# Collaborative Optimization of Aerodynamics and Wind Turbine Blades
## 3 Discipline Analysis
### Aerodynamic Discipline
#### 3.1.2 Aerodynamic Shape Design  
Based on the optimized airfoil, the aerodynamic shape of the wind turbine blade is further investigated. A wind turbine blade is typically a complicated, curved structure that generates lift and drag when the wind blows across the blade surface. In the study, the Wilson method is employed to obtain the aerodynamic shape of the blade. Based on BEM theory, the power coefficient with a blade element with a height of \(\mathrm{d}r\) can be calculated as  
\[\mathrm{d}C_{p}=\frac{8}{\lambda_{0}^{2}}b(1-a)F_{P}\lambda^{3}\mathrm{d}\lambda \tag{8}\]  
where \(\lambda_{0}\) and \(\lambda\) are the tip speed ratio and local speed ratio, respectively; \(a\) is the axial induction factor; \(b\) is the tangential induction factor; and \(F_{P}\) is the Prandtl tip loss factor [17], defined as  
\[F_{P}=\frac{2}{\pi}\mathrm{arccos}\Big{(}e^{-f}\Big{)} \tag{9}\]  
\[f=\frac{N_{b}}{2}\frac{R-r}{R\sin\varphi} \tag{10}\]  
where \(N_{b}\) is the number of blades; \(R\) is the wind rotor radius; \(r\) is the blade local radius; \(\varphi\) is the incoming flow direction angle, as shown in Figure 10, where \(\mathrm{d}F\) is the force impacted on the blade element; \(\mathrm{d}F_{n}\), \(\mathrm{d}F_{t}\), \(\mathrm{d}F_{L}\), \(\mathrm{d}F_{D}\) are projection components of \(\mathrm{d}F\) in different reference frames; \(\alpha\) is the angle of attack while \(\theta\) is the twist angle; and \(V\) is the wind \begin{table}
\begin{tabular}{c c c c} \hline
**Coefficient** & **Baseline Airfoil** & **Optimized Airfoil** & **Variation (\%)** \\ \hline \(C_{l}\) & 0.92564 & 0.93292 & 0.7 \\ \(C_{d}\) & 0.02163 & 0.0209 & \(-\)3.375 \\ \(C_{l}/C_{d}\) & 42.794 & 44.634 & 4.299 \\ \hline \end{tabular}
\end{table} 
Table 2: Optimization results of lift, drag, and lift-to-drag ratio. Figure 9: Optimization of airfoil of wind turbine blade. velocity far upstream while \(V_{0}\) is the relative wind velocity. Obviously, the relationship between velocity \(\overrightarrow{V_{0}}=((1+b)\omega r,(1-a)V)\) and incoming flow direction angle \(\varphi\) is  
\[\varphi=\arctan\frac{(1-a)}{(1+b)}\frac{V}{\omega r} \tag{11}\]  
and the relationship between twist angle \(\theta\), direction angle \(\varphi\), and angle of attack \(\alpha\) is  
\[\theta=\varphi-\alpha \tag{12}\]  
As can be seen from Figure 9, the force dF is closely related to the relative velocity \(V_{0}\). Furthermore, the \(V_{0}\) components projected in the axial and tangential directions depend on the induction factors \(a\) and \(b\). According to BEM theory, the variations in \(a\) and \(b\) can be obtained as  
\[a(1-aF_{P})=b(1+b)\lambda^{2} \tag{13}\]  
and the relationship between chord and axial induction factor is  
\[c=\frac{r}{N_{b}C_{l}}\frac{(1-aF_{P})aF_{P}}{(1-a)^{2}}\frac{8\pi\sin^{2} \varphi}{\cos\varphi} \tag{14}\]  
where \(c\) is the chord length and \(C_{l}\) is the lift coefficient, which is dependent on Reynolds number and angle of attack [19]. Based on the above calculations, the process of aerodynamic shape design of the blade is as shown in Figure 11. The main steps are as follows. 1. Global parameters including the number of blades, tip speed ratio, rated power and wind velocity, rotor angular velocity, and blade length are determined according to the specific needs. 2. One divides the blade into n parts alongside the length of the blade. Thus \(n+1\) sections can be obtained and each section can be viewed as an airfoil. The local radius \(r_{i}\) and speed ratio \(\lambda_{i}\) can also be obtained. 3. After initialization of induction factors \(a\) and \(b\), one can take the maximization of the power coefficient (8) as the objective function and expression (13) as the constraint and then perform an optimization calculation to obtain updated induction factors (\(a_{i}\), \(b_{i}\)) and tip loss factor \(F_{Pi}\):
4. Based on the optimized \(\{a_{i}\), \(b_{i}\), \(F_{Pi}\}\), one can obtain the direction angle \(\varphi_{i}\) according to (11) and chord length ci according to (14).