# Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow
## 2 Physical Problem and Numerical Methods  
### Physical Problem  
A 3D model of the flexible plates in a uniform flow is considered. Figure 1 shows the schematic of flow over two flexible plates in side-by-side arrangement. Both of the two plates are equal in geometry and mechanical property. As shown in Figure 1, the two plates with a fixed leading edge and a free trailing edge are immersed in a uniform flow. The flow's incoming velocity, dynamic viscosity and density are denoted as \(U_{\infty}\), \(\mu\) and \(\rho_{f}\), respectively. In this work, the aspect ratio is finite, defined as \(AR=W/L\), where \(W\) is the plate's width and \(L\) is the plate's length. Furthermore, the plate has a density \(\rho_{s}\), thickness \(h\) and Young's modulus \(E\). In the initial status, the upper and lower plates keep parallel in the field and have an inclined angle \(\gamma_{0}\) (\(\gamma_{0}\)=\(\pi\)/10) with the streamwise plane. The points \(C_{1}\) and \(B_{1}\) are located on the mid-point of the upper plate's leading edge and trailing edge, respectively.  
The line \(C_{1}B_{1}\) is called the upper plate's mid-line, as well as \(C_{2}B_{2}\) to the lower plate. The transverse gap distance \(G_{z}\) is defined as the distance from point \(C_{1}\) to point \(C_{2}\), denoted as \(G_{z}=|C_{1}C_{2}|\). For the purposes of simplification, the non-dimensional parameters \(L\), \(W\) and \(G_{z}\) will be used to denote the variants \(L/L\), \(W/L\) and \(G_{z}/L\), hereafter.  
Based on the dimensionless parameters \(U_{\infty},\rho_{f}\) and \(L\), the 3D flow field with flexible plates can be described as forced incompressible Navier-Stokes equations [31; 32; 40]:  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\
abla\mathbf{u}=-\
abla p +\frac{1}{Re}\
abla^{2}\mathbf{u}+\mathbf{f}_{s} \tag{1}\]  
\[\
abla\cdot\mathbf{u}=0 \tag{2}\]  
\[M_{s}\frac{\partial^{2}\mathbf{X}}{\partial t^{2}}=\sum_{i,j=1}^{2}\left[ \frac{\partial}{\partial s_{i}}\left(K_{s}\varphi_{ij}\left(\delta_{ij}-\left( \frac{\partial\mathbf{X}}{\partial s_{i}}\cdot\frac{\partial\mathbf{X}}{ \partial s_{j}}\right)^{-1/2}\right)\frac{\partial\mathbf{X}}{\partial s_{j} }-\frac{\partial}{\partial s_{j}}\left(K_{b}\gamma_{ij}\frac{\partial^{2} \mathbf{X}}{\partial s_{i}\partial s_{j}}\right)\right)\right]+\mathbf{F} \tag{3}\]  
where \(\mathbf{f}_{s}\) is the body force exerted on the fluid by the structure's boundary. Equation (3) is the plate's governing equation, which describes its deformation with large displacement [31; 32; 40]. In this equation, \(\mathbf{X}\) is the position of the Lagrangian node on the plate in global coordinates \(XYZ\); \(K_{b}\) and \(K_{s}\) are the bending and stretching stiffness of the plate, respectively. \(\mathbf{F}\) is the force density acted on the plate's boundary by the surrounding flow; \(\delta_{ij}\) is the Kronecker \(\delta\) function; and \(\gamma_{ij}\) and \(\varphi_{ij}\) are the out-of-plane and in-plane matrices, respectively.  
The mentioned above dimensionless variants concerned in the present model are given as follows:  
\[M_{s}=\rho_{s}h/(\rho_{f}L),\;\;K_{b}=EI/(\rho_{f}U_{\infty}^{2}L^{3}),\;\;K_{ s}=Eh/(\rho_{f}U_{\infty}^{2}L) \tag{4}\]  
Meanwhile, the Reynolds number in this work is defined based on the plate's length and incoming flow and can be expressed as:  
\[\mathrm{Re}=\rho_{f}U_{\infty}L/\mu \tag{5}\]  
The drag and lift force are also defined in a dimensionless form, as follows:  
\[C_{D}=F_{D}/(0.5\rho_{f}U_{\infty}^{2}LW),\;\;C_{L}=F_{L}/(0.5\rho_{f}U_{ \infty}^{2}LW) \tag{6}\]  
where \(F_{D}\) is the force exerted on the plate in the \(X\)-axis direction, while \(F_{L}\) is in the \(Z\)-axis direction.  
Figure 1: Schematic of the flow over two flexible plates in side-by-side arrangement: (**a**) axonometric view; (**b**) side view.