## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.2 Convergence of Sampling Period  
Another approach to controlling the total number of snapshots within the DMD analysis window is by adjusting the sampling period, where a longer sampling period results in a greater number of snapshots. In this section, we investigate the influence of the sampling period on DMD convergence criteria, including average and standard deviation of eigenvalue magnitudes \(|\lambda|\) and the relative error in field reconstruction, while maintaining a constant sampling frequency for a given cutoff frequency. Consistent with our previous analyses, two cutoff frequencies, \(f_{c}=100\) Hz and \(f_{c}=400\) Hz, with corresponding sampling frequencies of 1 kHz and 2 kHz, were examined. Figures 14 and 15, illustrate how these parameters vary with different sampling periods for the two low-pass-filtered cases. Note that all sampling periods were normalized by the maximum available period, \(T=4.05\) s.  
As shown in Figure 14, for \(f_{c}=100\) Hz, the eigenvalue magnitude \(|\lambda|\) converges to within 0.0001 of the ideal value of 1.0000 after \(T^{*}\approx 425\), corresponding to 3000 snapshots. In comparison, for \(f_{c}=400\) Hz, convergence is achieved at \(T^{*}\approx 230\), with 3250 snapshots. Despite both cases requiring a similar number of snapshots for convergence, the faster-sampled case (\(f_{c}=400\) Hz), which captures more high-frequency modes, requires a shorter time window. This is expected, as higher frequency modes converge more quickly due to their shorter time scales, providing more information per snapshot. As a result, the DMD algorithm stabilizes eigenvalues and mode shapes over fewer cycles. The dominance of short-wavelength phenomena in higher frequency modes also simplifies their resolution. In contrast, lower frequency modes, associated with larger, slower structures, demand longer time windows for full characterization, resulting in slower convergence. As we will discuss later, extending the sampling time window excessively can potentially destabilize the DMD process. As shown in Figure 15, for \(f_{c}=100\) Hz, the standard deviation of the eigenvalue magnitudes \((\sigma_{\lambda})\) for the pressure signal converges to below 0.01 after approximately \(T^{*}\approx 325\), corresponding to around 2350 snapshots. In contrast, the wall shear stress signal demonstrates less convergence, reaching a value below 0.02 after \(T^{*}=200\). For \(f_{c}=400\) Hz, convergence to below 0.01 occurs at \(T^{*}\approx 200\), with 2880 snapshots observed for both the pressure and wall shear stress signals. Figure 14: Convergence of the average for the magnitude of DMD eigenvalues with respect to sampling period for two conditioned signals: (**left**) \(f_{c}=100\) with \(f_{s}=1\) kHz; (**right**) \(f_{c}=400\) Hz with \(f_{s}=2\) kHz; \(P\) and \(WS\) stand for pressure and friction drag components, respectively. To determine the lowest frequency mode required for accurate DMD analysis, we examine the zoomed-in version of Figure 6, as shown in Figure 16. Based on the low-frequency asymptotic behavior, the lowest resolved mode should have a Strouhal number of approximately \(St\approx 0.002\) or \(f=0.28\) Hz. This indicates that the data sampling period \(T^{*}\) needs to be around 560 to capture at least one cycle of these large-scale modes, which is roughly half the time required to fully characterize the bimodality. However, a detailed investigation into bimodality is beyond the scope of this study and is reserved for future work. However, there are other implications of using a longer \(T_{w}\), particularly during the reconstruction time, which will discussed later in this section. It appears from the above discussion that the convergence of eigenvalues is somewhat independent of the sampling period and is instead a function of the number of snapshots (\(N\)). Figures 17 and 18 illustrate the convergence of the mean and standard deviation of \(|\lambda|\) for the pressure and wall shear stress signals in relation to the number of snapshots.