# An Effective Simulation Scheme for Predicting the Aerodynamic Heat of a Scramjet-Propelled Vehicle
## 3 Validation Tests
### Transonic Stage
#### 3.1.1 Grid Strategy
Next, we match this to the boundary layer thickness \(\delta\) to allow the inflation layers to cover the boundary layer thickness. Assume that the inflation layer has two layers, then the total thickness (\(y_{T}\)) is:  
Figure 1:1  
Similarly, the total thickness of four inflation layers is:  
Figure 1:2  
Therefore, the total thickness of \(N\) inflation layers is:  
Figure 1:3  
Equation (11) is the formula of a geometric series, which could be rewritten using the identity for convenience:  
Figure 1:4  
Hence, the total thickness of the inflation layer is:  
Figure 1:5  
Now, the error between the total thickness of the inflation layer \(y_{T}\) and the thickness of the boundary layer \(\delta\) is:  
Figure 1:6  
Equation (14) is a function of the growth ratio r and we need to find a value of \(r\) to make the error equal to \(0\), which is the total height of the inflation layer equal to the boundary layer thickness. The Newton-Raphson root-finding algorithm is adopted, and once the algorithm has converged, the solution \(r\) is the maximum growth ratio. Because, in mesh generators, the \(r\) is always greater than \(1\) and less than \(2\), the initial guess of \(1\) and \(2\) could be used to bound the root-finding algorithm to accelerate the convergence. Three different Mach numbers, \(0.768\), \(0.817\), and \(0.86\) were applied, respectively, in the validation tests at the subsonic stage. The static temperature of freestream \(T\) was \(300\) K and the dynamic viscosity \(\mu\) was \(1.7894\times 10^{-5}\) Pa-s; the density of air was ideal-gas, the specific heat at a constant air pressure was \(1006.43\) j/kg-k, and the characteristic length \(c\) was \(0.59696\) m. Initially, \(y^{+}\) was \(5\) to ensure the first layer cells of the boundary layer were in the sub-viscosity layer, and \(y_{H}\) was \(1.53\times 10^{-5}\), according to Equations (1)-(7). Next, the CFD analysis showed that the maximum value of \(y^{+}\) at the WALL surface reached \(6.5\). The simulation tests showed that the \(y^{+}\) value of \(1\) could guarantee that the first layer cells of the boundary layer are in the sub-viscosity layer during the simulation process. The turbulent viscosity ratio (TVR) was adopted to evaluate whether the boundary layer profile was captured well. Figure 3 shows the corresponding plots of the TVR for the mesh with \(y^{+}=1\) and \(N=20\), since the \(y^{+}\) value is 1, which means presence within the laminar sub-layer, and absence of turbulent viscosity. As the mesh moved through the buffer region and into the logarithmic region, the TVR value increased, and the maximum value occurred near the middle of the boundary layer. Subsequently, the turbulence gradually dissipated as the mesh approached the free stream condition, which was expected. Therefore, the values of \(y^{+}\) and N were sufficient, so we could calculate the maximum growth ratio r by Equation (14). The first layer height \(y_{H}\) and the maximum growth ratio were (1.49 \(\times\) 10\({}^{-6}\), 1.42), (1.38 \(\times\) 10\({}^{-6}\), 1.42) and (8.42 \(\times\) 10\({}^{-6}\), 1.34) respectively. To determine the best domain size, computational region analysis was performed. For this purpose, three sizes were chosen, as shown in Figure 4a [24]. The INLET boundary contained line a, the OUTPUT boundary contained line d, the FARDIELD boundary contained lines h, i, and j, the SYMMETRY boundary contained line e, and the WALL boundary contained lines f and g. Figure 4b offers a close view of the grids near the WALL region. Taking the numerical simulation adopting freestream Mach number of 0.768 and the Spalart-Allmaras model as an example, the values of the local Mach number \(m\), the pressure coefficient (\(c_{p}\)), and the temperature are as shown in Table 1, where \(x\) is the \(X\)-axis location, and the computational domain with 8c was chosen for low errors. Further, to be sure of grid independency, three levels of different mesh were applied.