# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 4 Conclusions  
In the present study, the transonic hump model designed by [PERSON] [14] was simulated and compared using RANS, URANS, IDDES, and IDDES-SLA methods, all within the \(k-\omega\) SST turbulent model, and the results were compared with the experimental data from [PERSON] [15]. The half model was used in the present work. And the tunnel wall was replaced by a far field boundary, which, due to the boundary of the shock wave, was far from the  
Figure 14: The spatial distribution of POD mode1, mode3, and mode5 of IDDES and IDDES-SLA flow field.  
Figure 15: The power spectral density (PSD) of Strouhal number (\(S_{t}\)) from 0.01 to 2 at the monitor point (\(x/c=1.0\), \(r/c=0.44\)).  
tunnel wall and the interaction was so weak that it can be ignored. A 16 million structured grid was applied in CFL3D solver at the condition that the Mach number was 0.875 and Reynolds number was 1 million. The grid independence was proved by comparing the field details and some specific data among three resolutions of grids.  
For method comparison, the separation location simulated by RANS and URANS had 10% difference from the experiment, while the IDDES and IDDES-SLA was 4.7% and 1.3%, respectively. For the reattachment location, there was approximately a 5% difference among the four methods. The most accurate results predicted by the IDDES-SLA method indicated that the flow separated at \(x/c=0.61\) and reattached at \(x/c=1.249\).  
Pressure and friction coefficients were analyzed, and the IDDES methods showed better accuracy in the wake region, possibly due to the high modeling of RANS methods. For velocity distribution, the results simulated by RANS and URANS matched the experiment well, except the separated region (\(x/c\) from 0.65 to 1.4). IDDES and IDDES-SLA were basically consistent with the experiment in the whole zone, especially the IDDES-SLA. It can be observed from the Q isosurface at the \(z-x\) section that IDDES methods had better resolution capacity and can capture the fine vortex structures, and the RANS methods only can obtain the macroscopic aerodynamic characteristics accurately.  
The temporal characteristics of URANS, IDDES, and IDDES-SLA were analyzed. It was shown that the flow field displayed time-independence beyond the height of 20% chord length. In the transient region, the variation was not obvious for URANS and the IDDES methods were opposite. The difference between IDDES and IDDES-SLA was basically in the detail data of the flow field. As for macroscopic characteristics or the resolution of vortex, the two methods performed with great prediction capacity.  
The deeper comparison work was conducted for IDDES and IDDES-SLA applying the POD analysis method. The results supposed that the original IDDES method was not sufficient in the resolution of small turbulence structures and would lose some period characteristics in the weak region. After changing the sub-grid method in IDDES-SLA, the high-frequency features would be more credible.  
This study primarily focused on comparing four CFD methods but conducted limited analyses of phenomena such as shock wave vibrations and the influence of periodic characteristics. These aspects will be explored further in future work.  
Conceptualization, M.Z. and Z.T.; methodology, M.Z.; software, Z.T.; validation, M.Z., Z.T. and S.T.; formal analysis, M.Z.; investigation, Z.W.; resources, Z.T.; data curation, M.Z.; writing--original draft preparation, M.Z. and Z.T.; writing--review and editing, M.Z., Z.W. and Z.T.; visualization, Z.T.; supervision, J.L.; project administration, J.L.; funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 92052203).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.