# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 3 Results and Discussion
### Discussion of Different Methods and Flow Field
It can be seen that, in the wake region at the 5% C monitoring position, the Mach number of Case2 fluctuated less with time, while both Case3 and Case4 exhibited strong unsteady fluctuations. At the 10%C monitoring location, Case2 showed no significant fluctuations over time, while the fluctuations in the wake regions of Case3 and Case4 decreased. Near the 20%C, the Mach number fluctuations over time for all three cases were not significant. It can be seen that, at least from the 20%C height outward, the entire flow field mainly exhibited steady characteristics. At the monitoring positions of 10%C and 20%C, there was a high Mach number region near \(x/c=0.8\) to \(1.0\) in Case2, but the region  
Figure 9: The velocity distribution along the z-axis on the hump surface (\(x/c\) from 0.05 to 0.5 (a), 0.55 to 1 (b) and 1.05 to 1.5 (c)). did not appear at the same position in Case3 and Case4. This region was compressed within a very small range near \(x/c=0.8\), which may be mainly due to the fact that, although the URANS method has transient characteristics, it cannot accurately simulate the fine flow field structure in the wake region. This also indicated that the IDDES method performs well in simulating the temporal and spatial characteristics of the flow field. Figure 11: The monitor position above the hump surface in the \(z-x\) section is displayed in **(a)**. The contour of Mach number of Case2 to Case4 is shown in **(b)**. Figure 10: The Q isosurface at the \(z-x\) section. The rigid vortex structures after shock wave were in the red square. For further comparison, the contour of time-averaged streamwise velocity (\(\rm u/u_{ref}\)), radial velocity(\(\rm v/u_{ref}\)), and Reynolds shear stress (\(\rm u^{\prime}v^{\prime}\)) for Case3 and Case4 are displayed in Figure 12. The difference between Case3 and Case4 in terms of streamwise velocity was mainly reflected in details, and there was not a significant difference in macro aspects. The shock structure shown in Case4 is more pronounced and enlarged, and the overall streamwise velocity in the wake region is slightly lower than that in Case3. In terms of radial velocity, there were significant differences between Case3 and Case4, but the range of the wake region remained basically the same. In terms of Reynolds stress, the macroscopic distribution of Case3 and Case4 was basically the same, but Case4 had a clearer boundary at the shock wave position. From the above comparison, it can be seen that there was no significant difference in the macroscopic distribution simulation results of flow field characteristics between IDDES and IDDES-SLA methods. The difference mainly lay in some details of the flow field, which was mainly due to the improvement of the original IDDES sub-grid discrimination method. POD method was applied to further compare the flow field resolution capability of IDDES and IDDES-SLA. The dimensionless density was used to perform the POD analysis of 500 flow field snapshots when the flow variation periods were stable. The gap of snapshots was 0.0001 s. Figure 13 showed the POD energy accumulation curve from mode 0 to mode 250 of two methods. It was obvious that the accumulation rate was faster for the IDDES method, where even the energy percent was close to 50% in the first 50 modes, while the IDDES-SLA was about 30%. It could be seen that the curves grow faster in the low modes of IDDES; however, the growth was moderate in the whole 250 modes of IDDES-SLA. This suggested that the change in the sub-grid determination method further improved the ability of the IDDES method to portray the details of the flow field and attenuated the degree of method modeling. The characteristics of low modes would be  
Figure 12: The contour of time-averaged streamwise velocity (\(\rm u/u_{ref}\)), radial velocity (\(\rm v/u_{ref}\)), and Reynolds shear stress (\(\rm u^{\prime}v^{\prime}\)) for Case3 and Case4. displayed and recognized more accurately in the IDDES-SLA.