# An Effective Simulation Scheme for Predicting the Aerodynamic Heat of a Scramjet-Propelled Vehicle
## 3 Validation Tests
### Hypersonic Stage  
#### 3.4.1 Grid Strategy  
The experimental data in Ref. [32] were used to validate the simulation parameters at this stage. The Mach number \(M\) was 7.05 and the wedge angle \(\alpha\) was 20\({}^{\circ}\). The shock angle s was 27.24\({}^{\circ}\), which was calculated by Equation (16). We then established and divided the external flow field model for the simulation, as shown in Figure 18. Line d is the flat plate and line g is the wedge section. The bold black dot is the (0,0) point of the coordinate system. The bold black square point is the origin of the wedge section (or the interaction of the plate and the wedge). At the hypersonic stage, the static temperature of the freestream \(T\) was 81.2K, the wall temperature \(T_{W}\) was 311 K, and the static pressure of the freestream \(p\) was 576 Pa. \(M\) was 7.05 and the velocity of the freestream \(U\) was 1274 m/s. The corresponding unit Reynolds number of the freestream was 5.8 \(\times\) 10\({}^{6}\), the characteristic length \(c\) was 2 m, the air density s was ideal-gas, the specific heat at constant air pressure was 1006.43 j/kg-k, and the viscosity adopted three-coefficient Sutherland Law. Figure 17: (a) error ratio of wall pressure coefficient distribution diagram at the wedge section adopting Spalart-Allmaras at \(T_{W}\) = 290 K; (b) error ratio of wall pressure coefficient distribution diagram at the wedge section adopting four different aspect ratios. Figure 18: The external flow field simulation model for the high supersonic stage. As in the previous stage, the computational region analysis was performed and three sizes were chosen, as shown in Figure 19a [24]. The INLET boundary contains lines a and h, the OUTPUT boundary contains lines c and j, the FARDIELD boundary contains lines k and l and the WALL boundary contains lines d and g. Figure 19b shows the close view of the refined mesh near the shock wave region. Taking the Spalart-Allmaras model as an example, the values of \(T/T_{INF}\) and the values of pressure at S = 0.155 m were as shown in Table 10. \(S\) is the distance to the origin point of the wedge along the wedge surface, \(y\) is the distance normal to the wedge surface, and \(T_{INF}\) is the freestream's static temperature ahead of interaction, which was 81.2 K. From Table 10, the computational domain with 8c was chosen for low ratio errors. Three levels of different mesh were tested to be sure of grid independency and from Table 11, the mesh with 650,000 cells was preferred. As in the high supersonic stage, keeping the total number of mesh cells unchanged, meshes with \(y^{+}\) = 0.7, 0.35, and 0.18 were applied to looking for the suitable cell Reynolds number for the wedge section at this stage, according to Ref. [14]. The corresponding \(y_{H}\) and \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Domains** \\ \end{tabular} } & \multicolumn{8}{c}{**y (cm)**} \\  & & & \multicolumn{8}{c}{**(U/U\({}_{\text{INF}}\), Pressure and Error Ratio)**} \\ \cline{2-13}  & & **0.2** & & **0.5** & & **0.8** & & **1.1** & & **1.4** & & **1.6** \\ \hline \multirow{3}{*}{6c} & 3.6211 & 6579.27 & 2.9443 & 6561.08 & 2.9117 & 6664.32 & 3.0921 & 6759.12 & 3.0671 & 6879.28 & 3.1098 & 6902.17 \\  & 3.79\% & 7.40\% & 7.82\% & 7.65\% & 3.87\% & 6.20\% & 8.92\% & 4.87\% & 8.03\% & 3.18\% & 7.79\% & 2.85\% \\ \hline \multirow{3}{*}{8c} & 3.5731 & 6648.16 & 2.9733 & 6629.71 & 2.9434 & 6731.84 & 3.0552 & 6849.32 & 3.0363 & 6975.39 & 3.0731 & 6989.37 \\  & 2.41\% & 6.43\% & 6.91\% & 6.69\% & 2.83\% & 5.25\% & 7.62\% & 3.60\% & 6.95\% & 1.82\% & 6.52\% & 1.63\% \\ \hline \multirow{3}{*}{12c} & 3.5892 & 6627.39 & 2.9628 & 6621.96 & 2.9274 & 6719.34 & 3.0729 & 6823.07 & 3.0541 & 6941.97 & 3.0913 & 6959.97 \\  & 2.87\% & 6.72\% & 7.24\% & 6.80\% & 3.35\% & 5.43\% & 8.24\% & 3.97\% & 7.58\% & 2.29\% & 7.15\% & 2.04\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Computational domain independency study of three levels of sizes with error ratios.