## 6 Conclusions  
This study investigated the convergence criteria for dynamic mode decomposition (DMD) parameters in relation to sampling frequency and period for flow past an Ahmed body at a high Reynolds number (\(Re_{H}=7.7\times 10^{5}\)). Over a total simulation time of 6 s, 40,000 snapshots were recorded at a 10 kHz sampling rate, with the first 2 s allocated for initialization and the remaining 4 s for capturing the flow dynamics.  
Spectral analysis of pressure and velocity signals indicated that the IDDES model effectively captured the flow dynamics up to 1 KHz corresponding to \(St=7.2\). Approximately 70% of the total drag was attributed to pressure drag on the rear fascia, with energy analyses showing that 90% of the energy in rear-fascia pressure and friction drag was concentrated at 26 Hz (\(St=0.18\)) and 84 Hz (\(St=0.60\)), respectively. The broader energy distribution in friction drag introduced challenges for DMD convergence.  
DMD eigenvalue convergence was achieved with a sampling frequency-to-bandwidth ratio (\(f_{s}/f_{c}\)) of 5-10, requiring approximately 2000 snapshots. The convergence of the reconstruction error was influenced by both sampling frequency and the number of snapshots, with velocity components needing more snapshots than pressure due to their broader energy distributions. Increasing the sampling period further improved the convergence of DMD eigenvalue magnitudes (\(|\lambda|\)), which stabilized around 1.0 after 2000-3000 snapshots. However, reconstruction errors remained unstable for pressure and wall shear stress.  
Mean-subtracted DMD demonstrated enhanced stability, generating modes with zero growth rates and reduced reconstruction errors. However, while mean-subtracted DMD  
Figure 27: Energy levels of selected modes (corresponding to \(St=0.08\), 0.13, 0.16, and 0.23) as a function of sampling periods for the data-centered pressure signal at the rear fascia. The signal was sampled at \(f_{s}=1\) kHz with a cut-off frequency of 100 Hz.  
performed well in identifying coherent structures, it may underestimate dissipative structures. Overall, this study provides valuable insights into DMD convergence and highlights the importance of optimizing sampling parameters for accurate modal decomposition in complex vehicle flow fields.  
This study provides an in-depth analysis of DMD convergence criteria, emphasizing the impact of sampling parameters on the accuracy and stability of modal decomposition. Although results are presented using only one vehicle shape, these findings contribute to a better understanding of DMD's application in fluid dynamics, particularly for complex flow fields around vehicle bodies. It is expected that the findings and observations can apply to other vehicle shapes, provided that the training data for DMD is obtained from high-fidelity, high-accuracy CFD simulations.  
Conceptualization, H.A. and M.U.; Methodology, H.A. and M.U.; Software, H.A.; Validation, H.A. and M.U.; Formal analysis, H.A. and M.U.; Investigation, H.A. and M.U.; Resources, M.U.; Data curation, H.A.; Writing--original draft, H.A. and M.U.; Writing--review & editing, H.A. and M.U.; Visualization, H.A.; Supervision, M.U.; Project administration, M.U.; Funding acquisition, M.U. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The raw data supporting the conclusions of this article can be made available by the authors on request.  
The authors declare no conflicts of interest.