# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 3 Results and Discussion  
### Grid Resolution and Field Validation  
The Q value was utilized to characterize the flow field details in the wake region [34], which was determined as follows:  
\[Q_{c}=-\frac{1}{2}\Bigg{[}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{2}+ \bigg{(}\frac{\partial v}{\partial y}\bigg{)}^{2}+\bigg{(}\frac{\partial w}{ \partial z}\bigg{)}^{2}\Bigg{]}-\bigg{(}\frac{\partial u}{\partial y}\bigg{)} \bigg{(}\frac{\partial v}{\partial x}\bigg{)}-\bigg{(}\frac{\partial u}{ \partial z}\bigg{)}\bigg{(}\frac{\partial w}{\partial y}\bigg{)} \tag{15}\]  
where \(x\), \(y\), and \(z\) represent the co-ordinates of the flow in the streamwise, lateral, and wall-normal directions, and u, \(v\), and w represent the velocity along the \(x\), \(y\), and \(z\) directions, respectively. For visualizing the vortex intensity, the Q value was colored by Mach number for three kinds of grid, as shown in Figure 5.  
The results of the IDDES-SLA method were visualized to validate the flow field characteristics. It is evident that the capability for vortex analysis improved with increasing grid resolution. A greater number of small-scale vortices were captured in the medium and fine grids, while the vortex structure was less defined in the coarse grid, potentially resulting in the loss of turbulent details and inaccuracies in aerodynamic characteristics. From the Q isosurface figures, the scales of vortex caught by the cases of medium grid and fine grid were similar, which qualitatively proved that the medium grid is enough to analyze the flow field. As for the Mach number, few differences could be found in the fine  
Figure 5: The Q isosurface of half model transient field colored by Mach number for coarse grid (**a**), medium grid (**b**), and fine grid (**c**). grid and medium grid. In the coarse grid, the distribution of the Mach number after shock wave was not symmetric and significantly different to the other two grids. A monitor point was set on the surface of the hump, as shown in Figure 6a. The point was located directly above the original point in the \(z-x\) plane, and the fluctuations of dimensionless density are plotted in Figure 6b. The dimensionless density was determined as \(\rho=\rho^{*}/\rho_{\infty}^{*}\), where the \(\rho^{*}\) was local dimensional density and \(\rho_{\infty}^{*}\) was dimensional density of inlet flow at the far field. It can be discovered visibly that density vibration at the monitor point has clear periodicity. The amplitude about the coarse grid was a bit lower than the other two grids in the whole period; in opposite, the fine grid and medium grid possessed a similar amplitude. Figure 6c shows the velocity distribution along the z-axis. The start position was across the monitor point and at the location \(z/c=0.4\), where \(z/c\) is the ratio of the z co-ordinate value and chord length \(c\). The experimental data were published by Sandia Wind Tunnel [15]. The results simulated by the three grid types exhibited good agreement with the experimental data, particularly when \(z/c>0.5\), in which \(u/u_{\infty}>0.9\) and the flow was beyond the boundary layer. In the boundary layer, there were minor discrepancies compared to the experimental results in all three grids. In particular, the results produced by coarse grids fit the experimental data poorly, which would obtain SWBLIs inaccurately and deeply influence the flow field in the wake region. For both the fine and medium grids, the most noticeable and acceptable difference with the experiment was in the area with the highest rate of velocity changes. In other regions, both grids simulated the flow accurately. Overall, the monitor point was situated in front of the shock wave, and the flow characteristics at this location significantly influenced the simulation results due to the interaction between the shock and the boundary layer. The cases of fine grid and medium grid performed well at this point. Figure 6: The position of monitor point on the surface of hump (**a**).