Proper Orthogonal Modes Research Articles

Experimental study on the unsteady behavior and frequency characteristics of high-speed submerged cavitating water jets

The frequency characteristics of cavitation fluctuations in high-speed cavitating jets are intricate due to the coupling mechanisms of unsteady behaviors. This study employs high-speed photography to experimentally investigate the relationship between frequency characteristics and the unsteady behavior of cavitating jets with various cavitation numbers. Temporal evolution patterns of the cavitating jets are analyzed through spatiotemporal (s-t) diagrams. The spatial distribution and temporal evolution of cavitation fluctuation frequencies are examined using fast Fourier transform (FFT) and continuous wavelet transform (CWT), respectively. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed to identify coherent structures and their corresponding frequencies. In results, the s-t diagrams reveal the distinct regions influenced by cavitation shedding and collapse. FFT results indicate that upstream of the jet trajectory, spectral energy is concentrated in the shedding band, shifting toward lower frequencies with increasing axial distance. The CWT spectrum exhibits a single peak in the upstream, identifying it as the shedding frequency. POD modes associated with shedding dominate the energy contribution at higher cavitation numbers, while they become less prominent at lower cavitation numbers. DMD extracts and identifies coherent structures associated with shedding through frequency-specific decomposition.

Experimental study on vortex-induced vibrations of double-hanger cable: Upstream wake flow induces downstream vibrations

This study is inspired by an experimental observation for vortex-induced vibrations (VIVs) of a double-hanger cable system, in which a smaller vibration of the downwind hanger cable is re-excited over a narrow range of wind speeds beyond the “lock-in” range. This phenomenon is known as the wake vortex-induced vibrations (WVIVs), where the upstream wake flow induces downstream vibrations. To further investigate the characteristics and reasons for WVIVs, a refined wind tunnel test of double-hanger cable was conducted to consider the influence of aerodynamic interaction. The double-hanger cable was modeled by the tandem-arranged spring-mounted cylinders vibrating in two dimensions. The vibration responses of hanger cables were obtained under various wind speeds to reproduce the lock-in phenomenon. In addition, the vibration trajectory, phase relationship, damping ratio, inter-cable correlation, and the wind pressure on the surface of cables were analyzed. Finally, the Stockbridge dampers were designed to suppress the vibrations. The results show that under the aerodynamic interaction of the cables, the onset wind speed of VIVs in the double cables increases, and the downstream cable WVIVs closely follow the VIVs. During the WVIV phase, the downstream cable behavior is characterized by increased negative aerodynamic damping and an inverse displacement correlation between the cables. The phase relationships between the cables are time-varying due to the aerodynamic interaction. The first proper orthogonal decomposition mode of wind pressure dominates the cross-wind of motions and is crucial in vibrations. Stockbridge dampers can effectively reduce the amplitude of VIVs and eliminate WVIVs in cables.

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.

Effects of axisymmetric confinement on vortical structures emanating from round turbulent jets

Confined jets occur in many engineering applications including combustion chambers, jet pumps, and chemical reactors. The effects of axisymmetric confinement on the vortical structures identified in a turbulent jet are investigated using large eddy simulation at a Reynolds number of 30 000 (based on nozzle exit conditions) and expansion ratio (chamber-to-nozzle diameter ratio) of five. The results obtained from the confined jet are compared with those of a free jet under the same nozzle exit flow conditions. A prominent recirculation zone forms between the expanding jet and the confining wall, resulting in early shear layer distortion and a shorter interaction length in the confined jet (0.85 jet diameters) compared to the free jet (1.15 jet diameters). Using the λ2 criterion for vortex identification, two dominant structural modes are identified in the near-exit region of the free jet: ring and helical modes. However, in the confined jet, the helical mode is absent, and the turbulent confined fluid accelerates the breakup of the ring vortices. The interaction of the secondary line vortices with the breaking structures leads to the formation of new hairpin-like vortices, which also contribute to further vortex breakup. These results explain the enhanced mixing performance of confined jets as the mixing is directly tied to the breakup of large vortical structures. Proper orthogonal decomposition modes are also presented to identify the structures/events with the highest contribution to the total turbulent kinetic energy in both flow fields.

Wake Dynamics of Complex Turning Vanes Using Time-Resolved Particle Image Velocimetry Measurements

Abstract The use of turning vanes spans multiple engineering disciplines such as aerospace, ocean, and biomedical to effectively turn an otherwise uniform flowfield and achieve desired downstream flow angles. The presented work investigates the wake dynamics generated by sets of complex turning vanes which contained nonaxisymmetric geometries, spanwise variations in turning angle, and multiple vane junctions. Time-resolved particle image velocimetry (TR-PIV) measurements were performed to collect three-component velocity data downstream of the vane pack geometries. As the vanes contained blunt trailing edges (TEs), large-scale periodic structures (von Kármán vortices) dominated the unsteady wakes. Two postprocessing methods, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), were employed to extract the wake energy or enstrophy content, corresponding spatial modes, and associated frequencies. This was completed for various parameters such as Reynolds number, vane turning angle, and vane trailing edge thickness. Spatial POD analyses showed that zero-turning vanes contained similar dynamics to that of a circular cylinder, and the total wake energy distributions were affected by freestream velocity. A spectral POD analysis in the wake of vane junctions found that the junction flow contained significant coherent content and gave some insight into the mean flow. Finally, vane parameters such as turning angle and TE thickness were found to play a large role in modifying the enstrophy content of the large-scale shedding modes.

Galerkin Model For Time Super-Sampling Of PIV Measurements

We propose a data-driven method for increasing the time resolution of snapshot Particle Image Velocimetry. The flow dynamics are reconstructed by integrating in time an empirical Galerkin model based on Proper Orthogonal Decomposition modes of the flow field, as developed by Noack et al. (2005). The main objective is to obtain a time-resolved description of the flow dynamics of an experimental non-time-resolved dataset. The proposed methodology is assessed both with numerical and experimental data of a wake and a jet flow. The results we obtained indicate that the method can effectively reconstruct the flow dynamics over a duration spanning several flow characteristic times.

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.

Modal decomposition of the shedding mechanism of partial cavitation in a Venturi Tube

Abstract The present paper studies the shedding mechanism of partial cavitation in a Venturi Tube, dominated by re-entrant jet and bubbly shock mechanisms, by using two data-driven modal decomposition methods: proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). According to the snapshot data series obtained by high-speed photography, the modal decomposition and reconstruction of the grey image are carried out. The POD results indicate that the main frequency of the re-entrant jet is higher than that of the shock under the sixth-order mode, and the energy amplitude of the latter is about 20 times that of the former. Furthermore, as the cavitation number increases, the condensation shock mechanism eventually replaces the re-entrant jet mechanism. The DMD results show that the shock behaves obvious traveling wave mode, because the frequency is higher and the phase of the spatial distribution changes evenly under the fourth-order mode. POD and DMD can help to understand the shedding mechanism of partial cavitation.

Reduced-order methods for neutron transport kinetics problem based on proper orthogonal decomposition and dynamic mode decomposition

This work proposes two reduced-order methods to address the high computing resource consumption of neutron transport kinetics problems. The first method is based on proper orthogonal decomposition (POD), which integrates the POD basis with the governing equation. The second method is based on dynamic mode decomposition (DMD), which constructs an approximate linear relationship to predict the time-dependent physical quantities. Numerical results demonstrate that both two reduced-order methods can achieve high accuracy and efficient prediction. For the selected cases, POD can obtain a speed-up ratio of 105 – 429, while DMD can obtain a speed-up ratio of 2300 – 12500. For step transient situations, POD and DMD predict power errors do not exceed 0.42% and 0.05%. For ramp transient situations, POD and DMD predict power errors do not exceed 0.69% and 1.9%. This work can provide some useful suggestions for applications and further development in neutron transport kinetics reduced-order model.

A closure mechanism for screech coupling in rectangular twin jets

The twin-jet configuration allows two different scenarios to close the screech feedback. For each jet, there is one loop involving disturbances which originate in that jet and arrive at its own receptivity point in phase (self-excitation). The other loop is associated with free-stream acoustic waves that radiate from the other jet, reinforcing the self-excited screech (cross-excitation). In this work, the role of the free-stream acoustic mode and the guided-jet mode as a closure mechanism for twin rectangular jet screech is explored by identifying eligible points of return for each path, where upstream waves propagating from such a point arrive at the receptivity location with an appropriate phase relation. Screech tones generated by these jets are found to be intermittent with an out-of-phase coupling as a dominant coupling mode. The instantaneous phase difference between the twin jets computed by the Hilbert transform suggests that a competition between out-of-phase and in-phase coupling is responsible for the intermittency. To model wave components of the screech feedback while ensuring perfect phase locking, an ensemble average of leading spectral proper orthogonal decomposition modes is obtained from several segments of large-eddy simulation data that correspond to periods of invariant phase difference between the two jets. Each mode is then extracted by retaining relevant wavenumber components produced via a streamwise Fourier transform. Spatial cross-correlation analysis of the resulting modes shows that most of the identified points of return for the cross-excitation are synchronised with the guided jet mode self-excitation, supporting that it is preferred in closing rectangular twin-jet screech coupling.

Compressed neural networks for reduced order modeling

Reduced order modeling (ROM) techniques, such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), have been widely used to analyze stationary flows. Neural networks such as autoencoders are effective to reduce the dimension of non-stationary flow, but their larger storage requirements compared to POD and DMD compromise the expectations on ROM. The present work aims at compressing the autoencoder model via two distinctively different approaches, i.e., pruning and singular value decomposition (SVD). The developed algorithm is then applied to reconstruct the flow fields of typical stationary (i.e., a laminar cylinder flow and two turbulent channel flows) and non-stationary (i.e., a laminar co-rotating vortex pair) examples. It is shown that pruning and SVD reduce the size of the autoencoder network to 6% and 3% for the two simple laminar cases (or 18% and 13%, 20%, and 10% for the two complex turbulent channel flow cases), respectively, with approximately the same order of accuracy. Therefore, the proposed autoencoders optimized by the network pruning and SVD lead to effective ROM of both stationary and non-stationary flows although they require more iterations to converge than conventional methods.

Investigation of Unsteady Combustion Regimes in a Kerosene-Fueled Scramjet with Air Throttling

This paper describes an experimental study investigating unsteady combustion regimes in a kerosene-fueled scramjet. The results are obtained under inflow conditions of a 2.9 MPa stagnation pressure, 1900 K stagnation temperature, and a Mach number of 3.0. The air throttling position is 240 mm downstream of the combustor entrance, with an air throttling flow rate (ratio of air throttling mass flux to inflow mass flux) of 38% and a fuel equivalence ratio of 0.37. Combustion is relatively stable when air throttling is applied and is dominated by auto-ignition. When air throttling is turned off, the combustion becomes more unsteady and is dominated by flame propagation. At the same time, the combustion mode changes, and the frequency of the combustion mode transition is 286 Hz. Schlieren images and one-dimension analysis show that the effect of air throttling is the coupling of cold throat (aerodynamic throat) and hot throat (thermal throat). The proper orthogonal decomposition and dynamic mode decomposition analysis present that when air throttling is applied or removed, the frequencies of injector–flame feedback are almost the same, while the frequencies of shock–flame feedback exhibit considerable variation, which is caused by the location of the precombustion shock affected by air throttling.

The reduced-order model of 5 × 5 fuel rod bundles

The fuel rod bundles are the core part of pressurized water reactors (PWRs), and its heat transfer characteristics directly impact the safety of PWRs. A computational fluid dynamics (CFD) model of 5 × 5 fuel rod bundles with a spacer grid is established, and the numerical simulation results are in excellent agreement with the experimental results. Then, the effects of four turbulence models, namely shear stress transport model, standard k–ε model, re-normalization group k–ε model, and realizable k–ε model on the thermal-hydraulic characteristics of the 5 × 5 fuel rod bundles are systematically investigated. Furthermore, two data-driven methods, namely proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), are used to analyze the flow fields of the 5 × 5 fuel rod bundles. The two methods can extract key modes or features to enhance the comprehension and description of the dynamic behaviors within the flow fields of 5 × 5 fuel rod bundles. Finally, two reduced-order models (ROMs), called the POD-radial basis function neural network surrogate model and DMD method, are constructed, which can enable rapid prediction of the flow fields for 5 × 5 fuel rod bundles with high accuracy. The CFD simulation results presented in this paper can provide valuable insights for studying the thermal-hydraulic characteristics of the 5 × 5 fuel rod bundles. The two ROMs proposed in this paper can significantly reduce the computational costs associated with studying the thermal-hydraulic characteristics of 5 × 5 fuel rod bundles.

Scale-adaptive simulation of the separated flow past a 90°-inclined prolate spheroid

The separated flow past a 6:1:1 spheroid is numerically investigated by means of the scale-adaptive simulation technique. The Reynolds number based on the free-stream velocity and the diameter at middle-section of the spheroid is located in the subcritical regime, i.e., Re = 3900. In comparison with the circular cylinder at the same Reynolds number, about 35% drag reduction is acquired by the spheroid, and the fluctuations of lift and drag are suppressed effectively. According to the detailed comparison, the satisfactory drag reduction and suppression of fluctuating force obtained by the spheroid are closely associated with the higher base-pressure and lower turbulent fluctuations in the near wake. Abundant contrasts of the different spanwise sections are presented to reveal the mechanism of constrained flow and apex effect of the spheroid. In addition, in order to provide reliable data for testing and developing turbulence models, a large number of turbulence statistics are computed and compared with previous data of the circular cylinder and sphere at comparable Reynolds numbers. Lower Reynolds stress peaks and less vigorous coherent structures indicate that the three-dimensional force and constrained flow caused by the spheroid can lead to the formation of steady shear layer and vortex separation. Furthermore, proper orthogonal decomposition and dynamic mode decomposition are employed to understand the large-scale wake flow structures behind the spheroid. The modal analysis results confirm that the wake of the spheroid is more stable than the circular cylinder, reconfirming the effective flow control.

Coupled Dynamics of Steady Jet Flow Control for Flexible Membrane Wings

We present a steady jet-flow-based flow control of flexible membrane wings for the adaptive and efficient motion of bat-inspired drones in complex flight environments. A body-fitted variational computational aeroelastic framework is adopted for the modeling of fluid–structure interactions. High-momentum jet flows are injected from the leading edge and transported to the wake flows to alter the aerodynamic performance and the membrane vibration. The coupled dynamic effect of active jet flow control on membrane performance is systematically explored. While the results indicate that the current active flow control strategy performs well at low angles of attack, its effectiveness degrades at high angles of attack with large flow separation. To understand the coupling mechanism, the variations of the vortex patterns are examined by the proper orthogonal decomposition modes, and the fluid transport process is studied by the Lagrangian coherent structures. Two scaling relations that quantitatively connect the membrane deformation with the aerodynamic loads presented in our previous work are verified even when active jet flow control is applied. A unifying feedback loop that reveals the fluid–membrane coupling mechanism is proposed. These findings can facilitate the development of next-generation bio-inspired drones that incorporate smart sensing and intelligent control.

An efficient online successive reanalysis method for dynamic topology optimization

In this study, an efficient reanalysis strategy for dynamic topology optimization is proposed. Compared with other related studies, an online successive dynamic reanalysis method and a Proper Orthogonal Decomposition (POD)-based approximate dynamic displacement strategy are integrated. In the dynamic reanalysis, the storage of the stiffness matrix decomposition can be avoided, and the reduced basis vectors should be successively updated according to the structural state in each iteration. Thus, the bottleneck of the combined approximation method for large-scale dynamic topology optimization can be overcome. Sequentially, POD is applied to obtain the approximate dynamic displacement, in which the Proper Orthogonal Mode (POM) of the displacement field is applied to obtain the approximate equivalent static loads of the Equivalent Static Load (ESL) method. Compared to the exact equivalent static loads at all the time intervals, the number of equivalent static loads is significantly reduced. Finally, the 2D and 3D test results indicate that the proposed method has remarkable speed-up effect under the premise of small relative error, support the strength of the proposed strategy.

Data-driven modal decomposition of R134a refrigerant cavitating flow in Venturi tube

This study utilized high-speed camera and large eddy simulation methods to explore the cavitating flow mechanisms and turbulence structures of R134a refrigerant inside a Venturi tube under varying cavitation numbers (CNs). Data-driven modal analysis approaches, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), were introduced to identify and extract the energy hierarchy and transient characteristics within the cavitating flow. The analysis of grayscale images indicated that the cavitating flow gradually transitioned from quasi-periodic to unsteady flow as the CN decreased, and the severity of cavitation correlates with lower peak frequencies. The POD analysis facilitated the extraction of coherent structures in the cavity's temporal evolution, and the results indicate that the quasi-ordering shedding and collapse of large-scale cavity clouds predominantly occur under low cavitation intensity conditions. As the CN increases, the influence of small-scale cavity shedding becomes more significant. The first 30 most energetic modes occupied over 75% of the entire energy, and they were used to reconstruct the cavitating flow, achieving good consistency with transient flow snapshots. Additionally, the DMD results of the cavitating flow yield three frequency spans, including several prominent characteristic frequencies. These spans are closely linked to the cavity cloud structures of varying scales, unveiling the structural characteristics of unsteady cavitating flow.

Reduced-Order Model Approaches for Predicting Airfoil Performance

This study delves into the construction of reduced-order models (ROMs) of a flow field over a NACA 0012 airfoil at a moderate Reynolds number and an angle of attack of 8∘. Numerical simulations were computed through the finite-volume solver OpenFOAM. The analysis considers two different reduction techniques: the standard Galerkin projection method, which involves projecting the governing equations onto proper orthogonal decomposition modes (POD−ROMs), and the cluster-based network model (CNM), a fully data-driven nonlinear approach. An analysis of the topology of the dominant POD modes was conducted, uncovering a traveling wave pattern in the wake dynamics. We compared the performances of both ROM techniques regarding their prediction of flow field behavior and integral quantities. The ROM framework facilitates the practical actuation of control strategies with significantly reduced computational demands compared to the full-order approach.

Influence mechanism and numerical reconstruction method of complex wind field acting on a large-MW floating wind turbine

The large-scale wind fields show a series of complex characteristics, including wind shear, turbulence intensity, and spatial coherence, which could significantly affect the coupled dynamic responses of floating wind turbines (FWTs). During numerical simulation, to obtain the coupled dynamic responses of large-scale FWTs more accurately, it is necessary to provide a more accurate simulation of the time scale and spatial scale characteristics of actual measured large-scale wind fields. This study investigated the influence mechanism and numerical reconstruction method of a complex wind field acting on a large-MW floating wind turbine. The influence mechanisms were concluded as follows: Wind shear was found to decrease the mean power generation and increase responses at 1P and 3P frequencies. Turbulent winds could significantly decrease the mean response and increase the low-frequency and 3P responses. The difference in spatial coherent structures between different wind fields was clarified by analyzing the proper orthogonal decomposition mode. As spatial coherence strengthened, the mean responses became smaller, the low-frequency responses became larger, and the 1P and 3P responses became smaller. On this basis, a numerical reconstruction method was proposed in this study to reconstruct large-scale numerical wind fields using wind speeds measured at finite spatial points in model tests or offshore measurements. This method accurately simulates the wind profile characteristics, turbulence intensity distribution along the vertical direction, and spatial coherence parameters of the wind field.

Data-driven identification and pressure fields prediction for parallel twin cylinders based on POD and DMD method

The mode analysis of parallel twin cylinders is conducted in this paper using two data-driven methods: proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). First, a high-fidelity computational fluid dynamics (CFD) model of parallel twin cylinders is established, and numerical simulations of the model are carried out. Subsequently, the fundamental principles of the POD and DMD algorithms are systematically introduced. Utilizing snapshots obtained from the high-fidelity CFD model, the POD and DMD methods are employed to extract the dominant flow structures. Furthermore, a comparison between the two data-driven methods is conducted by analyzing modal frequencies, pressure distribution, and the reconstruction errors of pressure fields. Finally, the pressure fields of non-sample points are predicted based on the POD–backpropagation neural network (BPNN) surrogate model and the DMD method, and the predicted results are compared with the CFD simulation results. It found that (i) the DMD method is capable of extracting the main coherent structures of the pressure fields, directly obtaining flow modes and their corresponding frequencies, and assessing the stability of flow modes; (ii) the DMD method can capture the main flow features of the pressure fields in both spatial and temporal dimensions, while the POD method is primarily efficient at capturing the spatial features of the pressure fields; (iii) in contrast to the frequency-ranked DMD method, the energy-ranked POD method can reconstruct the pressure fields using a smaller number of modes, indicating that the POD method has an advantage in terms of mode reduction; (iv) in contrast to the energy-ranked POD method, the frequency-ranked DMD method has a wider applicability to the range of flow types and has more advantages in stability analysis of complex dynamic systems; (v) the predicted pressure fields around the cylinder using the first five-order POD modes or DMD modes closely align with CFD calculation results. Additionally, the evolution of pressure fields predicted by the POD–BPNN surrogate model with the first five-order POD modes or the DMD method with the first 200-order DMD modes significantly agrees with CFD simulation results; (vi) the combined use of the POD–BPNN surrogate model and DMD methods allows efficient interpolation and extrapolation of samples, delivering exceptional predictive performance. This study offers insight into the coherent structures in parallel twin cylinders.