## 4 Conclusions  
In this study, the DDES method based on SST K-\(\omega\) turbulence model and overlapping mesh technique was used to numerically simulate the aerodynamic characteristics of a high-speed maglev train passing through a tunnel. The main analysis involved examining the effect of Helmholtz resonators on the pressure amplitude at the tunnel wall and change in the micro-pressure wave amplitude at the tunnel entrance during the operation of high-speed maglev trains. Specifically, the correlation between the increase in the wall pressure amplitude, decrease in the micro-pressure wave amplitude, and change in the Helmholtz resonator arrangement was emphasized. Based on the research in this study, the following main conclusions were obtained:  
1. In this study, the initial compressional wave frequency generated by the high-speed maglev train passing through the tunnel corresponded to 15 Hz in the main frequency. Furthermore, the resonant frequency of the Helmholtz resonator was 14.63 Hz, and the upper and lower limits of the resonant band were 13.76 Hz and 16.13 Hz, respectively. 2. The Helmholtz resonator effectively reduced the micro-pressure wave amplitude at the tunnel exit by reducing the initial compressional gradient. In the conventional tunnel, the micro-pressure wave amplitude at 20 m and 50 m from the tunnel exit was 641.26 Pa and 292.22 Pa, respectively. In the 94-cavity new tunnel with the Helmholtz resonator, the micro-pressure wave amplitude at 20 m and 50 m from the tunnel exit was 436.91 Pa and 195.12 Pa, respectively. The Helmholtz resonator provided 31.87% and 33.23% relief of the micro-pressure wave amplitude at the two locations. However, the presence of the Helmholtz resonator increased the tunnel blockage ratio by 18.9%, resulting in an 18.62% increase in the tunnel wall pressure. 3. The 72-cavity new tunnel 1 is an optimized solution for the 94-cavity new tunnel, which exhibited 29.28% and 30.65% relief of micro-pressure waves at 20-m and 50-m measurement points at the tunnel exit, respectively, and only a 4.36% increase in tunnel wall pressure. The 72-cavity new tunnel 1 is an optimized solution for the 94-cavity new tunnel and ensured that the micro-pressure waves are relieved while maintaining an increase in the train surface pressure within acceptable limits and in line with engineering requirements. 4. In the 72-cavity new tunnel 1 scheme, the micro-pressure wave amplitude at the exit of the tunnel was approximately inversely proportional to the 0.78 th power of the distance from the exit end of the tunnel, which approximates the fitted curve of the original tunnel scheme. This indicates that the presence of the Helmholtz resonator only reduces the initial compressional wave gradient and micro-pressure wave amplitude, and it does not affect the formation mechanism. Conceptualization, M.-Z.Y. and D.-Q.L. ; Methodology, T.-T.L. ; Formal analysis, P.Y. and D.-Q.L. ; Resources, M.-Z.Y. ; Writing--original draft preparation, D.-Q.L. ; Writing--review and editing, D.-Q.L. and S.Z. ; Supervision, S.Z. ; Funding acquisition, M.-Z.Y. All authors have read and agreed to the published version of the manuscript. The research was supported by the Graduate School-Enterprise Cooperation Project of Central South University, grant number 2022 XQLH066. Not applicable. Data Availability Statement: Data is contained within the article. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Tunnel Structure** & **Fitting Curves** & **Correlation Coefficient** \\ \hline Conventional tunnel & y =5657.39x\({}^{-0.74}\) & 0.996 \\
72-cavity new tunnel 1 & y =4471.37x\({}^{-0.78}\) & 0.997 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Micro-pressure wave fitting curves for different tunnel structures. **Acknowledgments:** The authors acknowledge the China CRRC Science and Technology Research and Development Program (approval no. 2021 CCA074), the National Numerical Wind Tunnel Project, and the High-Performance Computing Center of Central South University for providing computing resources.