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On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

Published online by Cambridge University Press:  05 June 2018

Nico Reinke ,
André Fuchs ,
Daniel Nickelsen  and
Joachim Peinke Open the ORCID record for Joachim Peinke [Opens in a new window]
Nico Reinke
Affiliation:
Institute of Physics, University of Oldenburg, Germany ForWind, University of Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany
André Fuchs
Affiliation:
Institute of Physics, University of Oldenburg, Germany ForWind, University of Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany
Daniel Nickelsen
Affiliation:
Institute of Physics, University of Oldenburg, Germany National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa Institute of Theoretical Physics, University of Stellenbosch, South Africa
Joachim Peinke*
Affiliation:
Institute of Physics, University of Oldenburg, Germany ForWind, University of Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

JFM classification

Turbulent Flows: Intermittency Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 848 , 10 August 2018 , pp. 117 - 153
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© 2018 Cambridge University Press 

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References

Arneodo, A., Baudet, C., Belin, F., Benzi, R., Castaing, B., Chabaud, B., Chavarria, R., Ciliberto, S., Camussi, R., Chillá, F., Dubrulle, B., Gagne, Y., Hebral, B., Herweijer, J., Marchand, M., Maurer, J., Muzy, J. F., Naert, A., Noullez, A., Peinke, J., Roux, F., Tabeling, P., van de Water, W. & Willaime, H. 1996 Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity. Europhys. Lett. 34 (6), 411–416.Google Scholar
Arneodo, A., Muzy, J. F. & Roux, S. G. 1997 Experimental anaylsis of self-similarity and random cascade process: application to fully developed turbulence data. J. Phys. II 7, 363–370.Google Scholar
Aronson, D. & Löfdahl, L. 1993 The plane wake of a cylinder: measurements and inferences on turbulence modeling. Phys. Fluids A 5, 1433–1437.Google Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge Science Classic.Google Scholar
Benzi, R., Biferale, L., Ciliberto, S., Struglia, M. V. & Tripiccione, R. 1996 Generalized scaling in fully developed turbulence. Physica D 96 (1–4), 162–181.Google Scholar
Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–R32.Google Scholar
Castaing, B. & Dubrulle, B. 1995 Fully developed turbulence: a unifying point of view. J. Phys. II 5, 895–899.Google Scholar
Chanal, O., Chabaud, B., Castaing, B. & Hébral, B. 2000 Intermittency in a turbulent low temperature gaseous helium jet. Eur. Phys. J. B 17 (2), 309–317.Google Scholar
Clay2000 Millennium problems, Clay Mathematics Institute http://www.claymath.org/millennium-problems.Google Scholar
Davidson, P. A. 2004 Turbulence: an Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davoudi, J., Tabar, M. & Rahimi, R. 2000 Multiscale correlation functions in strong turbulence. Phys. Rev. E 61 (6).Google Scholar
Dubrulle, B. 2000 Finite size scale invariance. Eur. Phys. J. B 14, 757–771.Google Scholar
Einstein, A. 1905 Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322 (8), 549–560.Google Scholar
Feller, W. 1968 An Introduction to Probability Theory and Its Applications, vol. 1. Wiley.Google Scholar
Friedrich, R. & Peinke, J. 1997a Description of a turbulent cascade by a Fokker–Planck equation. Phys. Rev. Lett. 78, 863.Google Scholar
Friedrich, R. & Peinke, J. 1997b Statistical properties of a turbulent cascade. Physica D 102, 147.Google Scholar
Friedrich, R. & Peinke, J. 2009 Fluid dynamics, turbulence. In Encyclopedia of Complexity and Systems Science (ed. Meyers, R. A.), pp. 3642–3661. Springer.Google Scholar
Friedrich, R., Peinke, J. & Naert, A. 1997 A new approach to characterize disordered structures. Z. Naturforsch. 52a, 588–592.Google Scholar
Friedrich, R., Peinke, J., Sahimi, M., Tabar, M. & Rahimi, R. 2011 Approaching complexity by stochastic processes: from biological systems to turbulence. Phys. Rep. 506, 87–162.Google Scholar
Friedrich, R., Zeller, J. & Peinke, J. 1998 A note on three-point statistics of velocity increments in turbulence. Europhys. Lett. 41 (2), 153–158.Google Scholar
Frisch, U. 2001 Turbulence: the Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Gagne, Y., Marchand, M. & Castaing, B. 1994 Conditional velocity pdf in 3-D turbulence. J. Phys. II 4 (1), 1–8.Google Scholar
Gledzer, E. 1997 On the Taylor hypothesis corrections for measured energy spectra of turbulence. Physica D 104, 163–183.Google Scholar
Gottschall, Julia & Peinke, Joachim 2008 On the definition and handling of different drift and diffusion estimates. New J. Phys. 10 (8), 083034.Google Scholar
Grauer, R., Homann, H. & Jean-Francois, P. 2012 Longitudinal and transverse structure functions in high-Reynolds-number turbulence. New J. Phys. 14 (6), 063016.Google Scholar
Gylfason, A. & Warhaft, Z. 2004 On higher order passive scalar structure functions in grid turbulence. Phys. Fluids 16, 4012–4019.Google Scholar
Honisch, C. & Friedrich, R. 2011 Estimation of Kramers–Moyal coefficients at low sampling rates. Phys. Rev. E 83, 066701.Google Scholar
Hurst, D. & Vassilicos, J. C. 2007 Scalings and decay of fractal-generated turbulence. Phys. Fluids 19 (3).Google Scholar
Keylock, C. J., Stresing, R. & Peinke, J. 2015 Gradual wavelet reconstruction of the velocity increments for turbulent wakes. Phys. Fluids 27 (2).Google Scholar
Kleinhans, D., Friedrich, R., Nawroth, A. P. & Peinke, J. 2005 An iterative procedure for the estimation of drift and diffusion coefficients of Langevin processes. Phys. Lett. A 346, 42–46.Google Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. SSSR 32, 16–18.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.Google Scholar
Kuczaj, A. K., Geurts, B. J. & McComb, W. D. 2006 Nonlocal modulation of the energy cascade in broadband-forced turbulence. Phys. Rev. E 74, 016306.Google Scholar
Laval, J.-P., Dubrulle, B. & Nazarenko, S. 2001 Nonlocality and intermittency in three-dimensional turbulence. Phys. Fluids 13, 1995.Google Scholar
Lemons, D. S. 1997 Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ (‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)). Am. J. Phys. 65 (11), 1079.Google Scholar
Lin, C. C. 1953 On Taylor’s hypothesis and the acceleration terms in the Navier–Stokes equation. Q. Appl. Maths 10 (4), 295–306.Google Scholar
Lück, St., Renner, Ch., Peinke, J. & Friedrich, R. 2006 The Markov–Einstein coherence length – a new meaning for the Taylor length in turbulence. Phys. Lett. A 359 (5), 335–338.Google Scholar
Lumley, J. L. 1965 Interpretation of time spectra measured in high intensity shear flows. Phys. Fluids 8 (6), 1056–1062.Google Scholar
Lundgren, T. S. 1967 Distribution functions in the statistical theory of turbulence. Phys. Fluids 10 (5), 969–975.Google Scholar
L’vov, V. & Procaccia, I. 1996 Fusion rules in turbulent systems with flux equilibrium. Phys. Rev. Lett. 76, 2898–2901.Google Scholar
Marcq, P. & Naert, A. 2001 A Langevin equation for turbulent velocity increments. Phys. Fluids 13, 2590–2595.Google Scholar
Melius, M. S., Tutkun, M. & Bayoán, C. R. 2014 Identification of Markov process within a wind turbine array boundary layer. J. Renew. Sustain. Energy 6 (2).Google Scholar
Monin, A. S. 1967 Equations of turbulent motion. J. Appl. Math. Mech. 31 (6), 1057–1068.Google Scholar
Monin, A. S., Iaglom, A. M. & Lumley, J. L. 2007 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. Courier Corporation.Google Scholar
Murzyn, F. & Bélorgey, M. 2005 Experimental investigation of the grid-generated turbulence features in a free surface flow. Exp. Therm. Fluid Sci. 29 (8), 925–935.Google Scholar
Mydlarski, L. & Warhaft, Z. 1996 On the onset of high-Reynolds-number grid-generated wind tunnel turbulence. J. Fluid Mech. 320, 331–368.Google Scholar
Naert, A., Friedrich, R. & Peinke, J. 1997 Fokker–Planck equation for the energy cascade in turbulence. Phys. Rev. E 56 (6), 6719–6722.Google Scholar
Nagata, K., Sakai, Y., Inaba, T., Suzuki, H., Terashima, O. & Suzuki, H. 2013 Turbulence structure and turbulence kinetic energy transport in multiscale/fractal-generated turbulence. Phys. Fluids 25 (6).Google Scholar
Nawroth, A. P., Peinke, J., Kleinhans, D. & Friedrich, R. 2007 Improved estimation of Fokker–Planck equations through optimisation. Phys. Rev. E 76, 056102.Google Scholar
Nelkin, M. 1992 In what sense is turbulence an unsolved problem? Science 255 (5044), 566–570.Google Scholar
Nickelsen, D. 2017 Master equation for She–Leveque scaling and its classification in terms of other Markov models of developed turbulence. J. Stat. Mech. 2017 (7), 073209.CrossRefGoogle Scholar
Nickelsen, D. & Engel, A. 2013 Probing small-scale intermittency with a fluctuation theorem. Phys. Rev. Lett. 110, 214501.Google Scholar
Novikov, E. A. 1994 Infinitely divisible distributions in turbulence. Phy. Rev. E (R) 50, R3303–R3305.Google Scholar
Pinton, J. F. & Labbé, R. 1994 Correction to the Taylor hypothesis in swirling flows. J. Phys. II 4 (9), 1461–1468.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Reinke, N., Fuchs, A., Hölling, M. & Peinke, J. 2014 Stochastic analysis of a fractal grid wake. Progress in Turbulence VI, Proceedings of the iTi Conference on Turbulence 2014, Springer Proceedings in Physics, vol. 6, pp. 165–177.Google Scholar
Reinke, N., Nickelsen, D., Engel, A. & Peinke, J. 2016 Application of an integral fluctuation theorem to turbulent flows. Springer Proceedings in Physics, vol. 165, pp. 19–25.Google Scholar
Renner, C.2002 Markowanalysen stochastisch fluktuierender Zeitserien. PhD thesis, Carl von Ossietzky Universität Oldenburg.Google Scholar
Renner, C., Peinke, J. & Friedrich, R. 2001 Experimental indications for Markov properties of small-scale turbulence. J. Fluid Mech. 433, 383–409.Google Scholar
Renner, Ch., Peinke, J. & Friedrich, R. 2002a Experimental indications for Markovproperties of small scale turbulence. PAMM 1 (1), 462–463.Google Scholar
Renner, C., Peinke, J., Friedrich, R., Chanal, O. & Chabaud, B. 2002b Universality of small scale turbulence. Phys. Rev. Lett. 89, 124502.Google Scholar
Risken, H. 1984 The Fokker–Planck Equation. Springer.Google Scholar
Seifert, U. 2005 Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95 (4), 040602.Google Scholar
Seifert, U. 2012 Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75 (12), 126001.Google Scholar
Sekimoto, K. 1998 Langevin equation and thermodynamics. Progr. Theoret. Phys. Suppl. 130, 17–27.Google Scholar
She, Z.-S. & Waymire, E. C. 1995 Quantized energy cascade and log-Poisson statistics in fully developed turbulence. Phys. Rev. Lett. 94, 262–899.Google Scholar
Siefert, M. & Peinke, J. 2004 Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence. Phys. Rev. E 70, 015302(R).Google Scholar
Siefert, M. & Peinke, J. 2006 Joint multi-scale statistics of longitudinal and transversal increments in small-scale wake turbulence. J. Turbul. 7, N50.Google Scholar
Sinhuber, M.2015 On the scales of turbulent motion at high Reynolds numbers, PhD thesis, Georg-August-Universität Göttingen.Google Scholar
Sinhuber, M., Bewley, G. O. & Bodenschatz, E. 2017 Dissipative effects on inertial-range statistics at high Reynolds numbers. Phys. Rev. Lett. 119, 134502.Google Scholar
Stresing, R., Kleinhans, D., Friedrich, R. & Peinke, J. 2012 Publisher’s note: different methods to estimate the Einstein–Markov coherence length in turbulence (Phys. Rev. E 83 046319 (2011)). Phys. Rev. E 85, 029907.Google Scholar
Stresing, R. & Peinke, J. 2010 Towards a stochastic multi-point description of turbulence. New J. Phys. 12 (10), 103046.Google Scholar
Stresing, R., Peinke, J., Seoud, R. E. & Vassilicos, J. C. 2010 Defining a new class of turbulent flows. Phys. Rev. Lett. 104, 194501.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164 (919), 476–490.Google Scholar
Tong, C. & Warhaft, Z. 1995 Passive scalar dispersion and mixing in a turbulent jet. J. Fluid Mech. 292, 1–38.Google Scholar
Tutkun, M. & Mydlarski, L. 2004 Markovian properties of passive scalar increments in grid-generated turbulence. New J. Phys. 6 (1), 49.Google Scholar
Valente, P. C. & Vassilicos, J. C. 2012 Universal dissipation scaling for nonequilibrium turbulence. Phys. Rev. Lett. 108, 214503.Google Scholar