Application of the dynamic mode decomposition to experimental data

The dynamic mode decomposition (DMD) is a data-decomposition technique that allows the extraction of dynamically relevant flow features from time-resolved experimental (or numerical) data. It is based on a sequence of snapshots from measurements that are subsequently processed by an iterative Krylov technique. The eigenvalues and eigenvectors of a low-dimensional representation of an approximate inter-snapshot map then produce flow information that describes the dynamic processes contained in the data sequence. This decomposition technique applies equally to particle-image velocimetry data and image-based flow visualizations and is demonstrated on data from a numerical simulation of a flame based on a variable-density jet and on experimental data from a laminar axisymmetric water jet. In both cases, the dominant frequencies are detected and the associated spatial structures are identified.     

Data-driven feature identification and sparse representation of turbulent flows

Identifying coherent structures in fluid flows is of great importance for reduced order modelling and flow control. However, extracting such structures from experimental or numerical data obtained from a turbulent flow can be challenging. A number of modal decomposition algorithms have been proposed in recent years which decompose time-resolved snapshots of data into spatial modes, each associated with a single frequency and growth-rate. Most prominently among them is dynamic mode decomposition (DMD). However, DMD-like algorithms create an arbitrary number of modes. It is common practice to then choose a smaller subset of these modes, for the purpose of model reduction and analysis, based on some measure of significance. In this work, we present a method of post-processing DMD modes for extracting a small number of dynamically relevant modes. We achieve this through an iterative approach based on the graph-theoretic notion of maximal cliques to identify clusters of modes and representing each cluster with a single representative mode.     

Visualization of coherent structure in scalar fields of unsteady jet flows with interferometric tomography and proper orthogonal decomposition

Proper orthogonal decomposition (POD) is a method of examining spatial coherence in unsteady flow fields from an ensemble of multidimensional measurements. When applied to experimental data, the proper orthogonal decomposition is generally restricted to data sets with low spatial resolution. This is because of the inherent difficulties in generating an ensemble of measurements that contain a large number of data points. In this paper, a system for obtaining a large ensemble of three-dimensional scalar measurements using interferometric tomography is presented. The proper orthogonal decomposition is applied in three spatial dimensions to experimental data of two jet-like flows. The coherent structure present in the near field of a neutrally buoyant, helium–argon jet and the far field of a buoyant helium jet into air is visualized. The POD results of the helium–argon jet clearly reveal the breakdown region of a sequence of vortex rings and a large-scale flapping motion in the jet far field. The POD of the buoyant helium jet shows a number of competing modes with varying degrees of helicity. Received: 14 January 2000/Accepted: 26 September 2000     

Three-dimensional flow structure in highly buoyant jet by scanning stereo PIV combined with POD analysis

The flow characteristics and the structure of highly buoyant jet of low density fluid issuing into a stagnant surrounding of high density fluid is studied by scanning stereo PIV combined with proper orthogonal decomposition (POD) analysis. The experiment is carried out at Froude number of 0.3 and Reynolds number of 200, which satisfies the inflow condition due to the unstable density gradient near the nozzle exit. An increase in the maximum mean velocity occurs and the vertical velocity fluctuation is highly amplified near the nozzle exit, which suggests the influence of inflow due to the unstable density gradient. The POD analysis indicates that the vertical velocity fluctuation is the major source of fluctuating energy contributing to the development of the highly buoyant jet. The examination of the POD modes show that the longitudinal structure of the vertical velocity fluctuation is generated along the jet axis having the opposite sign of velocity fluctuation on both sides of the jet axis. The vertical scale of the POD mode decreases with increasing the mode number and results in the frequent appearance of cross-flow across the buoyant jet. The reconstruction flow from the POD modes indicates that the vortex structure is caused by the highly sheared layer between the upward and downward velocity and the inflow is induced by the vortex structure. The magnitude of the vortex structure seems to be weakened with an increase in the distance from the nozzle and the buoyant jet approaches to an asymptotic state in the further downstream.     

Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes

We present a comparative analysis of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) computed from experimental data of a turbulent, quasi 2-D, confined jet with co-flow (Re?=?11,500, co-flow ratio inner-to-outer flow ≈2:1). The experimental data come from high-speed 2-D particle image velocimetry. The flow is fully turbulent, and it contains geometry-dependent large-scale coherent structures; thus, it provides an interesting benchmark case for the comparison between POD and DMD. In this work, we address issues related to snapshot selections (1), convergence (2) and the physical interpretation (3) of both POD and DMD modes. We found that the convergence of POD modes follows the criteria of statistical convergence of the autocovariance matrix. For the computation of DMD modes, we suggest a methodology based on two criteria: the analysis of the residuals to optimize the sampling parameters of the snapshots, and a time-shifting procedure that allows us to identify the spurious modes and retain the modes that consistently appear in the spectrum. These modes are found to be the ones with nearly null growth rate. We then present the selected modes, and we discuss the way POD and DMD rank them. POD analysis reveals that the most energetic spatial structures are related to the large-scale oscillation of the inner jet (flapping); from the temporal analysis emerges that these modes are associated with a low-frequency peak at St?=?0.02. At this frequency, DMD identifies a similar mode, where oblique structures from the walls appear together with the flapping mode. The second most energetic group of modes identified is associated with shear-layer oscillations, and to a recirculation zone near the inner jet. Temporal analysis of these modes shows that the flapping of the inner jet might be sustained by the recirculation. In the DMD, the shear-layer modes are separated from the recirculation modes. These have large amplitudes in the DMD. In conclusion, the DMD modes with eigenvalues on the unit circle are found to be similar to the most energetic POD modes, although differences appear due to the fact that DMD isolates structures associated with one frequency only.     

Experimental and numerical study on flow characteristics and heat transfer of an oscillating jet in a channel

A fluidic oscillator can produce self-induced and self-sustaining oscillating jet by fluid supply without moving parts. This device has attracted research interest in heat and mass transfer enhancement in recent years. In the current study, a double-feedback fluidic oscillator was numerically investigated based on three-dimensional unsteady Reynolds-averaged Navier-Stokes equations (3D-URANS) while the operating fluid is an incompressible flow. Then, the results were validated with experimental data by two-dimensional time-resolved particle image velocimetry (2D-TR-PIV) and thermographic phosphor thermometry (TPT) for the velocity and temperature field, respectively. A grid sensitivity study was done by comparison of instantaneous and time-averaged flow fields. Additionally, the proper orthogonal decomposition (POD) method was used to find the phase information of the oscillating jet, and fast Fourier transform (FFT) analysis was used to find the frequency of the oscillating jet to validate the numerical results. The effect of the working fluid was also studied. Finally, in order to determine the effect of the Reynolds number on heat transfer enhancement, the Q-criterion was calculated to provide detailed insight into the oscillating mechanism. The results show that the non-dimensional frequency of oscillation is independent of either the working fluid or mass flow rate. Additionally, for a given fluid, increasing Re causes strong vortices and increases the frequency of oscillation. However, the convection heat transfer did not change significantly when varying the mass flow rate because the convection velocity of vortices increases as the mass flow rate is enhanced. A comparison with a free jet reveals that the oscillating jet in a channel is useful in terms of covering a larger area.     

Meandering jets in shallow rectangular reservoirs: POD analysis and identification of coherent structures

The effect of the shallowness on meandering jets in a shallow rectangular reservoir is investigated. Four meandering flows were investigated in an experimental shallow rectangular reservoir. Their boundary conditions were chosen to cover a large range of friction numbers (defined with the sudden expansion width). Due to the unsteady characteristics of the flows, a proper orthogonal decomposition (POD) of the fluctuating part of the surface velocity fields measured using Large-Scale Particle Image Velocity was used for discriminating the flow structures responsible for the meandering of the jet. Less than 1 % of the calculated POD modes significantly contribute to the meandering of the jet, and two types of instability are in competition in such a flow configuration. The sinuous mode is the dominant mode in the flow, and it induces the meandering of the flow, while the varicose mode is a source of local mixing and weakly participates to the flow. The fluctuating velocity fields were then reconstructed using the POD modes corresponding to 80 % of the total mean fluctuating kinetic energy, and the coherent structures were identified using the residual vorticity, their centres being localised using a topology algorithm. The trajectories of the structures centres emphasise that at high friction number the coherent structures are small and laterally paired in the near, middle and far fields of the jet, while with decreasing friction number, the structures merge into large horizontal vortices in the far field of the jet, their trajectories showing more variability in space and time. The analysis of the stability regime finally reveals that the sinuous mode is convectively unstable and may become absolutely unstable at the end of the reservoir when the friction number is small.     

Flow Structures Associated with Turbulent Mixing Noise and Screech Tones in Axisymmetric Jets

A jet from an axisymmetric convergent nozzle is studied at ideal and underexpanded conditions using velocity and acoustic data. Two particle imaging velocimetry setups, a 10 kHz system and a multi-camera configuration, were used to capture near-field velocities while simultaneously sampled with far-field microphones. Proper orthogonal decomposition is performed on the velocity data to extract modes representative of physical processes in the flow. The decomposed velocity fields are then correlated with acoustic data to identify modes related to specific noise spectra. Specifically, four modes are associated with noise production in the sonic plume. Selective flow-field reconstruction is carried out, revealing interesting dynamics associated with loud flow states. In the supersonic case, screech-containing and turbulent mixing modes are isolated. The spatial modes of each data set are then compared for similarities in structures.     

Self-sustained global oscillations in a jet in crossflow

A jet in crossflow with an inflow ratio of 3, based on the maximum velocity of the parabolic jet profile, is studied numerically. The jet is modeled as an inhomogeneous boundary condition at the crossflow wall. We find two fundamental frequencies, pertaining to self-sustained oscillations in the flow, using full nonlinear direct numerical simulation (DNS) as well as a modal decomposition into global linear eigenmodes and proper orthogonal decomposition (POD) modes; a high frequency which is characteristic for the shear-layer vortices and the upright vortices in the jet wake, and a low frequency which is dominant in the region downstream of the jet orifice. Both frequencies can be related to a region of reversed flow downstream of the jet orifice. This region is observed to oscillate predominantly in the wall-normal direction with the high frequency, and in the spanwise direction with the low frequency. Moreover, the steady-state solution of the governing Navier?CStokes equations clearly shows the horseshoe vortices and the corresponding wall vortices further downstream, and the emergence of a distinct counter-rotating vortex pair high in the free stream. It is thus found that neither the inclusion of the jet pipe nor unsteadiness is necessary to generate the characteristic counter-rotating vortex pair.     

Flow regimes formed by a counterflow jet in a supersonic flow

Results on the effect of the dynamic pressure, Mach number, and temperature of a jet injected from a body upstream in a free supersonic flow on the formation of flow regimes are presented. Flow regimes that ensure the greatest decrease in the drag of the body are given, the mechanism of formation of the LPM flow structure is described, and an approximate criterion is found, which allows determination of the range of existence of various modes of jet penetration into the flow.     

Input-output reduced-order modeling of unsteady flow over an airfoil at a high angle of attack based on dynamic mode decomposition with control

Due to the damage caused by stall flutter, the investigation and modeling of the flow over a wind turbine airfoil at high angles of attack are essential. Dynamic mode decomposition (DMD) and dynamic mode decomposition with control (DMDc) are used to analyze unsteady flow and identify the intrinsic dynamics. The DMDc algorithm is found to have an identification problem when the spatial dimension of the training data is larger than the number of snapshots. IDMDc, a variant algorithm based on reduced dimension data, is introduced to identify the precise intrinsic dynamics. DMD, DMDc and IDMDc are all used to decompose the data for unsteady flow over the S809 airfoil that are obtained by numerical simulations. The DMD results show that the dominant feature of a static airfoil is the adjacent shedding vortices in the wake. For an oscillating airfoil, the DMDc results may fail to consider the effect of the input and have an identification problem. IDMDc can alleviate this problem. The dominant IDMDc modes show that the intrinsic flow for the oscillating case is similar to the unsteady flow over the static airfoil. Moreover, the input–output model identified by IDMDc can give better predictions for different oscillating cases than the identified DMDc model.     

Flow focusing and jet instability

流动聚焦是一种有效的微细射流产生方法,其原理可以描述为从毛细管流出的流体由另一种高速运动的流体驱动,经小孔聚焦后形成稳定的锥–射流结构,射流因不稳定性破碎成单分散的液滴.自从1998年流动聚焦被提出以来,陆续发展了单轴流动聚焦、电流动聚焦、复合流动聚焦和微流控流动聚焦等毛细流动技术.这些技术稳定、易操作、没有苛刻的环境条件的要求,能够制备单分散性较好的微纳米量级的液滴、颗粒和胶囊,在科学研究和实际应用中具有重要价值.流动聚焦涉及了多尺度、多界面和多场耦合的复杂流体力学问题,其中稳定的锥形是形成稳定射流的先决条件,过程参数是影响射流界面扰动发展的关键因素,而射流不稳定性分析是揭示射流破碎的最主要理论工具.该文回顾了近二十年来不同结构流动聚焦的研究进展,概述这些技术涉及的过程控制、流动模式、尺度律和不稳定性分析等关键力学问题,总结射流不稳定性的研究方法和已取得的成果,最后展望流动聚焦的研究方向和应用前景.     

流动聚焦及射流不稳定性

流动聚焦是一种有效的微细射流产生方法,其原理可以描述为从毛细管流出的流体由另一种高速运动的流体驱动,经小孔聚焦后形成稳定的锥–射流结构,射流因不稳定性破碎成单分散的液滴.自从1998年流动聚焦被提出以来,陆续发展了单轴流动聚焦、电流动聚焦、复合流动聚焦和微流控流动聚焦等毛细流动技术.这些技术稳定、易操作、没有苛刻的环境条件的要求,能够制备单分散性较好的微纳米量级的液滴、颗粒和胶囊,在科学研究和实际应用中具有重要价值.流动聚焦涉及了多尺度、多界面和多场耦合的复杂流体力学问题,其中稳定的锥形是形成稳定射流的先决条件,过程参数是影响射流界面扰动发展的关键因素,而射流不稳定性分析是揭示射流破碎的最主要理论工具.该文回顾了近二十年来不同结构流动聚焦的研究进展,概述这些技术涉及的过程控制、流动模式、尺度律和不稳定性分析等关键力学问题,总结射流不稳定性的研究方法和已取得的成果,最后展望流动聚焦的研究方向和应用前景.     

The method of dynamic mode decomposition in shallow water and a swirling flow problem

In this paper, a new vector‐filtering criterion for dynamic modes selection is proposed that is able to extract dynamically relevant flow features from dynamic mode decomposition of time‐resolved experimental or numerical data. We employ a novel modes selection criterion in parallel with the classic selection based on modes amplitudes, in order to analyze which of these procedures better highlight the coherent structures of the flow dynamics. Numerical tests are performed on two distinct problems. The efficiency of the proposed criterion is proved in retaining the most influential modes and reducing the size of the dynamic mode decomposition model. By applying the proposed filtering mode technique, the flow reconstruction error is shown to be significantly reduced. Copyright © 2016 John Wiley & Sons, Ltd.     

A CORRELATION EQUATION FOR CALCULATING INCLINED JET PENETRATION LENGTH IN A GAS-SOLID FLUIDIZED BED

Numerical simulation of gas-solid flow in a two-dimensional fluidized bed with an inclined jet was performed. The numerical model is based on the two-fluid model of gas and solids phase in which the solids constitutive equations are based on the kinetic theory of granular flow. The improved ICE algorithm, which can be used for both low and high-velocity fluid flow, were used to solve the model equations. The mechanism of jet formation was analyzed using both numerical simulations and experiments. The emergence and movement of gas bubbles were captured numerically and experimentally. The influences of jet velocity, nozzle diameter, nozzle inclination and jet position on jet penetration length were obtained. A semi-empirical expression was derived and the parameters were correlated from experimental data. The correlation equation, which can be easily used to obtain the inclined jet penetration length, was compared with our experimental data and published correlation equations.     

Mass and momentum transport in the near field of swirling turbulent jets. Effect of swirl rate

Local transport of the flow momentum and scalar admixture in the near-field of turbulent swirling jets (Re = 5,000) has been investigated by using a combination of the particle image velocimetry and planar laser-induced fluorescence methods. Advection and turbulent and molecular diffusions are evaluated based on the measured distributions of the mean velocity and concentration and the Reynolds stresses and fluxes. As has been quantified from the data, the flow swirl intensifies the entrainment of the surrounding fluid and promotes mass and momentum exchange in the outer mixing layer. A superimposed swirl results in the appearance of a wake/recirculation region at the jet axis and, consequently, the formation of an inner shear layer. In contrast to the scalar admixture, the momentum exchange in the inner shear layer is found to be strongly intensified by the swirl. For the jet with the highest considered swirl rate, a substantial portion of the surrounding fluid is found to enter the unsteady central recirculation zone, where it mixes with the jet that is issued from the nozzle. The contribution of the coherent velocity fluctuations, which are induced by large-scale vortex structures, to the turbulent transport has been evaluated based on triple decomposition, which was based on proper orthogonal decomposition analysis of the velocity data sets. For the considered domain of the jet with the highest swirl rate and vortex breakdown, the contributions of detected helical vortex structures, inducing pressing vortex core, to the radial fluxes of the flow momentum and the scalar admixture are found to locally exceed 65% and 80%, respectively.     

Dynamical mode decomposition of Gurney flap wake flow

The present work uses dynamic mode decomposition (DMD) to analyze wake flow of NACA0015 airfoil with Gurney flap. The physics of DMD is first introduced. Then the PIV-measured wake flow velocity field is decomposed into dynamical modes. The vortex shedding pattern behind the trailing edge and its high-order harmonics have been captured with abundant information such as frequency, wavelength and convection speed. It is observed that high-order dynamic modes convect faster than low-order modes; moreover the wavelength of the dynamic modes scales with the corresponding frequency in power law.     

Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density

Optical tomography is applied to the speckle photographic measurement of an asymmetric flow field with variable fluid density. The convolution back projection algorithm is used for obtaining the 3-D density distribution. Noise in the experimental data is reduced by spline smoothing. The method is verified with a steady, laminar, axisymmetric helium jet exhausting vertically into the ambient air, and then applied to a non-axisymmetric helium jet for determining the helium concentration. It is found that speckle photographic recordings are very adequate for tomographic reconstruction, because they provide a high number of data points from each projection. The influence of the limited number of projections on the reconstruction quality is particularly investigated. Dedicated to Professor Dr.-Ing. J. Zierep on the occasion of his 60th birthday