# An Effective Simulation Scheme for Predicting the Aerodynamic Heat of a Scramjet-Propelled Vehicle
## 3 Validation Tests  
### Transonic Stage  
#### 3.1.1 Grid Strategy  
The experimental data in Ref. [23] are used to validate the simulation parameters at this stage, and the simulation external flow field model is as shown in Figure 2. The bold  
Figure 1: (a) X51A; (b) The flight path of an RBCC scramjet-propelled vehicle. The initial conditions of flow at different phases are as follows: Climbing phase: (1) ejection phase, duration time is about 10 s. Flight altitude is about 11 km, Mach number is 0.8, pressure is 22,700 Pa, temperature is 216.77 K, air density is 0.3648 kg/m\({}^{3}\) and dynamic viscosity is 1.422 \(\times\) 10\({}^{-5}\) Pa-s. (2) sub-combustion stamping phase, duration time is about 20 s. Flight altitude is about 16 km, Mach number is 3, pressure is 10,350 Pa, temperature is 216.65 K, air density is 0.1663 kg/m\({}^{3}\) and dynamic viscosity is 1.422 \(\times\) 10\({}^{-5}\) Pa-s. (3) super-combustion stamping phase, duration time is about 4 s, flight altitude is about 26 km, Mach number is 6, pressure is 2188 Pa, temperature is 222.54 K, air density is 0.0339 kg/m\({}^{3}\) and dynamic viscosity is 1.454 \(\times\) 10\({}^{-5}\) Pa-s. Cruising phase: the flight altitude is about 30 km, Mach number is 6.5, pressure is 1197 Pa, temperature is 226.51 K, air density is 0.0181 kg/m\({}^{3}\) and dynamic viscosity is 1.475 \(\times\) 10\({}^{-5}\) Pa-s.  
black dot is the (0,0) point of the coordinate system. Line f is the wedge section, line g is the flat plate. Turbulent flows are affected by the presence of walls; therefore, the accurate representation of the flow in the near-wall region determines the successful prediction of wall-bounded turbulent flows, and it is necessary to ensure the first layer cells of the boundary layer are in the sub-viscosity layer. The calculation of the height of the first layer cells of the boundary layer begins with the flow Reynolds number (\(R_{e}\)):  
\[R_{e}=\frac{\rho UL}{\mu} \tag{1}\]  
where \(L\) is the characteristic length of the wedge, \(\mu\) is the dynamic viscosity of the air, \(\rho\) is the air density, and \(U\) is the velocity of freestream. Considering that the simulation model is wedge-shaped, an empirical correlation formulation to estimate the skin friction coefficient (\(C_{f}\)) is adopted, which is for the fully developed turbulent flow over a flat plate:  
\[C_{f}=\left[2\log_{10}(R_{e})-0.65\right]^{-2.3} \tag{2}\]  
Having computed the skin friction coefficient, the wall shear stress (\(\tau_{W}\)) is calculated:  
\[\tau_{W}=\frac{1}{2}\rho U^{2}C_{f} \tag{3}\]  
After that, we can calculate the friction velocity (\(u_{\tau}\)) from the wall shear stress (\(\tau_{W}\)):  
\[u_{\tau}=(\frac{\tau_{W}}{\rho})^{0.5} \tag{4}\]  
The equation for \(y^{+}\) is:  
\[y^{+}=\frac{\rho y_{P}u_{\tau}}{\mu} \tag{5}\]  
Therefore, the height of the wall-adjacent cell centroid from the wall (\(y_{P}\)) is:  
\[y_{P}=\frac{y^{+}\mu}{u_{\tau}\rho} \tag{6}\]  
Finally, the first height of the boundary layer cell (\(y_{H}\)) is double \(y_{P}\):  
\[y_{H}=2y_{P} \tag{7}\]  
Figure 2: The external flow field simulation model for the transonic stage. The result of the first height of the boundary layer cell is an estimate and needs to be updated using information from the CFD analysis. In addition to calculating the \(y_{H}\), the growth ratio (\(r\)) normal to the wall and inflation layers (\(N\)) may also be required by mesh generation, which allows the layers to inflate normally to the wall. For aerodynamic flows using RANS turbulence modeling, CFD often aims for \(N=15\)-\(30\) through the thickness of the boundary layer. By adopting this criterion, we can calculate r normally to the wall, which is started by applying the formulation to estimate the boundary layer thickness, where \(R_{e}\) is bigger than \(5\times 10^{5}\):  
Figure 1:0  
For a given number of \(r\) and \(N\), we can calculate the total thickness of the inflation layers.