## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.1 Convergence of Sampling Frequency
Figure 8 illustrates and compares the power spectral density (PSD) of both the raw and filtered signals (at \(f_{c}=100\) Hz and \(f_{c}=400\) Hz) for the pressure signal at the rear fascia and body friction. Figures 9-11 illustrate the effect of different sampling frequencies on eigenvalues averaged magnitude (\(|\lambda|\)), eigenvalues' standard deviation (\(\sigma_{\lambda}\)), and the reconstruction error of drag force, respectively. It is worth mentioning that \(|\lambda|\) is averaged over all calculated modes. There are two signals conditioned with cutoff frequencies (\(f_{c}\)) of 100 Hz and 400 Hz and a total window time of \(T_{w}=2\) s. As shown in Figure 9. increasing the sampling frequency (\(f_{s}\)) causes the mean of \(|\lambda|\) to converge toward the value near one. The magnitude of \(\lambda\) closer to one can cause more stability in the DMD process. When \(f_{s}\) is approximately ten times the cutoff frequency in the case of \(f_{c}=100\) Hz, and two and half times in the case of \(f_{c}=400\) Hz, \(|\lambda|\) begins to converge and becomes stable. The order of magnitude of \(|\lambda|\) has a direct relation with the total snapshot number. For both signals,  
with cutoff frequencies (\(f_{c}\)) of 100 and 400 Hz, \(\lambda\) converges to three decimal places after 2000 snapshots. This means the energy content of the signal in bandwidth within 100 and 400 Hz has very little impact on the order of magnitude of \(|\lambda|\) and it is highly dependent on the total number of snapshots. Figure 10 demonstrates that increasing the sampling frequency (\(f_{s}\)) leads to the convergence of \(\sigma_{\lambda}\) toward a value close to zero. The closer \(\sigma_{\lambda}\) is to zero, the greater the stability achieved in the DMD process. When \(f_{s}\) is approximately ten times the cut-off frequency for \(f_{c}=100\) Hz and three times for \(f_{c}=400\) Hz, \(\sigma_{\lambda}\) begins to stabilize. The magnitude of \(\sigma_{\lambda}\) is directly related to the total number of snapshots. For both signals with cut-off frequencies of 100 Hz and 400 Hz, \(\lambda\) converges after 2000 snapshots. This indicates that the energy content of the signal within the bandwidth of 100 to 400 Hz has minimal impact on the standard deviation of \(|\lambda|\), which is primarily influenced by the total number of snapshots. Figure 8: Comparison of the power spectral density (PSD) for the raw signal and filtered signals with cutoff frequencies of \(f_{c}=100\) Hz and \(f_{c}=400\) Hz for: (**left**) the rear fascia pressure signal and (**right**) body surface friction. Figure 9: Convergence of the average 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. Figure 11, on the left, illustrates the DMD reconstruction relative error as a function of sampling frequency. For this matter, the time series of pressure on the rear fascia and body friction were reconstructed using DMD at varying sampling frequencies (see Equation (6)), and then the pressure and friction drag were computed from these reconstructed signals. The relative error was calculated by comparing these values to the corresponding values from the true CFD solution. The error initially decreases but rises sharply to magnitudes of \(10^{2}\) and \(10^{3}\) near \(f_{s}/f_{c}=2\)-\(3\) for surface pressure. To achieve errors below 1%, a sampling ratio of \(f_{s}/f_{c}=4\)-\(5\) is required for pressure, and \(f_{s}/f_{c}=10\) for wall shear stress. This trend persists for signals filtered at \(f_{c}=400\) Hz, where the error drops below 1% around \(f_{s}=f_{c}=5\) for both pressure and friction signals. Overall, surface pressure converges more quickly than wall shear stress; this is probably due to the fact that the pressure signal has a much narrower band in terms of modes compared to the inertia terms.