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Review article. — Rev. Mod. Phys., 2001, Vol. 73, No. 4, p. 913–975.
The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scaleinvariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported elds. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.<strong><em>Contents <strong>Particles in fluid turbulence Single-particle diffusion Two-particle dispersion in a spatially smooth velocity Two-particle dispersion in a nonsmooth incompressible flow Two-particle dispersion in a compressible flow Multiparticle dynamics, statistical conservation laws and breakdown of scale invariance <strong>Passive Fields Unforced evolution of passive scalar and vector Fields Cascades of a passive scalar Passive fields in the inertial interval of turbulence Lagrangian numerics Inverse cascade in the compressible Kraichnan model Lessons for general scalar turbulence <strong>Burgers and Navier-Stokes equations Burgers turbulence Incompressible turbulence from a Lagrangian viewpoint
E-Book Content
Particles and Fields in Fluid Turbulence
arXiv:cond-mat/0105199 v1 9 May 2001
G. Falkovich
Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
K. Gaw¸edzki
CNRS, IHES, 91940 Bures-sur-Yvette and ENS-Lyon, 46 Alle d’Italie, 69364 Lyon, France
M. Vergassola
CNRS, UMR 6529 Observatoire de la Cˆ ote d’Azur, BP 4229, 06304 Nice, France
Abstract The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scaleinvariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.
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CONTENTS I. Introduction II. Particles in fluid turbulence A. Single-particle diffusion B. Two-particle dispersion in a spatially smooth velocity 1. General considerations 2. Solvable cases C. Two-particle dispersion in a nonsmooth incompressible flow 1. Richardson law 2. Breakdown of the Lagrangian flow 3. The example of the Kraichnan ensemble D. Two-particle dispersion in a compressible flow E. Multiparticle dynamics, statistical conservation laws and breakdown of scale invariance 1. Absolute and relative evolution of particles 2. Multiparticle motion in Kraichnan