## 5 Results and Discussion  
Implementing 3D dynamic mode decomposition (DMD) across the entire domain could provide deep insights into the flow dynamics and help explain how different flow regions--such as the wake, far wake, boundary layers, and shear layers--contribute to fluid fluctuations and their energy levels relative to the mean flow. However, storing and processing this 3D flow field data presents significant challenges. In this study, a transient simulation was carried out over 6 s or 833 large eddy turnover times (LETOTs) with a time-step of \(\Delta t=10^{-4}\) s (\(\Delta t^{*}=0.0139\)) for a case involving 27 million elements. Storing four seconds of data for DMD analysis would result in 40,000 snapshots, and storing all the time snapshots and grid points for the primary parameters (\(u\), \(v\), \(w\), and \(P\)) would require 36 TB of storage (9 TB per parameter). This also requires substantial RAM and extensive processing time beyond the capacity of our HPC system. Therefore, optimizing the amount of data to retain sufficient information about the system's dynamics is essential for constructing an efficient decomposition model.  
Dynamic mode decomposition (DMD) can potentially generate random and spurious modes. Extracting coherent and meaningful data from a high-dimensional, nonlinear  
Figure 4: Power spectral density of the velocity signals sampled with \(f_{s}\) = 2 kHz and \(T_{w}\) = 2.5 s: far wake, upper probes only (**left**), and close to the body upper and lower (**right**).  
Figure 3: Power spectral density of the pressure signals sampled at \(f_{s}\) = 2 kHz and \(T_{w}\) = 2.5 s: corner probes (**right**), center probes (**left**).  
system is challenging, which increases the likelihood of identifying spurious modes [54]. Furthermore, DMD spatial-temporal modes are not necessarily unique in space and orthogonal in time. [PERSON] et al. [33] showed that DMD produces a unique decomposition if and only if all modes are linearly independent and distinct, a somewhat unlikely condition for turbulent flows. DMD may also exhibit sensitivity to oversampling, where increasing the sampling frequency and time window can cause DMD to diverge or amplify both the intensity and number of spurious modes. This sensitivity can make DMD outcomes difficult to interpret and less reliable for flow control and for the development of reduced-order models, particularly in complex geometries at high Reynolds numbers. Finally, the total number of time snapshots used to extract temporal features can significantly impact the DMD outcome and its accuracy in providing the best high-dimensional linear fit.  
In the following sections, we will first examine how sensitive the drag force on the surface body is to sampling frequency and period. Ideally, for a statistically converged value, the variation should be within the error bounds if the sampling window is sufficiently large. Given that a time window of 160 s (\(T^{*}\) = 2.2 \(\times 10^{4}\)) is needed to statistically capture the impact of wake bistability, we expect some variance in the mean drag when averaged over a much shorter period, such as 2 s. Next, we will explore DMD convergence by analyzing how its parameters, such as eigenvalues, respond to changes in data sampling frequency and period. We will also evaluate DMD's consistency in reconstructing the flow field when the evolution matrix is determined using data with different sampling frequencies and periods. Finally, we will study the uniqueness of mode shapes, along with their growth or decay rates and energy levels, in detail. This analysis will help us develop a framework for decomposing the flow field that is computationally efficient, robust, and physically meaningful.