displacements of points \(B_{1}\) and \(B_{2}\) at \(G_{z}=3.0\). As shown in the figure, the two plates flap synchronously, but with the opposite phase. For instance, as the upper plate's displacement is moving upwards its peak, the lower plate's displacement is moving downwards its trough at the same time. Each of the plates' displacements are equal in absolute value, but with the opposite sign. Meanwhile, the two plates' flapping is periodical and sharing equivalent frequency and amplitude. Clearly, the motion of each plate is symmetrical about its local \(x_{b}\)-axis, and the two plates' flapping is symmetrical about the flow domain's transverse center-plane. Figure 11: Time evolutions of the vortical structures of two side-by-side flapping plates at \(G_{z}=0.1\). Figure 13 shows the envelops of the upper flapping plate at \(G_{z}=3.0\), including (a) the axonometric view and (b) the side view. As the plate's flapping is similar to the other, here just the envelops of the upper one are shown in a semi-apparent mesh.