## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.2 Convergence of Sampling Period
Interestingly, a time period of \(T^{*}=250\) corresponds to approximately half-cycle of the slowest dominant mode needed to resolve the flow. This, as discussed earlier, is sufficient to resolve based on the arguments of [PERSON] et al. [33], who demonstrated that DMD can capture modes with as little as one or half a cycle. Thus, this time period should theoretically be adequate to fully resolve the flow field using DMD. As shown in Equation (6), one of the key DMD output parameters is \(\Phi\), which represents the mode shape. To assess the effect of the sampling period on modal shape resolution, Figure 21 presents mode shapes obtained from rear fascia pressure data at a non-dimensional frequency, \(St=0.08\), for four different sampling periods \(T^{*}\) = 70, 140, 280, and 560. The data were sampled at 1 kHz with a low-pass cut-off frequency \(F_{c}=100\) Hz. The frequency \(St=0.08\) was chosen due to its significance in representing a coherent structure associated with bubble pumping [55]. The mode shapes show general consistency across all sampling periods, displaying two clear positive-negative regions. However, the boundaries between these regions become more distinct as the sampling window increases. It appears that spatial gradients in the mode intensity, while the energy levels decrease with  
Figure 19: DMD reconstruction relative error 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, respectively. longer sampling periods. This aligns with the convergence observed in Figure 14, where the growth rate stabilizes for sampling periods \(T^{*}=200\)-\(250\). Larger sampling periods enhance frequency resolution, improving the ability to distinguish temporal dynamics. Apart from convergence, it is essential that the growth rate (\(GR\)), as obtained from the complex-valued variable \(\omega_{k}\) in Equation (6), and corresponding mode energy levels, reach a physically interpretable state. For this reason, four significant coherent structures with \(St=0.08\), \(0.13\), \(0.17\), and \(0.23\) were selected for analysis. Coherent structures, in general, are expected to maintain their energy over extended periods; thus, it is reasonable to assume that the computed growth rate should approach zero as closely as possible. Figure 22 presents the growth rate and mode energy as a function of the sampling window. The results align well with the convergence criteria established in the sampling period study (as seen in Figure 14), with both parameters becoming stable after \(T^{*}=250\) (approximately \(T=1.8\) s), which corresponds to nearly \(20\) cycles of the slowest mode (\(St=0.08\)). This point also marks the onset of DMD divergence, as indicated in Figure 19. The DMD exhibits a promising trend where the growth rate of all modes approaches zero, suggesting that the modes conserve energy for a longer duration. This behavior was similarly observed in the study by [PERSON] et al. [56], where periodic components in a jet flow exhibited temporal modes with nearly zero growth rate. A growth rate near zero indicates that the mode is highly periodic. Additionally, The plot on the right in Figure 22 shows that the modes achieve consistent energy levels, with each mode containing approximately 1-2% of the energy of the mean mode. Slower modes tend to have slightly higher energy levels. It is important to note that the energy of each mode is calculated based on the RMS of the time dynamics of the corresponding mode [29]. Figure 20: Variation of reconstruction relative error with respect to the sampling period for pressure (‘P’) and wall shear stress (‘TW’) signals filtered at \(f_{c}=100\) Hz and \(f_{c}=400\) Hz. Cases 1–5 represent five different initial times, \(t_{0}^{*}=\{15,25,40,55,70\}\). The spatial modal analysis of DMD results has demonstrated its ability to distinguish and extract valuable coherent structures from chaotic and random patterns.