Skip to main content Accessibility help

Beyond Kolmogorov cascades

  • Bérengère Dubrulle (a1)

The large-scale structure of many turbulent flows encountered in practical situations such as aeronautics, industry, meteorology is nowadays successfully computed using the Kolmogorov–Kármán–Howarth energy cascade picture. This theory appears increasingly inaccurate when going down the energy cascade that terminates through intermittent spots of energy dissipation, at variance with the assumed homogeneity. This is problematic for the modelling of all processes that depend on small scales of turbulence, such as combustion instabilities or droplet atomization in industrial burners or cloud formation. This paper explores a paradigm shift where the homogeneity hypothesis is replaced by the assumption that turbulence contains singularities, as suggested by Onsager. This paradigm leads to a weak formulation of the Kolmogorov–Kármán–Howarth–Monin equation (WKHE) that allows taking into account explicitly the presence of singularities and their impact on the energy transfer and dissipation. It provides a local in scale, space and time description of energy transfers and dissipation, valid for any inhomogeneous, anisotropic flow, under any type of boundary conditions. The goal of this article is to discuss WKHE as a tool to get a new description of energy cascades and dissipation that goes beyond Kolmogorov and allows the description of small-scale intermittency. It puts the problem of intermittency and dissipation in turbulence into a modern framework, compatible with recent mathematical advances on the proof of Onsager’s conjecture.

Corresponding author
Email address for correspondence:
Hide All
Agafontsev, D. S., Kuznetsov, E. A. & Mailybaev, A. A. 2016 Development of high vorticity in incompressible 3d Euler equations: influence of initial conditions. J. Expl Theor. Phys. Lett. 104 (10), 685689.10.1134/S002136401622001X
Agafontsev, D. S., Kuznetsov, E. A. & Mailybaev, A. A. 2017 Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations. J. Fluid Mech. 813, R1.10.1017/jfm.2017.1
Arneodo, A., Baudet, C., Belin, F., Benzi, R., Castaing, B., Chabaud, B., Chavarria, R., Ciliberto, S., Camussi, R., Chill, F. et al. 1996 Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity. Europhys. Lett. 34 (6), 411416.10.1209/epl/i1996-00472-2
Arneodo, A., Benzi, R., Berg, J., Biferale, L., Bodenschatz, E., Busse, A., Calzavarini, E., Castaing, B., Cencini, M., Chevillard, L. et al. 2008 Universal intermittent properties of particle trajectories in highly turbulent flows. Phys. Rev. Lett. 100, 254504.10.1103/PhysRevLett.100.254504
Batterman, R. W. 2011 Emergence, singularities, and symmetry breaking. Foundations Phys. 41 (6), 10311050.10.1007/s10701-010-9493-4
Bec, J. & Khanin, K. 2007 Burgers turbulence. Phys. Rep. 447 (1), 166.10.1016/j.physrep.2007.04.002
Benzi, R. & Biferale, L. 2009 Fully developed turbulence and the multifractal conjecture. J. Stat. Phys. 135 (5), 977990.10.1007/s10955-009-9738-9
Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29R32.
Biferale, L., Boffetta, G., Celani, A., Devenish, B. J., Lanotte, A. & Toschi, F. 2004 Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93, 064502.10.1103/PhysRevLett.93.064502
Boffetta, G., Mazzino, A. & Vulpiani, A. 2008 Twenty-five years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin. J. Phys. A: Math. Theoret. 41 (36), 363001.10.1088/1751-8113/41/36/363001
Bradshaw, Z. & Tsai, T.-P. 2018 Self-similar solutions to the Navier–Stokes equations: a survey of recent results. In Nonlinear Analysis in Geometry and Applied Mathematics, Part 2, Harvard CMSA Series in Mathematics, vol. 2, p. 159. International Press.
Buckmaster, T., De Lellis, C., Székelyhidi, L. & Vicol, V. 2019 Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Maths 72 (2), 229274.10.1002/cpa.21781
Caffarelli, L., Kohn, R. & Nirenberg, L. 1982 Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Maths 35 (6), 771831.10.1002/cpa.3160350604
Campolina, C. S. & Mailybaev, A. A. 2018 Chaotic blowup in the 3d incompressible Euler equations on a logarithmic lattice. Phys. Rev. Lett. 121, 064501.10.1103/PhysRevLett.121.064501
Castaing, B., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46 (2), 177200.10.1016/0167-2789(90)90035-N
Chae, D. 2007 Nonexistence of self-similar singularities for the 3D incompressible euler equations. Commun. Math. Phys. 1 (273), 203215.10.1007/s00220-007-0249-8
Chevillard, L.2004 Description multifractale unifiée du phénomène d intermittence en turbulence Eulérienne et Lagrangienne. PhD thesis, Université Sciences et Technologies - Bordeaux I.
Chevillard, L., Castaing, B., Arneodo, A., Lévêque, E., Pinton, J.-F. & Roux, S. G. 2012 A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows. C. Rend. Phys. 13 (9), 899928; structures and statistics of fluid turbulence/Structures et statistiques de la turbulence des fluides.10.1016/j.crhy.2012.09.002
Clusel, M. & Bertin, E. 2008 Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics. Intl J. Mod. Phys. B 22 (20), 33113368.10.1142/S021797920804853X
Debue, P.2019 Experimental approach of the Euler and Navier–Stokes singularities problem. PhD thesis, Université Paris-Saclay, Paris.
Debue, P., Shukla, V., Kuzzay, D., Faranda, D., Saw, E.-W., Daviaud, F. & Dubrulle, B. 2018 Dissipation, intermittency, and singularities in incompressible turbulent flows. Phys. Rev. E 97, 053101.
Dombre, T. & Gilson, J.-L. 1998 Intermittency, chaos and singular fluctuations in the mixed Obukhov-Novikov shell model of turbulence.. Physica D 111 (1), 265287.10.1016/S0167-2789(97)80015-2
Drivas, T. D. 2019 Turbulent cascade direction and Lagrangian time-asymmetry. J. Nonlinear Sci. 29, 65.10.1007/s00332-018-9476-8
Drivas, T. D. & Eyink, G. L.2017 An onsager singularity theorem for leray solutions of incompressible Navier–Stokes. arXiv:1710.05205.
Duchon, J. & Robert, R. 2000 Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13 (1), 249255.10.1088/0951-7715/13/1/312
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1R44.10.1088/0951-7715/22/1/R01
Eyink, G. L.2007–2008 Turbulence Theory. Course notes, The Johns Hopkins University. Available at:
Eyink, G. L. & Drivas, T. D. 2015 Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158 (2), 386432.10.1007/s10955-014-1135-3
Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87135.10.1103/RevModPhys.78.87
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913975.10.1103/RevModPhys.73.913
Faranda, D., Lembo, V., Iyer, M., Kuzzay, D., Chibbaro, S., Daviaud, F. & Dubrulle, B. 2018 Computation and characterization of local subfilter-scale energy transfers in atmospheric flows. J. Atmos. Sci. 75 (7), 21752186.10.1175/JAS-D-17-0114.1
Farge, M. & Schneider, K. 2001 Coherent vortex simulation (cvs), a semi- deterministic turbulence model using wavelets. Flow Turbul. Combust. 66, 393426.10.1023/A:1013512726409
Frisch, U. 1996 Turbulence, the legacy of A. N. Kolmogorov. Cambridge University Press.
Frisch, U. 2016 The collective birth of multifractals. J. Phys. A: Math. Theoret. 49 (45), 451002.10.1088/1751-8113/49/45/451002
Frisch, U. & Parisi, G. 1985 On the singularity structure of fully developed turbulence. In Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (ed. Gil, M., Benzi, R. & Parisi, G.), pp. 8488. Elsevier.
Galtier, S. 2018 On the origin of the energy dissipation anomaly in (Hall) magnetohydrodynamics. J. Phys. A: Math. Theoret. 51 (20), 205501.10.1088/1751-8121/aabbb5
Granero-Belinchón, C., Roux, S. G. & Garnier, N. B. 2018 Kullback-Leibler divergence measure of intermittency: application to turbulence. Phys. Rev. E 97, 013107.
Jimenez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.10.1017/jfm.2018.144
Jucha, J., Xu, H., Pumir, A. & Bodenschatz, E. 2014 Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113, 054501.10.1103/PhysRevLett.113.054501
Kestener, P. & Arneodo, A. 2004 Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional turbulence velocity and vorticity data. Phys. Rev. Lett. 93 (4), 044501.10.1103/PhysRevLett.93.044501
Kimura, Y. & Moffatt, H. K. 2018 A tent model of vortex reconnection under Biot–Savart evolution. J. Fluid Mech. 834, R1.10.1017/jfm.2017.769
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluids for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301305.
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, 8285.10.1017/S0022112062000518
Kraichnan, R. H. 1974 On Kolmogorov’s inertial-range theories. J. Fluid Mech. 62 (2), 305330.10.1017/S002211207400070X
Kraichnan, R. H. 1975 Remarks on turbulence theory. Adv. Math. 16, 305.10.1016/0001-8708(75)90116-4
Kuzzay, D., Saw, E.-W., Martins, F. J. W. A., Faranda, D., Foucaut, J.-M., Daviaud, F. & Dubrulle, B. 2017 New method for detecting singularities in experimental incompressible flows. Nonlinearity 30 (6), 23812402.10.1088/1361-6544/aa6aaf
Laval, J.-P., Dubrulle, B. & Nazarenko, S. 2001 Nonlocality and intermittency in three-dimensional turbulence. Phys. Fluids 13 (7), 19952012.10.1063/1.1373686
Leberre, M. & Pomeau, Y.2018 Recording of Leray-type singular events in a high speed wind tunnel. arXiv:1801.01762.
Leray, J. 1934 Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193248.10.1007/BF02547354
Li, Y., Perlman, E., Wan, M., Yang, Y., Burns, R., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, N31.10.1080/14685240802376389
Mailybaev, A. A. 2012 Renormalization and universality of blowup in hydrodynamic flows. Phys. Rev. E 85, 066317.
Mailybaev, A. A. 2013 Blowup as a driving mechanism of turbulence in shell models. Phys. Rev. E 87, 053011.
Mandelbrot, B. B. 1972 Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence (ed. Rosenblatt, M. & Van Atta, C.), pp. 333351. Springer.10.1007/3-540-05716-1_20
Meneveau, C. & Sreenivasan, K. R. 1991 The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429484.10.1017/S0022112091001830
Monthus, C., Berche, B. & Chatelain, C. 2009 Symmetry relations for multifractal spectra at random critical points. J. Stat. Mech. 2009 (12), P12002.10.1088/1742-5468/2009/12/P12002
Mordant, N., Metz, P., Michel, O. & Pinton, J.-F. 2001 Measurement of Lagrangian velocity in fully developed turbulence. Phys. Rev. Lett. 87, 214501.10.1103/PhysRevLett.87.214501
Muzy, J. F., Bacry, E. & Arneodo, A. 1991 Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys. Rev. Lett. 67 (25), 35153518.10.1103/PhysRevLett.67.3515
Nazarenko, S. V. & Grebenev, V. N. 2017 Self-similar formation of the Kolmogorov spectrum in the Leith model of turbulence. J. Phys. A: Math. Theoret. 50 (3), 035501.10.1088/1751-8121/50/3/035501
Necas, J., Ruziicka, M. & Sverak, V. 1996 On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176 (2), 283294.10.1007/BF02551584
Nore, C., Castanon, Q. D., Cappanera, L. & Guermond, J.-L. 2018 Numerical simulation of the von Karman sodium dynamo experiment. J. Fluid Mech. 854, 164195.10.1017/jfm.2018.582
Onsager, L. 1949 Statistical hydrodynamics. Il Nuovo Cimento 6 (2), 279287.10.1007/BF02780991
Ott, S. & Mann, J. 2000 An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech. 422, 207223.10.1017/S0022112000001658
Paladin, G. & Vulpiani, A. 1987 Anomalous scaling laws in multifractal objects. Phys. Rep. 156 (4), 147225.10.1016/0370-1573(87)90110-4
Pereira, R. M., Garban, C. & Chevillard, L. 2016 A dissipative random velocity field for fully developed fluid turbulence. J. Fluid Mech. 794, 369408.10.1017/jfm.2016.166
Pumir, A. & Siggia, E. D. 1992 Finite-time singularities in the axisymmetric three-dimension Euler equations. Phys. Rev. Lett. 68, 15111514.10.1103/PhysRevLett.68.1511
Ravelet, F., Chiffaudel, A. & Daviaud, F. 2008 Supercritical transition to turbulence in an inertially driven von Kármán closed flow. J. Fluid Mech. 601, 339364.10.1017/S0022112008000712
Saint-Michel, B.2013 Von Karman flow as a paradigm for non-equilibrium statistical physics, Université Pierre et Marie Curie.
Saint-Michel, B., Herbert, E., Salort, J., Baudet, C., Bon Mardion, M., Bonnay, P., Bourgoin, M., Castaing, B., Chevillard, L., Daviaud, F. et al. 2014 Probing quantum and classical turbulence analogy in von Karman liquid helium, nitrogen, and water experiments. Phys. Fluids 26 (12), 125109.10.1063/1.4904378
Saw, E. W., Kuzzay, D., Faranda, D., Guittoneau, A., Daviaud, F., Wiertel-Gasquet, C., Padilla, V. & Dubrulle, B. 2016 Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow. Nature Comm. 7, 12466.10.1038/ncomms12466
She, Z.-S. & Waymire, E. C. 1995 Quantized energy cascade and log-poisson statistics in fully developed turbulence. Phys. Rev. Lett. 74, 262265.10.1103/PhysRevLett.74.262
da Silva, C. B. & Pereira, J. C. F. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20 (5), 055101.10.1063/1.2912513
Taylor, G. I. 1938 Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. 164 (916), 1523.
Touchette, H. 2009 The large deviation approach to statistical mechanics. Phys. Rep. 478 (1), 169.10.1016/j.physrep.2009.05.002
Xu, H., Bourgoin, M., Ouellette, N. T. & Bodenschatz, E. 2006 High order Lagrangian velocity statistics in turbulence. Phys. Rev. Lett. 96, 024503.10.1103/PhysRevLett.96.024503
Yeung, P. K., Zhai, X. M. & Sreenivasan, K. R. 2015 Extreme events in computational turbulence. Proc. Natl Acad. Sci. USA 112 (41), 1263312638.10.1073/pnas.1517368112
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed