# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 4 Results
### Comparison Study  
The comparison study between the analytical and CFD models was set to validate the CFD model used to develop the aerodynamic study. To do so, a CFD case with the aircraft geometry of Figure 1 was developed. Since the scramjet was neglected, few changes in the CFD domain were carried out: only the quadrilateral regions around the intrados of the aircraft were slightly modified, see Section 3. The structure of the mesh was maintained and its discretization was identical to that from Mesh 2 since it was the optimal one according to Section 4.1. All the other parameters such as the height of the cell in contact with the contours of the aircraft or the boundary conditions remained exactly the same. Furthermore, this case incorporated the same design angles as those from the analytical case and one single simulation per angle of attack was run, composing a total of 10 simulations. The pressure, density, temperature and Mach fields for \(AoA=0^{\circ}\) of the CFD case with the scramjet neglected are depicted in Figure 3. The different discontinuities due to the oblique shock and Prandtl-Meyer expansion waves are clearly visible as well as the drastic expansion at the trailing edge. These physical phenomena are precisely explained in Section 4.5. The comparison of the aerodynamic parameters is represented in Figure 4. Figure 4a shows that the CFD lift coefficient is lower than the analytical one in terms of absolute value; from \(AoA=-4^{\circ}\) to \(AoA=6^{\circ}\), the maximum deviation is under 1%, while from \(AoA=8^{\circ}\) to \(AoA=14^{\circ}\), a maximum deviation of 5% is observed, approximately. The CFD drag coefficient is around 22% higher (Figure 4b). These differences are mainly caused by the viscous stresses considered in the CFD model since the analytical model did not include them. The aerodynamic efficiency is plotted in Figure 4c where,  
Figure 3: Pressure field (**a**), density field (**b**), temperature field (**c**) and Mach field (**d**) of the CFD case with the scramjet neglected for \(AoA=0^{\circ}\). in terms of absolute value, the analytical aerodynamic efficiency is larger than the CFD for all the presented angles of attack since its drag coefficient is lower. In this case, the maximum efficiency in the analytical case is 18% higher than that of the CFD case (maximum deviation); such discrepancy is due to the viscous forces evaluated in the CFD simulations. Finally, according to Figure 4d, the pitching moment coefficient is similar in both types of studies and regardless of the angle of attack evaluated; only a maximum deviation of around 1% is observed. Table 5 shows the analysis of the zero pitching moment and Vertical Force Balance conditions of the CFD case with the scramjet neglected. For the analytical case studied, the same analysis is presented in the following section, Table 6. The characteristic angles of attack \(AoA_{(C_{MC_{CG}}=0)}\) and \(AoA_{(VFB)}\) are almost the same as those calculated in the analytical study, compare Tables 5 and 6. The CFD angle of attack that allows a zero pitching moment condition, \(C_{MC_{CG}}=0\), which is \(-3.89^{\circ}\), is 0.5% higher than that of the analytical case, shown in Table 6. For the CFD angle of attack in which the aircraft flies with the Vertical Force Balance condition, which is \(5.21^{\circ}\), it is exactly the same as the analytical one, shown in Table 6. As a result, for this last condition, the CFD lift coefficient is exactly the same as the analytical one and the CFD pitching moment coefficient \(C_{MC_{CG}}\) only differs by 3%. The CFD drag coefficient and CFD aerodynamic efficiency, however, differ by 21% and 17%, respectively, due to the consideration of the viscous effects in the CFD model. This study shows that the analytical and CFD implementations generate very similar results. Only the drag coefficient and aerodynamic efficiency show some deviations, around 20% between both models.