# Collaborative Optimization of Aerodynamics and Wind Turbine Blades
## 3 Discipline Analysis
### Aerodynamic Discipline
#### 3.1.1 Airfoil Design and Optimization
Figure 4: Computation domain of the airfoil. Figure 3: Optimization of airfoil based on Isght. Figure 2: CST approximation of NACA4412 airfoil. In meshing, a structured C-block grid is employed. The meshing results are shown in Figure 5. To analyze the effects of the computational mesh on the calculation results, a mesh independence check and quality check are performed. The results of the independence check are listed in Table 1, in which five schemes on grid number are tested. The relative errors with respect to the finest mesh scheme (i.e., Scheme 5) are calculated. As can be seen, the fourth scheme is reasonable since the accuracy approximates that of the fifth scheme while the calculation burden can be reduced. As a result, the number of grids is selected as 415,955. Figure 6 presents the results of the mesh quality check, based on the grid number adopted from Scheme 4. As can be seen, the mesh quality is over 0.85, which equates to a high quality of meshing. If we assume an inlet velocity \(v=8\) m/s and select the SST \(k-\omega\) turbulence model, the calculation results of the lift-to-drag ratio under the angle of attack \(\alpha\in[0,\,24^{\circ}]\) are shown in Figure 7. As can be seen, the optimal angle of attack is \(5.5^{\circ}\) since it relates to the maximal lift-to-drag ratio. Figure 8 shows the pressure and velocity distribution around the airfoil. As can be seen, the upper surface of the airfoil is at negative pressure and high velocity while the lower surface of the airfoil is at positive pressure and low velocity. This observation is consistent with both the Bernoulli equation and reality. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Scheme 1** & **Scheme 2** & **Scheme 3** & **Scheme 4** & **Scheme 5** \\ \hline Number of grids & 184,863 & 306,083 & 378,695 & 415,955 & 557,703 \\ Lif coefficient & 0.4037 & 0.39404 & 0.3933 & 0.3900 & 0.38917 \\ Error & 3.7\% & 1.2\% & 1.1\% & 0.2\% & N/A \\ \hline \end{tabular}
\end{table} 
Table 1: Mesh independence check. Figure 5: Airfoil meshing results. Figure 6: Mesh quality check. Based on the optimal angle of attack, optimization is performed to achieve the optimal airfoil profile. As presented in Figure 2, a 7 th-order polynomial can accurately describe the airfoil. Therefore, the coefficients of the shape function \(A_{i}(j)(i=U,L;j=0,1,\ldots 7)\) in (6) can be set as design variables. The optimization objectives are maximization of lift and lift-to-drag ratio:  
\[\left\{ \begin{array}{l}\max C_{l},\ \max\frac{C_{l}}{C_{d}},\\ \text{s.t.}\ C_{l}>C_{l0},C_{d}\leq C_{d0}\end{array} \right. \tag{7}\]  
where \(C_{l0},C_{d0}\) are the lift coefficient and drag coefficient, respectively, with respect to the baseline airfoil. An improved non-dominated sorting genetic algorithm, NSGA-II, is selected as the optimization algorithm. This algorithm is a very popular evolutionary algorithm for dealing with multi-objective optimization problems. Incorporating this algorithm into the \"optimization\" module shown in Figure 3, one can run an optimization loop. The optimization results are listed in Table 2. As can be seen, the lift and lift-to-drag ratio are both improved after optimization while drag is decreased, compared with the baseline airfoil. Figure 8: Pressure and velocity distribution in the case of optimal angle of attack. Figure 7: Variation of lift-to-drag ratio with angle of attack. A comparison between the baseline airfoil profile and the optimized airfoil profile is shown in Figure 9. As can be recognized, the bending curvature of the airfoil decreases after optimization, which accounts for the slight increase in lift, as shown in Table 2. However, the maximal airfoil thickness decreases and its corresponding location moves forward, which accounts for the obvious decrease in drag and the resulting increase in lift-to-drag ratio.