Turbulence: The Legacy of A. N. KolmogorovThis textbook presents a modern account of turbulence, one of the greatest challenges in physics. The state-of-the-art is put into historical perspective five centuries after the first studies of Leonardo and half a century after the first attempt by A.N. Kolmogorov to predict the properties of flow at very high Reynolds numbers. Such "fully developed turbulence" is ubiquitous in both cosmical and natural environments, in engineering applications and in everyday life. First, a qualitative introduction is given to bring out the need for a probabilistic description of what is in essence a deterministic system. Kolmogorov's 1941 theory is presented in a novel fashion with emphasis on symmetries (including scaling transformations) which are broken by the mechanisms producing the turbulence and restored by the chaotic character of the cascade to small scales. Considerable material is devoted to intermittency, the clumpiness of small-scale activity, which has led to the development of fractal and multifractal models. Such models, pioneered by B. Mandelbrot, have applications in numerous fields besides turbulence (diffusion limited aggregation, solid-earth geophysics, attractors of dynamical systems, etc). The final chapter contains an introduction to analytic theories of the sort pioneered by R. Kraichnan, to the modern theory of eddy transport and renormalization and to recent developments in the statistical theory of two-dimensional turbulence. The book concludes with a guide to further reading. The intended readership for the book ranges from first-year graduate students in mathematics, physics, astrophysics, geosciences and engineering, to professional scientists and engineers. |
Contents
CHAPTER 1 | 13 |
4 | 21 |
CHAPTER 3 | 27 |
Two experimental laws of fully developed turbulence | 57 |
CHAPTER 6 | 73 |
Phenomenology of turbulence in the sense of Kolmogorov 1941 | 100 |
Intermittency | 120 |
97 | 167 |
40 | 189 |
CHAPTER 9 | 195 |
45 | 227 |
| 255 | |
| 283 | |
| 289 | |
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Common terms and phrases
Anselmet argument assumed assumption Benzi blow-up closure correlation function cumulant defined derivation dimension discussed in Section dissipation range dynamical systems eddy turnover eddy viscosity energy dissipation energy flux energy spectrum enstrophy Euler equation experimental exponent h exponential finite fluid follows four-fifths law Fourier fractal Frisch fully developed turbulence function of order Gagne Galilean invariance Galilean transformations Hence high Reynolds numbers homogeneous Hopfinger and Antonia incompressible inertial range inertial-range scales integral scale intermittency invariant isotropic K41 theory Kolmogorov dissipation scale Kraichnan Landau Legendre transform lognormal model measure Meneguzzi Monin and Yaglom multifractal model Navier-Stokes equation nonlinear observed obtained order structure function Orszag phenomenology power-law predicted probabilistic r.m.s. velocity random cascade models random function random variable relation renormalization scaling exponent self-similar shell models simulations singularities solution spatial ẞ-model statistical structure functions Sulem symmetries three-dimensional transform turbulent flow velocity field velocity increments Vergassola vortex filaments vorticity wavenumbers zero
Popular passages
Page 258 - Transition to chaos in a shell model of turbulence, Physica D 80, 105-119.
Page 258 - Brachet, ME, Meiron, DI, Orszag, SA, Nickel, BG, Morf, RH & Frisch, U. 1983. Small-scale structure of the Taylor-Green vortex, J. Fluid Mech. 130, 411-451 Brachet, ME, Meneguzzi, M., Politano, H.
Page 259 - Tabeling, P. 1994. Quantitative experimental study of the free decay of quasi-two-dimensional turbulence, Phys. Rev. E 49, 454^61.
Page 256 - Aurell, E., Frisch, U., Lutsko, J. & Vergassola, M. 1992. On the multifractal properties of the energy dissipation derived from turbulence data, /. Fluid Mech.
Page 259 - Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion.