In this contribution, we consider the Dynamic Mode Decomposition (DMD) framework as a purely data-driven tool to investigate both standard and actuated turbulent channel databases via Direct Numerical Simulation (DNS). Both databases have comparable Reynolds number Re ≈ 3600. The actuation consists in the imposition of a streamwise-varying sinusoidal spanwise velocity at the wall, known to lead to drag reduction. Specifically, a composite-based DMD analysis is conducted, with hybrid snapshots composed by skin friction and Reynolds stresses. A small number of dynamic modes (∼3–9) are found to recover accurately the DNS Reynolds stresses near walls. Moreover, the DMD modes retrieved propagate at a range of phase speeds consistent with those reported in the literature. We conclude that composite DMD is an attractive, purely data-driven tool to study turbulent flows. On the one hand, DMD is helpful to identify features associated with the drag, and on the other hand, it reveals the changes in flow structure when actuation is imposed.

1.
Post-Processing of Numerical and Experimental Data, LS 2008-01
, edited by
P.
Millan
and
M. L.
Riethmuller
(
von Karman Institute for Fluid Dynamics
,
Sint-Genesius-Rode, Belgium
,
2008
).
2.
Recent Advances in Particle Image Velocimetry, LS 2009-01
, edited by
F.
Scarano
and
M. L.
Riethmuller
(
von Karman Institute for Fluid Dynamics
,
Sint-Genesius-Rode, Belgium
,
2009
).
3.
Large Eddy Simulation and Related Techniques, LS 2010-04
, edited by
U.
Piomelli
,
C.
Benocci
, and
J. P. A. J.
van Beeck
(
von Karman Institute for Fluid Dynamics
,
Sint-Genesius-Rode, Belgium
,
2010
).
4.
ERCOFTAC Direct and Large-Eddy Simulation X
, edited by
D. G. E.
Grigoriadis
,
B. J.
Geurts
,
H.
Kuerten
,
J.
Fröhlich
, and
V.
Armenio
(
Springer
,
2018
).
5.
J. L.
Lumley
,
Stochastic Tools in Turbulence
(
Academic Press
,
1970
).
6.
L.
Sirovich
, “
Turbulence and the dynamics of coherent structures
,”
Q. Appl. Math.
45
(
3
),
561
590
(
1987
).
7.
G.
Berkooz
,
P.
Holmes
, and
J. L.
Lumley
, “
The proper orthogonal decomposition in the analysis of turbulent flows
,”
Annu. Rev. Fluid Mech.
25
(
1
),
539
575
(
1993
).
8.
S.
Volkwein
, Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling: Lecture Notes,
2013
.
9.
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
).
10.
P. J.
Schmid
, “
Dynamic mode decomposition of numerical and experimental data
,”
J. Fluid Mech.
656
,
5
28
(
2010
).
11.
E.
Ferrer
,
J.
de Vicente
, and
E.
Valero
, “
Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools
,”
Int. J. Numer. Methods Fluids
76
(
3
),
169
184
(
2014
).
12.
N. E.
Huang
,
Z.
Shen
,
S. R.
Long
,
M. C.
Wu
,
H. H.
Shih
,
Q.
Zheng
,
N.-C.
Yen
,
C. C.
Tung
, and
H. H.
Liu
, “
The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis
,”
Proc. R. Soc. London, Ser. A
454
(
1971
),
903
995
(
1998
).
13.
L.
Agostini
and
M. A.
Leschziner
, “
On the influence of outer large-scale structures on near-wall turbulence in channel flow
,”
Phys. Fluids
26
(
7
),
075107
(
2014
).
14.
A.
Alt
ntaş
ı,
L.
Davidson
, and
S. H.
Peng
, “
A new approximation to modulation-effect analysis based on empirical mode decomposition
,”
Phys. Fluids
31
(
2
),
025117
(
2019
).
15.
P.
Moin
and
R. D.
Moser
, “
Characteristic-eddy decomposition of turbulence in a channel
,”
J. Fluid Mech.
200
,
471
509
(
1989
).
16.
N.
Aubry
,
P.
Holmes
,
J. L.
Lumley
, and
E.
Stone
, “
The dynamics of coherent structures in the wall region of the wall boundary layer
,”
J. Fluid Mech.
192
,
115
173
(
1988
).
17.
D.
Rempfer
and
H. F.
Fasel
, “
Evolution of three-dimensional coherent structures in a flat-plate boundary layer
,”
J. Fluid Mech.
260
,
351
375
(
1994
).
18.
B.
Podvin
, “
On the adequacy of the ten-dimensional model for the wall layer
,”
Phys. Fluids
13
(
1
),
210
224
(
2001
).
19.
B.
Podvin
and
J. L.
Lumley
, “
A low-dimensional approach for the minimal flow unit
,”
J. Fluid Mech.
362
,
121
155
(
1998
).
20.
B.
Podvin
, “
A proper-orthogonal-decomposition–based model for the wall layer of a turbulent channel flow
,”
Phys. Fluids
21
(
1
),
015111
(
2009
).
21.
F.
Waleffe
, “
On a self-sustaining process in shear flows
,”
Phys. Fluids
9
(
4
),
883
900
(
1997
).
22.
J.
Moehlis
,
H.
Faisst
, and
B.
Eckhardt
, “
A low-dimensional model for turbulent shear flows
,”
New J. Phys.
6
,
56
(
2004
).
23.
M.
Lagha
and
P.
Manneville
, “
Modeling transitional plane Couette flow
,”
Eur. Phys. J. B
58
(
4
),
433
447
(
2007
).
24.
M.
Lagha
, “
A comprehensible low-order model for wall turbulence dynamics
,”
Phys. Fluids
26
(
8
),
085111
(
2014
).
25.
M.
Lagha
,
J.
Kim
,
J. D.
Eldredge
, and
X.
Zhong
, “
A numerical study of compressible turbulent boundary layers
,”
Phys. Fluids
23
(
1
),
015106
(
2011
).
26.
M.
Lagha
,
J.
Kim
,
J. D.
Eldredge
, and
X.
Zhong
, “
Near-wall dynamics of compressible boundary layers
,”
Phys. Fluids
23
(
6
),
065109
(
2011
).
27.
J. N.
Kutz
,
S. L.
Brunton
,
B. W.
Brunton
, and
J. L.
Proctor
,
Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems
(
Society for Industrial and Applied Mathematics
,
USA
,
2016
).
28.
I.
Mezić
, “
Analysis of fluid flows via spectral properties of the Koopman operator
,”
Annu. Rev. Fluid Mech.
45
(
1
),
357
378
(
2013
).
29.
S.
Bagheri
, “
Koopman-mode decomposition of the cylinder wake
,”
J. Fluid Mech.
726
,
596
623
(
2013
).
30.
K.
Taira
,
S. L.
Brunton
,
S. T. M.
Dawson
,
C. W.
Rowley
,
T.
Colonius
,
B. J.
McKeon
,
O. T.
Schmidt
,
S.
Gordeyev
,
V.
Theofilis
, and
L. S.
Ukeiley
, “
Modal analysis of fluid flows: An overview
,”
AIAA J.
55
(
12
),
4013
4041
(
2017
).
31.
K. K.
Chen
,
J. H.
Tu
, and
C. W.
Rowley
, “
Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses
,”
J. Nonlinear Sci.
22
(
6
),
887
915
(
2012
).
32.
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
).
33.
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
).
34.
D.
Duke
,
J.
Soria
, and
D.
Honnery
, “
An error analysis of the dynamic mode decomposition
,”
Exp. Fluids
52
(
2
),
529
542
(
2012
).
35.
P. J.
Schmid
,
D.
Violato
, and
F.
Scarano
, “
Decomposition of time-resolved tomographic PIV
,”
Exp. Fluids
52
(
6
),
1567
1579
(
2012
).
36.
P. J.
Schmid
, “
Application of the dynamic mode decomposition to experimental data
,”
Exp. Fluids
50
(
6
),
1123
1130
(
2011
).
37.
P. J.
Schmid
,
L.
Li
,
M. P.
Juniper
, and
O.
Pust
, “
Applications of the dynamic mode decomposition
,”
Theor. Comput. Fluid Dyn.
25
(
1
),
249
259
(
2011
).
38.
S.
Le Clainche
,
J. M.
Vega
, and
J.
Soria
, “
Higher order dynamic mode decomposition of noisy experimental data: The flow structure of a zero-net-mass-flux jet
,”
Exp. Therm. Fluid Sci.
88
(
Suppl. C
),
336
353
(
2017
).
39.
T.
Sayadi
and
P. J.
Schmid
, “
Parallel data-driven decomposition algorithm for large-scale datasets: With application to transitional boundary layers
,”
Theor. Comput. Fluid Dyn.
30
(
5
),
415
428
(
2016
).
40.
T.
Sayadi
,
P. J.
Schmid
,
J. W.
Nichols
, and
P.
Moin
, “
Reduced-order representation of near-wall structures in the late transitional boundary layer
,”
J. Fluid Mech.
748
,
278
301
(
2014
).
41.
A.
Cassinelli
,
M.
de Giovanetti
, and
Y.
Hwang
, “
Streak instability in near-wall turbulence revisited
,”
J. Turbul.
18
(
5
),
443
464
(
2017
).
42.
T.
Grenga
,
J. F.
MacArt
, and
M. E.
Mueller
, “
Dynamic mode decomposition of a direct numerical simulation of a turbulent premixed planar jet flame: Convergence of the modes
,”
Combust. Theory Modell.
22
(
4
),
795
811
(
2018
).
43.
S.
Le Clainche
and
J. M.
Vega
, “
Higher order dynamic mode decomposition
,”
SIAM J. Appl. Dyn. Syst.
16
(
2
),
882
925
(
2017
).
44.
S.
Le Clainche
and
J. M.
Vega
, “
Analyzing nonlinear dynamics via data-driven dynamic mode decomposition-like methods
,”
Complexity
2018
,
1
.
45.
S.
Le Clainche
,
Z. H.
Han
, and
E.
Ferrer
, “
An alternative method to study cross-flow instabilities based on high order dynamic mode decomposition
,”
Phys. Fluids
31
(
9
),
094101
(
2019
).
46.
S.
Le Clainche
and
E.
Ferrer
, “
A reduced order model to predict transient flows around straight bladed vertical axis wind turbines
,”
Energies
11
(
3
),
566
(
2018
).
47.
M. R.
Jovanović
,
P. J.
Schmid
, and
J. W.
Nichols
, “
Sparsity-promoting dynamic mode decomposition
,”
Phys. Fluids
26
(
2
),
024103
(
2014
).
48.
F.
Guéniat
,
L.
Mathelin
, and
L. R.
Pastur
, “
A dynamic mode decomposition approach for large and arbitrarily sampled systems
,”
Phys. Fluids
27
(
2
),
025113
(
2015
).
49.
T.
Bui-Thanh
,
M.
Damodaran
, and
K. E.
Willcox
, “
Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition
,”
AIAA J.
42
(
8
),
1505
1516
(
2004
).
50.
T.
Braconnier
,
M.
Ferrier
,
J.-C.
Jouhaud
,
M.
Montagnac
, and
P.
Sagaut
, “
Towards an adaptive POD/SVD surrogate model for aeronautic design
,”
Comput. Fluids
40
(
1
),
195
209
(
2011
).
51.
P.
Luchini
and
M.
Quadrio
, “
A low-cost parallel implementation of direct numerical simulation of wall turbulence
,”
J. Comput. Phys.
211
(
2
),
551
571
(
2006
).
52.
C. W.
Rowley
and
S. T. M.
Dawson
, “
Model reduction for flow analysis and control
,”
Annu. Rev. Fluid Mech.
49
(
1
),
387
417
(
2017
).
53.
K.
Fukagata
,
K.
Iwamoto
, and
N.
Kasagi
, “
Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows
,”
Phys. Fluids
14
(
11
),
L73
(
2002
).
54.
C.
Viotti
,
M.
Quadrio
, and
P.
Luchini
, “
Streamwise oscillation of spanwise velocity at the wall of a channel for turbulent drag reduction
,”
Phys. Fluids
21
(
11
),
115109
(
2009
).
55.
J.
Kim
,
P.
Moin
, and
R.
Moser
, “
Turbulence statistics in fully developed channel flow at low Reynolds number
,”
J. Fluid Mech.
177
,
133
166
(
1987
).
56.
M.
Quadrio
and
P.
Ricco
, “
Critical assessment of turbulent drag reduction through spanwise wall oscillations
,”
J. Fluid Mech.
521
,
251
271
(
2004
).
57.
M.
Quadrio
,
P.
Ricco
, and
C.
Viotti
, “
Streamwise-traveling waves of spanwise wall velocity for turbulent drag reduction
,”
J. Fluid Mech.
627
,
161
178
(
2009
).
58.
S.
Ghebali
,
S. I.
Chernyshenko
, and
M.
Leschziner
, “
Can large-scale oblique undulations on a solid wall reduce the turbulent drag?
,”
Phys. Fluids
29
,
105102
(
2017
).
59.
G. E.
Karniadakis
and
K.-S.
Choi
, “
Mechanisms on transverse motions in turbulent wall flows
,”
Annu. Rev. Fluid Mech.
35
(
1
),
45
62
(
2003
).
60.
Y.
Saad
,
Numerical Methods for Large Eigenvalue Problems
(
Manchester University Press
,
Manchester
,
1992
).
61.
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
).
62.
M.
Quadrio
,
B.
Frohnapfel
, and
Y.
Hasegawa
, “
Does the choice of the forcing term affect flow statistics in DNS of turbulent channel flow?
,”
Eur. J. Mech.: B/Fluids
55
,
286
293
(
2016
), vortical structures and wall turbulence.
63.
S.
Pope
,
Turbulent Flows
, 2nd ed. (
CUP
,
2014
).
64.
J.
Jiménez
and
A.
Pinelli
, “
The autonomous cycle of near-wall turbulence
,”
J. Fluid Mech.
389
,
335
359
(
1999
).
65.
J.
Kim
and
F.
Hussain
, “
Propagation velocity of perturbations in turbulent channel flow
,”
Phys. Fluids A
5
(
3
),
695
706
(
1993
).
66.
W.
Schoppa
and
F.
Hussain
, “
Coherent structure generation in near-wall turbulence
,”
J. Fluid Mech.
453
,
57
108
(
2002
).
You do not currently have access to this content.