## 3 Numerical Results
### Test Case Description  
The test case focuses on the optimization of an Active Flow Control device, more precisely, a Synthetic Jet actuator. The device is tested on a SD7003 airfoil at an angle of attack of 14 degrees. For the comparison between the optimization algorithms, which is the main objective of this work, a single optimization problem with five design variables has been used. The objective function is to maximize the lift coefficient, and to do so the objective function is set to:  
\[f=-C_{l}\]  
At high angles of attack, the Synthetic Jet can greatly affect the flow structure, improving the lift coefficient. The Synthetic Jet actuator, if set properly, can help to reattach the flow to the airfoil or to almost avoid the detachment of the flow.  
The design variables are the same as the previous work by [PERSON] et al. [20]. For a full explanation and detail of the Synthetic Jet actuator design variables meaning refer to [20]. The five design variables are:  
\begin{tabular}{l l l} \(F^{+}\) & Non-dimensional frequency. \\ \(C_{\mu}\) & Momentum coefficient. \\ \(\theta\) & Jet inclination angle. \\ \(x/C\) & Non-dimensional jet position. \\ \(h/C\) & Non-dimensional jet width. \\ \end{tabular}  
The evaluation range of each design variable is shown in Table 3. The same ranges are used with all optimization algorithms.  
The flow has been solved with an unsteady Reynolds averaged Navier-Stokes model (URANS), using the OpenFOAM software. Other models, like direct numerical simulation (DNS) or large eddy simulation (LES) could be used to solve the Synthetic Jet simulation, but their computational cost is too high to perform so many optimizations with the avail  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Variable** & **Minimum Value** & **Maximum Value** \\ \hline \(F^{+}\) & 0.1 & 10 \\ \(C_{\mu}\) & 0.0001 & 0.02 \\ \(\theta^{o}\) & 5 & 175 \\ \(x/C\) & 0.001 & 0.3 \\ \(h/C\) & 0.005 & 0.015 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Active Flow Control design variables and their evaluation ranges.  
able resources. In addition, there is no need to use such precise solvers to evaluate the performance of the Hybrid Method. More details on the solver used can be consulted at [20], as this study uses the same model.  
The mesh used, see Figure 2a, is one of the meshes previously evaluated in [20], although having a smaller number of cells (34,448) than the final one employed in that paper, the maximum \(y^{+}\) after the simulation was \(y^{+}=1\). The mesh nearby the Synthetic Jet actuator is presented in Figure 2b. The run time of the simulation has been adjusted to 30 time units, which as shown in [20] is sufficient to reach convergence. It is important to note that this study is not about the Synthetic Jet actuator optimization, but to compare the optimization algorithms in a real world application with a significant computational cost and complexity. The study of the physical problem is not the main purpose of this study, which justifies reducing the precision of each CFD simulation in order to reduce the overall computational cost.  
In Figure 3, the temporal averaged streamlines and pressure field for the non-actuated case is presented. This configuration is called the baseline. From the streamlines, it can be seen that the flow is fully separated and the airfoil is under stall conditions.