Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The dynamic mode decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. It aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject—the turbulent free-shear flow over a prism—for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent.

This work develops a data analysis procedure, namely, proper orthogonal decomposition (POD)-dynamic mode decomposition (DMD) augmented analysis, to isolate the energy- and evolution-wise dominant features of flow field in a street canyon. This combination aims to extract modes imposing critical influence on pollutant dispersion from both energetic and dynamic perspectives. The two techniques were first conducted based on large-eddy simulation results. Subsequently, based on the POD and DMD ranking, the extracted modes were classified into three types: (1) type 1: energetically and dynamically significant mode; (2) type 2: energetically significant and dynamically insignificant mode; and (3) type 3: energetically insignificant and dynamically significant mode. Results show that mode type 1 contributes to the mainstream flow and the main vortex structures, which can be observed near the stagnation point, the separating point, and the fluid reattachment area. Mode type 2 throws light on where the turbulent kinetic energy is the largest, leading to periodically sudden pollutants increase on the building roof and the wake region. Mode type 3 contributes to the long-term reversed flow structures occurring near the stagnation point, inside the street canyon, and in the wake region. This technique can provide a systematic analysis of the flow field within a street canyon, and it also provides help for potential applications at a city scale, such as solving pollutant dispersion issues in urban areas.

The present work extends the parametric investigation on the sampling nuances of dynamic mode decomposition (DMD) under Koopman analysis. Through turbulent wakes, the study corroborated the generality of universal convergence states for all DMD implementations. It discovered implications of sampling range and resolution—determinants of spectral discretization by discrete bins and the highest resolved frequency range, respectively. The work reaffirmed the necessity of the convergence state for sampling independence, too. Results also suggested that the observables derived from the same flow may contain dynamically distinct information, thus altering the DMD output. Surface pressure and vortex fields are optimal for characterizing the structure and the flow field, respectively. Pressure, velocity magnitude, and turbulence kinetic energy also suffice for general applications, but Reynolds stresses and velocity components shall be avoided. Mean-subtraction is recommended for the best approximations of Koopman eigen tuples. Furthermore, the parametric investigation on truncation discovered some low-energy states that dictate a system's temporal integrity. The best practice for order reduction is to avoid truncation and employ dominant mode selection on a full-state subspace, though large-degree truncation supports fair data reconstruction with low computational cost. Finally, this work demonstrated synthetic noise resulting from pre-decomposition interpolation. In unavoidable interpolations to increase the spatial dimension n, high-order schemes are recommended for better retention of original dynamics. Finally, the observations herein, derived from inhomogeneous anisotropic turbulence, offer constructive references for DMD on fluid systems, if not also for others beyond fluid mechanics.

In this work, we introduce a novel data-driven formulation, the Koopman-Linearly Time-Invariant (Koopman-LTI) analysis, for analyzing Fluid-Structure Interactions (FSI). An implementation of the Koopman-LTI on a subcritical free-shear flow over a prism at Re = 22 000 corroborated a configuration-wise universal Koopman system, which approximated the configuration's nonlinear dynamics with stellar accuracy. The Koopman-LTI also successfully decomposed the entwined morphologies of raw measurement into a linear superposition of frequency-based constituents. Most importantly, with random and anisotropic turbulence, the Koopman-LTI yielded frequency-wise identical modes for structure response and fluid excitation, thus establishing direct constitutive relations between the phenomenology of fluid and structure.

This work augments a Linear-Time-Invariance (LTI) notion to the Koopman analysis, finding an invariant subspace on which consistent Koopman modes are expanded with fluid mechanics implications. The work also develops the Koopman-LTI architecture-a systematic procedure to associate fluid excitation and structure surface pressure by matching Koopman eigen tuples, establishing fluid-structure correspondences that examine fluid-structure interactions (FSI) at new angles. The data-driven, modular architecture also exhibits the potential to evolve with advances in Koopman algorithms. A pedagogical prism wake example demonstrated that the Koopman-LTI generated a near-perfect linearization of nonlinear FSI dynamics involving inhomogeneous anisotropic turbulence, with mean and root-mean-squared errors of O^-12 and O^-9, respectively; the infinite-dimensional Koopman modes were also approximated with O^-8 error. The subcritical wake during shear layer transition II was also reduced into only six dominant excitation-response Koopman duplets. The upstream and crosswind walls constitute a dynamically unified interface dominated by only two mechanisms. The downstream wall remains a distinct interface and is dominated by four other mechanisms. The complete revelation of the prism wake comes down to understanding the six mechanisms, which Part 2 (Li et al., 2022) will address by investigating the physics implications of the duplets' in-synch phenomenological features. Finally, the analysis revealed z-velocity's marginal role in the convection-dominated free-shear flow, Reynolds stresses' spectral description of cascading eddies, wake vortices' sensitivity to dilation and indifference to distortion, and structure responses' origin in vortex activities.

This serial work presents a linear-time-invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid mechanics: deterministically relating the fluid excitations and corresponding structure reactions. Part 1 (Li et al., Phys. Fluids, vol. 34, no. 12, p. 125136) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By a systematic analytical procedure, the Koopman-LTI generated sampling-independent linear models that captured all the recurring dynamics embedded in the input data, finding six corresponding, orthogonal, and in-synch fluid–structure mechanisms. This Part 2 analyses the six modal duplets to underpin their physical implications, providing a phenomenological analysis of the subcritical prism wake. Visualizing the newly proposed dynamic Koopman modes, results show that two mechanisms at St₁ = 0.1242 and St₅ = 0.0497 describe shear layer dynamics, the associated Bérnard–Kármán shedding and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of reaction. The on-wind walls’ dynamical similarity renders them a spectrally unified fluid–structure interface. Another four harmonic counterparts, namely the subharmonic at St₇ = 0.0683, the second harmonic at St₃ = 0.2422, and two ultra-harmonics at St₇ = 0.1739 and St₁₃ = 0.1935, govern the downstream wall. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in the wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structure reaction, benefiting future investigations into fluid–structure interactions and nonlinear, stochastic systems.