# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 3 Results and Discussion
### Discussion of Different Methods and Flow Field  
The analysis of grid resolution confirmed the accuracy of the medium grid. In this section, different methods were applied for further investigation based on the medium grid. The RANS, URANS, IDDES, and IDDES-SLA were named from Case1 to Case4, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Separation** & **Reattachment** \\  & **Location (x/c)** & **Location (x/c)** \\ \hline Experiment & 0.68 & 1.2 \\ Fine & 0.675 & 1.214 \\ Medium & 0.671 & 1.249 \\ Coarse & 0.664 & 1.324 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The specific location of separation and reattachment for experiment and three types of grids. Figure 7: The distribution of pressure coefficient (**a**) and friction coefficient (**b**) for three types of grids. Table 3 was inherited from [PERSON] [21] and the results of this paper were added. The table displays the specific separation and reattachment location and the difference from the experiment in different CFD methods or turbulent models. The difference in separation between the experimental and simulation results from Case1 to Case4 progressively decreased. The RANS method applied in Case1 produced steady results, different from this; from Case2 to Case4, the unsteady variation was considered and the averaged results in a whole period were used. Thus, Case1 to Case4 was closer to real flow. Theoretically, the accuracy of IDDES and IDDES-SLA was higher than URANS, and the separation results approved this opinion; even the difference was lower to 1.3% through the IDDES-SLA. Not similar to separation, the difference for reattachment was about 5% in Case1 to Case4; these differences were acceptable and close to other references. There were no significant differences in reattachment locations among the four methods. And Case4 was the closest to the experiment. Compared to other references, even though LES methods like HpMusic and WMLES were applied, the results did not meet expectations, particularly at the separation location. But the reattachment locations displayed in Table 3 were similar. These may approve that, although LES has strong analytical ability for flow fields, it is necessarily suitable for the current transonic hump structure. In contrast, the RANS methods (Case1, KE and SA turbulent model applied by [PERSON]) with weaker parsing ability can accurately obtain the separation position. The entire flow field did not separate before the shock wave and, due to the high viscosity of the RANS method, the simulation of this scenario was more accurate. However, the LES method was easily affected by the grid, so it may not be able to simulate well. In the separated flow field after the shock wave, the LES method becomes more accurate in calculating the fine structures of the field, thus obtaining better simulation results. The results of HpMusic model well illustrate this opinion. As for the IDDES method, especially after improving the sub-grid discriminant scale (IDDES-SLA), it could reduce the influence of the grid near the wall and capture sufficiently accurate turbulent structures in the wake region; the simulation results for the entire flow field are highly accurate. Pressure and friction coefficients on the hump surface intersected by \(z-x\) plane are shown in Figure 8a,b. From the perspective of \(C_{p}\), there were no significant differences in the distribution between Case1 and Case4 before the separation occurred. However, in the separation zone after the shock wave, the four methods began to show some differences. The results of Case1 and Case2 were relatively close, while the results of Case3 and Case4 were relatively close. After \(x/c\) exceeded 0.8, the trend of changes in Case3 and Case4 was basically consistent with the experiment, and the \(C_{p}\) at most positions correspond well with the experiment. Although Case1 and Case2 were close to the experimental trend, there was still a significant difference.