## 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. The intended readership for the book ranges from first-year graduate students in mathematics, physics, astrophysics, geosciences and engineering, to professional scientists and engineers. Elementary presentations of dynamical systems ideas, of probabilistic methods (including the theory of large deviations) and of fractal geometry make this a self-contained textbook. |

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### Contents

1 | |

12 Outline of the book | 11 |

Symmetries and conservation laws | 14 |

22 Symmetries | 17 |

23 Conservation laws | 18 |

24 Energy budget scalebyscale | 21 |

Why a probabilistic description of turbulence? | 27 |

32 A model for deterministic chaos | 31 |

74 Reynolds numbers and degrees of freedom | 106 |

75 Microscopic and macroscopic degrees of freedom | 109 |

76 The distribution of velocity gradients | 111 |

77 The law of decay of the energy | 112 |

finitetime blowup of ideal flow | 115 |

Intermittency | 120 |

82 Selfsimilar and intermittent random functions | 121 |

83 Experimental results on intermittency | 127 |

33 Dynamical systems | 36 |

34 The NavierStokes equation as a dynamical system | 37 |

Probabilistic tools a survey | 40 |

42 Random functions | 45 |

43 Statistical symmetries | 46 |

44 Ergodic results | 49 |

45 The spectrum of stationary random functions | 52 |

Two experimental laws of fully developed turbulence | 57 |

52 The energy dissipation law | 67 |

The Kolmogorov 1941 theory | 72 |

62 Kolmogorovs fourfifths law | 76 |

63 Main results of the Kolmogorov 1941 theory | 89 |

the lack of universality | 93 |

65 Historical remarks on the Kolmogorov 1941 theory | 98 |

Phenomenology of turbulence in the sense of Kolmogorov 1941 | 100 |

72 Basic tools of phenomenology | 101 |

73 The Richardson cascade and the localness of interactions | 103 |

84 Exact results on intermittency | 133 |

85 Intermittency models based on the velocity | 135 |

86 Intermittency models based on the dissipation | 159 |

87 Shell models | 174 |

88 Historical remarks on fractal intermittency models | 178 |

89 Trends in intermittency research | 182 |

Further reading a guided tour | 195 |

93 Mathematical aspects of fully developed turbulence | 199 |

94 Dynamical systems fractals and turbulence | 203 |

95 Closure functional and diagrammatic methods | 206 |

96 Eddy viscosity multiscale methods and renormalization | 222 |

97 Twodimensional turbulence | 240 |

255 | |

283 | |

289 | |

### Other editions - View all

Turbulence: The Legacy of A. N. Kolmogorov Uriel Frisch,Andreĭ Nikolaevich Kolmogorov Limited preview - 1995 |

Turbulence: The Legacy of A. N. Kolmogorov Uriel Frisch,Andreĭ Nikolaevich Kolmogorov No preview available - 1995 |

### Common terms and phrases

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