# An Effective Simulation Scheme for Predicting the Aerodynamic Heat of a Scramjet-Propelled Vehicle
## 3 Validation Tests
### Supersonic Stage  
3.2.1. Grid Strategy  
The experimental data in Ref. [27] were used to validate the simulation parameters at this stage. Given the existence of the shock wave in the airflow, the value of the shock wave angle was estimated firstly. The Mach number \(M\) was 2.85 and the wedge angle \(\alpha\) is 30\({}^{\circ}\), which meets the Equation (15), where \(\gamma\) = 1.4:  
\[\alpha<\frac{4}{3\sqrt{3}(\gamma+1)}\frac{(M^{2}-1)^{3/2}}{M^{2}} \tag{15}\]  
The shock angle \(s\) was 54\({}^{\circ}\), which was calculated by Equation (16):  
\[\cot\alpha=\tan\ s[\frac{(r+1)M^{2}}{2(M^{2}(\sin\ s)^{2}-1}-1] \tag{16}\]  
Next, we established and divided the external flow field model for the simulation, as shown in Figure 6. The bold black dot is the (0,0) point of the coordinate system. Lines o and q are the flat plate and line p is the wedge section. The bold black square point is the origin of the wedge section. At the supersonic stage, the total temperature of the freestream \(T_{t}\) was 270 K and the total pressure of the freestream \(p_{t}\) was 1.7 atm. The static temperature of the freestream \(T\) was calculated by Equation (17) and \(T\) was 102.88 K:  
\[\frac{T_{t}}{T}=1+\frac{\gamma-1}{2}M^{2} \tag{17}\]  
The static pressure of the freestream \(p\) was calculated by Equation (18) and \(p\) was 5881 Pa:  
\[\frac{p_{t}}{p}=(1+\frac{\gamma-1}{2}M^{2})^{\frac{\gamma}{T-1}} \tag{18}\]  
The speed of sound \(C_{air}\) was calculated by Equation (19) and \(C_{air}\) was 203.36348 m/s:  
\[C_{air}=20.05\sqrt{T} \tag{19}\]  
Figure 6: The external flow field simulation model for the supersonic stage. The Mach number \(M\) was 2.85 and the velocity of the freestream \(U\) was 579.5859 m/s. The corresponding unit Reynolds number of the freestream was \(18\times 10^{6}\), the characteristic length \(L\) was 1.08 m, 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. Initially, we referred to the value of \(y^{+}\) at the transonic stage to estimate \(y_{H}\) 3.25 \(\times\) 10\({}^{-6}\), while the test results showed that the maximum \(y^{+}\) of the wall surface exceeded 1 during the simulation. As described in the previous section, considering the maximum \(y^{+}\) of wall surface during the CFD simulation and the boundary layer profile through TVR, the \(y^{+}\) value was 0.8 and the \(y_{H}\) value was \(2.5\times 10^{-6}\). To determine the best domain size, a computational region analysis is performed. For this purpose, three sizes were chosen, as shown in Figure 7a [24]. The INLET boundary contains line a and h, the OUTPUT boundary contains lines d and k, the FARIELD boundary contains lines l, m, and n and the WALL boundary contains lines o, p, and q. Because of the shock wave, in order to obtain more accurate results, more attention was needed for the grids near the shock wave regions. Figure 7b demonstrates the close view of the grids near the shock wave region. Taking the SST k-omega model as an example, the values of the ratio between the wall surface pressure \(p\) and \(p_{t}\), and temperature are as shown in Table 4, where \(x\) is the distance to the origin point of the wedge section, and the computational domain with 8c was chosen for low ratio errors. Next, three levels of different mesh were applied to ensure grid independency. From Table 5, the mesh with 520,000 cells was applied at this stage.