# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 2 Analytical Model  
This section introduces the required methodology to calculate the mentioned aerodynamic parameters of the simplified shape of X-43A for a given initial geometry, freestream Mach number, flight altitude and different angles of attack (\(AoA\), from \(-4^{\circ}\) to \(14^{\circ}\) every \(0.1^{\circ}\)) as well as the way to calculate the flight conditions of Vertical Force Balance and zero pitching moment. The geometry of the simplified X-43A was defined by the design angles \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) and \(\alpha_{4}\) and the following main dimensions \(x_{1}=3.66\) m, \(x_{23}=1.83\) m, \(Y=0.2225\) m, \(x_{4}=0.762\) m and \(x_{5}=1.068\) m, which were extracted from [8; 11; 17]. In this case, the scramjet was neglected as well as the vertical and horizontal stabilising surfaces. Although the geometry is 2D, for physical calculations a thickness of 1 m is considered. Since an oblique shock wave or a Prandtl-Meyer expansion wave originated from each stage or surface depending on the angle of attack and the design angles, each one of them presented specific flow conditions. All these features are presented in Figure 1 where, for this particular angle of attack, two oblique shock waves with different shock wave angles (\(\beta_{2}\) and \(\beta_{3}\)) appears at the beginning of stages 2 and 3 and three Prandtl-Meyer expansion waves at the beginning of stages 1, 4 and 5. Moreover, \(\theta_{1}\) and \(\theta_{2}\) are characterising the deflection angles between the unperturbed flow given by \(M_{\infty}\) and stages 1 and 2. The oblique shock wave and Prandtl-Meyer expansion wave theories employed to find the flow conditions downstream the waves, which are the Mach number, static pressure and temperature as well as the fluid density, start from the flow conditions upstream. Their generic equations and the procedure used to determine the physical parameters are presented afterwards. In addition, steady state flow conditions were considered, the viscous and heat transfer effects were neglected, the air was set to be a perfect gas and the flow conditions at the trailing edge (small vertical surface) were neglected. The oblique shock wave theory, according to [18; 19], defines the angle \(\beta\) as the angle between the shock wave and the upstream flow direction (the unknown angle that needs to be determined at each stage), the deflection angle between the upstream flow and the solid wall surface known as \(\theta\) and the generic upstream Mach number \(M_{u}\). Equation (1) relates these parameters:  
Figure 1: Oblique shock waves and Prandtl–Meyer expansion waves on the simplified shape of X-43A. \[\tan\theta=\frac{{{M_{u}}^{2}\sin 2\beta-2\cot\beta}}{{{M_{u}}^{2}\left({\gamma+ \cos 2\beta}\right)+2}} \tag{1}\]  
where \(\gamma\) is the adiabatic index of the flow. To obtain the Mach number after the oblique shock wave \(M_{d}\), initially Equation (2) is required. From this equation, the upstream Mach number perpendicular to the oblique shock wave \(M_{u_{n}}\) is gathered. The downstream Mach number perpendicular to the shock wave \(M_{d_{n}}\) is to be obtained from Equation (3), and, introducing it in Equation (4), the downstream Mach number is finally determined:  
\[M_{u_{n}}=M_{u}\sin\beta \tag{2}\]  
\[{M_{d_{n}}}^{2}=\frac{{1+\frac{1}{2}\left({\gamma-1}\right){M_{u_{n}}}^{2}}}{{ \gamma{M_{u_{n}}}^{2}}-\frac{1}{2}\left({\gamma-1}\right)} \tag{3}\]  
\[M_{d}=\frac{{M_{d_{n}}}}{{\sin\left({\beta-\theta}\right)}} \tag{4}\]  
The relations between the downstream and upstream static pressure \(p_{d}\) and \(p_{u}\), respectively, and static temperature \(T_{d}\) and \(T_{u}\), respectively, are presented in Equations (5) and (6).