Impact of Key DMD Parameters on Modal Analysis of High-Reynolds-Number Flow Around an Idealized Ground Vehicle

Abstract

This study provides a detailed analysis of the convergence criteria for dynamic mode decomposition (DMD) parameters, with a focus on sampling frequency and period in high-Reynolds-number flows. The analysis is based on flow over an idealized road vehicle, the Ahmed body ($Re=7.7\times {10}^5$), using computational fluid dynamics (CFD) data from improved delayed detached eddy simulation (IDDES). The pressure and velocity spectrum analysis validated IDDES’s ability to capture system dynamics, consistent with existing studies. For a comprehensive understanding of the contributions of different components of the circle, the Ahmed body was divided into three regions: (a) front; (b) side, lower, and upper surfaces; and (c) rear fascia. Both pressure and skin-friction drag were analyzed in terms of frequency spectra and cumulative energy. Key findings show that a 90% contribution to the pressure drag comes from modes with a frequency of less than 26 Hz ($St$ = 0.187), while the friction drag requires 84 Hz ($St$ = 0.604) for similar energy capture. This study highlights the significance of accounting for intermittency and non-stationary behavior in turbulent flows for DMD convergence. A minimum of 3000 snapshots is necessary for the convergence of DMD eigenvalues, and sampling frequency ratios between 5 and 10 are needed to achieve a reconstruction error of less than 1%. The sampling period’s convergence showed that $T^{*}=250$ (equivalent to 20 cycles of the slowest coherent structures) stabilizes coherent mode shapes and energy levels. Beyond this, DMD may become unstable. Additionally, mean subtraction was found to improve DMD stability. These results offer critical insights into the effective application of DMD in analyzing complex vehicle flow fields.

1. Introduction

Turbulence is characterized by chaotic fluid motion as exhibited by apparently random fluctuations in pressure and velocity and represents a highly non-linear dynamic system with variations across both temporal and spatial domains. Thus, modal decomposition of the flow field is crucial for researchers to better understand the nature and dynamics of turbulent flows, particularly the spatiotemporal evolution of coherent structures within the flow, which are responsible for the generation of turbulence [1]. Over the last century, numerous analytical approaches and mathematical tools have been developed to gain deeper insights into the physics of turbulence. Among these, Fourier analysis stands out as one of the oldest and most frequently used methods for the modal decomposition of fluid flows. The literature on this approach is extensive, with numerous papers exploring its applications. To illustrate the longevity and continued relevance of this method, consider two studies that delineate the timeline: Taylor’s 1938 paper [2], which utilized Fourier analysis for the time and length scale analysis of flow past a prism, and the more recent work by Klotz in 2021 [3], which investigated the dynamics of large and small-scale motions in plane Couette–Poiseuille flow.

Proper orthogonal decomposition (POD) is another widely adopted decomposition technique that was introduced to the fluid dynamics community by Lumley [4] in 1967. POD decomposes the flow field into spatially orthogonal modes, ordered by energy level, each with a time history characterized by fluctuating components and a frequency spectrum. Since its introduction, POD has been extensively applied in various contexts, including identifying coherent structures [5], controlling flow by manipulating dominant flow structures [6,7], and developing reduced-order models for complex, computationally intensive flow problems [8,9].

The other later-developed yet very popular approach involves the use of “Wavelet Transform” (WT), originally developed by Morlet in 1980 for seismic data analysis [10]. WTs are mathematical tools used to decompose signals into their constituent frequencies, offering both time and frequency localization, which makes them particularly useful for analyzing non-stationary turbulent fluid flows. The use of wavelets in turbulence was first reported by Farge and Rabreau [11] in their work to analyze a two-dimensional homogeneous turbulent flow field obtained from numerical simulations. Interested readers are directed to the 1992 review paper by Marie Farge [12], which highlights the potential of wavelet transforms (WTs) in fluid dynamics. The paper demonstrates their effectiveness in identifying coherent structures within turbulent flows and discusses some of the initial pioneering works. Papers from Farge’s group laid the groundwork for subsequent research, which has expanded the application of WTs in fluid flow modal decomposition. Recent studies have further advanced the application of wavelet-based techniques to analyze complex flow patterns and enhance the understanding of flow dynamics in various engineering applications (cf. [13,14,15] to cite but a few). These contemporary works demonstrate the ongoing evolution and usefulness of the WT approach in extracting meaningful insights from fluid flow data, particularly in addressing the challenges posed by the multi-scale nature of turbulence, turbulence modeling, data-driven turbulence, and the development of reduced order modeling (ROM). In addition, there are efforts that combine the features of both POD and WT to further our understanding of the dynamics of turbulence, as well as applications in the data-driven turbulence approaches. For example, Zheng et al. [16] developed a hybrid decomposition technique to decompose the multi-scale structures of a semi-cylinder wake using a 2D orthogonal wavelet first and then further analyzed the decomposed structures with POD to extract distinctive flow patterns from the dominating POD modes of intermediate-scale and small-scale structures.

In recent years, a dimensional reduction algorithm, developed by Schmid and coworkers [17], called dynamic mode decomposition(DMD), has emerged as a popular alternative to the modal decomposition techniques discussed earlier. DMD decomposes the time series of flow field data into spatial modes with dynamic time histories that exhibit decaying or growing amplitudes with fixed oscillatory frequencies. Unlike POD, DMD modes are not necessarily orthogonal in the spatial domain, although their individual time dynamics might. The spatial–temporal modal decomposition with individual temporal characteristics has given DMD a unique advantage over other decomposition techniques, leading to its widespread application in both fundamental and applied fluid dynamics research [18,19,20,21,22]. These advantages, alongside its inherent challenges, make DMD the primary decomposition technique employed in this study. The DMD decomposition is closely related to Koopman analysis and has led to numerous extensions and applications across fluid dynamics and other fields like video surveillance, epidemiology, neurobiology, and financial engineering. Since its first deployment in fluid dynamics research, DMD has evolved into various extensions, generalizations, and improvements. These efforts have resulted in an extensive list of published research in the field of fluid dynamics, and for brevity, we will limit our discussions to the works that relate very closely to ours, that is, those involving primarily ground vehicles. However, interested readers are directed to the recently published review paper by Schmid [23], which provides an in-depth discussion on the practical aspects of DMD and its variants and their effectiveness as quantitative tools for analyzing complex fluid flows.

The dynamics of turbulent flow play a crucial role in determining the characteristics of fluid flows around ground vehicles. Ground vehicles experience three aerodynamic forces (drag, lift, side) and three moments (rolling, pitching, and yawing). Dynamic turbulent structures and their interactions in the wake region, far wake region, and boundary layer near the car body significantly contribute to the six aerodynamic component forces. Therefore, modal decomposition techniques, including dynamic mode decomposition (DMD), are valuable tools for enhancing fuel economy and vehicle maneuverability. They achieve this by identifying dynamic body force features, designing passive and active control systems, and constructing reduced-order models (ROMs).

DMD’s ability to compute spatiotemporal modes, isolate individual components in the time domain, and facilitate future predictions makes it highly promising for automotive applications. For instance, Ikeda et al. [24,25] used DMD to extract flow structures responsible for lift forces with an oscillation frequency of 5 Hz around a realistic car body, revealing that high-energy fluctuations were concentrated on the trunk, spare tire pan, and underbody. Similarly, Edwige et al. [26] employed sparse promoting dynamic mode decomposition to capture an asymmetric structure with ${St}_W=0.22$ in the flow regime of a 47° Ahmed body, designing a closed-loop controller to symmetrize the mode. Here, f, U, and W represent the dimensional frequency, reference velocity, and vehicle width, respectively, with the Strouhal number based on the width defined as ${St}_W\equiv fW/U$. Note that, in this paper, we will be using the height of the Ahmed body as the characteristic length, and the corresponding Strouhal number will be denoted simply as $St$. In another study, Evstafyeva et al. [27] used DMD to analyze the shedding mode frequency at $Re$ = 345 of a square-back Ahmed body and developed a controller to reduce base pressure drag and re-symmetrize the wake. The Reynolds number, ${Re}_W$, with the width of the geometry (W) as the characteristic length, was defined as ${Re}_W\equiv UW/\nu$. Building on this, Dalla et al. [28] applied DMD to a simplified truck model at ${Re}_H$ = 20,000, identifying toroidal vortex structures near the wake and large hairpin vortices shed from the longest part of the wake, which were associated with bistability triggers. Ahani et al. [29] conducted a DMD analysis on a simplified fastback generic car body, known as DrivAer [30], capturing the most dominant and energetic dynamic modes of the flow. They identified modes with $St$ = 0.08 and 0.13 as the most energetic, with their shapes and frequencies aligning well with existing literature, despite differences in geometry and Reynolds number. In an experimental study, Siddiqui and Agelin-Chaab [31] examined the flow around an elliptical Ahmed body using DMD. They found that the most energetic mode decays faster in the elliptical body compared to the standard Ahmed body. This effect shifts the high-drag critical angle from 30° to a lower-drag angle of 25°. Recently, Misar et al. [32] successfully applied DMD to develop a reduced-order model (ROM) that predicts force and moment coefficients for flow past an idealized ground vehicle at a Reynolds number of $2.7\times {10}^6$. Most research studies that used dynamic mode decomposition (DMD) to analyze flow dynamics in road vehicle aerodynamics have employed sampling frequencies ($f_s$) ranging from 10 Hz to 10 kHz, sampling periods ($T_w$) from ${10}^{-2}$ to ${10}^3$ s, and Reynolds numbers between ${10}^3$ and ${10}^6$.

The choice of the sampling window length can significantly affect the trade-off between frequency and time resolution, making the selection of an appropriate windowing strategy challenging for non-stationary signals. In this study, where DMD is the selected decomposition technique, there is a very limited understanding of the relationship between sampling parameters and the consistency and uniqueness of DMD outputs, especially in applications such as ground vehicles that deal with complex geometry at high Reynolds numbers. Most DMD studies discussing DMD output uniqueness involve more periodic flows, such as jets or flow past 2D cylinders at low Reynolds numbers. For example, two years after the introduction of DMD by Schmid, Chen et al. [33], in their numerical study of the flow past a circular cylinder, stated that, unlike the discrete Fourier transform (DFT), DMD does not require a specific number of cycles for the target frequency to capture dynamic components. They also mentioned that individual modes with decaying or growing rates enable DMD to handle non-periodic flows as well. To the best of the authors’ knowledge, one of the few research studies that has conducted a parametric investigation into the effect of data sampling on DMD performance is by Li et al. [34]. They identified four stages of DMD convergence with respect to cycle numbers: initialization, transition, stabilization, and divergence. They also discovered that convergence of the sampling frequency is mode-specific, suggesting that an oscillatory structure might be resolved by 15 frames. However, their case study, limited to flow past a square prism dominated by Kaŕmań vortex street, did not provide deep insights into the uniqueness of DMD performance in mode shapes and energy levels. Nevertheless, since flow past a vehicle involves a mix of wall-bounded, free shear, and wake flows at very high Reynolds numbers, the interactions between these flows can have a significant impact on the complexity level of DMD decomposition. Hence, establishing a criterion for sampling frequency and period is crucial to ensure that the dynamic mode decomposition (DMD) output accurately reflects the system dynamics and its energy levels.

In this paper, we study the convergence criteria of dynamic mode decomposition (DMD) parameters with respect to sampling frequency and window size. We conduct an in-depth analysis of mode responses in terms of shape and energy level. Additionally, we evaluate the performance of DMD in decomposing the flow field into meaningful modes with consistent energy levels and mode shapes. Our case study examines the flow past a square-back Ahmed body at a high Reynolds number, ${Re}_H=7.7\times {10}^5$. Initially, we discuss the numerical setup, followed by a brief overview of the DMD algorithm. Subsequently, we evaluate whether the flow field resolved by improved delayed detached eddy simulation (IDDES) is suitable for modal decomposition. In the results and discussion sections, we investigate the convergence criteria of drag force in different regions on the Ahmed body surfaces concerning sampling frequency and period. We then examine DMD parameter convergence (eigenvalues, growth/decay rates, and reconstruction relative error) regarding sampling frequency and time window. Furthermore, we analyze mode shapes across different sampling periods, leading to various frequency resolutions, and assess the consistency and uniqueness of DMD responses. Finally, we repeat all convergence studies with the mean value subtracted from the feeding matrix and compare the results to the classic DMD with the mean value included.

2. Numerical Setup

The simulations performed in this study were carried out using a commercial finite volume solver computational fluid dynamics (CFD) package, STAR-CCM+ version 20.2., where a full-scale square-back Ahmed body model was placed in a computational domain with dimensions of 30 L × 30 W × 30 H, where L, W, and H denote the length, width, and height of the Ahmed body, respectively. The dimensions of the virtual wind tunnel resulted in a blockage ratio of 0.1% with respect to the frontal area of the geometry. The body was positioned $10L$ downstream of the inlet plane. The freestream velocity at the inlet was 40 m/s, corresponding to a Reynolds number based on the Ahmed body height, $R{e}_H$ of $7.7\times {10}^5$. A velocity of 40 m/s was chosen as the free-stream velocity because the flow becomes Reynold-independent at this speed [35]. The turbulence intensity and length scale at inlet boundary conditions were 0.25% and 10 mm, respectively. The pressure outlet condition was imposed as the boundary condition at the outlet. The top and side planes were set to zero-gradient (symmetry) conditions. A no-slip moving ground plane is defined, with the tangential velocity at the ground surface corresponding to the freestream velocity.

The computational domain was discretized using an unstructured hexahedral cell mesh created by the “trimmed cell” mesher option of STAR-CCM+. The “trimmed cell” algorithm generates cells that are multiples of $2n$, either greater or smaller than the selected base size; hence, it enables us to refine the wake region and to better capture the unsteadiness and turbulent characteristics of flow caused by boundary layer separation. A base size of 4 mm was used for both the geometry surface and volume mesh around the vehicle body and underbody; however, a wake region with a size of 2 mm was generated in order to resolve the flow field in the wake region. The 28-prism layer cells with an overall thickness of 10 mm and a first layer height of 0.005 mm were used along the vehicle surface to properly compute the high gradient boundary layer and achieve a wall-$y^{+}$ of less than $99\%$ for most of the cells. Note that $y^{+}\equiv y{u}_{\tau }/\nu$, where $\nu$ denotes the fluid kinematic viscosity; ${\tau }_w$ and $\rho$ represent the wall shear stress and fluid density, respectively, and the friction velocity is denoted by $u_{\tau }\equiv \sqrt{{\tau }_w/\rho }$. This setting resulted in a total cell count of 27 million cells for the volume mesh.

Scmid et al. [17] recommended using data from particle image velocimetry (PIV) and direct numerical simulation (DNS) as inputs for DMD analyses, due to the capabilities of these approaches in capturing the dynamics of fluid flow fields. However, in most real-world automotive flow problems, these methods are impractical. Specifically, in computational fluid dynamics (CFD), coarse-grained simulations that rely on turbulence modeling approaches are often the only viable options. Muld et al. [36] investigated the potential of DES methods to capture the flow field properly for POD and DMD analyses and concluded that DES data can be used as input for both POD and DMD algorithms. Therefore, in this paper, we used the improved delayed detached eddy simulation (IDDES) [37] methodology. It takes advantage of the less expensive shear stress transport (SST) $k-\omega$ [38] unsteady Reynolds-averaged Navier–Stokes (RANS) turbulence model in the boundary layers and uses LES in the wake region. In their study on ground vehicle platooning involving two Ahmed bodies, Bounds et al. [39] demonstrated that IDDES is an effective methodology for investigating fluid flow problems characterized by complex aerodynamic interactions, including flow over an isolated Ahmed body. For a more in-depth discussion on the suitability of IDDES for CFD simulations of the Ahmed body, readers are directed to the works of Ashton et al. [40], Guilmineau et al. [41]. Furthermore, see Fu et al. [42] and Misar et al. [43] for comprehensive analyses of the effectiveness of RANS and IDDES approaches in elucidating turbulence flow structures and assessing the veracity of flow predictions for high-Reynolds-number ground vehicle flows under various operating conditions.

The IDDES methodology evolved from the detached eddy simulation (DES) approach proposed by Spalart and his coworkers [44]. One known limitation of the detached eddy simulation (DES) methodology involves predicting artificial flow separation, termed grid-induced separation (GIS). This occurs when the cell size within the boundary layer falls below a critical threshold. To mitigate GIS, the RANS model can be shielded from the DES formulation near the wall using a parameter that depends on eddy viscosity and wall distance [45]. This approach, known as delayed detached eddy simulation (DDES), introduces a delay in the transition from RANS to LES. The improved delayed detached eddy simulation (IDDES) model, employed in this study, represents an advancement over DDES. It incorporates various enhancements to improve its versatility for high-Reynolds-number flows. These include mechanisms to prevent excessive dissipation due to premature switching from RANS to LES and the capability to operate in wall-modeled LES (WMLES) mode for more efficient resolution of wall-bounded flows (see Shur et al. [46], Gritskevich et al. [47]). For a comprehensive understanding of IDDES, interested readers are encouraged to refer to the original publications cited above.

Simulations were performed using an implicit, unsteady, and segregated incompressible solver on an unstructured grid with a time step of ${10}^{-4}$ s using second-order temporal discretization. A second-order upwind scheme was applied to discretize the momentum and turbulence equations, while an algebraic multigrid (AMG) method with a V-cycle was employed for a faster iterative solution of the momentum equations. A two-layer, all-$y^{+}$ wall treatment was applied to the simulation to ensure the boundary layer improved the predictive accuracy even in cases where $y^{+}$ was not sufficiently small due to the complexity of geometry. The total physical simulation time was approximately 6.2 s. To ensure the washout of initial effects, the data corresponding to $t=2.2$ s were not considered and the last 4 s of data were used for subsequent analyses. This provided sufficient time for the average flow field to become temporally converged and to capture low-frequency coherent structures. However, all averages are represented using an ensemble average with a sliding window of 2 s. This sets the lowest resolved frequency (f) at 0.5 Hz, which corresponds to a non-dimensional frequency $St\equiv f\ast L/U_{\infty }$ of $3.6\times {10}^{-3}$; here, L and $U_{\infty }$ represent the characterized length and velocity scales, respectively. Note that the time-step $\Delta t={10}^{-4}$ corresponds to a Nyquist frequency of 5 KHz.

3. DMD Algorithm

The dynamic mode decomposition (DMD) method is used in this article. We decompose the two instantaneous parameters—wall shear stress and pressure on the Ahmed body surface—into spatial and temporal modes. The spatial mode helps us observe the coherent structures on the car surface, while the temporal modes provide valuable information about the time evolution of these structures as the flow progresses. In addition, by transferring these modes to the frequency domain, we can study the time-dependent nature of the flow. This aids in understanding the correlation between different coherent structures and their contribution to the aerodynamic characteristics. For these decomposition purposes, the collected data are arranged into a vector of interest, described as follows:

$$X_i=\left[x_i^1,x_i^2,x_i^3,x_i^4.\dots ,x_i^n\right]$$

(1)

In Equation (1), the subscript i represents the ith element of the grid in the flow field. The superscript represents the snapshot sequence number, and n is the total number of time snapshots for the grid element $X_i$. In this way, the quantity $x_i^j$ in the row vector implies the value of a scalar x at the ith element corresponding to time-sequence j. If we combine the row vectors of all grid elements, the resulting matrix will take the form as shown in Equation (2):

$$\mathbf{X}=\left[\begin{array}{ccc}{x}_1^1& \dots & x_1^n\\ .& \dots & .\\ .& \dots & .\\ .& \dots & .\\ x_m^1& \dots & x_m^n\end{array}\right]$$

(2)

DMD decomposes a scalar x into a set of spatial modes that have a fixed oscillation frequency with a decay/growth rate; scalar x can be any fluid property, such as the i-component of the velocity, pressure, vorticity, turbulent kinetic energy (TKE), etc. The algorithm for finding the modal frequency and decay rate involves splitting the matrix X into two matrices. One matrix contains values from time-sequence 1 to $n-1$, and the other matrix is formed by shifting the first matrix forward by one time-sequence. Thus, we obtain two matrices, $X^k$ and $X^{k+1}$, as shown in Equation (3):

$${\mathbf{X}}^{\left(k\right)}=\left[\begin{array}{ccc}{x}_1^1& \dots & x_1^{N-1}\\ .& \dots & .\\ .& \dots & .\\ .& \dots & .\\ x_m^1& \dots & x_m^{N-1}\end{array}\right]\mathrm{and}{\mathbf{X}}^{(k+1)}=\left[\begin{array}{ccc}{x}_1^2& \dots & x_1^N\\ .& \dots & .\\ .& \dots & .\\ .& \dots & .\\ x_m^2& \dots & x_m^N\end{array}\right]$$

(3)

Schmid [17] suggested that, for a small time difference, the flow evolution can be linearized as follows:

$${\mathbf{X}}^{(k+1)}\approx A{\mathbf{X}}^{\left(k\right)}$$

(4)

where the matrix $\mathbf{A}$ is a linear operator that describes the temporal evolution of the system’s output in a linear way, even though the system itself is nonlinear, i.e., it is a Koopman operator. Since $\mathbf{X}$ is generally not a square matrix, the standard matrix inversion cannot be applied, and the pseudoinverse must be used instead. Thus, A can be determined by multiplying ${\mathbf{X}}^{(k+1)}$ by the pseudoinverse of ${\mathbf{X}}^{\left(k\right)}$. For simplicity, matrix ${\mathbf{X}}^{(k+1)}$ will be denoted as ${\mathbf{X}}^{\prime }$ and should not be confused with the adjoint of the matrix, which is represented as ${\mathbf{X}}^{*}$.

The eigenvalue $\lambda$ can be expressed in complex form as $\lambda =a+ib$, where a is the real part, which relates to the growth or decay of the mode over time, and b is the imaginary part, which corresponds to the oscillation frequency of the mode. Equation (5) relates the eigenvalue $\lambda$ of the system to its corresponding temporal growth (real component) rate and frequency (imaginary component).

$$\omega ={\displaystyle \frac{ln\left(\lambda \right)}{dt}}={\displaystyle \frac{ln(a+ib)}{dt}}$$

(5)

By calculating the growth rate and frequency of modes, the DMD algorithm can provide a local, high-dimensional linear approximation of the solution to the underlying nonlinear dynamical system.

$${\displaystyle \frac{d\mathbf{X}}{dt}}=\mathbf{A}x\to x_{(i,t)}=\sum _{k=1}^n{\Phi }_k{e}^{{\omega }_{k}t}{b}_k$$

(6)

where ${\Phi }_k$ and $b_k$ represent the complex-valued spatial distributions that correspond to a particular dynamic mode and coefficients of the initial conditions, respectively. The term $x_{(i,t)}$ denotes the state of the flow scalar x corresponding to the ith grid element at time t. The real part of ${\Phi }_k$ represents the physical displacement or shape of the mode at different points in space while the imaginary part corresponds to the phase shifts, indicating oscillatory motion over time if there is an oscillatory component in the eigenvalue.

Matrix $\mathbf{A}$ is an $m\times m$ matrix that can become very large. For example, in a 3D flow field with millions of grid points and thousands of time snapshots, matrix $\mathbf{A}$ could contain on the order of ${10}^{12}$ elements. Storing such a matrix requires terabytes of memory. Additionally, the eigendecomposition of this large, sparse matrix using classical methods may become unstable and costly, and there is no guarantee that useful dominant dynamic modes can be extracted. Therefore, solutions are needed to address these issues.

First, we decompose matrix $\mathbf{X}$ using the singular value decomposition (SVD). In linear algebra, SVD is a factorization method that decomposes a real or complex matrix into an orthogonal eigenbasis. The orthogonal basis functions from SVD provide more stable solutions (see Equation (7)).

$${\mathbf{X}}^{\mathbf{k}}\approx \mathbf{U}\Sigma {\mathbf{V}}^{*}$$

(7)

where $\mathbf{U}$, $\Sigma$, and $\mathbf{V}$ are the left unitary matrix (rotation), singular values (rescaler), and right unitary matrix (rotation), respectively. This guarantees the existence of eigenvalues and eigenvectors. To avoid the $m\times m$ matrix, we use reduced SVD, also known as economy SVD. This approach provides a unitary matrix $\mathbf{U}$ of size $m\times (n-1)$, singular values of size $(n-1)\times (n-1)$, and a unitary matrix $\mathbf{V}$ of size $(n-1)\times (n-1)$. By substituting these decomposed matrices into Equation (1), we can obtain matrix $\mathbf{A}$ as follows:

$$\mathbf{A}={\mathbf{X}}^{(k+1)}\mathbf{V}{\Sigma }^{-1}{\mathbf{U}}^{*}$$

(8)

The size of matrix $\mathbf{A}$ remains $m\times m$. For the final rank reduction, we can add another step by choosing a reduction rank manually, which can vary depending on the application. We can select this rank r to be less than the total number of snapshots, or it may be equal to the total number of snapshots.

$$\tilde{\mathbf{A}}={\mathbf{U}}^{*}{\mathbf{X}}^{(k+1)}\mathbf{V}{\Sigma }^{-1}$$

(9)

The eigenvalues and eigenvectors of matrix $\tilde{\mathbf{A}}$ provide useful information about spatial phases and their unique temporal characteristics.

4. Numerical Validation

In this section, we present the results of the numerical validation of the generic Ahmed body model, evaluating the accuracy of the IDDES turbulence model in predicting the aerodynamic drag coefficient. In addition, we assess the model’s capability to capture both coherent and incoherent dynamic flow structures. The simulation results have been compared with experimental works from the literature.

4.1. Aerodynamic Coefficients

The full body drag coefficient of the Ahmed body is computed and compared to the experimental work of [48]. The drag coefficient $C_D$ is defined as follows:

$$C_D={\displaystyle \frac{F_D}{0.5\rho A{U}_{\infty }^2}}$$

(10)

where $F_D$, $\rho$, $U_{\infty }$, and A stand for the drag force (force in the positive x or stream direction), fluid density, reference velocity, and characteristic area (normally the frontal area of the vehicle), respectively. In this study, we use $A=0.115$ m² and $U_{\infty }=40$ m/s. Fluid properties are evaluated in a standard atmosphere. With the vehicle height, H, used as the reference length, these correspond to a Reynolds number, $R{e}_H=7.7\times {10}^5$.

Full body and base drag, as well as pressure drag at the rear fascia, obtained by the numerical model are 0.299 and 0.216; this shows good agreement with the reported value of $C_D=0.315$ in experimental work [48]. The relative error is approximately 5% and it has about a 170 count difference. It is worth mentioning that the reported global drag in the literature for square-back Ahmed body varies within the range of 0.250–0.364 and there is significant disagreement in the literature about the Ahmed body reported drag [49,50,51,52]. The computed drag coefficient lies within the reported range in the literature.

To better understand the contributions of different body sections to the drag force components, the geometry was divided into three parts: (a) the nose (frontal surface), (b) the side, upper, and lower surfaces, and (c) the rear fascia. The drag coefficient for each section was then calculated separately, as shown in Figure 1.

As shown in

Table 1. Drag coefficient components; $C_{D,p}$ and $C_{D,f}$ stand for pressure and friction drag, respectively.

	Rear Fascia	Body	Nose
$C_{D,p}$	0.216 (71%)	0.031 (10.5%)	0.014 (4.5%)
$C_{D,f}$	0	0.033 (10.8%)	0.005 (1.5%)

, the rear fascia pressure drag ($C_{D,p}$), also known as base drag, makes the largest contribution, accounting for nearly 70%. Following this, the friction drag ($C_{D,f}$) on the side body contributes around 11%. The remaining contributions come from the pressure drags on the body and nose. Notably, the pressure drag on the body is entirely due to the slits; otherwise, the pressure drag on the rest of the assigned body area is zero, as the pressure is perpendicular to the flow direction.

4.2. 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. Muld 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. Volpe’s experiment [48], conducted at a Reynolds number of $R{e}_H=7.7\times {10}^5$ and a free-stream velocity of 40 m/s, serves as a reference. For this analysis, eight probes (P1–P8) were positioned on the rear fascia of the Ahmed 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 Ahmed 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 Volpe [48], with Strouhal numbers ($St$) of 0.08, 0.13, and 0.3, are present in the current numerical model (see Figure 3 and Figure 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 Volpe et al. [48] and Grandemange et al. [50], corresponding to approximately 8–10 s, or a non-dimensional time range of $T^{*}$ = 1000–1400 ($T^{*}\equiv T{U}_{\infty }/H$). 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

Lahaye 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, Liu et al. [53] reported high-frequency structures near $St$ = 2.12, corresponding to $St$ = 0.58 relative to height, which they attributed to vortex shedding from upstream struts. They also noted the observation of second harmonics of this mode, with a frequency twice that of $St$ = 2.12. Hence, the high-frequency components observed in Figure 3 and Figure 4 can be related to coherent structures emanating from struts. Further investigation is necessary to fully interpret the physical meaning of these high-frequency components.

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 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. Chen 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.

5.1. Drag Force Sensitivity to Sampling Frequency and Period

In this section, we aim to determine the minimum sampling frequency ($f_s$) and sampling period ($T_w$) required to acquire sufficient data for modal decomposition and ensure the convergence of key aerodynamic parameters, such as the drag coefficient, $C_D$. Initially, we analyzed the power spectral density (PSD) of the global drag on the Ahmed body, as well as the pressure and friction drag on the body, nose, and rear fascia. This FFT spectral analysis was performed with a sampling frequency of $f_s$ = 10 kHz and a time window of 4 s. Figure 5 shows the PSD of the drag force without any smoothing. The raw spectrum indicates that the most energetic components are within below a non-dimensional frequency or Strouhal number ($St$) of 0.36 (i.e., below 50 Hz). Beyond this, energy decreases consistently up to 1 kHz ($St$ = 7.2), with an exception around 350 Hz ($St$ = 2.52). In Section 4, we identify significant oscillatory components at 245, 294, and 400 Hz ($St$ = 1.76, 2.11, and 2.88). Although the pressure signal is sampled at 10 kHz, giving a Nyquist frequency of 5 kHz, Figure 5 (marked in the red square) shows that the maximum resolvable structure by this IDDES simulation is about 1 kHz ($St$ = 7.2), beyond which, the signal flattens and carries little useful information. This is primarily because the mesh resolution used in this study is incapable of spatially resolving smaller-scale flow structures of about 1 kHz.

Next, we study the frequency spectrum and cumulative sum of friction and pressure drag individually for the nose, body, and rear fascia, which are plotted in Figure 6a–f, where we can clearly see that the PSD values of the pressure drag for all three body-regions exhibit similar trends. Note that in these figures, to plot pressure and friction drag components—which have different orders of magnitude—on the same graph, these quantities are normalized. Each drag component, whether pressure or friction, is scaled so that its cumulative sum equals one over the range of non-dimensional frequency considered. Similarly, the spectra of the pressure and friction drags are normalized by their respective total energy to bring them to the same order of magnitude. In these figures, we can observe a few very low-frequency structures (1–10 Hz, corresponding to $St$ = 0.007–0.07), which are not well-explained in the literature. Additionally, significant high-frequency components appear near 349 Hz ($St$ = 2.54) and 600 Hz ($St$ = 4.32). As frequency increases, the amplitude decreases drastically, with minimal notable peaks.

As discussed earlier, the turbulence model and grid resolution used in this study were not capable of resolving turbulent structures beyond 1 kHz, which is corroborated by the cumulative sum of frequency spectrum components. Figure 6b,d,f show that the system’s energy does not change significantly beyond 400 Hz ($St$ = 2.88). In Figure 6b, the pressure drag ($C_{D,p}$ at the rear fascia reaches 90% of its energy within the range of $St$ = 0.18 (approximately 26 Hz). There is a spike at 349 Hz $St$ = 2.51), which is likely caused by the interaction of coherent vortex shedding structures from the side, lower, and upper edges of the Ahmed body, representing an averaged effect. This peak may represent the combined high-frequency components from the side and lower edges, which typically fall within the range of $St$ = 1.44 to 4.5 (200–400 Hz). This frequency range accounts for approximately 4% of the total energy in the signal. In contrast, the body and nose pressure drag exhibit fewer dynamic structures, with significantly slower slopes compared to the rear fascia. The component at 349 Hz contains the largest portion of energy, indicating that dynamic components are more prominent at the rear fascia compared to the nose and body. The nose is primarily subject to stagnation, with minimal dynamic structures, and the energetic fluctuating components develop from areas near the edges toward the Ahmed body. Additionally, the assigned body section contributes almost zero drag force; only the slits of the Ahmed body generate pressure drag, while the remaining sections are perpendicular to the drag axis, playing a key role in lift and side forces, which are beyond the scope of this paper.

Figure 6c,d, respectively, present the frequency spectrum and the cumulative sum of components for the friction drag ($C_{D,f}$) on the body and nose. The friction drag of the rear fascia is zero because the surface is normal to the drag force axis and is, therefore, excluded from the analysis. As expected, the body contributes more to friction drag compared to the nose. Although the peaks in the spectrum for low-frequency components are less prominent than those in the pressure signal, they are still present. The body friction drag reaches 90% of its energy within the range of $St$ = 0.60 (approximately 84 Hz). Similar to the $C_{D,p}$ spectrum in Figure 6a, a notable high-frequency structure at 349 Hz ($St$ = 2.51) was observed for $C_{D,f}$ in Figure 6a, indicating a good correlation between pressure and stream-wise velocity in quasi-periodic structures. Similar to the $C_{D,p}$ spectrum but with a wider range, ($C_{D,f}$ maintains energy in the frequency range below 84 Hz ($St$ = 0.60), with a pronounced peak at $St$ = 2.51 (f = 349 Hz).

It is also worth noting that the $C_{D,f}$ spectrum for the body shows a more gradual energy distribution compared to the nose. This is primarily due to the largely stagnant flow in the nose region. Finally, Figure 6e,f compare the pressure and friction drag for the full Ahmed body. The results show that the energy reduction slope for pressure drag is steeper than for friction drag, consistent with trends reported in the literature [48].

In order to capture all low-frequency dominant modes and ensure the convergence of $C_D$, it is essential to identify a sufficiently large sampling time window. In order to ascertain an optimum time window, we calculated the dependence of the mean and RMS of the pressure and friction drag on the sampling periods, as shown in Figure 7. According to the literature [48,50], for the flow over an Ahmed Body, the lowest oscillatory structure is $St$ = 0.08, corresponding to f = 11 Hz for the present study. However, the spectral analysis of pressure and wall shear stress also revealed some dominant frequencies in the lower range, near $St$ = 0.036 and 0.018.

As discussed in previous sections, Chen et al. [33] demonstrated that DMD can capture a mode with as little as one or even half a cycle, whereas classical FFT may require up to five cycles. Based on this, a sampling window of 0.2 s ($T^{*}=28$) should be sufficient to capture low-energy modes, such as the one at $St$ = 0.08 ($f_s=11.1$ Hz). Figure 7a shows that the full-body pressure drag requires approximately 2.0 s ($T^{*}\approx 280$) to reach a quasi-stationary trend, with the mean $C_D$ converging within $\pm 0.0001$. It is important to note that the pressure at the rear fascia exhibits bimodality, switching phases between positive and negative values every $T^{*}=1000$, known as the scale of bistability [48]. The RMS of pressure drag coefficients also reflects these phase shifts, indicating their influence on the results (see Figure 7b). In general, achieving stability or convergence in fluctuation measurements is challenging due to the chaotic and random nature of turbulent flow, which would require the ensemble averaging of multiple datasets. However, the mean value of friction drag converges more quickly than that of pressure drag. After approximately 1 s ($T^{*}\approx 140$), the friction drag signal shows some stability, with fluctuations reduced to the fifth decimal place. Both the RMS of friction and pressure drag display a switching behavior, where the RMS values initially decrease with $T^{*}$, then abruptly increase, and subsequently decrease again, eventually becoming asymptotic. These fluctuations, highlighted in red boxes in the figures, are likely related to bimodality, although a more detailed explanation requires further investigation.

5.2. DMD Convergence Criterion

5.2.1. Convergence of Sampling Frequency

In this section, we will discuss the dependence of the DMD-based eigendecomposition, and subsequent signal reconstruction on the sampling frequency for two flow parameters: surface pressure at the rear fascia and wall shear stress (WS) on the body (as shown in Figure 1). DMD is a high-dimensional linear regression technique that relies on time derivatives ($\mathrm{d}y/\mathrm{d}x$) to capture oscillatory structures across different time scales. To resolve these structures accurately, the temporal resolution of the snapshots must be fine enough to distinguish each time scale. In analogy to Fourier analysis, where the Nyquist–Shannon theorem dictates that the sampling frequency must be at least twice the maximum frequency to avoid aliasing (for example, a 20 Hz structure requires a 40 Hz sampling frequency), DMD poses additional challenges. Unlike Fourier transforms, DMD modes are not necessarily orthogonal in space or time and approximate fields using decaying or growing oscillatory modes. This sensitivity to mode growth/decay rates, energy, and time derivatives makes DMD more demanding in terms of temporal resolution. It may require higher sampling rates than those prescribed by the Nyquist criterion to accurately resolve local temporal events and variations in energy levels, ensuring proper dynamic capture.

To illustrate the complexities of DMD, let us revisit Figure 6, which shows that the rear-fascia pressure signal and body friction reach 90% of their respective total energy at two distinct frequencies: $St$ = 0.18 (f = 26 HZ) for the pressure signal and $St$ = 0.60 (f = 84 Hz) for the body friction. Beyond this range, both signals display several high-energy spikes, with a notable spike at $St$ = 2.51 (f = 349 Hz) 349, contributing ≈5% of the total energy. We believe this phenomenon is not due to numerical methods or simulation quality; rather, it likely stems from physical characteristics of the flow structures associated with this particular geometry or possibly an artifact of the DMD approach. However, based on the prior discussion, it is more probable that these spikes are inherent to the geometry, and DMD has effectively revealed them. Clearly, improper characterization of this spike, specifically in terms of frequency, amplitude, and growth rate, will lead to significant errors in flow-field reconstruction when using lower-dimensional representations.

To better understand how these structure frequencies and energy levels influence DMD convergence, we examined it using two frequency bandwidths: one spanning $St$ = 0–0.72 (f = 0–100 Hz) and another covering $St$ = 0–2.88 (f = 0–400 Hz). The data were sampled at frequencies higher than each bandwidth to allow comparison, assessing the impact of range and signal energy on DMD convergence. Note that the original simulation data were sampled at 10 kHz, making both cases subject to down-sampling. This down-sampling, performed by an integer factor k (decimation), can introduce aliasing, where high-frequency components are folded into the lower frequency range. To mitigate this, signals were filtered with cutoff frequencies ($f_c$) of 100 Hz and 400 Hz, reducing the effects of aliasing from down-sampling. For the 0–100 Hz bandwidth, sampling frequencies ($f_s$) ranged from 100 Hz to 10 kHz, with a constant sampling window of $T_w=2$ s ($T^{*}=278$). For the 0–400 Hz bandwidth, sampling frequencies varied from 400 Hz to 10 kHz, maintaining the same $T_w=2$ s. The 2-s window was chosen based on the observed convergence condition (Figure 7). Figure 8 illustrates and compares the power spectral density (PSD) of both the raw and filtered signals (at $f_c=100$ Hz and $f_c=400$ Hz) for the pressure signal at the rear fascia and body friction.

Figure 9, Figure 10 and Figure 11 illustrate the effect of different sampling frequencies on eigenvalues averaged magnitude ($\left|\lambda \right|$), eigenvalues’ standard deviation (${\sigma }_{\lambda }$), and the reconstruction error of drag force, respectively. It is worth mentioning that $\left|\lambda \right|$ is averaged over all calculated modes. There are two signals conditioned with cutoff frequencies ($f_c$) of 100 Hz and 400 Hz and a total window time of $T_w$ = 2 s. As shown in Figure 9. increasing the sampling frequency ($f_s$) causes the mean of $\left|\lambda \right|$ to converge toward the value near one. The magnitude of $\lambda$ closer to one can cause more stability in the DMD process. When $f_s$ is approximately ten times the cutoff frequency in the case of $f_c$ = 100 Hz, and two and half times in the case of $f_c$ = 400 Hz, $\left|\lambda \right|$ begins to converge and becomes stable. The order of magnitude of $\left|\lambda \right|$ has a direct relation with the total snapshot number. For both signals, with cutoff frequencies ($f_c$) of 100 and 400 Hz, $\lambda$ converges to three decimal places after 2000 snapshots. This means the energy content of the signal in bandwidth within 100 and 400 Hz has very little impact on the order of magnitude of $\left|\lambda \right|$ and it is highly dependent on the total number of snapshots.

Figure 10 demonstrates that increasing the sampling frequency ($f_s$) leads to the convergence of ${\sigma }_{\lambda }$ toward a value close to zero. The closer ${\sigma }_{\lambda }$ is to zero, the greater the stability achieved in the DMD process. When $f_s$ is approximately ten times the cut-off frequency for $f_c=100$ Hz and three times for $f_c=400$ Hz, ${\sigma }_{\lambda }$ begins to stabilize. The magnitude of ${\sigma }_{\lambda }$ is directly related to the total number of snapshots. For both signals with cut-off frequencies of 100 Hz and 400 Hz, $\lambda$ converges after 2000 snapshots. This indicates that the energy content of the signal within the bandwidth of 100 to 400 Hz has minimal impact on the standard deviation of $\left|\lambda \right|$, which is primarily influenced by the total number of snapshots.

Figure 11, on the left, illustrates the DMD reconstruction relative error as a function of sampling frequency. For this matter, the time series of pressure on the rear fascia and body friction were reconstructed using DMD at varying sampling frequencies (see Equation (6)), and then the pressure and friction drag were computed from these reconstructed signals. The relative error was calculated by comparing these values to the corresponding values from the true CFD solution. The error initially decreases but rises sharply to magnitudes of ${10}^2$ and ${10}^3$ near $f_s/f_c$ = 2–3 for surface pressure. To achieve errors below 1%, a sampling ratio of $f_s/f_c$ = 4–5 is required for pressure, and $f_s/f_c=10$ for wall shear stress. This trend persists for signals filtered at $f_c=400$ Hz, where the error drops below 1% around $f_s=f_c=5$ for both pressure and friction signals. Overall, surface pressure converges more quickly than wall shear stress; this is probably due to the fact that the pressure signal has a much narrower band in terms of modes compared to the inertia terms. These findings are in line with the work of Misar et al. [32], which successfully captured the full pressure signal spectrum on the surface of an Ahmed body with a 35-degree slant using 2000 snapshots over 0.2 s at a sampling rate of 10 kHz. This is consistent with our results where we found that DMD requires a minimum of 2000 snapshots for eigenvalue convergence. While their setup used a shorter time window and higher sampling frequency, we observed similar convergence for longer time periods and bandwidths of 100 Hz and 400 Hz. These findings highlight that DMD convergence requires a sufficiently high sampling frequency relative to the cutoff frequency to resolve individual time scales and energy distributions across the bandwidth.

To further illustrate this, spectral analyses (using fast Fourier transform of FFT) of the DMD-reconstructed pressure and friction drag signals are compared to the spectra obtained from the low-pass-filtered CFD solution data in Figure 12 and Figure 13 for the two cut-off frequencies. These figures demonstrate the effect of sampling on the signals that were conditioned using low-pass filters at $f_c=100$ Hz and $f_c=400$ Hz but sampled at various frequencies. In Figure 12a, as the sampling frequency increases, the power spectral density (PSD) of the DMD-reconstructed signal progressively aligns with that of the true CFD signal. When the sampling frequency ($f_s$) is equal to twice the bandwidth (100 Hz), the reconstructed signal exhibits significant deviations, with a relative error on the order of ${10}^3$. As $f_s$ increases, the relative error decreases, reaching approximately 2% at $f_s=400$ Hz and dropping below 1% at $f_s=1$ kHz. A similar trend is observed for wall shear stress, as shown in Figure 12b, where the PSD spectrum approaches the original as the sampling frequency increases. The relative error for the wall shear stress starts at 150% at $f_s=200$ Hz, rises to around ${10}^3$ at $f_s=400$ Hz, and decreases to about 3% at $f_s=1$ kHz. The same process was applied to the pressure and friction signals with a bandwidth of $f_c=400$ Hz. As seen in Figure 13, increasing the sampling frequency significantly reduces the relative reconstruction error. For instance, at $f_s=1$ kHz, the friction signal shows a large deviation with an error on the order of ${10}^4$. However, increasing the sampling frequency to 5 kHz (eight times the bandwidth) reduces the error to below 1%, achieving a nearly perfect match in the spectrum.

Li et al. [34] investigated DMD convergence criteria for free-shear flow around an infinite square prism at $Re$ = 22,000. They found that sampling frequency convergence is mode-specific, with each mode requiring a different number of frames to converge. They proposed a rule of thumb: approximately 15 frames are needed per mode, based on its individual oscillatory frequency. In terms of energy levels, higher-frequency modes require fewer frames for accurate capture. Consequently, as the frequency bandwidth increases from 100 Hz to 400 Hz, the required resolution to capture these modes decreases slightly, particularly for pressure drag. Their results indicate that DMD convergence depends heavily on the energy distribution across frequency bandwidths, although the structure of individual high-frequency, high-energy modes can significantly affect DMD performance. Our study reaches a similar conclusion but offers a less restrictive guideline, particularly for high-Reynolds-number flows with higher-frequency structures. For simulations resolving structures up to 2 kHz, five frames per mode should be sufficient, and this can likely be extended safely to cases where modes up to 10 kHz are present.

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 $\left|\lambda \right|$ 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. Figure 14 and Figure 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 $\left|\lambda \right|$ 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 $\left({\sigma }_{\lambda }\right)$ 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.

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). Figure 17 and Figure 18 illustrate the convergence of the mean and standard deviation of $\left|\lambda \right|$ for the pressure and wall shear stress signals in relation to the number of snapshots. In one instance, we maintained a constant sampling period ($T_w$ = 2 s) while increasing N by raising the sampling frequency. In another instance, we kept the sampling frequency constant and increased the sampling period. As shown in Figure 17, the mean of $\left|\lambda \right|$ begins to converge around N = 2000–2500 and reaches a plateau after 4000. Figure 18 shows that the standard deviation of $\left|\lambda \right|$ begins to converge around 2000–2500. However, it does not achieve a plateau comparable to the mean of $\left|\lambda \right|$. In general, the results presented in Figure 17 and Figure 18 indicate that the convergence of eigenvalue decomposition depends on the sampling period or frequency in relation to the total number of snapshots.

Next, we studied the impact of the sampling period on flow reconstruction. As shown in Figure 19, increasing the sampling period helps reduce the reconstruction relative error up to a ratio of $T^{*}=250$ for both cases, $f_c$ = 100 and 400 Hz. However, beyond this sampling period, in the case of wall shear stress, DMD tends to diverge. The equivalent cycle number is about 20 corresponding to the slowest global modes with $St$ = 0.08 (f = 11.1 Hz). Li et al. [34] found that beyond 20 cycles of the slowest mode DMD starts to divergence. According to their study, four states of mode convergence were identified: initialization, transition, stabilization, and divergence, depending on the sampling range. Between 1 and 8 cycles, DMD experiences an initialization state; from 8 to 15 cycles, it passes through a transition phase; after 15 cycles, it stabilizes up to 20 cycles, and then it begins to diverge. Therefore, while increasing the sampling period can aid in achieving the convergence of eigenvalues, it is crucial to maintain a high time resolution ($\Delta t$) to avoid oversampling and the generation of spurious modes. DMD’s performance is sensitive to spurious modes. Le Clainche and Vega [19] suggested that the double generation of a mode at a specific frequency can be spurious. Higher frequency domain resolution ($\Delta f$) increases the risk of this vulnerability. If DMD assigns a large growth rate to a mode due to increased cycles from an extended sampling period, it can result in a significantly large growth or damping ratio, causing a deviation from the true nature of the structure.

For further investigation, we selected five different time windows ($T^{*}$= 210, 245, 275, 315, 345). For each window, five different initial times ($t_0^{*}=15,25,40,55,70$) were chosen to calculate the reconstruction relative error. Figure 20a–d show the reconstruction relative error for two pressure and wall shear stress signals at filtering cut-off frequencies of 100 Hz and 400 Hz. The results demonstrate instability in DMD behavior across the cases, with instability typically appearing around $T^{*}=250$ and worsening beyond $T^{*}=300$. This instability is random and does not follow a consistent trend in increasing relative error. 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 Chen 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 intensify, while the energy levels decrease with 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 Le Clainche 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].

The spatial modal analysis of DMD results has demonstrated its ability to distinguish and extract valuable coherent structures from chaotic and random patterns. However, not all modes in DMD decomposition exhibit coherence. The nature of modes with higher growth rate magnitudes, whether they are growing (positive) or decaying (negative), and their relationship to either dissipative or productive turbulent structures remains unclear. Furthermore, DMD is likely to generate spurious and non-physical signals, which poses challenges in physically interpreting its outputs. Berizzi et al. [57] showed that oversampling results in the appearance of spurious modes, while Hemati et al. [58] observed that under-truncation in DMD inevitably produces spurious eigenvalues, as an artifact of “fitting the noise”. In their real-time identification study of electromechanical oscillation systems, Berizzi et al. [57] found that physical modes typically have higher energy than spurious ones and that the damping factor influences the energy level. According to Hemati et al. [58], in cases where no truncation is applied, as in the present study, there is a likelihood that many of the spurious eigenvalues may lie on the unit circle, making it difficult to distinguish relevant modes from irrelevant ones. This may be in contrast with the observations of Le Clainche and Vega [19] who showed that modes with the same frequency but smaller amplitudes are often spurious. Moreover, according to Colbrook et al. [59], DMD modes with $\left|\lambda \right|$ far from the unit circle are more likely to be spurious, especially in turbulent flows with continuous time and length scales from large to small structures. Interestingly, our initial investigations into applying truncation in the DMD process align with the findings of Colbrook et al. [59] and Le Clainche and Vega [19]. Consequently, we decided to remove the truncation steps from our DMD algorithm altogether. Based on the analyses presented, we conclude that there is a strong correlation between the growth/decay rate and the nature of the mode, whether it is coherent, dissipative/productive, or spurious.

Next, we investigate the impact of the sampling period and frequency resolution on the energy contents of the DMD-resolved modes. As we know, increasing the sampling period ($T_w$) provides higher resolution in the frequency domain and decreases frequency intervals ($\Delta f$). In DMD, the frequency intervals may not be constant over the frequency spectrum. DMD identifies temporal modes based on the underlying dynamics of the flow rather than through predefined bins, as in a discrete Fourier transform (DFT). In this study, we examined how the choice of sampling period affects the frequency spectrum and the resolution between adjacent frequency bins. We used three different sampling periods of 0.5, 1, and 2 s, each sampled at $f_s$ = 1 kHz. Then we calculated frequency intervals between adjacent frequency bins using the equation $\Delta f_n=f_{i+1}-f_i$. Figure 23 illustrates the normal distribution of coagulated frequency intervals for each sampling period. In the case of $T_w$ = 0.5 s, the mean of frequency intervals is around 2 Hz with a standard deviation $\sigma$ = 0.47. Increasing $T_w$ to 1 s shifts the mean value to 1 Hz and almost halves the standard deviation ($\sigma$ = 0.21). The trend remains the same as the mean of intervals decreases to 0.5 Hz with $\sigma$ = 0.12 in the case of $T_w$ = 2 s. This variability can be considered as an advantage of DMD due to its flexibility in capturing physical modes at exact frequencies, although it can also introduce difficulties in achieving convergence in DMD.

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 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 24. 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.

However, in the context of Tennekes and Lumley [60] (p. 259), this is not unexpected and may represent the real flow physics. Tennekes and Lumley [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”. The Strouhal number associated with the pumping as mentioned in the existing literature which is obtained through spectral analysis, may not be associated with a unique frequency but may be associated with eddies with a winder-range of frequency band and some phase relationship between them, and the DMD was able to predict them. However, this makes the identification of spurious modes more difficult and as well as necessitates the establishment of a new rule of how to in DMD of dealing with a mode that may represent the same structure but consists of motions with varying frequencies and phase relations between them.

5.3. Mean-Subtracted DMD

One key takeaway from the discussion in Section 5.2 is that low-frequency, slowly evolving modes are a primary contributor to instability in the DMD reconstruction algorithm. In this context, Li et al. [61] demonstrated that mean subtraction significantly improved the accuracy and stability of DMD-generated Koopman [62] models. By subtracting the mean flow, DMD becomes more effective at detecting and accurately resolving these modes, which are critical to flow instabilities. This process also enhances convergence by reducing bias toward stationary structures. Additionally, mean-subtracted DMD isolates unsteady, dynamic structures in the flow by removing the steady-state component prior to decomposition. This approach allows for a clearer focus on transient and oscillatory phenomena, such as vortex shedding or wake dynamics, which may otherwise be obscured by the mean flow. With this in mind, in what follows next, we explore the impact of the mean subtraction on DMD convergence.

The literature suggests that DMD with data centering, such as mean subtraction, can reliably compute the eigenvalue spectra of system dynamics. Without centering, DMD often fails to accurately represent dynamics, particularly in systems with high effective rank or complex structures [63]. However, there is a risk that mean-subtracted DMD outputs may resemble those of the discrete Fourier transform (DFT). Hirsh et al. [63] found that DMD with data centering generally produces distinct results from DFT, except in cases of regular periodicity, which is unlikely in complex turbulent flows around vehicles at high Reynolds numbers [61]. Several data-centering methods are available in the literature, including subtracting the mean flow field, a reference point, or an equilibrium state. Bohon et al. [64] demonstrated that mean subtraction helps DMD avoid converging to the zero-growth rate unit circle and enables the capture of quasi-periodic standing waves in complex systems like rotating detonation combustors (RDCs). However, this process can sometimes mislead the analysis by equating dissipative structures with coherent ones and focusing on slowly evolving components, potentially missing critical dynamic behaviors. In the following sections, we compare mean-subtracted and non-mean-subtracted DMD to observe how relative error trends vary with sampling frequency ($f_s$) and time window ($T_w$) when the mean is removed from the original signal.

When mean-subtracted data are input into the DMD algorithm, the magnitude of the eigenvalues is constrained to one, and the growth rate becomes zero. As a result, we exclude $\left|\lambda \right|$ from the convergence analysis for mean-subtracted data and focus instead on the effect of the mean subtraction on the convergence of DMD reconstruction relative error. Figure 25 shows the relative error with respect to sampling frequency for two filtered signals at $f_c=100$ Hz and $f_c=400$ Hz. The results indicate that mean-subtracted data converges more rapidly, exhibiting two to four orders of magnitude lower relative error compared to non-centered data. Furthermore, no instability was observed in the mean-subtracted case, whereas non-centered data showed instability.

In the case of incremental $T_w$, as shown in Figure 26 for two filtered signals at $f_c=100$ Hz and $f_c=400$ Hz, mean-subtracted DMD not only performed better but also demonstrated greater robustness compared to non-centered data. Increasing the sample number by extending $T_w$ further reduced the relative error. Since the growth rate and $\left|\lambda \right|$ remain close to zero and one, respectively, the risk of divergence or the generation of spurious modes with unrealistic decay/growth ratios is negligible. This makes the mean-subtracted DMD particularly suitable for applications like coherent structure extraction, where quasi-periodic oscillations and energy conservation over longer periods are expected. Additionally, the presence of non-growing and decaying modes enhances the stability of DMD for future state predictions in reduced-order modeling.

The variation of the modes’ energy ($St$ = 0.08, 0.13, 0.16, and 0.23) with respect to the sampling period ($T_w$) is shown in Figure 27. The results demonstrate a reliable convergence in energy levels for all selected modes. Energy convergence is observed after $T^{*}$ = 250, corresponding to T = 1.8 s. Additionally, modes with lower $St$ have higher energy levels. For instance, $St$ = 0.08 (f = 11.1 Hz) converges to an energy level of 1.5% of the energy of the mean mode. The energy level drops for higher frequencies, for example, mode with $St$ = 0.23 having an energy level near 0.5% of the mean mode.

6. Conclusions

This study investigated the convergence criteria for dynamic mode decomposition (DMD) parameters in relation to sampling frequency and period for flow past an Ahmed body at a high Reynolds number (${Re}_H=7.7\times {10}^5$). Over a total simulation time of 6 s, 40,000 snapshots were recorded at a 10 kHz sampling rate, with the first 2 s allocated for initialization and the remaining 4 s for capturing the flow dynamics.

Spectral analysis of pressure and velocity signals indicated that the IDDES model effectively captured the flow dynamics up to 1 KHz corresponding to $St$ = 7.2. Approximately 70% of the total drag was attributed to pressure drag on the rear fascia, with energy analyses showing that 90% of the energy in rear-fascia pressure and friction drag was concentrated at 26 Hz ($St$ = 0.18) and 84 Hz ($St$ = 0.60), respectively. The broader energy distribution in friction drag introduced challenges for DMD convergence.

DMD eigenvalue convergence was achieved with a sampling frequency-to-bandwidth ratio ($f_s/f_c$) of 5–10, requiring approximately 2000 snapshots. The convergence of the reconstruction error was influenced by both sampling frequency and the number of snapshots, with velocity components needing more snapshots than pressure due to their broader energy distributions. Increasing the sampling period further improved the convergence of DMD eigenvalue magnitudes ($\left|\lambda \right|$), which stabilized around 1.0 after 2000–3000 snapshots. However, reconstruction errors remained unstable for pressure and wall shear stress.

Mean-subtracted DMD demonstrated enhanced stability, generating modes with zero growth rates and reduced reconstruction errors. However, while mean-subtracted DMD performed well in identifying coherent structures, it may underestimate dissipative structures. Overall, this study provides valuable insights into DMD convergence and highlights the importance of optimizing sampling parameters for accurate modal decomposition in complex vehicle flow fields.

This study provides an in-depth analysis of DMD convergence criteria, emphasizing the impact of sampling parameters on the accuracy and stability of modal decomposition. Although results are presented using only one vehicle shape, these findings contribute to a better understanding of DMD’s application in fluid dynamics, particularly for complex flow fields around vehicle bodies. It is expected that the findings and observations can apply to other vehicle shapes, provided that the training data for DMD is obtained from high-fidelity, high-accuracy CFD simulations.