# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 3 Results and Discussion
### Discussion of Different Methods and Flow Field
It was not until x/c exceeded 1.6 that the experimental results gradually became consistent. This may be due to the fact that Case1 uses the RANS method, which cannot simulate the separated flow well. Case2 added transient factors on \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**CaseID**} & **Separation Location** & **Difference From** & **Reattachment Location** & **Difference From** \\  & **(v/c)** & **Experiment (\%)** & **(v/c)** & **Experiment (\%)** \\ \hline Experiment & 0.68 & / & 1.2 & / \\ CASE1: RANS & 0.61 & 10.2 & 1.137 & 5.3 \\ CASE2: URANS & 0.611 & 10.1 & 1.253 & 4.4 \\ CASE3: IDDES & 0.712 & 4.7 & 1.275 & 6.3 \\ CASE4: IDDES-SLA & 0.671 & 1.3 & 1.249 & 4.1 \\ HpMusic (\(p=2\)) & 0.54 & 22.9 & 1.23 & 2.5 \\ [PERSON] et al., SA & 0.72 & 5.7 & / & / \\ [PERSON] et al., KE & 0.7 & 2.9 & / & / \\ [PERSON] et al., WMLES & 0.9 & 27.8 & / & / \\ [PERSON] et al., Medium & 0.76 & 11.1 & 1.14 & 5.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Separation and reattachment location. the basis of Case1, which could improve the accuracy of simulating the flow in the wake region to a certain extent. Case3 and Case4 used RANS method near the wall and LES method far away from the wall in the separation zone, so the simulation of the flow field structure in the wake zone was more accurate and closer to real flow. Therefore, the average results matched the experimental results more closely. For \(C_{f}\), similar to the \(C_{p}\) results, Case1 and Case2 based on RANS and Case3 and Case4 based on IDDES methods remained consistent but, overall, there were significant differences between these two methods. In the range of \(x/c\) from 0.6 to 0.8, Case1 and Case2 showed a clear trend of first decreasing and then increasing, while Case3 and Case4 did not exhibit this characteristic, which may be due to the method itself. As \(x/c\) increased, although the calculation results of the four methods were consistent with the experimental trend, the errors became progressively larger. Among them, the errors of Case3 and Case4 are smaller than those of Case1 and Case2, indicating that the IDDES method has stronger simulation ability for separated flow than the RANS method. Based on other references, current simulations of \(C_{f}\), regardless of the method used, have significant differences from experiments, and this is not an isolated case. The distribution of \(C_{f}\) on the surface of Case1 was described in Figure 8c, and the separation position and reattachment position were clearly observed. These two positions had certain changes along the circumference, especially the reattachment position. This may be mainly due to the unsteady characteristics of the entire flow along the circumferential direction reflected in Case1, and the division of grid blocks may also have a certain impact on the results. Figure 8: The distribution of pressure coefficient (**a**) and friction coefficient (**b**) from Case1 to Case4, and the \(C_{f}\) distribution on the surface of transonic hump of Case1 (**c**). The velocity distribution along the z-axis on the hump surface from \(x/c=0.05\) to \(1.5\) is shown in Figure 9. The \(r_{Surf}\) was the distance from the \(z\) co-ordinate to the hump surface in the \(z-x\) plane, where the \(r_{Surf}\) of the hump surface was \(0\). From the whole view of the velocity distribution, Case1 to Case4 matched all the experimental values well when the \(r_{Surf}/c\) was greater than \(0.08\), but there was a certain difference in the range of \(0\) to \(0.08\). Especially between the separation point and the reattachment point, the difference was very obvious. Within the range of \(x/c\) from \(0.055\) to \(0.6\), the velocity distribution of Case1 and Case2 remained almost unchanged but, as \(x/c\) increased, there was a significant difference between the velocity distribution near the wall and the experimental values. Case3 and Case4 were in good agreement with experimental values within this range.