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The linear-time-invariance notion to the Koopman analysis is a recent advance in fluid mechanics [Li et al., “The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid-structure association,” Phys. Fluids 34(12), 125136 (2022c) and Li et al., “The linear-time-invariance notion of the Koopman analysis—Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake,” J. Fluid Mech. 959, A15 (2023a)], targeting the long-standing issue of correlating nonlinear excitation and response phenomena in fluid-structure interactions (FSI), or, in the simplified case, flow over rigid obstacles. Continuing the serial research, this work presents a data-driven, Koopman-inspired methodology to decouple nonlinear FSI by establishing cause-and-effect correspondences between structure surface pressure and the flow field. Exploiting unique features of the Koopman operator, the new methodology renders dynamic visualizations of in-sync, fluid-structure-coupled Koopman modes possible, fostering phenomenological analysis and statistical quantifications of FSI energy transfers. Instantaneous contribution contours and densities offer new angles to evaluate pathways of energy amplification and diminution. The methodology enables better descriptions and interpretations of phenomena occurring in the flow and on the boundary (walls) of an FSI domain and readily applies to a broad spectrum of engineering problems given its data-driven nature. © 2024 Author(s).

This work presents a post-data analysis procedure, namely, proper orthogonal decomposition (POD)–dynamic mode decomposition (DMD)–discrete Fourier transform analysis, for evaluating the dominant features of the flow fields from both energetic and dynamic perspectives. The large-eddy simulation (LES) was first employed to reproduce the flow field surrounding three types of building layouts. Subsequently, both POD and DMD were conducted according to LES simulation results. The extracted modes were classified into three types based on the POD and DMD: Type-1 mode: energetically and dynamically significant mode, Type-2 mode: energetically significant and dynamically insignificant mode, and Type-3 mode: energetically insignificant and dynamically significant mode. The findings indicate that Type-1 mode governs the primary velocity field and the predominant vortex patterns observed at the rear of the building arrays, as the reduction of inter-building widths leads to a shorter flow separation region. Type-2 mode is characterized by the presence of small-scale vortices and the high turbulent kinetic energy region, which periodically triggers pollutant increase in the vicinity of structures. Type-3 mode demonstrates a minimal energetic influence on the flow field; nevertheless, it significantly contributes to the consistent build-up of pollutants within the far-wake region. The present study also investigates the predominant coherent structures of flow fields concerning various building layouts and highlights the influence of passage widths on the efficiency of pollutant removal. This comprehensive analysis enables a systematic exploration of flow patterns within various building layouts, offering potential solutions for pollutant dispersion challenges in metropolitan areas.

This work investigates the subcritical free-shear prism wake at Re=22,000 by the Koopman analysis using the Dynamic Mode Decomposition (DMD) algorithm. The Koopman model linearized nonlinearities in the stochastic, homogeneous anisotropic turbulent wake, generating temporally orthogonal eigen tuples that carry meaningful, coherent structures. Phenomenological analysis of dominant modes revealed their physical interpretations: Mode 1 renders the mean-field dynamics, Modes 2 describes the roll-up of the Strouhal vortex, Mode 3 describes the Bloor-Gerrard vortex resulting from the Kelvin-Helmholtz instability inside shear layers, its superposition onto the Strouhal vortex, and the concurrent flow entrainment, Modes 6 and 10 describe the low-frequency shedding of turbulent separation bubbles (TSBs) and turbulence production, respectively, which contribute to the beating phenomenon in the lift time history and the flapping motion of shear layers, Modes 4, 5, 7, 8, and 9 are the relatively trivial harmonic excitations. This work demonstrates the Koopman analysis’ ability to provide insights into free-shear flows. Its success in subcritical turbulence also serves as an excellent reference for applications in other nonlinear, stochastic systems. Copyright © 2023 Techno-Press, Ltd.

This paper distills the serial work of Li et al., 2020a, Li et al., 2021, Li et al., 2022a, Li et al., 2022b, Li et al., 2022c, Li et al., 2023a, Li et al., 2023b into a practice guide for the Dynamic Mode Decomposition (DMD) with a particular emphasis on the quick learning and adaptation of the technique from a wind engineering perspective. The guide begins with discussions of the DMD's connections with the Koopman, Fourier & Laplace, the Proper Orthogonal Decomposition, and machine learning theories. Then, based on 1.5 × 106 core hours of parametric investigations on paradigmatic cases, recommendations that broadly apply to bluff-body wind engineering problems—especially regarding the input sequence—are formulated. In the temporal aspect, results highlight the DMD's preference for steady, stationary, equilibrium, or quasi-equilibrium systems, and suggest a sampling resolution 15–20 times the frequency of the dynamics of interest. Emphasis is also placed on the necessity of the Stabilization state for sampling convergence while avoiding the input of overdetermined systems. In the spatial aspect, high-dimensional, mean-subtracted data with minimal interpolation and noise proved to yield better Koopman models. Different input variables, though derived from the same system, may also result in non-unique Koopman modes depending on their nonlinear richness. Finally, two open-source MATLAB algorithms have been discussed and made open-source to facilitate the easy dissemination and usage for wind engineers. Via pedagogical examples, Algorithm 1 outlines a standard procedure to check for sampling convergence, and Algorithm 2 performs the similarity-expression DMD and visualizes the outputs.

Yu Jiang,1,&ast; Xisheng Lin,2,&ast; Yilin Wang,1 Jia Li,1,3 Guoqi Wang,4 Yutong Meng,5 Ming Li,1,3 Yi Li,1,3 Yan Luo,1,3 Zefu Gao,1 Pengbin Yin,1,3 Licheng Zhang,1,3 Houchen Lyu,1,3 Peifu Tang1,3 1Department of Orthopedics, Chinese PLA General Hospital, Beijing, People’s Republic of China; 2Department of Rehabilitation, the Second Medical Center of Chinese PLA General Hospital, Beijing, People’s Republic of China; 3National Clinical Research Center for Orthopedics, Sports Medicine & Rehabilitation, Beijing, People’s Republic of China; 4Department of Pediatrics, Chinese PLA General Hospital, Beijing, 100853, People’s Republic of China; 5Department of Orthopedics, Beijing Chaoyang Hospital, Capital Medical University, Beijing, People’s Republic of China&ast;These authors contributed equally to this workCorrespondence: Houchen Lyu; Peifu Tang, Department of Orthopedics, Chinese PLA General Hospital, No. 28, Fuxing Road, Beijing, 100853, People’s Republic of China, Tel +86-13501149301, Email houchenlyu@301hospital.com.cn; pftang301@126.comPurpose: To evaluate the impact of preoperative anemia on postoperative complications after hip fracture surgery.Patients and Methods: We conducted a retrospective study including hip fracture patients at a teaching hospital between 2005 and 2022. We defined preoperative anemia as the last hemoglobin measurement level before surgery < 130 g/L for men and < 120 g/L for women. The primary outcome was a composite of in-hospital major complications, including pneumonia, respiratory failure, gastrointestinal bleeding, urinary tract infection, incision infection, deep venous thrombosis, pulmonary embolism, angina pectoris, arrhythmia, myocardial infarction, heart failure, stroke, and death. Secondary outcomes were cardiovascular events, infection, pneumonia, and death. We used multivariate negative binomial or logistic regression to evaluate the impact of anemia and its severity, categorized as mild (90–

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.

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. © 2023 Author(s).

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. Demonstrated via a generic street canyon flow simulated through large-eddy simulation (LES), this systematic procedure identified three types of flow field modes according to energetic and dynamic significance, providing useful guidance for pollutant dispersion phenomenon analysis.

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.

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. © 2022 Author(s).

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 (FSIs) 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 II [Li et al., "A parametric and feasibility study for data sampling of the dynamic mode decomposition: Range, resolution, and universal convergence states, " Nonlinear Dyn. 107(4), 3683-3707 (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. Published under an exclusive license by AIP Publishing.

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. © 2021 Author(s).