# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 4 Results
### Analytical Case and Geometric Modifications  
The dependence of the lift coefficient, drag coefficient, aerodynamic efficiency and pitching moment coefficient on the angle of attack of the analytical case studied at Mach 7 and an altitude of 30 km are depicted in Figure 5. In the analytical study, it was verified that the aerodynamic coefficients did not depend on the altitude, but the main parameters in which the aircraft flew with the Vertical Force Balance condition did. Within the angles of attack from \(-4^{\circ}\) to \(14^{\circ}\), the lift coefficient increases almost linearly with the angle of attack (Figure 5a), while the drag coefficient does it in a parabolic way (Figure 5b). The stall in \(C_{L}\) vs. \(AoA\) is not contemplated since the effects of the boundary layer detachment were not considered. Efficiency \(E\) increases with the increase in the angle of attack, reaches its maximum value of 4.02 at \(AoA=4.94^{\circ}\), and then decreases (Figure 5c). Finally, the pitching moment coefficient, which was considered to be positive in the pitch up of the aircraft, also increases approximately in a linear way with the \(AoA\) (Figure 5d). As its gradient, known as the longitudinal static stability index, is positive, the aircraft is unstable with regard to any disturbance in the angle of attack; when \(\Delta AoA>0\), then \(\Delta C_{M_{CG}}>0\) and the aircraft tends to raise its nose. Table 6 shows these coefficients in addition to the shock wave angles \(\beta_{2}\) and \(\beta_{3}\) for the zero pitching moment and Vertical Force Balance conditions as well as the corresponding \(AoA_{(C_{M_{CG}}=0)}\) and \(AoA_{(VFB)}\). Both parameters \(\beta_{2}^{noc}\) and \(\beta_{3}^{noc}\) are the shock wave angles of stages 2 and 3, respectively, in which the aircraft would fly with the nose optimised configuration according to the defined angles of attack. This configuration can be considered if the deviations between them and \(\beta_{2}\) and \(\beta_{3}\) are under 10%. The zero pitching moment condition is achieved for \(AoA_{(C_{M_{CG}}=0)}<0\), which leads to a negative lift coefficient and aerodynamic efficiency, and a Prandtl-Meyer expansion wave is originated at the beginning of stage 2. Moreover, since \(\beta_{3}\) is very close to \(\beta_{3}^{noc}\), the aircraft can be considered to fly with the nose optimised configuration. In case of the Vertical Force Balance condition, since \(AoA_{(VFB)}=5.21^{\circ}\), the lift coefficient is positive and the aerodynamic efficiency is \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Condition** & \(AoA\) **(\({}^{\circ}\))** & \(C_{L}\) & \(C_{D}\) & \(E\) & \(C_{M_{CG}}\) & \(\beta_{2}\) **(\({}^{\circ}\))** & \(\beta_{3}\) **(\({}^{\circ}\))** & \(\beta_{2}^{noc}\) **(\({}^{\circ}\))** & \(\beta_{3}^{noc}\) **(\({}^{\circ}\))** \\ \hline \(C_{M_{CG}}=0\) & \(-3.87\) & \(-0.0486\) & \(0.0116\) & \(-4.19\) & \(0\) & - & \(14.7\) & - & \(15.5\) \\ VFB & \(5.21\) & \(0.0899\) & \(0.0224\) & \(4.01\) & \(0.0268\) & \(14.7\) & \(16.9\) & \(15.5\) & \(15.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Aerodynamic parameters and shock wave angles for the zero pitching moment (\(C_{M_{CG}}=0\)) and Vertical Force Balance (VFB) conditions of the analytical case. almost maximum, but \(C_{M_{CC}}>0\). As \(\beta_{2}\) differs by 5% from \(\beta_{2}^{noc}\) and \(\beta_{3}\) by 8% from \(\beta_{3}^{noc}\), the aircraft can also be considered to fly with the shock waves of stages 2 and 3 focusing on the tip of the lower edge of the scramjet. The variations of the lift coefficient, drag coefficient, aerodynamic efficiency and pitching moment coefficient as a consequence of the changes in the design angles are evaluated next.