## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.2 Convergence of Sampling Period
When the time window is short, frequency intervals are shorter. Hence, DMD modes might represent a superposition of frequencies within the range defined by \(\Delta f\). For example, with a time window \(T_{w}=0.5\) s, the frequency resolution is 2 Hz, meaning a mode identified at 11 Hz (\(St=0.08\)) may include contributions from the entire frequency band between 10 and 12 Hz. Therefore, each DMD mode can be interpreted as capturing the dominant dynamics within its corresponding frequency bin, blending nearby frequencies that cannot  
Figure 16.1 Probability density function (PDF) of DMD frequency resolution at different \(T_{w}=\{0.5,1,2\}\) s; pressure signal at the rear fascia sampled with \(f_{s}=1\) kHz. be individually resolved. To improve frequency resolution and more accurately capture specific modes, increasing the time window is necessary. However, this must be balanced against the potential introduction of nonphysical growth or decay rates, particularly in turbulent flows where we have a continuous cascade of energy from larger scales to smaller ones. In order to assess the potential for generating spurious modes in the DMD process, the sampling period was increased to \(T_{w}\) = 4 s, while maintaining a constant sampling frequency of 1 kHz. This analysis focused on the modal shapes within a narrow frequency band of 11-12 Hz, providing a frequency resolution \(\Delta f\) of 1 Hz (\(\Delta St\) = 0072). With a 1 kHz sampling rate and a 4 s sampling window, we anticipated detecting four distinct modes within this frequency bandwidth. As expected, four modes were identified at \(St\) = 0.081, 0.082, 0.083, and 0.0860, each exhibiting negative growth rates of \(-\)0.02, \(-\)0.48, \(-\)0.13, and \(-\)0.29, respectively, as shown in Figure 16.2 Notably, the mode closest to the globally reported pumping mode, at \(St\) = 0.081, displayed the smallest decay rate, which was nearly zero; note that, as mentioned earlier multiple times, the global pumping mode has a non-dimensional frequency of 0.08. This suggests that this mode persists for a longer duration and remains coherent over time, in line with previous findings in the literature. Interestingly, two other modes, at \(St\) = 0.082 and 0.083, exhibited very similar shapes but distinct decay rates, hinting at a possible duality in the system's behavior. While the mode at \(St\) = 0.082 decays much faster, the mode at \(St\) = 0.083 persists longer, suggesting that they may represent different physical phenomena despite their similar spatial structures. A gradual increase in decay rates was observed across the modes, with the exception of the mode at \(St\) = 0.083, which behaves as an outlier. One of these two modes raises concerns about potential DMD divergence, it is tempting to think of this as a spurious mode as modes with higher decay or growth rates exacerbate instability. Figure 16.3 Mode duality in DMD in the case of decomposition of the filtered pressure signal at the rear fascia with \(f_{c}\) = 100 Hz, \(f_{s}\) = 1 kHz, and \(T_{w}\) = 4 s within the frequency range of \(f\) = 11–12 Hz (\(St\) = 0.08–0.09); GR stands for growth rate. (**a–d**) represent four captured modes within the specified frequency range. However, in the context of [PERSON] and [PERSON] [60] (p. 259), this is not unexpected and may represent the real flow physics. [PERSON] and [PERSON] [60] wrote: \"The Fourier Transform of the velocity field is a decomposition into waves of different wavelengths; each move is associated with a single Fourier coefficient. An eddy, however, is associated with many Fourier coefficients and the phase relations among them. Fourier transforms are used for convenience (spectra can be measured conveniently) and more sophisticated transforms are needed if one wants to decompose the fields into eddies instead of waves\".