# Airfoil Optimization Design Based on the Pivot Element Weighting Iterative Method
## 4 Results  
There are 15 design variables of the upper airfoil. The range of them is from \(-\)0.0339 to +0.0339. The lower airfoil also has 15 design variables whose range is from \(-\)0.0263 to +0.0263. 300 different-shaped airfoil samples are generated by the Opt LHD. The lift and drag coefficients of every sample airfoil are calculated by the numerical simulation using the SA model. These aerodynamic data are trained as the samples of the RBF approximation model. Moreover the errors of RBF are analyzed using 50 additional Opt LHD samples. The results are listed in Table 2.  
Figure 19: Lift and lift to drag polar curves of E387 airfoil at \(Re\) = 2 \(\times\) 10\({}^{5}\). S-A: Spalart–Allmaras model. Figure 18: Grids around the E387 airfoil. The initial and optimized airfoils are shown in Figure 20. The leading edge radius of S1223 airfoil decreases and the camber of the whole airfoil increases. It leads to the increase of the lift coefficient and decrease of the drag coefficient. Optimization results are represented in Table 3.  
Aerodynamic characteristics of the optimized airfoil are compared with the initial airfoil under the condition of \(Ma=0.1\) and \(Re=1.86\times 10^{5}\). The results are shown in Figure 21. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aerodynamic Coefficient** & **RMSE** & \(R^{2}\) \\ \hline \(C_{\text{L}}\) & \(0.00485\) & \(0.99955\) \\ \(C_{\text{D}}\) & \(0.03\) & \(0.98619\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation of the approximation model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Upper Airfoil** & \multirow{2}{*}{**Min**} & \multirow{2}{*}{**Max**} & \multirow{2}{*}{**Optimum**} & \multicolumn{2}{c}{**Lower Airfoil**} & \multirow{2}{*}{**Min**} & \multirow{2}{*}{**Max**} & \multirow{2}{*}{**Optimum**} \\
**Variables** & & & & & & & \\ \hline \(A_{\text{up1}}\) & \(-0.0339\) & \(0.0339\) & \(-0.02188\) & \(A_{\text{low1}}\) & \(-0.0263\) & \(0.0263\) & \(0.01346\) \\ \(A_{\text{  
It can be seen from Figure 21 that the lift coefficient of optimized airfoil maximumly increases by about 1%, the drag coefficient maximumly decreases by about 1.57% and the lift-drag ratio maximumly increases by about 2.57% when angle of attack changes from 0\({}^{\circ}\) to 8\({}^{\circ}\). When angle of attack is greater than 8\({}^{\circ}\), the lift coefficient of the optimized airfoil reduces and drag coefficient increases. Because the leading edge radius of the optimized airfoil decreases and the trailing edge thickness becomes thin, compared with the initial airfoil. Therefore, it leads to the expansion of the laminar flow separation area for the optimized airfoil. Velocity streamlines of the optimized and initial airfoils under the condition of 12\({}^{\circ}\) are presented in Figure 22. It indicates that the laminar flow area of the initial airfoil is larger than the optimized airfoil. Figure 21: Comparisons of aerodynamic characteristics between the optimized airfoil and the initial S1223 airfoil. Figure 22: Velocity streamlines at the 12\({}^{\circ}\) angle of attack. In the case of \(12^{\circ}\), the wall shear stresses in \(x\) axis direction of the initial and optimized airfoil are shown in Figure 23. \(\tau_{x}\) of the initial airfoil is less than 0 in the location of 0.7196 chord. So the coordinate of the laminar flow separation point is 0.7196 chord of the initial S1223 airfoil. And \(\tau_{x}\) of the optimized airfoil is less than 0 in the location of 0.6695 chord. It indicates that the coordinate of the laminar flow separation point is 0.6695\(c\) for the optimized airfoil. Therefore, the laminar flow separation area of the optimized airfoil is bigger than the initial airfoil. And the lift coefficient of the optimized airfoil reduces and drag coefficient increases at the large angle of attack compared with the initial S1223 airfoil.