# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 3 CFD Model
The resulting mesh had approximately \(3\times 10^{6}\) elements with a maximum aspect ratio of 26.03, a maximum non-orthogonality of 46.24\({}^{\circ}\) and a maximum skewness ratio of 1.814, and was extruded one element since OpenFOAM only accepted 3D models. According to the OpenFOAM User's Guide, see [22], the mesh employed can be considered to be a good one, since OpenFOAM recommends not exceeding a non-orthogonality of 65\({}^{\circ}\) and a skewness ratio of 4. An overview of the resulting mesh is depicted in Figure 2.  
The same incident Mach number and altitude as in the analytical model was considered. Table 2 shows the main characteristics of the incoming flow, which was also considered to be a perfect gas. The initial data for the computation where the Reynolds and Prandtl numbers, their respective values were \(Re_{\infty}=9.16\times 10^{6}\) and \(Pr_{\infty}=7.27\times 10^{-1}\). The solver rhoCentralFoam was used, which is a density-based compressible flow solver based on central-upwind schemes of Kurganov and Tadmor. Although the flow was highly turbulent according to \(Re_{\infty}\), the authors decided to directly use the continuity, Navier-Stokes and energy equations for compressible flow conditions without employing any turbulence model, in order to avoid the use of the parameters associated with a given model and their possible tuning. Since a maximum Courant number of 0.85 was imposed, a time step between \(10^{-7}\) s and \(10^{-6}\) s was automatically displayed by the software. Although the air at an altitude of 30 km and at Mach number 7 should be considered as a real gas, in the present paper, it was assumed to be an ideal gas. Furthermore, the fluid was assumed to be Newtonian and the viscous stresses were considered. The following boundary conditions were specified along the domain. * The Dirichlet boundary conditions were used for:
* Velocity at the inlet, up, down and walls faces. * Pressure at the inlet and outlet faces. * Temperature at the inlet face. * The Neumann boundary conditions were used for:
* Velocity at the outlet face. * Pressure at the up, down and wall faces. * Temperature at the outlet, up, down and wall faces. The residuals, which were the scalar components of the velocity and the specific enthalpy, were considered to be tightly converged as they were under magnitude orders of \(10^{-6}\). The aerodynamic parameters were directly determined by the simulation, while the mass flow rate throughout the scramjet and parallel to its walls \(m_{S}\) involved a post-processing step based on numerical integration. It was also checked that these aerodynamic parameters converged properly to a constant value. The analysis of the zero pitching moment and Vertical Force Balance conditions was carried out by approximating all the discretized values of the aerodynamic parameters and mass flow rate into polynomials of degree \(n=6\). Three different cases where each one of them was characterised by specific design angles presented in Table 3 were considered. For each case, one simulation per angle of attack was performed where, in fact, the velocity vector was the only modified parameter. Therefore, 10 simulations were done per case studied, composing a total of 30 CFD simulations. \begin{table}
\begin{tabular}{l l} \hline \hline Velocity vector & \(\overrightarrow{L_{\infty}}=2135.3\) (\(\cos AoA,\ \sin AoA,\ 0\)) ms\({}^{-1}\) \\ Angle of attack & \(AoA=[-4,\ -2,\ 0,\ 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14]^{\circ}\) \\ Pressure & \(p_{\infty}=1172\) Pa \\ Temperature & \(T_{\infty}=231.49\) K \\ Density & \(\rho_{\infty}=0.0176\) kgm\({}^{-3}\) \\ Dynamic viscosity & \(\mu_{\infty}=1.502\times 10^{-5}\) kgm\({}^{-1}\)s\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Oncoming flow characteristics and initial data.