A phase proper orthogonal decomposition (phase POD) method is demonstrated utilizing phase averaging for the decomposition of spatiotemporal behavior of statistically non-stationary turbulent flows in an optimized manner. The proposed phase POD method is herein applied to a periodically forced statistically non-stationary lid-driven cavity flow, implemented using the snapshot proper orthogonal decomposition algorithm. Space-phase modes are extracted to describe the dynamics of the chaotic flow, in which four central flow patterns are identified for describing the evolution of the energetic structures as a function of phase. The modal building blocks of the energy transport equation are demonstrated as a function of the phase. The triadic interaction term can here be interpreted as the convective transport of bi-modal interactions. Non-local energy transfer is observed as a result of the non-stationarity of the dynamical processes inducing triadic interactions spanning across a wide range of mode numbers.

1.
G.
Rosi
and
D.
Rival
, “
A Lagrangian perspective towards studying entrainment
,”
Exp. Fluids
59
,
19
(
2018
).
2.
T.
Lacassagne
,
J.
Vatteville
, and
C.
Degouet
, “
PTV measurements of oscillating grid turbulence in water and polymer solutions
,”
Exp. Fluids
61
,
165
(
2020
).
3.
K.
Steiros
, “
Balanced nonstationary turbulence
,”
Phys. Rev. E
105
,
035109
(
2022
).
4.
A.
Kolmogorov
, “
The local structure of turbulence in incompressible viscous fluid for very large Reynolds' numbers
,”
Akad. Nauk SSSR Dokl.
30
,
301
305
(
1941
).
5.
A.
Kolmogorov
, “
On the degeneration of isotropic turbulence in an incompressible viscous fluid
,”
Akad. Nauk SSSR Dokl.
31
,
319
323
(
1941
).
6.
A.
Kolmogorov
, “
Dissipation of energy in locally isotropic turbulence
,”
Akad. Nauk SSSR Dokl.
32
,
16
(
1941
).
7.
G. K.
Batchelor
,
The Theory of Homogeneous Turbulence
(
Cambridge University Press
,
Cambridge
,
1982
).
8.
J.
Lumley
, “
The structure of inhomogeneous turbulent flows
,” in
Atmospheric Turbulence and Radio Wave Propagation
, edited by
A. M.
Yaglom
and
V. I.
Tatarski
(
Nauka
,
Moskow
,
1967
), pp.
166
178
.
9.
J.
Lumley
,
Stochastic Tools in Turbulence
(
Academic Press
,
New York
,
1972
).
10.
M. N.
Glauser
and
W. K.
George
, “
Orthogonal decomposition of the axisymmetric jet mixing layer including azimuthal dependence
,” in
Advances in Turbulence
, edited by
G.
Comte-Bellot
and
J.
Mathieu
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
,
1987
), pp.
357
366
.
11.
W. K.
George
, “
Insight into the dynamics of coherent structures from a proper orthogonal decomposition
,” in
Symposium on Near Wall Turbulence
(
Springer Berlin Heidelberg
,
Dubrovnik, Yugoslavia
,
1988
).
12.
M. N.
Glauser
,
S. J.
Leib
, and
W. K.
George
, “
Coherent structures in the axisymmetric turbulent jet mixing layer
,” in
Turbulent Shear Flows 5
, edited by
F.
Durst
,
B. E.
Launder
,
J. L.
Lumley
,
F.
Schmidt
, and
J. H.
Whitelaw
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
,
1987
), pp.
134
145
.
13.
J. H.
Citriniti
and
W. K.
George
, “
Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition
,”
J. Fluid Mech.
418
,
137
166
(
2000
).
14.
D.
Jung
,
S.
Gamard
, and
W. K.
George
, “
Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. I. The near-field region
,”
J. Fluid Mech.
514
,
173
204
(
2004
).
15.
S.
Gamard
,
D.
Jung
, and
W. K.
George
, “
Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. II. The far-field region
,”
J. Fluid Mech.
514
,
205
230
(
2004
).
16.
A.
Towne
,
O. T.
Schmidt
, and
T.
Colonius
, “
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
,”
J. Fluid Mech.
847
,
821
867
(
2018
).
17.
O. T.
Schmidt
and
T.
Colonius
, “
Guide to spectral proper orthogonal decomposition
,”
AIAA J.
58
,
1023
1033
(
2020
).
18.
S.
Derebail Muralidhar
,
B.
Podvin
,
L.
Mathelin
, and
Y.
Fraigneau
, “
Spatio-temporal proper orthogonal decomposition of turbulent channel flow
,”
J. Fluid Mech.
864
,
614
639
(
2019
).
19.
A.
Hodžić
,
P. J.
Olesen
, and
C. M.
Velte
, “
On the discrepancies between POD and Fourier modes on aperiodic domains
,” arXiv:2207.02550 (
2022
).
20.
S. V.
Gordeyev
and
F. O.
Thomas
, “
A temporal proper decomposition (TPOD) for closed-loop flow control
,”
Exp. Fluids
54
,
1477
(
2013
).
21.
O. T.
Schmidt
and
P. J.
Schmid
, “
A conditional space–time POD formalism for intermittent and rare events: Example of acoustic bursts in turbulent jets
,”
J. Fluid Mech.
867
,
R2
(
2019
).
22.
A. S.
Monin
and
A. M.
Yaglom
,
Statistical Fluid Mechanics: Mechanics of Turbulence
(
The MIT Press
,
Massachusetts
,
1971
).
23.
H. A.
Carlson
,
R.
Verberg
, and
C. A.
Harris
, “
Aeroservoelastic modeling with proper orthogonal decomposition
,”
Phys. Fluids
29
,
020711
(
2017
).
24.
G.
Riches
,
R.
Martinuzzi
, and
C.
Morton
, “
Proper orthogonal decomposition analysis of a circular cylinder undergoing vortex-induced vibrations
,”
Phys. Fluids
30
,
105103
(
2018
).
25.
G.
Charalampous
,
C.
Hadjiyiannis
, and
Y.
Hardalupas
, “
Proper orthogonal decomposition of primary breakup and spray in co-axial airblast atomizers
,”
Phys. Fluids
31
,
043304
(
2019
).
26.
X.
He
,
Z.
Fang
,
G.
Rigas
, and
M.
Vahdati
, “
Spectral proper orthogonal decomposition of compressor tip leakage flow
,”
Phys. Fluids
33
,
105105
(
2021
).
27.
N.
Blanc
,
T.
Boutin
,
I.
Bendaoud
,
F.
Soulié
, and
C.
Bordreuil
, “
Proper orthogonal decomposition analysis of variable temperature field during gas tungsten arc welding
,”
Phys. Fluids
33
,
125123
(
2021
).
28.
Z.
Zhang
,
H.
Chen
,
J.
Yin
,
Z.
Ma
,
Q.
Gu
,
J.
Lu
, and
H.
Liu
, “
Unsteady flow characteristics in centrifugal pump based on proper orthogonal decomposition method
,”
Phys. Fluids
33
,
075122
(
2021
).
29.
M.
Schiødt
,
A.
Hodžić
,
F.
Evrard
,
M.
Hausmann
,
B.
van Wachem
, and
C. M.
Velte
, “
Characterizing Lagrangian particle dynamics in decaying homogeneous isotropic turbulence using proper orthogonal decomposition
,”
Phys. Fluids
34
,
063303
(
2022
).
30.
P. J.
Olesen
,
A.
Hodžić
,
S. J.
Andersen
,
N. N.
Sørensen
, and
C. M.
Velte
, “
Dissipation-optimized proper orthogonal decomposition
,”
Phys. Fluids
35
,
015131
(
2023
).
31.
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
).
32.
M.
Couplet
,
P.
Sagaut
, and
C.
Basdevant
, “
Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow
,”
J. Fluid Mech.
491
,
275
284
(
2003
).
33.
M. J.
Balajewicz
,
E. H.
Dowell
, and
B. R.
Noack
, “
Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation
,”
J. Fluid Mech.
729
,
285
308
(
2013
).
34.
M. W.
Lee
and
E. H.
Dowell
, “
Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases
,”
Nonlinear Dyn.
101
,
1457
1471
(
2020
).
35.
R.
Rubini
,
D.
Lasagna
, and
A. D.
Ronch
, “
The l1-based sparsification of energy interactions in unsteady lid-driven cavity flow
,”
J. Fluid Mech.
905
,
A15
(
2020
).
36.
R.
Rubini
,
D.
Lasagna
, and
A. D.
Ronch
, “
A priori sparsification of Galerkin models
,”
J. Fluid Mech.
941
,
A43
(
2022
).
37.
P.
Holmes
,
J. L.
Lumley
,
G.
Berkooz
, and
C. W.
Rowley
,
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
,
2nd ed.
, Cambridge Monographs on Mechanics No. 2 (
Cambridge University Press
,
2012
).
38.
L.
Sirovich
, “
Turbulence and the dynamics of coherent structures
,”
Q. Appl. Math.
45
,
561
571
(
1987
).
39.
U.
Frisch
,
Turbulence: The Legacy of A. N. Kolmogorov
(
Cambridge University Press
,
Cambridge
,
1995
).
40.
M.
Toda
,
R.
Kubo
, and
N.
Saito
,
Statistical Physics I: Equilibrium Statistical Mechanics
(
Springer-Verlag
,
Berlin, Heidelberg
,
1983
).
41.
W.
Parry
,
Topics in Ergodic Theory
(
Cambridge University Press
,
Cambridge
,
1981
).
42.
R. V.
Hogg
,
J. W.
McKean
, and
A. T.
Craig
,
Introduction to Mathematical Statistics
(
Pearson Education, Inc
.,
Boston
,
2019
).
43.
F.
Denner
,
F.
Evrard
, and
B. G.
van Wachem
, “
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
).
44.
M. W.
Lee
,
E. H.
Dowell
, and
M. J.
Balajewicz
, “
A study of the regularized lid-driven cavity's progression to chaos
,”
Commun. Nonlinear Sci. Numer. Simul.
71
,
50
72
(
2019
).
45.
M. J.
Vogel
,
A. H.
Hirsa
, and
J. M.
Lopez
, “
Spatio-temporal dynamics of a periodically driven cavity flow
,”
J. Fluid Mech.
478
,
197
226
(
2003
).
46.
S. S.
Mendu
and
P.
Das
, “
Fluid flow in a cavity driven by an oscillating lid-a simulation by lattice Boltzmann method
,”
Eur. J. Mech., B: Fluids
39
,
59
70
(
2013
).
47.
J.
Zhu
,
L. E.
Holmedal
,
H.
Wang
, and
D.
Myrhaug
, “
Vortex dynamics and flow patterns in a two-dimensional oscillatory lid-driven rectangular cavity
,”
Eur. J. Mech., B
79
,
255
269
(
2020
).
48.
R.
Iwatsu
,
J. M.
Hyun
, and
K.
Kuwahara
, “
Numerical simulation of flows driven by a torsionally oscillating lid in a square cavity
,”
J. Fluids Eng.
114
,
143
151
(
1992
).
49.
H. M.
Blackburn
and
J. M.
Lopez
, “
The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow
,”
J. Fluid Mech.
497
,
289
317
(
2003
).
50.
T. A.
Oliver
,
N.
Malaya
,
R.
Ulerich
, and
R. D.
Moser
, “
Estimating uncertainties in statistics computed from direct numerical simulation
,”
Phys. Fluids
26
,
035101
(
2014
).
51.
R. J.
Moffat
, “
Describing the uncertainties in experimental results
,”
Exp. Therm. Fluid Sci.
1
,
3
17
(
1988
).
52.
H. W.
Coleman
and
W. G.
Steele
, “
Engineering application of experimental uncertainty analysis
,”
AIAA J.
33
,
1888
1896
(
1995
).
53.
Y.
Zhang
,
A.
Hodzic
,
F.
Evrard
,
B.
Wachem
, and
C. M.
Velte
, see https://doi.org/10.11583/DTU.21444684
Phase POD Modes and Triadic Interactions
(
DTU
,
2022
).”
54.
N. E.
Huang
,
Z.
Shen
,
S. R.
Long
,
M.-L. C.
Wu
,
H. H.
Shih
,
Q.
Zheng
,
N.-C.
Yen
, and
C. C.
Tung
, “
The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis
,”
Proc. R. Soc. A
454
,
903
995
(
1998
).
55.
P.
Flandrin
and
P.
Goncalvès
, “
Empirical mode decomposition as data-driven wavelet-like expansions
,”
Int. J. Wavelets, Multiresolution Inf. Process.
2
,
477
496
(
2004
).
56.
Y.
Huang
,
F. G.
Schmitt
,
Z.
Lu
, and
Y.
Liu
, “
Analysis of daily river flow fluctuations using empirical mode decomposition and arbitrary order Hilbert spectral analysis
,”
J. Hydrol.
373
,
103
111
(
2009
).
57.
W. K.
George
,
Lectures in Turbulence for the 21st Century
(
2013
); available at http://www.turbulence-online.com/Publications/Lecture_Notes/Turbulence_Lille/TB_16January2013.pdf.
58.
A.
Hodzic
, “
A tensor calculus formulation of the Lumley decomposition applied to the turbulent axi-symmetric jet far-field
,” Ph.D. thesis (
Technical University of Denmark
,
Denmark
,
2018
).
59.
B. J.
McKeon
, “
The engine behind (wall) turbulence: Perspectives on scale interactions
,”
J. Fluid Mech.
817
,
P1
(
2017
).
60.
R. H.
Kraichnan
, “
Inertial-range transfer in two- and three-dimensional turbulence
,”
J. Fluid Mech.
47
,
525
535
(
1971
).
61.
R. H.
Kraichnan
, “
Eddy viscosity in two and three dimensions
,”
J. Atmos. Sci.
33
,
1521
1536
(
1976
).
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