## 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. |

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### Contents

CHAPTER | 5 |

CHAPTER | 13 |

1 | 19 |

CHAPTER 3 | 27 |

3 | 87 |

4 | 100 |

Intermittency | 120 |

Experimental results on intermittency | 127 |

Exact results on intermittency | 133 |

6 | 146 |

CHAPTER 9 | 195 |

255 | |

283 | |

289 | |

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### Common terms and phrases

Anselmet argument assumed assumption average B-model Benzi closure cumulative defined denote derivation diffusivity dimension discussed in Section dissipation range dissipation scale dynamical systems eddy viscosity energy flux energy spectrum enstrophy Euler equation experimental exponent h filtering finite fluid follows four-fifths law Fourier fractal Frisch fully developed turbulence Gagne Galilean invariance Galilean transformations high Reynolds numbers homogeneous Hopfinger and Antonia incompressible inertial range inertial-range integral scale intermittency invariant isotropic K41 theory Kolmogorov Kraichnan Landau Legendre transform lognormal measure Monin and Yaglom multifractal multifractal model Navier–Stokes equation nonlinear observed obtained order structure function Orszag phenomenology power-law probabilistic random function random variable relation renormalization scalar scaling exponent shell models shown in Fig simulations singularities solution space spatial statistical structure functions Sulem symmetries tent map three-dimensional transformations turbulent flow two-dimensional turbulence velocity field velocity increments Vergassola vortex filaments vorticity wavenumber 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.