# Shock Reduction through Opposing Jets--Aerodynamic Performance and Flight Stability Perspectives
## 4 Aerodynamic Performance and Stability Analysis
At low angles of attack, no secondary body is present for low reduced frequency (\(k=0.05\)), but the tertiary body (jet with recirculation regions) still produce effects on the stability derivatives as shown in Figure 15b. The comparison of without and with jet cases at \(k=0.05\) shows that the without jet case depicts linear trend whereas the nonlinearity produces for increasing angles of attack by the ejection of jet. As clear from above Figure 12a-d, by increasing angles the jet disturbed from the leading edge causes instability in flow which leads the disturbance in the dynamic stability parameters as illustrated in Figure 15b. By increasing the reduced frequency, a similar linear trend starts building for higher angles of attack as that of without ejection of jet. The reason behind this linearity is reduction of opposing jet (tertiary body) effectiveness on the movement of the airfoil (primary body) but the secondary body (adverse pressure gradient) still exists as they were for without opposing jet. The other parameter of interest is the influence of the angles of attack on the stability derivatives. The analysis was carried out at several angles of attack (\(\alpha_{o}=0^{\circ},2^{\circ},4^{\circ},6^{\circ}\) & \(8^{\circ}\)) with fixed amplitude of \(4^{\circ}\) and reduced frequency of \(0.2\). By the application of jet, the slope of stability derivatives is damped as compared to without jet cases (Figure 16a,b). With increasing angles of attack, a negative slope starts developing for pitching moment of simple perturbed airfoil (as shown in Figure 16a), while the ejection of opposing jet from the leading edge leads the negative slope towards the positivity as depicts in Figure 16b. At higher angle (\(8^{\circ}\)) the effect of opposing jet starts reducing (or vanished) which leads the positive slope towards the negative value. The number of calculations was made to investigate the influence of mean angles of attack on airfoil stability. The analysis were carried out at several mean angles of attack (\(\alpha_{o}=0^{\circ},2^{\circ},4^{\circ},6^{\circ}\) & \(8^{\circ}\)) with fixed reduced frequency \(k=0.2\) at mid chord movement. In this analysis the amplitude of motion was also varied and selected as, \(\alpha_{A}=2^{\circ},4^{\circ},6^{\circ}\) & \(8^{\circ}\). The comparison in Figure 17a,b shows that by implementing jet on the stagnation point of an airfoil the dynamic stability enhances. It was noticed that for jet case the pitch damping derivatives decreases as the angle of attack increases. But beyond the mean angle \(6^{\circ}\) the dynamic pitch coefficient  
Figure 15: Effects of reduced frequency on the pitch damping of airfoil with constant \(\alpha_{o}=0^{\circ}\) and \(\alpha_{A}=4^{\circ}\). value starts increasing again because at higher mean angle the recirculation regions of jet attached to the free stream flow and destroyed the jet mode, so the value of \(C_{m_{q}}+C_{m_{\Delta}}\) starts increasing and reached close to the without jet case. The reduction at the lower angles are due to the abrupt decrease in the pressure on the upper surface at windward side as shown in Figure 18a-c. This reduction brings the restoring moment in the dynamic system of airfoil, which ultimately decreases the overall dynamic derivatives of stability at lower angles of attack. The positive coefficient of pressure distribution of leading edge is reduced with the ejection of jet. The vicinity on the upper surface contributes a destabilizing to damping coefficient that suppress with introduction of jet. Figure 16: Comparison of pitch moment coefficient \(C_{m}\) for \(\alpha_{o}=2^{\circ}\) to \(8^{\circ}\) with constant amplitude (\(\alpha_{A}=4^{\circ}\)).