# An Effective Simulation Scheme for Predicting the Aerodynamic Heat of a Scramjet-Propelled Vehicle
## 3 Validation Tests
### High Supersonic Stage  
#### 3.3.1 Grid Strategy  
The experimental data in Ref. [28] were used to validate the simulation parameters at this stage. The Mach number \(M\) was 5 and the wedge angle \(\alpha\) was 15\({}^{\circ}\). The shock angle s was 24.32\({}^{\circ}\), which was calculated by Equation (16). Next, we established and divided the external flow field model for the simulation, as shown in Figure 10. The bold black dot is the (0,0) point of the coordinate system. Line m is the flat plate and line n is the wedge section. The bold black square point is the origin of the wedge section. \(L\) is the distance between the flat plate's leading edge and the wedge section origin, which was 0.25 m.  
Figure 8: (**a**) surface-pressure distributions from experiment and SST k-omega simulation; (**b**) surface-pressure distributions from experiment and Spalart-Allmaras simulation. Figure 9: (**a**) density profile at x = 0 from experiment and SST k-omega simulation; (**b**) density profile at x = 0 from experiment and Spalart-Allmaras simulation. At the high supersonic stage, the total temperature of the freestream \(T_{t}\) was 475 K and the total pressure of the freestream \(p_{t}\) was 4.5 \(\times\) 10\({}^{5}\) Pa. The thermal conditions at the wall were \(T_{w}\) = 100 K leading to a ratio \(T_{W}/T_{r}\) = 0.24 and \(T_{w}\) = 290 K leading to a ratio \(T_{W}/T_{r}\) = 0.72, where \(T_{r}\) is the recovery temperature. The static temperature of the freestream \(T\) was calculated by Equation (17), and \(T\) was 79.17 K; the static pressure of the freestream \(p\) was calculated by Equation (18), and \(p\) is 850.5173 Pa; the speed of sound \(C_{air}\) was calculated by Equation (19), and \(C_{air}\) was 178.3999 m/s. The Mach number \(M\) was 5 and the velocity of the freestream \(U\) was 891.99969 m/s. The density of air was ideal-gas, the specific heat at constant air pressure was 1006.43 j/kg-k, and the viscosity adopted three-coefficient Sutherland Law. The corresponding unit Reynolds number of the freestream was 6 \(\times\) 10\({}^{6}\) and the characteristic length \(c\) was about 0.34659 m. At this stage, shock waves occurred and play important roles that affected the simulation. As in the previous stage, the computational region analysis was performed and three sizes were chosen, as shown in Figure 11a [24]. The INLET boundary contains lines a, e and o, the OUTPUT boundary contains lines d and h, the FARDIELD boundary contains lines i, j, k, and p, and the WALL boundary contains lines m, n, and q. Figure 11b shows the close view of the refined mesh near the shock wave region. Taking the SST k-omega model at \(T_{W}\) = 290 K as an example, the values of the wall pressure coefficient and wall Stanton number (_st_) were as shown in Table 7, where x is the \(X\)-axis location, and the computational domain with 8c was chosen for low ratio errors. Further, three levels of different mesh were applied to be sure of grid independency and, from Table 8, the mesh with 540,000 cells was preferred. Figure 11: (**a**) computational domain; (**b**) close view of grids in near shock wave region. Figure 10: The external flow field simulation model for the high supersonic stage. Ref. [9] suggests that for predicting the aerodynamic heat of flow around a 2D cylinder, the grid with the cell Reynolds number that represents the minimum normal grid distance to the wall should be less than 8. Therefore, keeping the total number of mesh cells unchanged, meshes with \(y^{+}=\) 0.3, 0.15 and 0.08 were applied when looking for the suitable cell Reynolds number for the wedge section at this stage. The corresponding \(y_{H}\) and the cell Reynolds number were about (2.46 \(\times\) 10\({}^{-6}\), 16), (1.23 \(\times\) 10\({}^{-6}\), 8) and (6.56 \(\times\) 10\({}^{-7}\), 4), respectively. Ref.