We introduce a novel way to extract information from turbulent datasets by applying an Auto Regressive Moving Average (ARMA) statistical analysis. Such analysis goes well beyond the analysis of the mean flow and of the fluctuations and links the behavior of the recorded time series to a discrete version of a stochastic differential equation which is able to describe the correlation structure in the dataset. We introduce a new index Υ that measures the difference between the resulting analysis and the Obukhov model of turbulence, the simplest stochastic model reproducing both Richardson law and the Kolmogorov spectrum. We test the method on datasets measured in a von Kármán swirling flow experiment. We found that the ARMA analysis is well correlated with spatial structures of the flow, and can discriminate between two different flows with comparable mean velocities, obtained by changing the forcing. Moreover, we show that the Υ is highest in regions where shear layer vortices are present, thereby establishing a link between deviations from the Kolmogorov model and coherent structures. These deviations are consistent with the ones observed by computing the Hurst exponents for the same time series. We show that some salient features of the analysis are preserved when considering global instead of local observables. Finally, we analyze flow configurations with multistability features where the ARMA technique is efficient in discriminating different stability branches of the system.
REFERENCES
1.
A. N.
Kolmogorov
, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers
,” Dokl. Akad. Nauk SSSR
30
, 299
–303
(1941
).2.
S.
Kato
, H.
Ito
, T.
Ichikawa
, and M.
Kagami
, “Laser doppler velocimeter
,” U.S. patent 5,587,785 (24 December 1996
).3.
R. J.
Adrian
and J.
Westerweel
, Particle Image Velocimetry
(Cambridge University Press
, 2010
), Vol. 30
.4.
5.
F.
Ravelet
, A.
Chiffaudel
, and F.
Daviaud
, “Supercritical transition to turbulence in an inertially driven von Kármán closed flow
,” J. Fluid Mech.
601
, 339
–364
(2008
).6.
P.-P.
Cortet
, P.
Diribarne
, R.
Monchaux
, A.
Chiffaudel
, F.
Daviaud
, and B.
Dubrulle
, “Normalized kinetic energy as a hydrodynamical global quantity for inhomogeneous anisotropic turbulence
,” Phys. Fluids
21
(2
), 025104
(2009
).7.
Y.
Zhou
, “Interacting scales and energy transfer in isotropic turbulence
,” Phys. Fluids A: Fluid Dyn.
5
(10
), 2511
–2524
(1993
).8.
A.
Obukhov
, “On the distribution of energy in the spectrum of turbulent flow
,” Dokl. Akad. Nauk SSSR
32
, 22
–24
(1941
).9.
D. J.
Thomson
, “Criteria for the selection of stochastic models of particle trajectories in turbulent flows
,” J. Fluid Mech.
180
, 529
–556
(1987
).10.
U.
Frisch
, Turbulence
(Cambridge University Press
, Cambridge, UK
, 1996
), Vol. 1
, p. 310
.11.
J.-P.
Laval
, B.
Dubrulle
, and J.
McWilliams
, “Langevin models of turbulence: Renormalization group, distant interaction algorithms or rapid distortion theory?
,” Phys. Fluids
15
, 1327
(2003
).12.
R.
Friedrich
, “Statistics of lagrangian velocities in turbulent flows
,” Phys. Rev. Lett.
90
(8
), 084501
(2003
).13.
A.
Baule
and R.
Friedrich
, “Joint probability distributions for a class of non-Markovian processes
,” Phys. Rev. E
71
(2
), 026101
(2005
).14.
A.
Aringazin
and M.
Mazhitov
, “Stochastic models of lagrangian acceleration of fluid particle in developed turbulence
,” Int. J. Mod. Phys. B
18
(23n24
), 3095
–3168
(2004
).15.
A.
Aringazin
and M.
Mazhitov
, “One-dimensional Langevin models of fluid particle acceleration in developed turbulence
,” Phys. Rev. E
69
(2
), 026305
(2004
).16.
P. H.
Franses
and D.
Van Dijk
, Non-linear Time Series Models in Empirical Finance
(Cambridge University Press
, 2000
).17.
J.
Guiot
, “ARMA techniques for modelling tree-ring response to climate and for reconstructing variations of paleoclimates
,” Ecol. Modell.
33
(2
), 149
–171
(1986
).18.
L.
Marié
and F.
Daviaud
, “Experimental measurement of the scale-by-scale momentum transport budget in a turbulent shear flow
,” Phys. Fluids
16
, 457
(2004
).19.
P.-P.
Cortet
, A.
Chiffaudel
, F.
Daviaud
, and B.
Dubrulle
, “Experimental evidence of a phase transition in a closed turbulent flow
,” Phys. Rev. Lett.
105
(21
), 214501
(2010
).20.
A.
Naso
, R.
Monchaux
, P.-H.
Chavanis
, and B.
Dubrulle
, “Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined
,” Phys. Rev. E
81
(6
), 066318
(2010
).21.
B.
Saint-Michel
, B.
Dubrulle
, L.
Marié
, F.
Ravelet
, F.
Daviaud
et al, “Evidence for forcing-dependent steady states in a turbulent swirling flow
,” Phys. Rev. Lett.
111
, 234502
(2013
).22.
N.
Leprovost
, L.
Marié
, and B.
Dubrulle
, “A stochastic model of torques in von kármán swirling flow
,” Eur. Phys. J. B
39
(1
), 121
–129
(2004
).23.
J. H.
Titon
and O.
Cadot
, “The statistics of power injected in a closed turbulent flow: Constant torque forcing versus constant velocity forcing
,” Phys. Fluids
15
(3
), 625
–640
(2003
).24.
H.
Tennekes
, “Similarity relations, scaling laws and spectral dynamics
,” in Atmospheric Turbulence and Air Pollution Modeling
, edited by F. T. M.
Nieuwstadt
and H.
van Dop
(Reidel
, 1982
), pp. 37
–68
.25.
J.-F.
Pinton
and R.
Labbé
, “Correction to the taylor hypothesis in swirling flows
,” J. Phys. II
4
(9
), 1461
–1468
(1994
).26.
P. J.
Brockwell
and R. A.
Davis
, Time Series: Theory and Methods
, 2nd ed. (Springer
, 1990
).27.
G. E.
Box
and G. M.
Jenkins
, Time Series Analysis: Forecasting and Control
(Holden-Day
, 1970
).28.
G.
Casella
and R. L.
Berger
, Statistical Inference
(Duxbury Press
, Belmont, CA
, 1990
), Vol. 70
.29.
B.
Saint-Michel
, F.
Daviaud
, and B.
Dubrulle
, “A zero-mode mechanism for spontaneous symmetry breaking in a turbulent von Kármán flow
,” New J. Phys.
16
(1
), 013055
(2014
).30.
P.-P.
Cortet
, E.
Herbert
, A.
Chiffaudel
, F.
Daviaud
, B.
Dubrulle
, and V.
Padilla
, “Susceptibility divergence, phase transition and multistability of a highly turbulent closed flow
,” J. Stat. Mech.: Theory Exp.
, P07012
(2011
).31.
G. I.
Taylor
, “Statistical theory of turbulence
,” Proc. R. Soc. London, Ser. A
151
(873
), 421
–444
(1935
).32.
F.
Ravelet
, L.
Marié
, A.
Chiffaudel
, and F.
Daviaud
, “Multistability and memory effect in a highly turbulent flow: Experimental evidence for a global bifurcation
,” Phys. Rev. Lett.
93
(16
), 164501
(2004
).33.
D.
Faranda
, D. B.
Pons
, F. M.
Emanuele
, and F.
Daviaud
, “Probing turbulence intermittency via auto-regressive moving-average model
,” preprint arXiv:1407.5478 (2014
).34.
V.
Nikora
, D.
Goring
, and R.
Camussi
, “Intermittency and interrelationships between turbulence scaling exponents: Phase-randomization tests
,” Phys. Fluids
13
(5
), 1404
–1414
(2001
).35.
D.
Prichard
and J.
Theiler
, “Generating surrogate data for time series with several simultaneously measured variables
,” Phys. Rev. Lett.
73
(7
), 951
(1994
).36.
L.
Biferale
, I.
Daumont
, A.
Lanotte
, and F.
Toschi
, “Anomalous and dimensional scaling in anisotropic turbulence
,” Phys. Rev. E
66
(5
), 056306
(2002
).37.
A.
Celani
and M.
Vergassola
, “Statistical geometry in scalar turbulence
,” Phys. Rev. Lett.
86
(3
), 424
(2001
).38.
39.
B. B.
Mandelbrot
and J. W.
Van Ness
, “Fractional Brownian motions, fractional noises and applications
,” SIAM Rev.
10
(4
), 422
–437
(1968
).40.
R.
Weron
, “Estimating long-range dependence: finite sample properties and confidence intervals
,” Phys. A (Amsterdam, Neth.)
312
(1
), 285
–299
(2002
).41.
A.
Arneodo
, C.
Baudet
, F.
Belin
, R.
Benzi
, B.
Castaing
, B.
Chabaud
, R.
Chavarria
, S.
Ciliberto
, R.
Camussi
, F.
Chilla
et al, “Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity
,” EPL (Europhys. Lett.)
34
(6
), 411
(1996
).42.
J.
Laval
, B.
Dubrulle
, and S.
Nazarenko
, “Nonlocality and intermittency in three-dimensional turbulence
,” Phys. Fluids
13
(7
), 1995
–2012
(2001
).43.
B.
Saint-Michel
, E.
Herbert
, J.
Salort
, C.
Baudet
, M. B.
Mardion
, P.
Bonnay
, M.
Bourgoin
, B.
Castaing
, L.
Chevillard
et al, “Probing quantum and classical turbulence analogy through global bifurcations in a von Kármán liquid Helium experiment
,” preprint arXiv:1401.7117 (2014
).44.
C.
Connaughton
, A. C.
Newell
, and Y.
Pomeau
, “Non-stationary spectra of local wave turbulence
,” Phys. D
184
(1
), 64
–85
(2003
).45.
46.
P. J.
Brockwell
and R. A.
Davis
, Introduction to Time Series and Forecasting
, 2nd ed. (Springer
, 2002
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
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