# Shock Reduction through Opposing Jets--Aerodynamic Performance and Flight Stability Perspectives
## 4 Aerodynamic Performance and Stability Analysis  
The aerodynamic efficiency (lift-to-drag ratio) increases with the increase in angle of attack and maximizes till \(8^{\circ}\). The advantage of reduction in coefficient of drag reduces with the increase in angle of attack as shown in Figure 13b. The overall performance of airfoil increases by using counter flow jet as depicted from Figure 13a. The re-circulation regions tilt in the windward direction and fill the wake areas of airfoil at higher angles of attack. It is evident from the Figure 13a that after the stall of an airfoil, the performance of jet is no longer as efficient as it is for lower angles of attack. Figure 14a show the comparison of lift coefficients of with and without cases at range of angles of attack (\(\alpha=0^{\circ}\)-\(8^{\circ}\)). It is evident from Figure 14a that by implementing jet the lift increases, drag reduces and static stability remains unchanged at positive angles of attack (Figure 14b). The opposing jet certainly influence stability parameters as well. These stability parameters were calculated at constant free stream Mach 1.4 for both with and without jet cases with a pivot point at mid (\(0.5c\)) of chord of airfoil. The equation of linear line is assumed analogously for the calculation of static stability derivatives. \[y=mx+c \tag{12}\]  
however, \(m\) is the slope of linear line. The adopted analogy for the hysteresis loops of pitching moment coefficient (\(C_{m}\)) is changing as  
\[y=C_{ij}x+C_{i\ell}, \tag{13}\]  
Figure 12: Pressure distribution of free stream flow with increasing angles of attack (\(\alpha=0^{\circ}\) to \(8^{\circ}\)) at constant \(PR=3.0\) and \(M_{\infty}=1.4\). whereas, \(i\) and \(j\) represents the moment (\(m\)) and angle of attack (\(\alpha\)) of perturbed airfoil. So, the equation 13 becomes  
\[y=C_{m_{\alpha}}x+C_{m_{\alpha}}, \tag{14}\]  
and \(C_{m_{\alpha}}\) is the pitching moment slope coefficient of hysteresis loops. To evaluate dynamic stability derivatives, fully converged static cases were initialized with forced oscillations. The sensitivity of dynamic stability derivatives is checked against reduced frequency, mean angle of attack and amplitude of forced oscillations. The reduced frequency contributes a lot in the longitudinal pitch damping derivatives. The dynamic derivatives of with and without opposing jet are shown in Figure 15a,b for several values of reduced frequencies with constant amplitude \(\alpha_{A}=4^{\circ}\). Figure 15a shows that by increasing reduced frequency the body moves towards the unstable behavior (The value of \((C_{m_{\alpha}}+C_{m_{\alpha}})\) start increasing). Although it shows a linear behavior along increasing angles of attack. But in the case of ejection of jet these linear behaviors drastically change for low frequencies (\(k=0.05\), \(0.1\)). However, for higher frequencies in opposing jet shows linear behavior for increasing angles of attack as it was for no jet case. Figure 14: Effects of opposing jet on the lift and moment coefficients of airfoil by increasing the static angles of attack (\(\alpha=0^{\circ}\) to \(8^{\circ}\)). Figure 13: Effects of opposing jet by increasing the \(\alpha\). Basically, there are three types of bodies (primary, secondary and tertiary) that affect the moment coefficient and its damping forces as shown in Figure 15a,b. The most important body is the primary body that is an airfoil. As the angles of attack starts increasing from \(0^{\circ}\) the secondary body (generation of adverse pressure gradient) starts developing and effects on the overall moment coefficient \(C_{m}\). This adverse pressure gradient (secondary body) brings stall for the body at high angles of attack. But in case of jet ejection, a jet with recirculation regions represents a third body (tertiary body) generated which reduces drag from the leading edge of an airfoil but also effects the stability derivatives.