## 4 Results and Discussion
### Verification and Validation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**NREL S-809**} & \multicolumn{2}{c}{**Relative Variation**} \\ \cline{2-5}  & **Experimental** & **Optimized Airfoil** & **Optimized Airfoil** & **CST with** & **PARSEC with** \\  & **Data (OSU)** & **(CST Method)** & **(PARSEC Method)** & **Experimental Data** & **Experimental Data** \\ \hline \(C_{I}\) & 0.79 & 0.883 & 0.87 & +11.8\% & +10.1\% \\ \(C_{d}\) & 0.0131 & 0.0134 & 0.0148 & \(-\)2.2\% & \(-\)12.1\% \\ \(L\)/\(D\) & 60.3 & 65.9 & 58.8 & +9.6\% & \(-\)2.0 \% \\ AOA & 6.2\({}^{\circ}\) & 6.2\({}^{\circ}\) & 6.2\({}^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance comparison of the original and optimized airfoils (NREL S-809). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **CST Parameters** & **Variables** & **PARSEC Parameters** \\ \hline \(A_{1}^{0}\) & \(-\)0.1112 & P\({}^{1}\) = rleup & 0.0216 \\ \(A_{1}^{1}\) & \(-\)0.4286 & P\({}^{2}\) = rleup & 0.010 \\ \(A_{1}^{2}\) & \(-\)0.2461 & P\({}^{3}\) = Xup & 0.3826 \\ \(A_{1}^{3}\) & 0.0525 & P\({}^{4}\) = Yup & 0.1018 \\ \(A_{u}^{0}\) & 0.1682 & P\({}^{5}\) = YXXup & \(-\)1.201 \\ \(A_{1}^{1}\) & 0.3288 & P\({}^{6}\) = Xlow & 0.3633 \\ \(A_{u}^{2}\) & 0.2567 & P\({}^{7}\) = Ylow & \(-\)0.1081 \\ \(A_{u}^{3}\) & 0.1788 & P\({}^{8}\) = YXXlow & 1.526 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{P\({}^{9}\) = yte} & 0 \\ \hline \hline  & \multicolumn{2}{c}{P\({}^{10}\) = delta yte} & 0 \\ \hline \hline  & \multicolumn{2}{c}{P\({}^{11}\) = alpha te} & 8.5 \\ \hline \hline  & \multicolumn{2}{c}{P\({}^{12}\) = beta te} & 8.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The design parameters for the NREL S809 airfoil. gave a better performance than the experimental and PARSEC methods. For the blade design, \(C_{l}\) was kept at a high level, which implied a good off-design property of the small wind turbine. Figure 11 shows the coefficient of drag, which was a very difficult quantity to calculate numerically as well as experimentally. By the CST methodology, the drag was minimized the most compared with the other drag coefficients calculated experimentally by OSU and by the PARSEC method. For the lift coefficient and \(L\)/\(D\) ratio, the present results gave a slightly higher value than the experimental results provided by OSU [37]. The comparisons and calculations used in this paper validate our numerical methodology. Figure 11: Variation of drag coefficient with the angle of attack for the S809 airfoil. Figure 10: Variation of lift coefficient with the angle of attack for the S809 airfoil. Figure 9: Shape comparison of the baseline and optimized airfoils (NREL S-809). The pressure distribution over the surface of the baseline airfoil and the optimized airfoil obtained by the CST and PARSEC methods at a 6.2\({}^{\circ}\) angle of attack were compared by drawing a pressure coefficient (\(Cp\)) graph, shown in Figure 12. It was found that the \(Cp\) values obtained from XFOIL simulation were relatively high for both the optimized airfoil and the baseline airfoil. XFOIL uses a panel technique method to calculate the pressure distribution over the upper and lower surfaces of the airfoil. These are the aerodynamic properties of an airfoil which are the direct result obtained from the design objective. Figure 13 shows the convergence graph of an optimized airfoil for the CST and PARSEC parametric methods with 8 and 12 basic design parameters, respectively. Figure 13a displays error bars with respect to the number of iterations and the pressure distribution curve at a 6.2\({}^{\circ}\) angle of attack, while Figure 13b shows the iterations of function evaluations used to converge the airfoil with the lift coefficient. The residual pressure at the surface of the optimized airfoil generated by CST was very small, and the geometry residuals were below 3 \(\times\) 10\({}^{-3}\). The time consumed by the algorithm of the CST and PARSEC optimization processes was 210 and 272 min, respectively.