# Numerical Simulation of Fluid-Structure Coupling for a Multi-Blade Vertical-Axis Wind Turbine
## 2 Theoretical Background and Modeling Methods
Dynamic grid governing equation:  
The equation for the conservation of a general scalar \(\varphi\) on any control volume \(V\) of boundary movement can be expressed as:  
\[\frac{d}{dt}\int\limits_{V}\rho\varphi dV+\int\limits_{\partial V}\rho\varphi( \stackrel{{\rightarrow}}{{u}}-\stackrel{{\rightarrow}}{{u}}_{g})d \stackrel{{\rightarrow}}{{A}}=\int\limits_{\partial V}\Gamma \
abla\varphi d\stackrel{{\rightarrow}}{{A}}+\int\limits_{V}S_{ \varphi}dV \tag{6}\]  
In the formula, \(V\) (\(t\)) represents the control volume that undergoes changes in size and shape over time, \(\partial V(t)\) represents the boundary that controls the motion of the volume, \(\stackrel{{\rightarrow}}{{u}}_{g}\) represents the motion velocity of the moving grid, \(\rho\) represents the velocity of the fluid, \(\stackrel{{\rightarrow}}{{u}}\) represents the velocity vector of the fluid, \(\Gamma\) represents the dissipation coefficient, and \(S_{\varphi}\) is the source of the scalar \(\varphi\). In Equation (6), the time derivative term can be obtained by using the first-order backward difference formula:  
\[\frac{d}{dt}\int\limits_{V}\rho\varphi dV=\frac{(\rho\varphi V)^{n+1}-(\rho \varphi V)^{n}}{\Delta t} \tag{7}\]  
In the formula, superscripts n and \(n+1\) denote the current and next-time layers. The control volume at the (\(n+1\))th time step, denoted as \(V^{n+1}\), can be obtained using the following formula:  
\[V^{n+1}=V^{n}+\frac{dV}{dt}\Delta t \tag{8}\]  
In the formula, \(\frac{dV}{dt}\) is the derivative of the control with respect to time. To satisfy the conservation law of the grid, the derivative of the control body with respect to time can be calculated by the following formula:  
\[\frac{dV}{dt}=\int\limits_{\partial p}\stackrel{{\rightarrow}}{{u} }_{g}d\stackrel{{\rightarrow}}{{A}}=\sum\limits_{f}^{n}\stackrel{{ \rightarrow}}{{u}}_{gj}\stackrel{{\rightarrow}}{{A}}_{j} \tag{9}\]In the formula, \(n_{f}\) is the number above the control volume and \(\overset{\rightharpoonup}{A}_{j}\) is the \(j\)th area vector. The dot product \(\overset{\rightharpoonup}{u}_{g,j}\overset{\rightharpoonup}{A}_{j}\) on the faces of each control volume can be calculated by the following formula:  
\[\overset{\rightharpoonup}{u}_{g,j}\overset{\rightharpoonup}{A}_{j}=\frac{ \delta V_{j}}{\Delta t} \tag{10}\]  
In the formula, \(\delta V_{j}\) is the volume swept out by surface \(j\) on the control body in the time step \(\Delta t\). The governing equation of the fluid has been discussed above in this paper. The governing equation of the structure is analyzed. Considering the effect of the outflow field, the dynamic response of the blade structure is described by the following equation of motion [35]:  
\[Mu^{\prime\prime}+Cu^{\prime}+Ku=F(t) \tag{11}\]  
In the formula: \(M\) is the mass matrix, \(C\) is the damping matrix, \(K\) is the stiffness matrix, \(F\) is the load exerted by the fluid on the blade, u is the structural displacement of a wind turbine, \(u^{\prime}\) is the blade velocity, and \(u^{\prime\prime}\) is the acceleration. For the analysis of fluid and solid, using the same time step, the three-dimensional transient flow field is obtained first, and then the transient dynamic characteristics of the structure are analyzed.