# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 2 Analytical Model
As the air is considered to be a perfect gas, the expression \(\rho=\frac{p}{RT}\) is used to calculate the downstream density, where \(R\) is the gas constant in JK\({}^{-1}\) kg\({}^{-1}\):  
\[\frac{p_{d}}{p_{u}}=\frac{{1+\gamma{M_{u_{n}}}^{2}}}{{1+\gamma{M_{d_{n}}}^{2}}} \tag{5}\]  
\[\frac{T_{d}}{T_{u}}=\frac{{1+\frac{1}{2}\left({\gamma-1}\right){M_{u_{n}}}^{2} }}{{1+\frac{1}{2}\left({\gamma-1}\right){M_{d_{n}}}^{2}}} \tag{6}\]  
In the Prandtl-Meyer expansion wave theory, according to [18; 19], the flow is considered to be isentropic. The theory starts with the definition of the Prandtl-Meyer function, which depends on the unknown downstream Mach number according to Equation (7):  
\[\
u_{PM}\left({M_{d}}\right)=\sqrt{\frac{{\gamma+1}}{{\gamma-1}}}\tan^{-1} \sqrt{\frac{{\gamma+1}}{{\gamma-1}}\left({{M_{d}}^{2}-1}\right)}-\tan^{-1} \sqrt{{M_{d}}^{2}-1} \tag{7}\]  
This equation defines the wall inclination angle \(\
u_{PM}\) that the fluid particles need to turn, considering that the upstream flow is having a Mach number associated with one \(M_{u}=1\), and in order to obtain a desired downstream Mach number \(M_{d}\). Since the upstream Mach number is not \(M_{u}=1\), Equation (7) needs to be applied twice, to initially obtain the wall inclination angle \(\
u_{PM}\left({M_{u}}\right)\) needed to accelerate the fluid from an hypothetical upstream Mach number \(M_{u}=1\) to the real existing upstream Mach number \(M_{u}\). As the real wall angle \(\theta\) is known, Equation (8) will be employed to obtain the flow turning angle \(\
u_{PM}\left({M_{d}}\right)\) to determine the downstream real Mach number providing the upstream Mach number is one \(M_{u}=1\). Once this angle is obtained, it can be substituted in Equation (7) to determine the required downstream Mach number \(M_{d}\):  
\[\theta=\
u_{PM}\left({M_{d}}\right)-\
u_{PM}\left({M_{u}}\right) \tag{8}\]  
As the process is considered isentropic, the ratio of variables across the expansion is calculated by assuming conservation of total (or stagnation) quantities, see Equations (9) and (10):  
\[\frac{T_{d}}{T_{u}}=\frac{{1+\frac{1}{2}\left({\gamma-1}\right){M_{u}}^{2}}} {{1+\frac{1}{2}\left({\gamma-1}\right){M_{d}}^{2}}} \tag{9}\]\[\tan\theta=\frac{{{M_{u}}^{2}\sin 2\beta-2\cot\beta}}{{{M_{u}}^{2}\left({\gamma+ \cos 2\beta}\right)+2}} \tag{1}\]0  
Starting from the freestream flow conditions (stage \(\infty\), upstream conditions), where \(M_{\infty}=7\), and considering the International Standard Atmosphere (ISA) [20] at an altitude of \(z=30\) km, in which \(p_{\infty}\), \(T_{\infty}\) and \(\rho_{\infty}\) were previously defined, the flow conditions of stages 1 and 2 (downstream conditions) were calculated by using the equations presented above. Afterwards, stage 2 became the new upstream stage and stage 3 the downstream stage, and so on until stage 5. Table 1 shows the deflection angle between the different stages and the physical phenomenon (oblique shock wave or Prandtl-Meyer expansion wave) that occurs depending on the angle of attack and the design angles. Lift \(L\) and drag \(D\) are the aerodynamic forces calculated from the constant pressure distribution of each stage in the perpendicular and streamwise flow directions, respectively. A thrust force \(F_{T}\) was added in the calculation, its application point being located at the centre of the scramjet engine, not shown in Figure 1 (point located in the middle of stage 4 and separated downwards a vertical distance half of the engine width). Since the aircraft position remained constant, the thrust force was always pointing horizontally, regardless of the angle of attack evaluated. The pitching moment \(M_{CG}\) is, therefore, the sum of all moments evaluated about the Centre of Gravity (CG).