# Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow
## 2 Physical Problem and Numerical Methods
### Numerical Methods
#### 2.2.2 Validation  
The validation of the present numerical algorithm will be briefly described in this section. Considering the typical case, we simulate a square \((AR=1)\) flexible plate in a uniform flow. Compared to the twin plate model shown in Figure 1, there is only an isolated flexible plate immersed in the flow field in this case. Meanwhile, the leading edge of the plate is located in the mid-plane of computational domain \((z=1/2L_{z})\). Figure 2 shows the transverse displacement of the mid-point on the trailing edge; the black solid and red dashed line are the present result and given by [PERSON] and [PERSON] [31], respectively. As it shows, the transverse displacement varies periodically with time increasing, meaning that the plate is flapping passively in the transverse direction. Furthermore, the flapping amplitude of the trailing edge's mid-point is about 0.77. The present result is almost coincident with the previous studies [31; 32; 33]. The good agreement indicates that the present numerical algorithm is accurate. To further analyze the motion of the plate, Figure 3 shows the envelop of an isolated flapping flexible plate, including (a) the 3D envelop of the flapping plate and (b) the envelop of the plate's mid-line. In the plate's 3D flapping envelop, the black solid line denotes its mid-line, the segmentation before and after which are displayed in semi-transparent mesh and solid surface style, respectively. As shown in Figure 3a, the plate's passive flapping is around its fixed leading edge. In the process of passive flapping, the plate loses its initial flat status, which is obviously different from the rigid plate's flapping [44; 45]. Moreover, the motion of the flexible plate apparently contains the deformation streamwise and the deformation spanwise. Due to its flexibility, the plate's motion in the uniform flow is entirely passive, induced by the interaction between surrounding flow and the structure itself. This deformation characteristic is similar to flapping wings or fins in some extent, especially in spanwise flexibility. According to our previous study [46], the deformation of the tuna's caudal fin spanwise can improve its propulsive efficiency. Compared to imposing on the fin the prescribed motion equation spanwise, here the deformation is the result of fluid and inertial force, which is much closer to fish swimming in nature. In Figure 3b, point \(B\) is the mid-point on the pate's trailing-edge. The trajectory of point Bloods like the number \"8\", which is formed by the free flapping of the plate's trailing-edge in the transverse direction with the longitudinal traveling wave. Meanwhile, the trajectory of point \(B\) is symmetrical about the plate's local \(x_{b}\)-axis, indicating that passive flapping of  
Figure 2: Comparison of the present result with previous data reported by [PERSON] and [PERSON] [31]. the plates keeps symmetry in the positive and negative \(z_{b}\)-axis direction. Due to the phenomenon of asymmetrical flapping occurring in the side-by-side arrangement, we define \(Z_{B,out}\) and \(Z_{B,in}\) as the absolute value of the plate's maximum transverse displacement in the positive and negative \(z_{b}\)-axis direction, respectively. Additionally, the flapping amplitude \(A_{f}\) of an isolated plate is defined as \(A_{f}=Z_{B,out}+Z_{B,in}\). The hydrodynamics and motion of the flexible plate are closely associated with the vortex in the wake. Figure 4 shows the instantaneous vortical structure of flow over an isolated square flexible plate at \(Re=200\). Here, the vortical structure is identified by the \(\lambda_{2}\) criterion [47], and an iso-surface of \(\lambda_{2}=-0.2\) is taken to plot the vorticity contour in 3D. From top to bottom, the three snapshots are chosen at the same instant, but with different projection directions. It can be clearly seen from the figure that a chain of hairpin-like vortical structures is arrayed in order around the flexible plate.