Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T10:11:15.469Z Has data issue: false hasContentIssue false
  • English
  • Français

Beyond Kolmogorov cascades

Published online by Cambridge University Press:  22 March 2019

Bérengère Dubrulle Open the ORCID record for Bérengère Dubrulle [Opens in a new window]
Bérengère Dubrulle*
Affiliation:
SPEC, CEA, CNRS, Université Paris-Saclay, F-91191 CEA Saclay, Gif-sur-Yvette, France
*
†Email address for correspondence: berengere.dubrulle@cea.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Turbulent Flows: Intermittency Turbulent Flows: Turbulence theory
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , P1
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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), 685–689.Google Scholar
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.Google Scholar
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), 411–416.Google Scholar
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.Google Scholar
Batterman, R. W. 2011 Emergence, singularities, and symmetry breaking. Foundations Phys. 41 (6), 1031–1050.Google Scholar
Bec, J. & Khanin, K. 2007 Burgers turbulence. Phys. Rep. 447 (1), 1–66.Google Scholar
Benzi, R. & Biferale, L. 2009 Fully developed turbulence and the multifractal conjecture. J. Stat. Phys. 135 (5), 977–990.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
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.Google Scholar
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.Google Scholar
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.Google Scholar
Buckmaster, T., De Lellis, C., Székelyhidi, L. & Vicol, V. 2019 Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Maths 72 (2), 229–274.Google Scholar
Caffarelli, L., Kohn, R. & Nirenberg, L. 1982 Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Maths 35 (6), 771–831.Google Scholar
Campolina, C. S. & Mailybaev, A. A. 2018 Chaotic blowup in the 3d incompressible Euler equations on a logarithmic lattice. Phys. Rev. Lett. 121, 064501.Google Scholar
Castaing, B., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46 (2), 177–200.Google Scholar
Chae, D. 2007 Nonexistence of self-similar singularities for the 3D incompressible euler equations. Commun. Math. Phys. 1 (273), 203–215.Google Scholar
Chevillard, L.2004 Description multifractale unifiée du phénomène d intermittence en turbulence Eulérienne et Lagrangienne. PhD thesis, Université Sciences et Technologies - Bordeaux I.Google Scholar
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), 899–928; structures and statistics of fluid turbulence/Structures et statistiques de la turbulence des fluides.Google Scholar
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), 3311–3368.Google Scholar
Debue, P.2019 Experimental approach of the Euler and Navier–Stokes singularities problem. PhD thesis, Université Paris-Saclay, Paris.Google Scholar
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.Google Scholar
Dombre, T. & Gilson, J.-L. 1998 Intermittency, chaos and singular fluctuations in the mixed Obukhov-Novikov shell model of turbulence.. Physica D 111 (1), 265–287.Google Scholar
Drivas, T. D. 2019 Turbulent cascade direction and Lagrangian time-asymmetry. J. Nonlinear Sci. 29, 65.Google Scholar
Drivas, T. D. & Eyink, G. L.2017 An onsager singularity theorem for leray solutions of incompressible Navier–Stokes. arXiv:1710.05205.Google Scholar
Duchon, J. & Robert, R. 2000 Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13 (1), 249–255.Google Scholar
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1–R44.Google Scholar
Eyink, G. L.2007–2008 Turbulence Theory. Course notes, The Johns Hopkins University. Available at: http://www.ams.jhu.edu/eyink/Turbulence/notes/.Google Scholar
Eyink, G. L. & Drivas, T. D. 2015 Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158 (2), 386–432.Google Scholar
Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135.Google Scholar
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
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), 2175–2186.Google Scholar
Farge, M. & Schneider, K. 2001 Coherent vortex simulation (cvs), a semi- deterministic turbulence model using wavelets. Flow Turbul. Combust. 66, 393–426.Google Scholar
Frisch, U. 1996 Turbulence, the legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Frisch, U. 2016 The collective birth of multifractals. J. Phys. A: Math. Theoret. 49 (45), 451002.Google Scholar
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. 84–88. Elsevier.Google Scholar
Galtier, S. 2018 On the origin of the energy dissipation anomaly in (Hall) magnetohydrodynamics. J. Phys. A: Math. Theoret. 51 (20), 205501.Google Scholar
Granero-Belinchón, C., Roux, S. G. & Garnier, N. B. 2018 Kullback-Leibler divergence measure of intermittency: application to turbulence. Phys. Rev. E 97, 013107.Google Scholar
Jimenez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.Google Scholar
Jucha, J., Xu, H., Pumir, A. & Bodenschatz, E. 2014 Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113, 054501.Google Scholar
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.Google Scholar
Kimura, Y. & Moffatt, H. K. 2018 A tent model of vortex reconnection under Biot–Savart evolution. J. Fluid Mech. 834, R1.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluids for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301–305.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
Kraichnan, R. H. 1974 On Kolmogorov’s inertial-range theories. J. Fluid Mech. 62 (2), 305–330.Google Scholar
Kraichnan, R. H. 1975 Remarks on turbulence theory. Adv. Math. 16, 305.Google Scholar
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), 2381–2402.Google Scholar
Laval, J.-P., Dubrulle, B. & Nazarenko, S. 2001 Nonlocality and intermittency in three-dimensional turbulence. Phys. Fluids 13 (7), 1995–2012.Google Scholar
Leberre, M. & Pomeau, Y.2018 Recording of Leray-type singular events in a high speed wind tunnel. arXiv:1801.01762.Google Scholar
Leray, J. 1934 Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193248.Google Scholar
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.Google Scholar
Mailybaev, A. A. 2012 Renormalization and universality of blowup in hydrodynamic flows. Phys. Rev. E 85, 066317.Google Scholar
Mailybaev, A. A. 2013 Blowup as a driving mechanism of turbulence in shell models. Phys. Rev. E 87, 053011.Google Scholar
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. 333–351. Springer.Google Scholar
Meneveau, C. & Sreenivasan, K. R. 1991 The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484.Google Scholar
Monthus, C., Berche, B. & Chatelain, C. 2009 Symmetry relations for multifractal spectra at random critical points. J. Stat. Mech. 2009 (12), P12002.Google Scholar
Mordant, N., Metz, P., Michel, O. & Pinton, J.-F. 2001 Measurement of Lagrangian velocity in fully developed turbulence. Phys. Rev. Lett. 87, 214501.Google Scholar
Muzy, J. F., Bacry, E. & Arneodo, A. 1991 Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys. Rev. Lett. 67 (25), 3515–3518.Google Scholar
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.Google Scholar
Necas, J., Ruziicka, M. & Sverak, V. 1996 On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176 (2), 283–294.Google Scholar
Nore, C., Castanon, Q. D., Cappanera, L. & Guermond, J.-L. 2018 Numerical simulation of the von Karman sodium dynamo experiment. J. Fluid Mech. 854, 164–195.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Il Nuovo Cimento 6 (2), 279–287.Google Scholar
Ott, S. & Mann, J. 2000 An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech. 422, 207–223.Google Scholar
Paladin, G. & Vulpiani, A. 1987 Anomalous scaling laws in multifractal objects. Phys. Rep. 156 (4), 147–225.Google Scholar
Pereira, R. M., Garban, C. & Chevillard, L. 2016 A dissipative random velocity field for fully developed fluid turbulence. J. Fluid Mech. 794, 369–408.Google Scholar
Pumir, A. & Siggia, E. D. 1992 Finite-time singularities in the axisymmetric three-dimension Euler equations. Phys. Rev. Lett. 68, 1511–1514.Google Scholar
Ravelet, F., Chiffaudel, A. & Daviaud, F. 2008 Supercritical transition to turbulence in an inertially driven von Kármán closed flow. J. Fluid Mech. 601, 339–364.Google Scholar
Saint-Michel, B.2013 Von Karman flow as a paradigm for non-equilibrium statistical physics, Université Pierre et Marie Curie.Google Scholar
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.Google Scholar
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.Google Scholar
She, Z.-S. & Waymire, E. C. 1995 Quantized energy cascade and log-poisson statistics in fully developed turbulence. Phys. Rev. Lett. 74, 262–265.Google Scholar
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.Google Scholar
Taylor, G. I. 1938 Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. 164 (916), 15–23.Google Scholar
Touchette, H. 2009 The large deviation approach to statistical mechanics. Phys. Rep. 478 (1), 1–69.Google Scholar
Xu, H., Bourgoin, M., Ouellette, N. T. & Bodenschatz, E. 2006 High order Lagrangian velocity statistics in turbulence. Phys. Rev. Lett. 96, 024503.Google Scholar
Yeung, P. K., Zhai, X. M. & Sreenivasan, K. R. 2015 Extreme events in computational turbulence. Proc. Natl Acad. Sci. USA 112 (41), 12633–12638.Google Scholar