Examining the spatiotemporal coherence structures constitutes a significant issue in investigating the hydrodynamics of spouted fluidized beds. This study employed data-driven methods of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) to identify and analyze the coherent structures presented in a spouted fluidized bed. It was found that the POD modes have higher defined energy and richer structure than DMD modes. The mean modes of POD and DMD exhibit reflectional symmetry, similar to the time-averaged flow field from the computational fluid dynamics–discrete element method simulation. Under the same energy ratio criterion, fewer spatial modes are required for POD reconstruction than for DMD reconstruction. POD is more suitable for low-dimensional reconstruction of the instantaneous flow field for the spouted fluidized bed, while DMD excels at constructing the mean flow field and the detailed instantaneous flow field. This study provides valuable insights into the coherent structures and flow field reconstruction in spouted fluidized beds.
REFERENCES
1.
J.
Link
, L.
Cuypers
, N.
Deen
, and J.
Kuipers
, “Flow regimes in a spout–fluid bed: A combined experimental and simulation study
,” Chem. Eng. Sci.
60
, 3425
–3442
(2005
).2.
J.
Wang
, “Continuum theory for dense gas-solid flow: A state-of-the-art review
,” Chem. Eng. Sci.
215
, 115428
(2020
).3.
Y.
Tsuji
, T.
Kawaguchi
, and T.
Tanaka
, “Discrete particle simulation of two-dimensional fluidized bed
,” Powder Technol.
77
, 79
–87
(1993
).4.
M. A.
van der Hoef
, M.
Ye
, M.
van Sint Annaland
, A.
Andrews
, S.
Sundaresan
, and J.
Kuipers
, “Multiscale modeling of gas-fluidized beds
,” Adv. Chem. Eng.
31
, 65
–149
(2006
).5.
W.
Bian
, X.
Chen
, and J.
Wang
, “A critical comparison of two-fluid model, discrete particle method and direct numerical simulation for modeling dense gas-solid flow of rough spheres
,” Chem. Eng. Sci.
210
, 115233
(2019
).6.
W.
Ge
, L.
Guo
, X.
Liu
, F.
Meng
, J.
Xu
, W. L.
Huang
, and J.
Li
, “Mesoscience-based virtual process engineering
,” Comput. Chem. Eng.
126
, 68
–82
(2019
).7.
J.
Xu
, P.
Zhao
, Y.
Zhang
, J.
Wang
, and W.
Ge
, “Discrete particle method for engineering simulation: Reproducing mesoscale structures in multiphase systems
,” Resour. Chem. Mater.
1
, 69
–79
(2022
).8.
P. J.
Schmid
, “Data-driven and operator-based tools for the analysis of turbulent flows
,” in Advanced Approaches in Turbulence
(Elsevier
, 2021
), pp. 243
–305
.9.
J. L.
Lumley
, “The structure of inhomogeneous turbulent flows
,” Atmos. Turbul. Radio Wave Propag.
1
, 166
–178
(1967
).10.
11.
G.
Berkooz
, P.
Holmes
, and J. L.
Lumley
, “The proper orthogonal decomposition in the analysis of turbulent flows
,” Annu. Rev. Fluid Mech.
25
, 539
–575
(1993
).12.
R.
Reichert
, F.
Hatay
, S.
Biringen
, and A.
Huser
, “Proper orthogonal decomposition applied to turbulent flow in a square duct
,” Phys. Fluids
6
, 3086
–3092
(1994
).13.
P.
Druault
, J.
Delville
, and J.-P.
Bonnet
, “Proper orthogonal decomposition of the mixing layer flow into coherent structures and turbulent Gaussian fluctuations
,” C. R. Mec.
333
, 824
–829
(2005
).14.
S.
Roudnitzky
, P.
Druault
, and P.
Guibert
, “Proper orthogonal decomposition of in-cylinder engine flow into mean component, coherent structures and random Gaussian fluctuations
,” J. Turbul.
7
, N70
–20
(2006
).15.
P.
Holmes
, J. L.
Lumley
, G.
Berkooz
, and C. W.
Rowley
, Turbulence, Coherent Structures, Dynamical Systems and Symmetry
(Cambridge University Press
, Cambridge
, 2012
).16.
B.
Podvin
and Y.
Fraigneau
, “A few thoughts on proper orthogonal decomposition in turbulence
,” Phys. Fluids
29
, 020709
(2017
).17.
Y.
Zheng
, D.
Zhang
, H.
Rinoshika
, and A.
Rinoshika
, “Continuous wavelet analysis and proper orthogonal decomposition on particle dynamics in a horizontal self-exited gas-solid two-phase pipe flow
,” Powder Technol.
408
, 117746
(2022
).18.
S.
Li
, G.
Duan
, and M.
Sakai
, “Development of a reduced-order model for large-scale Eulerian–Lagrangian simulations
,” Adv. Powder Technol.
33
, 103632
(2022
).19.
A.
Palacios
, C.
Finney
, P.
Cizmas
, S.
Daw
, and T. O.
Brien
, “Experimental analysis and visualization of spatiotemporal patterns in spouted fluidized beds
,” Chaos
14
, 499
–509
(2004
).20.
J.
Sun
and Y.
Yan
, “Characterization of flow intermittency and coherent structures in a gas–solid circulating fluidized bed through electrostatic sensing
,” Ind. Eng. Chem. Res.
55
, 12133
–12148
(2016
).21.
M. R.
Haghgoo
, D. J.
Bergstrom
, and R. J.
Spiteri
, “Analyzing dominant particle-flow structures inside a bubbling fluidized bed
,” Int. J. Heat Fluid Flow
77
, 232
–241
(2019
).22.
J.
Higham
, M.
Shahnam
, and A.
Vaidheeswaran
, “Using a proper orthogonal decomposition to elucidate features in granular flows
,” Granular Matter
22
, 1
–13
(2020
).23.
J. R.
Williams
and N.
Rege
, “Coherent vortex structures in deforming granular materials
,” Mech. Cohesive-Frict. Mater.
2
, 223
–236
(1997
).24.
J.
Higham
, W.
Brevis
, and C.
Keylock
, “A rapid non-iterative proper orthogonal decomposition based outlier detection and correction for PIV data
,” Meas. Sci. Technol.
27
, 125303
(2016
).25.
C. W.
Rowley
, I.
Mezić
, S.
Bagheri
, P.
Schlatter
, and D. S.
Henningson
, “Spectral analysis of nonlinear flows
,” J. Fluid Mech.
641
, 115
–127
(2009
).26.
P. J.
Schmid
, “Dynamic mode decomposition of numerical and experimental data
,” J. Fluid Mech.
656
, 5
–28
(2010
).27.
J. N.
Kutz
, S. L.
Brunton
, B. W.
Brunton
, and J. L.
Proctor
, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems
(SIAM
, Philadelphia
, 2016
).28.
P. J.
Schmid
, “Dynamic mode decomposition and its variants
,” Annu. Rev. Fluid Mech.
54
, 225
–254
(2022
).29.
J. L.
Proctor
, S. L.
Brunton
, and J. N.
Kutz
, “Dynamic mode decomposition with control
,” SIAM J. Appl. Dyn. Syst.
15
, 142
–161
(2016
).30.
P. J.
Baddoo
, B.
Herrmann
, B. J.
McKeon
, J. N.
Kutz
, and S. L.
Brunton
, “Physics-informed dynamic mode decomposition
,” Proc. R. Soc. A
479
, 20220576
(2023
).31.
D.
Li
, B.
Zhao
, S.
Lu
, and J.
Wang
, “Physics-informed dynamic mode decomposition for short-term and long-term prediction of gas-solid flows
,” Chem. Eng. Sci.
289
, 119849
(2024
).32.
D.
Li
, B.
Zhao
, S.
Lu
, and J.
Wang
, “A data-driven method for fast predicting the long-term hydrodynamics of gas–solid flows: Optimized dynamic mode decomposition with control
,” Phys. Fluids
36
, 103332
(2024
).33.
P.
Cizmas
, A.
Palacios
, T.
O'Brien
, and M.
Syamlal
, “Proper-orthogonal decomposition of spatio-temporal patterns in fluidized beds
,” Chem. Eng. Sci.
58
, 4417
–4427
(2003
).34.
T.
Yuan
, P.
Cizmas
, and T. O.
Brien
, “A reduced-order model for a bubbling fluidized bed based on proper orthogonal decomposition
,” Comput. Chem. Eng.
30
, 243
–259
(2005
).35.
P. G.
Cizmas
, B. R.
Richardson
, T. A.
Brenner
, T. J. O.
Brien
, and R. W.
Breault
, “Acceleration techniques for reduced-order models based on proper orthogonal decomposition
,” J. Comput. Phys.
227
, 7791
–7812
(2008
).36.
D.
Li
, B.
Zhao
, and J.
Wang
, “Data-driven identification of coherent structures in gas–solid system using proper orthogonal decomposition and dynamic mode decomposition
,” Phys. Fluids
35
, 013321
(2023
).37.
A.
Chatterjee
, “An introduction to the proper orthogonal decomposition
,” Curr. Sci.
78
, 808
–817
(2000
).38.
G. W.
Stewart
, “On the early history of the singular value decomposition
,” SIAM Rev.
35
, 551
–566
(1993
).39.
I.
Mezić
, “Analysis of fluid flows via spectral properties of the Koopman operator
,” Annu. Rev. Fluid Mech.
45
, 357
–378
(2013
).40.
S. L.
Brunton
, M.
Budišić
, E.
Kaiser
, and J. N.
Kutz
, “Modern Koopman theory for dynamical systems
,” SIAM Rev.
64
, 229
–340
(2022
).41.
J. H.
Tu
, “Dynamic mode decomposition: Theory and applications
,” Ph.D. thesis (Princeton University
, Princeton
, 2013
).42.
J.
Kou
and W.
Zhang
, “An improved criterion to select dominant modes from dynamic mode decomposition
,” Eur. J. Mech. B/Fluids
62
, 109
–129
(2017
).43.
T.
Holzmann
, see https://holzmann-cfd.com/ for “Mathematics, Numerics, Derivations and OpenFOAM®
” (2025
; last accessed on April 06, 2025).44.
L.
Lu
, J.
Xu
, W.
Ge
, Y.
Yue
, X.
Liu
, and J.
Li
, “EMMS-based discrete particle method (EMMS–DPM) for simulation of gas–solid flows
,” Chem. Eng. Sci.
120
, 67
–87
(2014
).45.
L.
Lu
, J.
Xu
, W.
Ge
, G.
Gao
, Y.
Jiang
, M.
Zhao
, X.
Liu
, and J.
Li
, “Computer virtual experiment on fluidized beds using a coarse-grained discrete particle method—EMMS-DPM
,” Chem. Eng. Sci.
155
, 314
–337
(2016
).46.
Y.
Zhang
, Y.
Zhao
, L.
Lu
, W.
Ge
, J.
Wang
, and C.
Duan
, “Assessment of polydisperse drag models for the size segregation in a bubbling fluidized bed using discrete particle method
,” Chem. Eng. Sci.
160
, 106
–112
(2017
).47.
J.
Xu
, X.
Liu
, S.
Hu
, and W.
Ge
, “Virtual process engineering on a 3D circulating fluidized bed with multi-scale parallel computation
,” J. Adv. Manuf. Process.
1
, e10014
(2019
).48.
Y.
Zhang
, Y.
Zhao
, Z.
Gao
, C.
Duan
, J.
Xu
, L.
Lu
, J.
Wang
, and W.
Ge
, “Experimental and Eulerian-Lagrangian-Lagrangian study of binary gas-solid flow containing particles of significantly different sizes
,” Renewable Energy
136
, 193
–201
(2019
).49.
B.
Lan
, J.
Xu
, P.
Zhao
, Z.
Zou
, Q.
Zhu
, and J.
Wang
, “Long-time coarse-grained CFD-DEM simulation of residence time distribution of polydisperse particles in a continuously operated multiple-chamber fluidized bed
,” Chem. Eng. Sci.
219
, 115599
(2020
).50.
X.
Liu
, J.
Xu
, W.
Ge
, B.
Lu
, and W.
Wang
, “Long-time simulation of catalytic MTO reaction in a fluidized bed reactor with a coarse-grained discrete particle method—EMMS-DPM
,” Chem. Eng. J.
389
, 124135
(2020
).51.
Y.
Zhang
, Y.
Jia
, J.
Xu
, J.
Wang
, C.
Duan
, W.
Ge
, and Y.
Zhao
, “CFD intensification of coal beneficiation process in gas-solid fluidized beds
,” Chem. Eng. Process. Process Intensif.
148
, 107825
(2020
).52.
J.
Wang
, P.
Zhao
, and B.
Zhao
, “Supersonic and near-equilibrium gas-driven granular flow
,” Phys. Fluids
32
, 113302
(2020
).53.
P.
Zhao
, J.
Xu
, W.
Ge
, and J.
Wang
, “A CFD-DEM-IBM method for Cartesian grid simulation of gas-solid flow in complex geometries
,” Chem. Eng. J.
389
, 124343
(2020
).54.
P.
Zhao
, J.
Xu
, X.
Liu
, W.
Ge
, and J.
Wang
, “A computational fluid dynamics-discrete element-immersed boundary method for Cartesian grid simulation of heat transfer in compressible gas–solid flow with complex geometries
,” Phys. Fluids
32
, 103306
(2020
).55.
P.
Zhao
, J.
Xu
, Q.
Chang
, W.
Ge
, and J.
Wang
, “Euler-Lagrange simulation of dense gas-solid flow with local grid refinement
,” Powder Technol.
399
, 117199
(2022
).56.
P.
Zhao
, J.
Xu
, B.
Zhao
, D.
Li
, and J.
Wang
, “Cartesian grid simulation of reacting gas-solid flow using CFD-DEM-IBM method
,” Powder Technol.
407
, 117651
(2022
).57.
S.
Lu
, B.
Lan
, J.
Xu
, B.
Zhao
, Z.
Zou
, J.
Wang
, H.
Li
, and Q.
Zhu
, “Optimization of multiple-chamber fluidized beds using coarse-grained CFD-DEM simulations: Regulation of solids back-mixing
,” Powder Technol.
428
, 118886
(2023
).58.
B.
Lan
, J.
Xu
, S.
Lu
, Y.
Liu
, F.
Xu
, B.
Zhao
, Z.
Zou
, M.
Zhai
, and J.
Wang
, “Direct reduction of iron-ore with hydrogen in fluidized beds: A coarse-grained CFD-DEM-IBM study
,” Powder Technol.
438
, 119624
(2024
).59.
S.
Lu
, B.
Lan
, D.
Li
, C.
Fan
, J.
Xu
, B.
Zhao
, Z.
Zou
, H.
Li
, J.
Wang
, and Q.
Zhu
, “Regulation characteristics and law of residence time distribution of polydisperse particles in numbered-up multiple-chamber fluidized bed reactors
,” Powder Technol.
439
, 119733
(2024
).60.
W.
Li
, Y.
Liu
, J.
Luo
, Q.
Zhou
, B.
Zhao
, and J.
Wang
, “Fast simulation of industrial-scale bubbling fluidized beds using mesoscience-based structural model
,” Chem. Eng. Sci.
288
, 119770
(2024
).61.
Y.
Yue
, T.
Wang
, M.
Sakai
, and Y.
Shen
, “Particle-scale study of spout deflection in a flat-bottomed spout fluidized bed
,” Chem. Eng. Sci.
205
, 121
–133
(2019
).62.
B. R.
Noack
, K.
Afanasiev
, M.
Morzyński
, G.
Tadmor
, and F.
Thiele
, “A hierarchy of low-dimensional models for the transient and post-transient cylinder wake
,” J. Fluid Mech.
497
, 335
–363
(2003
).63.
L.
Sirovich
, “Turbulence and the dynamics of coherent structures. I-III. Coherent structures
,” Q. Appl. Math.
45
, 561
–571
(1987
).64.
D.
Gidaspow
, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions
(Academic Press
, New York
, 1994
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
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