## 4 Numerical Validation
### Signal Spectrum  
Implementing a modal decomposition algorithm, such as dynamic mode decomposition (DMD), requires acquiring signals that provide sufficient, clean, and undistorted information about turbulent system dynamics. There are only a few studies in the literature that have used data generated by IDDES for modal decomposition. [PERSON] et al. [36] demonstrated that IDDES effectively captures flow dynamics, particularly in the wake region of a surface-mounted cube. However, it remains uncertain whether IDDES, which relies on the RANS model to simulate wall boundaries, can accurately capture the dynamics within boundary layers. As such, to assess the suitability of the IDDES turbulence model for dynamic decomposition, spectral analyses of pressure and velocity at various locations were performed, with results compared to experimental studies from the literature. [PERSON]'s experiment [48], conducted at a Reynolds number of \(Re_{H}=7.7\times 10^{5}\) and a free-stream velocity of 40 m/s, serves as a reference. For this analysis, eight probes (P1-P8) were \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Rear Fascia** & **Body** & **Nose** \\ \hline \(C_{D,p}\) & 0.216 (71\%) & 0.031 (10.5\%) & 0.014 (4.5\%) \\ \(C_{D,f}\) & 0 & 0.033 (10.8\%) & 0.005 (1.5\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Drag coefficient components; \(C_{D,p}\) and \(C_{D,f}\) stand for pressure and friction drag, respectively. Figure 1: The [PERSON] body geometry is split into three distinctive regions. positioned on the rear fascia of the [PERSON] body--with four at the corners and four in the center--to capture surface pressure signals for spectral analysis. Seven additional probes (P9-P15) were placed near the upper edge along the wake of the [PERSON] body, with one probe (P16) near the bottom edge, to record the velocity signal. Figure 2 shows the locations of the pressure and velocity probes used to capture these time histories. Overall, IDDES successfully captured both low- and high-frequency coherent structures. All quasi-steady coherent structures identified by [PERSON] [48], with Strouhal numbers (\(St\)) of 0.08, 0.13, and 0.3, are present in the current numerical model (see Figures 3 and 4). As shown in Figure 3, the two major coherent structures, with \(St\) = 0.08 and \(St\) = 0.13, are clearly prominent in the spectral analysis, each displaying distinct peaks. The structure with \(St\) = 0.08 is associated with global pumping, while the \(St\) = 0.13 structure corresponds to transverse vortex shedding. Figure 3(left) reveals significant differences between the probes on the left (P3) and right (P7) at the mid-height of the vehicle body, highlighting a strong asymmetry in the vortices shed by the body's left and right sides. This asymmetry can be attributed to wake bistability, where two mirrored states alternate due to fluctuations in the base pressure gradients. These fluctuations result in positive pressure values in one state, known as the positive phase, and negative values in the other, referred to as the negative phase. Interestingly, the vortices shed from the top seem to be somewhat in sync with those from the right side, while the vortices from the left and bottom exhibit similar behavior. Capturing the dynamic behavior of bistability requires a time scale three orders of magnitude larger than the wake time scale (\(H/U_{\infty}\)), as reported by [PERSON] et al. [48] and [PERSON] et al. [50], corresponding to approximately 8-10 s, or a non-dimensional time range of \(T^{*}\) = 1000-1400 (\(T^{*}\equiv TU_{\infty}/H\)). Since this study's sampling window only covers one cycle of phase switching, at least 20 cycles would be needed to statistically capture the full bistability behavior [50]. Given the limited sampling window, a detailed analysis of bimodality behavior will not be discussed further  
[PERSON] et al. [35] observed dominant oscillatory structures in the \(St\) = 1.5-4.5 range, which they related to Kelvin-Helmholtz instabilities in shear layers. Additionally, [PERSON] et al.