# Numerical Simulation of Fluid-Structure Coupling for a Multi-Blade Vertical-Axis Wind Turbine
## 2 Theoretical Background and Modeling Methods  
In this example, an unidirectional coupling approach is employed, neglecting the influence of solid deformation on the flow field. Figure 2 shows the flowchart of the fluid-structure coupling analysis, which mainly involves utilizing the Fluid Flow module for  
Figure 1: Graphical abstract of this paper. flow-field analysis and the Static Structure module for mechanical analysis of the wind turbine blades. The analysis was carried out through the coupled module constructed in ANSYS Workbench 17.0. In this paper, the governing equation describing the motion of wind turbines is established for analysis. The focus is primarily on simulating the rotation of wind turbine blades under low-speed flow conditions, considering both blade rotation and external wind speed. Energy transfer is not considered in the analysis, and the process is primarily controlled by the continuity equation and momentum equation [26; 27; 28; 29]. Continuity equation [30; 31]  
\[\frac{\partial}{\partial t}\iiint\limits_{V}\rho dxdydz+\iiint\limits_{A}\rho v \cdot\overset{\rightarrow}{n}dA=0 \tag{1}\]  
In the formula, \(V\) represents the control volume; \(A\) represents the control surface; and \(\overset{\rightarrow}{n}\) is to control the external normal direction of the control surface. The initial term on the left side of the equation is to control volume mass gain. The second term represents the net mass flowing into the control body through its surface. According to the O-Gao formula in mathematics, the above equation can be converted into a differential form in the rectangular coordinate system [32; 33]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{2}\]  
In the formula, \(\rho\) is the density (kg/m\({}^{3}\)), \(t\) is the time(s), \(u\), \(v\), and \(w\) are velocity components (m/s) in the \(x\), \(y\), and \(z\) directions, respectively. Figure 2: Process of fluid-solid coupling analysis. Momentum conservation equation [34]:  
Momentum equation in x direction:  
\[ \begin{array}{l}\frac{\partial\rho u}{\partial\tau}+\frac{\partial\rho uu}{ \partial x}+\frac{\partial\rho uv}{\partial y}+\frac{\partial\rho w}{\partial z }=-\frac{\partial P}{\partial x}+\frac{\partial}{\partial x}[\mu\frac{ \partial u}{\partial x}]+\frac{\partial}{\partial y}[\mu\frac{\partial u}{ \partial y}]+\frac{\partial}{\partial z}[\mu\frac{\partial u}{\partial z}]+\\ \mu[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+ \frac{\partial^{2}w}{\partial z^{2}}]\end{array}  \tag{3}\]  
Momentum equation in y direction:  
\[ \begin{array}{l}\frac{\partial\rho v}{\partial\tau}+\frac{\partial\rho uu}{ \partial x}+\frac{\partial\rho wv}{\partial y}+\frac{\partial\rho ww}{ \partial z}=-\frac{\partial P}{\partial y}+\frac{\partial}{\partial x}[\mu \frac{\partial w}{\partial x}]+\frac{\partial}{\partial y}[\mu\frac{\partial v }{\partial y}]+\frac{\partial}{\partial z}[\mu\frac{\partial v}{\partial z}]+ \\ \mu[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+ \frac{\partial^{2}v}{\partial z^{2}}]\end{array}  \tag{4}\]  
Momentum equation in z direction:  
\[ \begin{array}{l}\frac{\partial\rho w}{\partial\tau}+\frac{\partial\rho w}{ \partial x}+\frac{\partial\rho wv}{\partial y}+\frac{\partial\rho ww}{ \partial z}=-\frac{\partial P}{\partial z}+\frac{\partial}{\partial x}[\mu \frac{\partial w}{\partial x}]+\frac{\partial}{\partial y}[\mu\frac{\partial w }{\partial y}]+\frac{\partial}{\partial z}[\mu\frac{\partial w}{\partial z}]+ \\ \mu[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+ \frac{\partial^{2}w}{\partial z^{2}}]-\rho_{0}g^{\beta}\end{array}  \tag{5}\]  
In the formula: \(u\), \(v\), and \(w\) is the velocity in the \(x\), \(y\), and \(z\) directions(m/s); \(\mu\) is viscosity (Pa-s); \(P\) is pressure (N/m\({}^{2}\)).