# Aerodynamic Study of the NASA's X-43A Hypersonic Aircraft
## 3 CFD Model  
The current section encompasses the pre-process of the CFD model in which the real 2D design of X-43A is considered. Therefore, the definition of the geometric and physical conditions, the finite volume discretization of the computational domain and the OpenFOAM set up are precisely described. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ups.** & **Downst.** & **Physical Phenomenon** & **Deflection Angle** \\
**Stage** & **Stage** & **If \(AoA<\alpha_{1}\), oblique shock wave** & \(\theta_{1}=\alpha_{1}-AoA\) \\ \hline Stage \(\infty\) & Stage 1 & If \(AoA<\alpha_{1}\), Prandtl–Meyer expansion wave & \(\theta_{1}=AoA-\alpha_{1}\) \\ Stage \(\infty\) & Stage 2 & If \(AoA<0^{\circ}\), oblique shock wave & \(\theta_{2}=AoA+\alpha_{2}\) \\ Stage 2 & Stage 3 & Oblique shock wave & \(\theta_{3}=\alpha_{2}+\alpha_{3}\) \\ Stage 3 & Stage 4 & Prandtl–Meyer expansion wave & \(\theta_{4}=\alpha_{3}\) \\ Stage 4 & Stage 5 & Prandtl–Meyer expansion wave & \(\theta_{5}=\alpha_{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Deflection angle and physical phenomenon between the different stages. The OpenFOAM version 6 software enabled to carry out a more complex geometry of X-43A based on the implementation of the scramjet, which, in fact, only consisted of a trapezoidal shape extending along stage 4 at a distance of \(y_{S}=0.1101\) m from it. Therefore, the CFD geometry of the aircraft was characterised by the same parameters as those from Figure 1 but with the addition of the scramjet. The CFD domain, which is visualised in Figure 2a, had a rectangular shape, the same orientation as the aircraft and was divided into different quadrilateral regions. Between the left side (inlet) and the leading edge of X-43A, a distance of 1.5c was set in order to guarantee an undisturbed flow stream; a distance of 5c was chosen between the trailing edge and the right side (outlet) of the domain to ensure that the oblique shock and Prandtl-Meyer expansion waves dissipated properly. A distance of 3c was selected between the up or down sides and the chord line of the aircraft for the purpose of avoiding the impact of those physical phenomena into the domain limits. In addition, different small regions in contact with the contours of the aircraft (walls), named boundary layer domain, were designed with the objective of facilitating the boundary layer capture. The other regions further away from the aircraft which were in contact with the domain perimeter zones were called the outside domain. The CFD domain, as well as the computational mesh, were created by the Gmsh software (version 3.0.6). The quadrilateral regions of the domain allowed the construction of a structured mesh since the solution converged earlier and there was a higher control in the distance between the solid boundaries and the first mesh layer (Y) than in an unstructured mesh. Two different structured meshes were characterised: the boundary layer mesh, which corresponded to the discretization of the boundary layer domain, and the far field mesh, which was the mesh of the outside such domain. In case of the boundary layer mesh, the height of the cells in contact with the walls was selected to capture the effects of the buffer layer (\(y^{+}\) between 5 and 30) according to the turbulent boundary layer equations from [21]. Despite a \(y^{+}\) lower than 1 would have permitted a more precise visualisation of the boundary layer effects, in the current study, it was not possible due to the limitations in the computational power. Therefore, a relatively moderate growth rate (1.5%) in the perpendicular direction of the walls was chosen along the domain except from the regions surrounding the scramjet, which was about 1%. For the far field mesh, only a moderate growth rate between 2% and 4% was  
Figure 2: Grid used in the present study, (**a**) entire mesh domain, (**b**) mesh nearby the body, (**c**) mesh details at several body locations. selected.