Abstract
The hydrodynamic time evolution is Hamiltonian in the inertial range (i.e., in the absence of viscosity). From this we obtain that the macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow in a nonstandard geometry. In the absence of fluctuations this means that the Kolmogorov theory of turbulence is equivalent to a heat flow for a suitable mechanical system. Turbulent fluctuations (intermittency) correspond to thermal fluctuations for the heat flow. A relatively crude estimate of the thermal fluctuations, based on standard ideas of nonequilibrium statistical mechanics is presented: this agrees remarkably well with what is observed in several turbulence experiments. A logical relation with the lognormal theory of Kolmogorov and Obukhov is also indicated, which shows what fails in this theory, and what can be rescued.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Dimensional analysis says how various quantities (like velocity or energy) depend on certain variables (like spatial distance, and time): velocity is spatial distance divided by time, energy is mass times velocity squared, etc. Dimensional analysis appears somewhat trivial, but for the turbulent energy cascade it has led to spectacular predictions.
References
D. Ruelle, Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics. PNAS 109, 20344–20346 (2012)
D. Ruelle, Non-equilibrium statistical mechanics of turbulence. J. Stat. Phys. 157, 205–218 (2014)
G. Gallavotti, G. Garrido, Non-equilibrium statistical mechanics of turbulence: comments on Ruelle’s intermittency theory, in The Foundations of Chaos Revisited: From Poincaré to Recent Advancements, ed. by C. Skiadas (Springer, Heidelberg, 2016). doi:10.1007/978-3-319-29701-9
D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971) and 23, 343–344 (1971)
J.P. Gollub, H.L. Swinney, Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930 (1975)
A. Libchaber, From chaos to turbulence in Benard convection. Proc. R. Soc. Lond. A413, 63–69 (1987)
E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
P. Cvitanović (ed.), Universality in Chaos, 2nd edn. (Adam Hilger, Bristol, 1989)
B.-L. Hao (ed.), Chaos II (World Scientific, Singapore, 1990)
L.-S. Young, What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108, 733–754 (2002)
C. Bonatti, L.J. Díaz, M. Viana, Dynamics Beyond Uniform Hyperbolicity (Springer, Berlin, 2005)
G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)
D. Dolgopyat, C. Liverani, Energy transfer in a fast-slow Hamiltonian system. Commun. Math. Phys. 308, 201–225 (2011)
D. Ruelle, A mechanical model for Fourier’s law of heat conduction. Commun. Math. Phys. 311, 755–768 (2012)
A.N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301–305 (1941)
A.N. Kolmogorov, On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540 (1941)
A.N. Kolmogorov, Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (1941)
G. Parisi, U. Frisch, On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics, ed. by M. Ghil, R. Benzi, G. Parisi (North-Holland, Amsterdam, 1985), pp. 84–88
R. Benzi, G. Paladin, G. Parisi, A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17, 3521–3531 (1984)
F. Anselmet, Y. Gagne, E.J. Hopfinger, R.A. Antonia, High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63–89 (1984)
J. Schumacher, J. Scheel, D. Krasnov, D. Donzis, K. Sreenivasan, V. Yakhot, Small-scale universality in turbulence. Proc. Natl. Acad. Sci. USA 111(30), 10961–10965 (2014)
A.N. Kolmogorov, 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 (1962)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ruelle, D. (2016). Hydrodynamic Turbulence as a Nonstandard Transport Phenomenon. In: Skiadas, C. (eds) The Foundations of Chaos Revisited: From Poincaré to Recent Advancements. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-29701-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-29701-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29699-9
Online ISBN: 978-3-319-29701-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)