Dynamic Mode Decomposition Method Research Articles

Oscillation search robust dynamic mode decomposition method and its application in rolling bearing fault diagnosis

Oscillation search robust dynamic mode decomposition method and its application in rolling bearing fault diagnosis

Assessing the Effect of Coriolis Acceleration on the Coherent Structures in the Wake of a Wind Turbine using DMD

Abstract The present work aims to study the effect of veer, namely, the height-dependent lateral deflection of wind velocity due to Coriolis acceleration, on the coherent structures in the wake of the NREL 5-MW reference wind turbine using the sparsity-promoting dynamic mode decomposition (SP-DMD) method for the detection of dynamically-relevant flow structures. Large eddy simulation of the flow impacting the wind turbine is carried out by solving the filtered Navier-Stokes equations for incompressible flows. The effects of both tower and nacelle are included in the simulation using the immersed boundary method. Simulations are performed at Re ≈ 108, generating the inlet velocity profile by a precursor simulation of the atmospheric boundary layer subjected to Coriolis acceleration. The analysis of the DMD results allows to study the effects of the Coriolis acceleration on the most relevant dynamic modes in the turbine wake and to understand the basic mechanisms by which the wind veer significantly affects the wake recovery rate. Moreover, as a result of the SP-DMD methodology, the most relevant modes are extracted from the wake and a limited subset of relevant flow features that optimally approximates the original data sequence is identified. This small number of modes represent the kernel that can be employed for developing an accurate reduced order model of the wake.

The structure and dynamics of the laminar separation bubble

A novel selective mode decomposition, proper orthogonal decomposition and dynamic mode decomposition methods are used to analyse large-eddy simulation data of the flow field about a NACA0012 airfoil at low Reynolds numbers of $5\times 10^4$ and $9\times 10^4$ , and at near-stall conditions. The objective of the analysis is to investigate the structure of the laminar separation bubble (LSB) and its associated low-frequency flow oscillation (LFO). It is shown that the flow field can be decomposed into three dominant flow modes: two low-frequency modes (LFO-Mode-1 and LFO-Mode-2) that govern an interplay of a triad of vortices and sustain the LFO phenomenon, and a high-frequency oscillating (HFO) mode featuring travelling Kelvin–Helmholtz waves along the wake of the airfoil. The structure and dynamics of the LSB depend on the energy content of these three dominant flow modes. At angles of attack lower than the stall angle of attack and above the angle of a full stall, the flow is dominated by the HFO mode. At angles of attack above the stall angle of attack the LFO-Mode-2 overtakes the HFO mode, triggers instability in the LSB and initiates the LFO phenomenon. Previous studies peg the structure, stability and bursting conditions of the separation bubble to local flow parameters. However, the amplitude of these local flow parameters is dependent on the energy content of the three dominant flow modes. Thus, the present work proposes a more robust bursting criterion that is based on global eigenmodes.

Numerical study of wave resonance characteristics in gaps of a floating array

Wave resonance in the gaps formed by a four-float array for various drafts and incident wave frequencies is investigated using a numerical wave tank based on OpenFOAM. In the gap perpendicular to the direction of wave propagation, the resonant wave height is higher than that between two side-by-side floats under the same draft, and the resonant frequency is also different. Significant variations in wave height distribution are observed along the gap parallel to the wave propagation direction under different incident wave frequencies. When the incident wave frequencies are higher than the resonant frequency, the lateral force amplitude on the front floats increases, while the force amplitude on the rear floats does not show this effect. Using the dynamic mode decomposition method, we discover that the irregular distribution of wave heights across different frequencies leads to an increase in the lateral force amplitude on the front floats at non-resonant frequencies.

Study of unsteady shear flow near stall in a shrouded supercritical carbon dioxide centrifugal compressor

This study investigates the unsteady shear flow patterns and their relation with the aerodynamic instability of a shrouded supercritical carbon dioxide (SCO2) centrifugal compressor impeller using the dynamic mode decomposition method combined with detached eddy simulation. The results demonstrate that discrete shedding vortex array in the blade tip region is the dominant mode of the SCO2 shrouded centrifugal compressor impeller, with a frequency of 0.5 times the blade passing frequency. In contrast, the main flow structures in the identical impeller using ideal air consist of small-scale shedding vortices near the suction surface and the separated secondary flow. Further analysis of flow field details indicates that the shedding vortex structure inside the shrouded centrifugal impeller originates from the Kelvin–Helmholtz instability generated on the shear interface between the mainstream and the recirculation flow. The greater shear intensity within the SCO2 centrifugal impeller is the reason for the evident occurrence of vortex shedding phenomena in its flow field. Additionally, during the transition from near-stall to stall, vortex pairing phenomena happens at the throat of the centrifugal impeller due to evolutional behaviors of the vortexes. The occurrence of vortex pairing leads to faster blockage of the blade tip region and then the secondary flow spillage over to adjacent passage, ultimately resulting in reverse flow in the blade tip region and extensive blockage at the impeller inlet. The unsteady shear flow evolution clearly demonstrates the processes of stall in SCO2 shrouded impellers.

Data-Driven Control Method Based on Koopman Operator for Suspension System of Maglev Train

The suspension system of the Electromagnetic Suspension (EMS) maglev train is crucial for ensuring safe operation. This article focuses on data-driven modeling and control optimization of the suspension system. By the Extended Dynamic Mode Decomposition (EDMD) method based on the Koopman theory, the state and input data of the suspension system are collected to construct a high-dimensional linearized model of the system without detailed parameters of the system, preserving the nonlinear characteristics. With the data-driven model, the LQR controller and Extended State Observer (ESO) are applied to optimize the suspension control. Compared with baseline feedback methods, the optimization control with data-driven modeling reduces the maximum system fluctuation by 75.0% in total. Furthermore, considering the high-speed operating environment and vertical dynamic response of the maglev train, a rolling-update modeling method is proposed to achieve online modeling optimization of the suspension system. The simulation results show that this method reduces the maximum fluctuation amplitude of the suspension system by 40.0% and the vibration acceleration of the vehicle body by 46.8%, achieving significant optimization of the suspension control.

Passive control of thermoacoustic instability in a high-efficiency domestic gas stove with fine-tuning burner

Thermoacoustic oscillation is a significant challenge in the development of advanced thermal power and propulsion systems. The present work focuses on the whistling phenomenon in the practical burners of a developing high-efficiency domestic gas stove. The acoustic analysis is conducted by a Helmholtz acoustic solver, identifying the 1/2 wave longitudinal mode in the outer ring or center nozzle and its upstream pipe and estimating the modal eigenfrequencies fa=599Hz and fb=647Hz which are close to the main frequency of the whistling measured in experiments. The sequential flame images captured by a high-speed camera are processed by a dynamic mode decomposition method to illustrate flame structure oscillations at the peak frequency mode. The results verify that thermoacoustic oscillations are the generation mechanism of the whistling and display the periodic fluctuation characteristics of the flame and the spatial distribution of heat release. Moreover, both structural fine-tuning designs help to achieve better suppression for thermoacoustic oscillations and avoid the whistling phenomenon. These structural fine-tuning strategies have been researched for several decades but rarely reported to be applied to industrial burners in previous work. Motivated by this, it will provide a new insight into the investigation of burner structures on thermoacoustic behaviors and the application of these passive control strategies.

Prediction of Turbulent Boundary Layer Flow Dynamics with Transformers

Time-marching of turbulent flow fields is computationally expensive using traditional Computational Fluid Dynamics (CFD) solvers. Machine Learning (ML) techniques can be used as an acceleration strategy to offload a few time-marching steps of a CFD solver. In this study, the Transformer (TR) architecture, which has been widely used in the Natural Language Processing (NLP) community for prediction and generative tasks, is utilized to predict future velocity flow fields in an actuated Turbulent Boundary Layer (TBL) flow. A unique data pre-processing step is proposed to reduce the dimensionality of the velocity fields, allowing the processing of full velocity fields of the actuated TBL flow while taking advantage of distributed training in a High Performance Computing (HPC) environment. The trained model is tested at various prediction times using the Dynamic Mode Decomposition (DMD) method. It is found that under five future prediction time steps with the TR, the model is able to achieve a relative Frobenius norm error of less than 5%, compared to fields predicted with a Large Eddy Simulation (LES). Finally, a computational study shows that the TR achieves a significant speed-up, offering computational savings approximately 53 times greater than those of the baseline LES solver. This study demonstrates one of the first applications of TRs on actuated TBL flow intended towards reducing the computational effort of time-marching. The application of this model is envisioned in a coupled manner with the LES solver to provide few time-marching steps, which will accelerate the overall computational process.

Complex flow field analysis in Multi-Shaft stirred Reactors: Dynamics of Wave-Vortex coupling revealed by POD and DMD methods

Insufficient understanding of complex flow structures in multi-shaft stirred reactors hinders industrial adoption. In this work, Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) methods were used to analyze the Large-Eddy Simulation (LES) results of the stationary and rotating zones for three types of stirred reactors, including multi-shaft configurations. The results indicate that the flexible spatial distribution of the impellers in multi-shaft stirred reactors enhances fluid interactions, leading to the superior performance in energy cascading, flow field self-regulation, and wave-vortex coupling. Frequency analysis of single modes underscores DMD’s efficacy in interpreting complex flow phenomena. Based on this, the DMD modal coefficients were fitted to equations, clarifying the development process of wave-vortex coupling within the stirred reactors. Additionally, by integrating fundamental fluid mechanics equations with the outcomes of the DMD method, a wave-vortex coupling model was developed, which is anticipated to yield a more accurate representation of the flow field.

Entropic Regression Dynamic Mode Decomposition (ERDMD) Discovers Informative Sparse and Nonuniformly Time Delayed Models

In this work, we present a method that determines optimal multistep Dynamic Mode Decomposition (DMD) models via Entropic Regression (ER), which is a nonlinear information flow detection algorithm. Motivated by the Higher-Order DMD (HODMD) method of [Le Clainche & Vega, 2017], and the ER technique for network detection and model construction found in [Sun et al., 2015; AlMomani et al., 2020], we develop a method that we call ERDMD, which produces high fidelity time-delay DMD models that allow for nonuniformity in the delays. This optimal choice of delays is discovered by maximizing informativity as measured through ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of DMD.

Data-driven linearization of dynamical systems

Dynamic mode decomposition (DMD) and its variants, such as extended DMD (EDMD), are broadly used to fit simple linear models to dynamical systems known from observable data. As DMD methods work well in several situations but perform poorly in others, a clarification of the assumptions under which DMD is applicable is desirable. Upon closer inspection, existing interpretations of DMD methods based on the Koopman operator are not quite satisfactory: they justify DMD under assumptions that hold only with probability zero for generic observables. Here, we give a justification for DMD as a local, leading-order reduced model for the dominant system dynamics under conditions that hold with probability one for generic observables and non-degenerate observational data. We achieve this for autonomous and for periodically forced systems of finite or infinite dimensions by constructing linearizing transformations for their dominant dynamics within attracting slow spectral submanifolds (SSMs). Our arguments also lead to a new algorithm, data-driven linearization (DDL), which is a higher-order, systematic linearization of the observable dynamics within slow SSMs. We show by examples how DDL outperforms DMD and EDMD on numerical and experimental data.

Research on the Flow-Induced Vibration of Cylindrical Structures Using Lagrangian-Based Dynamic Mode Decomposition

An oscillating flow past a structure represents a complex, high-dimensional, and nonlinear flow phenomenon, which can lead to the failure of structures due to material fatigue or constraint relaxation. In order to better understand flow-induced vibration (FIV) and coupled flow fields, a numerical simulation of a two-degrees-of-freedom FIV in a cylinder was conducted. Based on the Lagrangian-based dynamic mode decomposition (L-DMD) method, the vorticity field and motion characteristics of a cylinder were decomposed, reconstructed, and predicted. A comparison was made to the traditional Eulerian-based dynamic mode decomposition (E-DMD) method. The research results show that the first-order mode in the stable phase represents the mean flow field, showcasing the slander tail vortex structure during the vortex-shedding period and the average displacement in the in-line direction. The second mode predominantly captures the crossflow displacement, with a frequency of approximately 0.43 Hz, closely matching the corresponding frequency observed in the CFD results. The higher dominant modes mainly capture outward-spreading, smaller-scale vortex structures with detail displacement characteristics. The motion of the cylinder in the in-line direction was accompanied by symmetric vortex structures, while the motion of the cylinder in the crossflow direction was associated with anti-symmetric vortex structures. Additionally, crossflow displacement will cause a symmetrical vortex structure that spreads laterally along the axis behind the cylinder. Finally, when compared with E-DMD, the L-DMD method demonstrates a notable advantage in analyzing the nonlinear characteristics of FIV.

Experimental investigation on aerodynamic noise of a small-scale multi-blade centrifugal fan

In this study, the aerodynamic noise of a small-scale centrifugal fan was experimentally investigated using acoustic testing techniques and time-resolved stereoscopic particle image velocimetry (SPIV). During the acoustic experiments, both far-field noise and near-field pressure fluctuations of the test fan were measured. The overall far-field noise towards the fan inlet side was found to be higher than that of the back side. The pressure fluctuations on the fan upper casing exceeded those on the side wall due to the uncontracted volute tongue, indicating pronounced flow-to-wall interactions. Moreover, based on a simultaneous measurement, the coherence between the near-field pressure fluctuations and far-field noise highlighted the significant contributions of impeller rotation to noise radiation. SPIV measurements uncovered the time-averaged and transient flow fields at the fan's inlet and outlet. The time-averaged results demonstrated the concentrated inlet flow and outlet flow separation, leading to high flow unsteadiness. Transient flow fields displayed an asymmetric jet-wake region characterized by both quasi-steady flow and rotational flow behaviours. The instantaneous flow results were analyzed using the dynamic mode decomposition (DMD) method, which clearly recognized the jet-wake patterns with frequencies corresponding to the rotational frequency. The observed consistency in frequency characteristics among noise, pressure fluctuations, and unsteady flow affirms that flow dynamics are crucial to the primary noise mechanisms.

Investigation of Reynolds number effects on flow stability of a transonic compressor using a dynamic mode decomposition method

With the continuous increase in aircraft flight altitude, low Reynolds number effects in high-altitude environments significantly impact the stability of compressor operation. In this study, dynamic mode decomposition (DMD) of the flow field at various time instances near stall conditions in a transonic compressor was performed to investigate the influence of Reynolds number (Re) effects on critical flow characteristics and instability mechanisms. The results indicate that under different Reynolds number index (RNI) conditions near stall conditions, the frequency of tip leakage flow varies. Under high RNI condition, the influence of tip leakage flow is concentrated mainly from the leading edge to the mid-chord. The stall mode is primarily caused by the interaction of shock with the Tip Leakage Vortex, resulting in the breakdown of the vortex. The development of vortex structures inside the passage leads to a delay in the flow field response to blade passing. Under low RNI condition, the tip clearance leakage flow is stronger, with a relatively reduced influence from the leading edge to the mid-chord and an increased impact from the mid-chord to the trailing edge. The main cause of the stall mode is the circumferential movement of vortex structures, causing secondary leakage over adjacent blade tip clearances. The blockage formed by vortex structures within the passage is weaker, and the flow field response to blade passing remains consistent with the blade passing frequency. However, as stall develops further, the flow field response significantly decreases.

PCDMD: Physics-constrained dynamic mode decomposition for accurate and robust forecasting of dynamical systems with imperfect data and physics

The dynamic mode decomposition (DMD) method has attracted widespread attention as a representative modal-decomposition method and can build a predictive model. However, DMD may give predicted results that deviate from physical reality in some scenarios, such as dealing with translation problems or noisy data. Here, we propose a physics-constrained DMD (PCDMD) method to address this issue. The proposed PCDMD method first employs a data-driven model using DMD, then calculates the residual of the physical equations, and finally corrects the predicted results using Kalman filter and gain coefficients. In this way, the PCDMD method can integrate the physics-informed equations with the data-driven model generated by DMD. Numerical experiments are conducted using PCDMD, including the Allen–Cahn, advection-diffusion, Burgers' equations and lid-driven cavity flow. The results demonstrate that the proposed PCDMD method can reduce the reconstruction and prediction errors by 1%-10% by incorporating physical constraints. Regarding noisy datasets and imperfect physical constraints, PCDMD can still ensure that the predicted results satisfy the physical constraints, thereby reducing errors. Program summaryProgram title: PCDMDDataset link:https://github.com/YinYuhuiTJU/PCDMDLicensing provisions: MITProgramming language: Python

Non-intrusive reduced-order model for time-dependent stochastic partial differential equations utilizing dynamic mode decomposition and polynomial chaos expansion

In this study, we present a novel non-intrusive reduced-order model (ROM) for solving time-dependent stochastic partial differential equations (SPDEs). Utilizing proper orthogonal decomposition (POD), we extract spatial modes from high-fidelity solutions. A dynamic mode decomposition (DMD) method is then applied to vertically stacked matrices of projection coefficients for future prediction of coefficient fields. Polynomial chaos expansion (PCE) is employed to construct a mapping from random parameter inputs to the DMD-predicted coefficient field. These lead to the POD–DMD–PCE method. The innovation lies in vertically stacking projection coefficients, ensuring time-dimensional consistency in the coefficient matrix for DMD and facilitating parameter integration for PCE analysis. This method combines the model reduction of POD with the time extrapolation strengths of DMD, effectively recovering field solutions both within and beyond the training time interval. The efficiency and time extrapolation capabilities of the proposed method are validated through various nonlinear SPDEs. These include a reaction–diffusion equation with 19 parameters, a two-dimensional heat equation with two parameters, and a one-dimensional Burgers equation with three parameters.

Experimental study on dynamic characteristics of shock train in an isolator with different types of incident shocks

The hypersonic mixed-compression inlet, operating in an off-design state, can be categorized into low-speed and over-speed regimes based on whether the external compression shock is incident into the internal flow channel. In this study, we investigate the movement process of the shock train within an inlet/isolator under both low-speed and over-speed conditions by generating various incident shocks using wedges installed in a direct-connected ground wind tunnel. Experimental investigations are conducted to examine the dynamic characteristics of the shock train in an isolator subjected to different types of incident shocks at an incoming Mach number of 2.7. The findings reveal that varying levels of backpressure resistance for the shock train are observed with different types of incident shocks. Through the movement trajectory of the shock train leading shock (STLS) and power spectral density analysis, it is found that unilateral incident shocks result in a more intense oscillation process for the shock train with a lower dominant oscillation frequency. The dynamic mode decomposition method identifies different oscillation structures within the unsteady shock train flow field and highlights that dominant mode energy primarily concentrates at the STLS, while its symmetry is influenced by the type of incident shock. Specifically, the symmetric bilateral incident shock tends to promote a higher degree of symmetry in the STLS structure while reducing its oscillation strength; however, when the STLS passes over the reflection point of the incident shock, the rapid upstream movement of the shock train still occurs in this situation, thereby inducing inlet unstart and compromising engine safety.

Mode decomposition of core dynamics transients using higher-order DMD method

Accurately predicting three-dimensional power distribution within a reactor core during reactivity transients is crucial for optimizing reactor operational control. This paper proposes a higher-order dynamic mode decomposition (DMD) method for analyzing highly nonlinear dynamic systems in nuclear reactors. Through analysis of the 3-D LWR Core Transient Benchmark (3DLWRCT) with rod ejection accident, traditional DMD reconstruction techniques were found inadequate in accurately capturing the system’s dynamic evolution using the given simulation data, primarily due to the nonlinear features and local high-order characteristics of the reactor system. In contrast, the HODMD approach demonstrated its effectiveness in predicting future variations in the core’s state. These findings establish the feasibility of employing the HODMD method for transient analysis of core dynamics, leading to promising prospects for dynamic analysis in nuclear energy systems. Future research directions for improving the reconstructive accuracy of HODMD include capturing nonlinear characteristics, automating parameter selection, enabling multi-scale analysis, enhancing anomaly detection capabilities, and providing a broader context for analysis.

Assessment of ZWD field predictions using the dynamic mode decomposition method

Assessment of ZWD field predictions using the dynamic mode decomposition method

Effect of micro-clearance structure on the collapse of individual liquid hydrogen bubbles

Cavitation occurs in the micro-clearance of liquid-hydrogen-lubricated bearings owing to the pressure drop caused by high-speed shearing. The pressure undulation caused by cavitation collapse results in bearing surface erosion and significantly affects the bearing performance. In this study, a modified Z-G-B cavitation model was used to study the crushing process of a single liquid hydrogen bubble in a shear micro-clearance. Fast Fourier transform (FFT) and wavelet transform (WT) were applied to study the frequency characteristics of the pressure, mass transfer rate, and vapor mass fraction during bubble rupture in shearing micro-clearance. To obtain a deeper insight into the details of the effect of the shear micro-clearance structure on bubble collapse, the relationship between the flow field energy, attenuation rate, and frequency was investigated. The proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) methods were used to analyze the energy of each order mode of the flow field. The analysis results of the bubble vibration intensity with respect to time and frequency provide a theoretical basis for the optimization of the bearing structure.