# Aerodynamic Efficiency Improvement on a NACA-8412 Airfoil via Active Flow Control Implementation
## 4 Results
### Optimal \(\theta\) and \(C_{\mu}\) Configuration  
Figure 9 introduces the different values of the temporal averaged \(C_{l}\), \(C_{d}\) and aerodynamic efficiency \(E=C_{l}/C_{d}\) as a function of the different jet injection angles \(\theta\) and momentum coefficients \(C_{\mu}\) studied. Observing Figure 9a, it can be stated that regardless of the injection angle studied, high \(C_{l}\) are obtained for momentum coefficients ranging between \(0.0005\leq C_{\mu}\leq 0.05\). In general the optimum jet inclination angle is \(\theta=40^{\circ}\), although two maximum lift coefficients are obtained at \(\theta=40^{\circ}\)\(C_{\mu}=0.0005\) and \(\theta=20^{\circ}\)\(C_{\mu}\) = 0.001, respectively. Regarding the results at \(\theta=30^{\circ}\), the graphic presents a much flatter shape than for \(\theta=20^{\circ}\), with an efficiency maximum at about \(C_{\mu}=0.005\). In fact, regardless of the injection angle studied, the maximum lift coefficients are obtained for momentum coefficients in the range \(0.0005\leq C_{\mu}\leq 0.001\).  
For \(C_{\mu}\) values larger than 0.015, regardless of the injection angle, the lift coefficient values significantly decrease. Finally, the \(C_{l}\) curve obtained for \(\theta=40^{\circ}\) is the most flat one, indicating the lift obtained at this inclination angle is good and pretty stable for a large range of momentum coefficients. Note that as \(\theta\) increases, \(C_{l}\) tends to be more stable in the range of \(C_{\mu}\) between 0.0005 and 0.015, indicating that the effectiveness of \(C_{\mu}\) is higher for growing \(\theta\) values. When considering the drag coefficient in Figure 9b, minimum values are obtained for \(\theta=40^{\circ}\), although for \(\theta=30^{\circ}\) relatively low drag coefficients are observed. The minimum \(C_{d}\) is obtained at \(\theta=40^{\circ}\) and \(C_{\mu}=0.003\). Comparing Figure 9a with Figure 9b it can be concluded that to maximize lift and minimize drag the optimum jet injection angle is \(\theta=40^{\circ}\), yet, the momentum coefficient to maximize lift is \(C_{\mu}=0.0005\) while the one minimizing drag needs to be \(C_{\mu}=0.003\). In order to solve this dilemma the airfoil efficiency must be considered.  
Airfoil efficiency \(E\) is presented in Figure 9c. Maximum aerodynamic efficiencies are obtained for \(\theta=40^{\circ}\) followed by \(\theta=30^{\circ}\). The maximum aerodynamic efficiency has been obtained for \(\theta=40^{\circ}\) and \(C_{\mu}=0.003\), with \(E=12.3291\). Comparing this case with the baseline case, it supposes a \(\Delta C_{l}=0.51154\), \(\Delta C_{d}=-0.098\), and \(\Delta E=7.2043\). Consequently, these results have been considered satisfactory in order to proceed with the frequency study.  
For completeness, Table A1 is presented in Appendix A, where the results obtained from the different simulations previously introduced in Figure 9 are presented. Note that all values presented in this table correspond to the time averaged values obtained during the last 10 seconds of each simulation.  
Figure 9: Aerodynamic coefficients comparison for the different \(C_{\mu}\) and \(\theta\) studied. (**a**) Lift coefficient comparison for the different \(C_{\mu}\) and \(\theta\) studied. (**b**) Drag coefficient comparison for the different \(C_{\mu}\) and \(\theta\) studied. (**c**) Aerodynamic efficiency comparison for the different \(C_{\mu}\) and \(\theta\) studied.