In particle-laden turbulence, the Fourier Lagrangian spectrum of each phase is regularly computed, and analytically derived response functions relate the Lagrangian spectrum of the fluid and the particle phase. However, due to the periodic nature of the Fourier basis, the analysis is restricted to statistically stationary flows. In the present work, utilizing the bases of time-focalized proper orthogonal decomposition (POD), this analysis is extended to temporally non-stationary turbulence. Studying two-way coupled particle-laden decaying homogeneous isotropic turbulence for various Stokes numbers, it is demonstrated that the temporal POD modes extracted from the dispersed phase may be used for the expansion of both fluid and particle velocities. The POD Lagrangian spectrum of each phase may thus be computed from the same set of modal building blocks, allowing the evaluation of response functions in a POD frame of reference. Based on empirical evaluations, a model for response functions in non-stationary flows is proposed. The related energies of the two phases is well approximated by simple analytical expressions dependent on the particle Stokes number. It is found that the analytical expressions closely resemble those derived through the Fourier analysis of statistically stationary flows. These results suggest the existence of an inherent spectral symmetry underlying the dynamical systems consisting of particle-laden turbulence, a symmetry which spans across stationary/non-stationary particle-laden flow states.

1.
Abdelsamie
,
A. H.
and
Lee
,
C.
, “
Decaying versus stationary turbulence in particle-laden isotropic turbulence: Turbulence modulation mechanism
,”
Phys. Fluids
24
,
015106
(
2012
).
2.
Aubry
,
N.
, “
On the hidden beauty of the proper orthogonal decomposition
,”
Theor. Comput. Fluid Dyn.
2
,
339
352
(
1991
).
3.
Aubry
,
N.
,
Guyonnet
,
R.
, and
Lima
,
R.
, “
Spatiotemporal analysis of complex signals: Theory and applications
,”
J. Stat. Phys.
64
,
683
739
(
1991
).
4.
Aubry
,
N.
,
Holmes
,
P.
,
Lumley
,
J. L.
, and
Stone
,
E.
, “
The dynamics of coherent structures in the wall region of a turbulent boundary layer
,”
J. Fluid Mech.
192
,
115
173
(
1988
).
5.
Ayyalasomayajula
,
S.
,
Warhaft
,
Z.
, and
Collins
,
L.
, “
Modeling inertial particle acceleration statistics in isotropic turbulence
,”
Phys. Fluids
20
,
095104
(
2008
).
6.
Berk
,
T.
and
Coletti
,
F.
, “
Dynamics of small heavy particles in homogeneous turbulence: A Lagrangian experimental study
,”
J. Fluid Mech.
917
,
A47
(
2021
).
7.
Boivin
,
M.
,
Simonin
,
O.
, and
Squires
,
K. D.
, “
Direct numerical simulation of turbulence modulation by particles in isotropic turbulence
,”
J. Fluid Mech.
375
,
235
263
(
1998
).
8.
Brandt
,
L.
and
Coletti
,
F.
, “
Particle-laden turbulence: Progress and perspectives
,”
Annu. Rev. Fluid Mech.
54
,
159
189
(
2022
).
9.
Capecelatro
,
J.
and
Desjardins
,
O.
, “
An Euler–Lagrange strategy for simulating particle-laden flows
,”
J. Comput. Phys.
238
,
1
31
(
2013
).
10.
Citriniti
,
J. H.
and
George
,
W. K.
, “
Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition
,”
J. Fluid Mech.
418
,
137
166
(
2000
).
11.
Crowe
,
C. T.
,
Sharma
,
M. P.
, and
Stock
,
D. E.
, “
The particle-source-in cell (PSI-CELL) model for gas-droplet flows
,”
J. Fluids Eng.
99
,
325
332
(
1977
).
12.
Csanady
,
G.
, “
Turbulent diffusion of heavy particles in the atmosphere
,”
J. Atmos. Sci.
20
,
201
208
(
1963
).
13.
Delville
,
J.
,
Ukeiley
,
L.
,
Cordier
,
L.
,
Bonnet
,
J.-P.
, and
Glauser
,
M.
, “
Examination of large-scale structures in a turbulent plane mixing layer. part 1. proper orthogonal decomposition
,”
J. Fluid Mech.
391
,
91
122
(
1999
).
14.
Denner
,
F.
,
Evrard
,
F.
, and
van Wachem
,
B. G.
, “
Conservative finite-volume framework and pressure-based algorithm for flows of incompressible, ideal-gas and real-gas fluids at all speeds
,”
J. Comput. Phys.
409
,
109348
(
2020
).
15.
Druzhinin
,
O.
and
Elghobashi
,
S.
, “
On the decay rate of isotropic turbulence laden with microparticles
,”
Phys. Fluids
11
,
602
610
(
1999
).
16.
Elghobashi
,
S.
, “
On predicting particle-laden turbulent flows
,”
Appl. Sci. Res.
52
,
309
329
(
1994
).
17.
Elghobashi
,
S.
and
Truesdell
,
G.
, “
Direct simulation of particle dispersion in a decaying isotropic turbulence
,”
J. Fluid Mech.
242
,
655
700
(
1992
).
18.
Evrard
,
F.
,
Denner
,
F.
, and
van Wachem
,
B.
, “
Euler-Lagrange modelling of dilute particle-laden flows with arbitrary particle-size to mesh-spacing ratio
,”
J. Comput. Phys.: X
8
,
100078
(
2020
).
19.
Evrard
,
F.
,
Denner
,
F.
, and
van Wachem
,
B.
, “
Quantifying the errors of the particle-source-in-cell Euler-Lagrange method
,”
Int. J. Multiphase Flow
135
,
103535
(
2021
).
20.
Ferrante
,
A.
and
Elghobashi
,
S.
, “
On the physical mechanisms of two-way coupling in particle-laden isotropic turbulence
,”
Phys. Fluids
15
,
315
329
(
2003
).
21.
Glauser
,
M. N.
and
George
,
W. K.
, “
Application of multipoint measurements for flow characterization
,”
Exp. Therm. Fluid Sci.
5
,
617
632
(
1992
).
22.
Good
,
G.
,
Ireland
,
P.
,
Bewley
,
G.
,
Bodenschatz
,
E.
,
Collins
,
L.
, and
Warhaft
,
Z.
, “
Settling regimes of inertial particles in isotropic turbulence
,”
J. Fluid Mech.
759
,
R3
(
2014
).
23.
Gustavsson
,
K.
and
Mehlig
,
B.
, “
Statistical models for spatial patterns of heavy particles in turbulence
,”
Adv. Phys.
65
,
1
57
(
2016
).
24.
Hekmati
,
A.
,
Ricot
,
D.
, and
Druault
,
P.
, “
About the convergence of pod and EPOD modes computed from CFD simulation
,”
Comput. Fluids
50
,
60
71
(
2011
).
25.
Hinze
,
J.
,
Turbulence
, McGraw-Hill Classic Textbook Reissue Series (
McGraw-Hill
,
1975
).
26.
Hodžić
,
A.
,
Olesen
,
P. J.
, and
Velte
,
C. M.
, “
On the discrepancies between pod and Fourier modes on aperiodic domains
,” arXiv:2207.02550 (
2022
).
27.
Iqbal
,
M.
and
Thomas
,
F.
, “
Coherent structure in a turbulent jet via a vector implementation of the proper orthogonal decomposition
,”
J. Fluid Mech.
571
,
281
326
(
2007
).
28.
Ireland
,
P. J.
,
Bragg
,
A. D.
, and
Collins
,
L. R.
, “
The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects
,”
J. Fluid Mech.
796
,
617
658
(
2016a
).
29.
Ireland
,
P. J.
,
Bragg
,
A. D.
, and
Collins
,
L. R.
, “
The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects
,”
J. Fluid Mech.
796
,
659
711
(
2016b
).
30.
Johansson
,
P. B.
,
George
,
W. K.
, and
Woodward
,
S. H.
, “
Proper orthogonal decomposition of an axisymmetric turbulent wake behind a disk
,”
Phys. Fluids
14
,
2508
2514
(
2002
).
31.
Letournel
,
R.
,
Laurent
,
F.
,
Massot
,
M.
, and
Vié
,
A.
, “
Modulation of homogeneous and isotropic turbulence by sub-Kolmogorov particles: Impact of particle field heterogeneity
,”
Int. J. Multiphase Flow
125
,
103233
(
2020
).
32.
Lumley
,
J. L.
, “
The structure of inhomogeneous turbulent flows
,” in
Atmospheric Turbulence and Radio Wave Propagation
(
Nauka
,
1967
), pp.
166
178
.
33.
Lumley
,
J. L.
,
Stochastic Tools in Turbulence
(
Courier Corporation
,
2007
).
34.
Mallouppas
,
G.
,
George
,
W.
, and
van Wachem
,
B.
, “
New forcing scheme to sustain particle-laden homogeneous and isotropic turbulence
,”
Phys. Fluids
25
,
083304
(
2013
).
35.
Mallouppas
,
G.
,
George
,
W.
, and
van Wachem
,
B.
, “
Dissipation and inter-scale transfer in fully coupled particle and fluid motions in homogeneous isotropic forced turbulence
,”
Int. J. Heat Fluid Flow
67
,
74
85
(
2017
).
36.
Maxey
,
M.
, “
Simulation methods for particulate flows and concentrated suspensions
,”
Annu. Rev. Fluid Mech.
49
,
171
193
(
2017
).
37.
Muralidhar
,
S. D.
,
Podvin
,
B.
,
Mathelin
,
L.
, and
Fraigneau
,
Y.
, “
Spatio-temporal proper orthogonal decomposition of turbulent channel flow
,”
J. Fluid Mech.
864
,
614
639
(
2019
).
38.
Salazar
,
J. P.
and
Collins
,
L. R.
, “
Inertial particle acceleration statistics in turbulence: Effects of filtering, biased sampling, and flow topology
,”
Phys. Fluids
24
,
083302
(
2012
).
39.
Schiller
,
L.
and
Naumann
,
A.
, “
Über die grundlegenden berechnungen bei der schwerkraftaufbereitung
,”
Z. Ver. Dtsch. Ing.
77
,
318
320
(
1933
).
40.
Schiødt
,
M.
,
Hodzic
,
A.
,
Evrard
,
F.
,
Hausmann
,
M.
,
van Wachem
,
B.
, and
Velte
,
C. M.
, “
Characterizing Lagrangian particle dynamics in decaying homogeneous isotropic turbulence using proper orthogonal decomposition
,”
Phys. Fluids
34
,
063303
(
2022
).
41.
Schneiders
,
L.
,
Meinke
,
M.
, and
Schröder
,
W.
, “
Direct particle–fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence
,”
J. Fluid Mech.
819
,
188
227
(
2017
).
42.
Sirovich
,
L.
, “
Turbulence and the dynamics of coherent structures. I. Coherent structures
,”
Quart. Appl. Math.
45
,
561
571
(
1987
).
43.
Squires
,
K.
and
Eaton
,
J.
, “
Effect of selective modification of turbulence on two-equation models for particle-laden turbulent flows
,”
J. Fluids Eng.
116
,
778
784
(
1994
).
44.
Sundaram
,
S.
and
Collins
,
L. R.
, “
A numerical study of the modulation of isotropic turbulence by suspended particles
,”
J. Fluid Mech.
379
,
105
143
(
1999
).
45.
Tchen
,
C. M.
, “
Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid
,” Ph.D. thesis (
Delft University
,
1947
).
46.
Toschi
,
F.
and
Bodenschatz
,
E.
, “
Lagrangian properties of particles in turbulence
,”
Annu. Rev. Fluid Mech.
41
,
375
404
(
2009
).
47.
Towne
,
A.
,
Schmidt
,
O. T.
, and
Colonius
,
T.
, “
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
,”
J. Fluid Mech.
847
,
821
867
(
2018
).
48.
Xu
,
Y.
and
Subramaniam
,
S.
, “
Consistent modeling of interphase turbulent kinetic energy transfer in particle-laden turbulent flows
,”
Phys. Fluids
19
,
085101
(
2007
).
49.
Zhang
,
Z.
,
Legendre
,
D.
, and
Zamansky
,
R.
, “
Model for the dynamics of micro-bubbles in high-Reynolds-number flows
,”
J. Fluid Mech.
879
,
554
578
(
2019
).
You do not currently have access to this content.