# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 3 Results and Discussion
### Discussion of Different Methods and Flow Field
When \(x/c\) was greater than \(0.65\), it was basically near the shock wave and separation region. At this point, all cases from Case1 to Case4 showed significant differences from the experimental values, but the overall trend of velocity distribution remained similar. As \(x/c\) increased, in the wake region, there was a significant difference between Case1 and Case2 and the experiment. The prediction of the reverse velocity distribution was significantly greater than the experiment itself, and there was a clear bulge in the figure. However, Case3 and Case4 can well match the experiment. Case4, which specifically used the IDDES-SLA method, had a better fit with the experiment than Case3, which also demonstrated the advantages of improving the sub-grid discriminant scale. At the flow reattachment, the difference between Case1 and Case2 in the experiment gradually decreased. After guiding \(x/c\) to greater than \(1.4\), Case1 to Case4 remained basically consistent with the experimental values. It can be seen that the four methods of RANS, URANS, IDDES, and IDDES-SLA had good agreement with experimental values outside the wake region. However, in the wake region, RANS and URANS methods cannot accurately simulate the surface velocity distribution of the hump. This was mainly due to the high modeling of the RANS method, which cannot accurately analyze the complex flow in the wake region. Although there was a certain gap between the IDDES method and experimental values, the overall resolution of the wake region is significantly better than the RANS method. Figure 10 displays the Q isosurface at the \(z-x\) section. In four cases, the shock wave was obvious and the \(\lambda\)-structure appeared at the root of the shock wave, which was fit to other research. The main reason for the appearance of structure may be due to the appearance of secondary waves after the shock wave, but fusion occurred above the two shock waves, ultimately forming this structure. After the shock wave, all four cases generated separated vortices. The difference is that Case1 and Case2 had larger vortex structures, but Case2 had a slightly smaller volume than Case1. On the other hand, Case3 and Case4 had similar vortex structures at the same location but with significantly smaller volumes. In the separation zone, there were a large number of finely divided vortex structures, which were not present in Case1 and Case2. Comparing Case3 and Case4, there were no significant differences between the two methods in capturing vortex structures in the entire wake region, and both demonstrated the ability to simulate fine flow field structures. Through spatial slicing, it can be seen that RANS and URANS methods have a high degree of modeling for separated flow fields and cannot accurately capture small vortex structures, while IDDES and IDDES-SLA methods had a higher degree of refinement. According to the vortex details in four methods, the RANS and URANS were worse in predicting the field details and the instability state of the shear layer. One of the most obvious phenomena was that the size of the rigid vortex after shock wave in RANS and URANS was bigger than the other two DES methods. Although there still was a small degree of modeling in predicting the \(K-H\) instability phenomena of the shear layer in IDDES and IDDES-SLA, the description of field details was better. Theoretically, the vortex viscosity of the sub-grid model \(\Delta_{SLA}\) was smaller than \(\Delta_{max}\); thus, the detailed structures of IDDES-SLA in the field were characterized well. The results from Case2 to 4 exhibited transient characteristics. Monitoring areas were set up at 5%, 10%, and 20% of chord height positions on the surface of the hump in \(z-x\) section to monitor the changes in Mach number over time in the \(x/c\) range from \(-0.8\) to \(2.0\). Figure 11a shows the monitor position and Figure 11b shows the contour of Mach number of Case2 to Case4.