# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 4 Results
### CFD Cases
The density, temperature and Mach fields of the domain for the same angle of attack are depicted in Figure 11 in addition to the \(y^{+}\) distribution along the walls of the aircraft. The density behaviour (Figure 11a) is almost the same as the pressures: a high density in some regions inside the scramjet and an almost zero one along the trailing edge. In addition, as the flight altitude is high, the density values are relatively small as well as those from pressure. Since the air was considered to be a perfect gas, the density along the surfaces is reduced due to the high temperatures there. In case of the temperature field (Figure 11b), two remarkable things are given in addition to the discontinuities caused by the oblique shock and Prandtl-Meyer expansion waves. Firstly, some aircraft contours are illuminated or, equivalently, the temperature is higher. This means that these regions capture the thermal boundary layer; the surface temperature is higher and decreases with the normal direction until arriving to the freestream temperature, which it keeps constant. From a theoretical point of view, this high temperature is caused by the viscous stresses, which, in turn, are caused by the enormous velocity gradients along the boundary layer. According to Figure 11d, these regions are the same as  
Figure 10: Pressure field of case 1 for \(AoA=0^{\circ}\), (**a**) all the aircraft, (**b**) attached leading edge (different colour scale), (**c**) tip of the lower edge of the scramjet, (**d**) scramjet region. those with low \(y^{+}\) values, which, evidently, translates into a more precise capture of the boundary layer. Contrary to that, other regions where the pressure is relatively high (e.g., some parts of the walls inside the scramjet or the wall of stage 3), the maximum \(y^{+}\) values are around 50 or, in other words, are relatively high and, consequently, the temperature gradients can not be properly visualised. In fact, the \(y^{+}\) of the different surfaces of the aircraft are extremely different due to the drastic changes of the air properties that are caused by the shock and Prandtl-Meyer expansion waves phenomena. Secondly, the temperature around the trailing edge is extremely high and decreases gradually downstream producing two lines around it. These lines represent discontinuities such as expansion waves, forming a triangle that dissipates with the distance. Finally, for the Mach field (Figure 11c), a zero Mach number is found in some aircraft contours, which translates into a zero flow velocity. In fact, these regions are also the same as those with low \(y^{+}\), meaning that some part of the boundary layer is captured. The Mach number starts being zero on the surface and increases gradually along its normal direction until arriving to the Mach number downstream the wave. Overall, the Mach field has the same behaviour as the pressure and temperature fields. The oblique and Prandtl-Meyer expansion waves are clearly delimited, and the regions where the pressure increases, the Mach number decreases and vice versa. As well as in the temperature field, a large fluid expansion occurs at the trailing edge; the Mach number falls to zero and increases progressively downstream. The main fields for \(AoA=14^{\circ}\) of case 1 are also evaluated, see Figures 12 and 13. Under these conditions, a Prandtl-Meyer expansion wave is originated at the beginning of extrados, thus the pressure (Figure 12a), density (Figure 12b) and temperature (Figure 13a) decrease and the Mach number (Figure 13b) increases. The increase of the angle of attack causes a displacement to the left side of shock 3, which cuts the one coming from stage 2 due to its superiority in pressure strength and incapacitates it to reach the scramjet. A third reflection takes place at the end of the tip of the scramjet, which is almost imperceptible due to its weakness.