## 3 Numerical Results
### Results Based on the Fluid Flow Performance  
This subsection provides a comparison between the characteristics of the flow field corresponding to each of the optimal solutions labeled in Figure 4. The lift coefficient and design variables of each of the optimal solutions are presented in Table 5. For a full explanation of the flow structure and details on the Synthetic Jet actuator performance, the reader is directed to [20].  
As explained in the Section 3.1, the flow without the Synthetic Jet actuator is fully detached, see Figure 3. The objective of the Synthetic Jet actuator is to prevent or minimize flow separation. The averaged streamlines and pressure field of the optimized cases are presented and discussed in this section. The flow field corresponding to Case 1 is presented in Figure 5. Despite the fact that the flow separation is slightly delayed versus the baseline case, a large vortical structure is still observed over the airfoil.  
In Figure 6, the averaged streamlines and pressure field obtained from case 2 is presented. It shows a late reattachment of the flow, which improves the lift coefficient of the baseline by 74%. This solution has been obtained by one of the Genetic Algorithm's runs, with around 400 evaluations of the objective function. The resulting lift coefficient is \(C_{l}=1.39\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Genetic Algorithm** & **Hybrid Method** & **Conjugate Gradient** \\ \hline \multirow{4}{*}{\(C_{l}\)} & \multirow{4}{*}{1.39} & \multirow{4}{*}{1.52} & 0.81 \\  & & & 1.26 \\  & & & 1.38 \\  & & & 1.42 \\  & & & 1.42 \\  & & & 1.46 \\ \hline \(\overline{C_{l}}\) & 1.44 & 1.53 & 1.29 \\ \hline \(\sigma\) & 0.07071 & 0.01414 & 0.2458 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Best lift coefficient of each optimization case, means and standard deviation.  
Figure 5: Averaged streamlines and pressure field of the Case 1 (\(C_{l}=0.81\)).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(\mathbf{C_{l}}\) & \(\mathbf{F^{+}}\) & \(\mathbf{C_{\mu}}\) & \(\boldsymbol{\theta^{o}}\) & \(\boldsymbol{x/C}\) & \(\boldsymbol{h/C}\) \\ \hline
1 & 0.81 & 8.6 & \(1\times 10^{-4}\) & 136 & \(3\times 10^{-1}\) & \(5\times 10^{-3}\) \\
2 & 1.39 & 0.3 & \(1.47\times 10^{-2}\) & 24 & \(1\times 10^{-3}\) & \(1.4\times 10^{-2}\) \\
3 & 1.49 & 3.2 & \(1.92\times 10^{-2}\) & 8 & \(2\times 10^{-2}\) & \(5\times 10^{-3}\) \\
4 & 1.52 & 6.7 & \(2\times 10^{-2}\) & 7 & \(2.6\times 10^{-2}\) & \(5\times 10^{-3}\) \\
5 & 1.54 & 9.9 & \(2\times 10^{-2}\) & 5 & \(1.97\times 10^{-2}\) & \(5\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Values of the lift coefficient and design variables of the five labeled cases.  
The averaged streamlines and pressure fields of cases 3, 4 and 5 are presented in Figures 7-9, respectively. All of them show a complete flow reattachment, with very minor differences in the size of the laminar bubble appearing close to the airfoil leading edge. The bubbles are in fact located near the Synthetic Jet position, just downstream of it, as can be seen in the above mentioned figures. The optimization of the flow control actuation parameters has managed to successfully reattached the flow along the entire airfoil chord.