## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.1 Convergence of Sampling Frequency
These findings are in line with the work of [PERSON] et al. [32], which successfully captured the full pressure signal spectrum on the surface of an Ahmed body with a 35-degree slant using 2000 snapshots over 0.2 s at a sampling rate of 10 kHz. This is consistent with our results where we found that DMD requires a minimum of 2000 snapshots for eigenvalue convergence. While their setup used a shorter time window and higher sampling frequency, we observed similar convergence for longer time periods and bandwidths of 100 Hz and 400 Hz. These findings highlight that DMD convergence requires a sufficiently high sampling frequency relative to the cutoff frequency to resolve individual time scales and energy distributions across the bandwidth. To further illustrate this, spectral analyses (using fast Fourier transform of FFT) of the DMD-reconstructed pressure and friction drag signals are compared to the spectra obtained from the low-pass-filtered CFD solution data in Figures 12 and 13 for the two cut-off frequencies. These figures demonstrate the effect of sampling on the signals that were conditioned using low-pass filters at \(f_{c}=100\) Hz and \(f_{c}=400\) Hz but sampled at various frequencies. In Figure 12a, as the sampling frequency increases, the power spectral density (PSD) of the DMD-reconstructed signal progressively aligns with that of the true CFD signal. When the sampling frequency (\(f_{s}\)) is equal to twice the bandwidth (100 Hz), the reconstructed signal exhibits significant deviations, with a relative error on the order of \(10^{3}\). As \(f_{s}\) increases, the relative error decreases, reaching approximately 2% at \(f_{s}=400\) Hz and dropping below 1% at \(f_{s}=1\) kHz. A similar trend is observed for wall shear stress,  
Figure 10: Convergence of the standard deviation for the magnitude of DMD eigenvalues with respect to sampling resolution with a 2-s sampling window (\(T_{w}=2\) s) for two filtered signals: (**left**) \(f_{c}=100\) Hz, (**right**) \(f_{c}=400\) Hz; note that ’P’ and ’WS’ refer to pressure and friction, respectively. as shown in Figure 12b, where the PSD spectrum approaches the original as the sampling frequency increases. The relative error for the wall shear stress starts at 150% at \(f_{s}=200\) Hz, rises to around \(10^{3}\) at \(f_{s}=400\) Hz, and decreases to about 3% at \(f_{s}=1\) kHz. The same process was applied to the pressure and friction signals with a bandwidth of \(f_{c}=400\) Hz. As seen in Figure 13, increasing the sampling frequency significantly reduces the relative reconstruction error. For instance, at \(f_{s}=1\) kHz, the friction signal shows a large deviation with an error on the order of \(10^{4}\). However, increasing the sampling frequency to 5 kHz (eight times the bandwidth) reduces the error to below 1%, achieving a nearly perfect match in the spectrum. [PERSON] et al. [34] investigated DMD convergence criteria for free-shear flow around an infinite square prism at \(Re=\) 22,000. They found that sampling frequency convergence is mode-specific, with each mode requiring a different number of frames to converge. They proposed a rule of thumb: approximately 15 frames are needed per mode, based on its individual oscillatory frequency. In terms of energy levels, higher-frequency modes require fewer frames for accurate capture. Consequently, as the frequency bandwidth increases from 100 Hz to 400 Hz, the required resolution to capture these modes decreases slightly, particularly for pressure drag. Their results indicate that DMD convergence depends heavily on the energy distribution across frequency bandwidths, although the structure of individual high-frequency, high-energy modes can significantly affect DMD performance. Our study reaches a similar conclusion but offers a less restrictive guideline, particularly for high-Reynolds-number flows with higher-frequency structures. For simulations resolving structures up to 2 kHz, five frames per mode should be sufficient, and this can likely be extended safely to cases where modes up to 10 kHz are present.