# Aerodynamic Efficiency Improvement on a NACA-8412 Airfoil via Active Flow Control Implementation
## 4 Results
### Forcing Frequency Analysis  
As it has been previously mentioned, maintaining constant the groove position and width as well as the AFC parameters optimized in Section 4.1 (\(\theta=40^{\circ}\) and \(C_{\mu}=0.003\)), a set of different \(F_{*}^{+}\) values (\(0.5\leq F_{*}^{+}\leq 7\)) have been applied in the present subsection. After simulating each \(F_{*}^{+}\) case for a computational time of \(30\,s\), the resulting time averaged \(C_{l}\) and \(C_{d}\) values, as well as the airfoil efficiency are presented in Table 3. The optimum case corresponds to \(F_{*}^{+}=4\), with a resulting \(C_{l}=1.8147\), \(C_{d}=0.0942\) and \(E=19.2649\). It can be concluded that the modification of the pulsating flow frequency has brought an efficiency improvement of around 56% versus the one obtained with the optimum \(C_{\mu}\) and \(\theta\). The increase in lift and decrease in drag corresponds to \(\Delta C_{l}=0.0726\) and \(\Delta C_{d}=-0.0471\), respectively. When comparing the final AFC optimum properties with the baseline case ones, it is observed an airfoil efficiency increase of \(275.8\%\), being the lift increase and drag decrease respectively of \(\Delta C_{l}=0.588\) and \(\Delta C_{d}=-0.1451\). From Table 3 there is another condition which is worth to report, this is the maximum lift condition, which happens for \(F_{*}^{+}=3\). It is interesting to note that the AFC parameters to obtain maximum efficiency and maximum lift are almost the same, just the pulsating frequency is slightly different, being four times and three times the natural vortex shedding frequency, respectively. The drastic increase in airfoil efficiency is clearly understood when observing the evolution of the pressure coefficient over the airfoil when the optimized AFC parameters are considered. In Figure 10, the pressure coefficient distribution along the chord is presented for the baseline, maximum efficiency and maximum lift cases. For both maximum conditions, a substantial increase of \(C_{P}\) is observed along the airfoil. From \(x/C=0\) up to \(x/C=0.6\), a huge decrease of the pressure coefficient is seen on the airfoil upper surface, while on the last 40% of the chord the pressure is a bit higher than the one obtained in the baseline case. To further understand the flow structure around the airfoil, streamlines of the averaged flowfield for the maximum efficiency and maximum lift cases are presented in Figure 11a,b, respectively. A drastic reduction of the vorticular structure generated over the airfoil upper surface with respect to the baseline case solution (see Figure 8) can be observed for both configurations presented. Clearly, for the maximum efficiency condition vortical structures have nearly disappeared over the airfoil. Just a small laminar bubble which appears at about \(x/C=0.7\) and disappears before \(x/C=0.8\) can be observed, indicating a corresponding boundary layer separation and reattachment at these points. The separation of the boundary layer is delayed to about \(x/C=0.7\) for the case of maximum lift, reattaching close to the trailing edge. An elongated vortical structure is therefore generated around the airfoil trailing edge. \begin{table}
\begin{tabular}{c c c c c} \hline \(C_{\mu}\) & \(F_{*}^{+}\) & \(C_{l}\) & \(C_{d}\) & \(E\) \\ \hline \multirow{8}{*}{0.003} & 0.5 & 1.6473 & 0.2063 & 7.9853 \\  & 1 & 1.7421 & 0.1413 & 12.3331 \\  & 2 & 1.8075 & 0.1289 & 14.0214 \\ \cline{1-1}  & 3 & 1.8272 & 0.1120 & 16.3160 \\ \cline{1-1}  & 4 & **1.8147** & **0.0942** & **19.2649** \\ \cline{1-1}  & 5 & 1.8220 & 0.1054 & 17.2829 \\ \cline{1-1}  & 7 & 1.7724 & 0.1190 & 14.8941 \\ \hline \end{tabular}
\end{table} 
Table 3: Time averaged aerodynamic coefficients and efficiency obtained for the set of \(F_{*}^{+}\) studied. A good method to understand the flow evolution over the airfoil is via plotting the boundary layer thickness along the airfoil chord.