Turbulence is a huge subject of ongoing research. This book bridges the modern development in dynamical systems theory and the theory of fully developed turbulence. Many solved and unsolved problems in turbulence have equivalencies in simple dynamical models, which are much easier to handle analytically and numerically. This book gives a modern view of the subject by first giving the essentials of the theory of turbulence before moving on to shell models. These show much of the same complex behaviour as fluid turbulence, but are much easier to handle analytically and numerically. Any necessary maths is explained and self-contained, making this book ideal for advanced undergraduates and graduate students, as well as researchers and professionals, wanting to understand the basics of fully developed turbulence.
Author(s): Peter D. Ditlevsen
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 164
Tags: Механика;Механика жидкостей и газов;Турбулентность;
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1 Introduction to turbulence......Page 13
1.1 The Navier–Stokes equation......Page 15
1.2 Kolmogorov's 1941 theory (K41)......Page 19
1.3 The spectral Navier–Stokes equation......Page 21
1.4 The spectral energy density......Page 22
1.5 The spectral energy flux......Page 24
1.6 The closure problem......Page 25
1.7 The four-fifth law......Page 26
1.8 Self-similarity of the energy spectrum......Page 30
1.8.1 K — epsilon models......Page 34
1.10 The vorticity equation......Page 35
1.11 Intermittency in turbulence......Page 36
1.12 Finite time singularities......Page 37
1.13 Problems......Page 38
2.1 2D turbulence......Page 42
2.2 Atmospheric turbulence......Page 46
2.3 The governing equations......Page 47
2.3.2 Hydrostatic balance......Page 49
2.5 Stratification and rotational Froude number......Page 50
2.6 Quasi-geostrophy......Page 51
2.7 Observations of the atmosphere......Page 53
2.8 Problems......Page 56
3 Shell models......Page 58
3.1 The Obukhov shell model......Page 59
3.2 Liouville's theorem......Page 60
3.3 The Gledzer shell model......Page 62
3.4 Scale invariance of the shell model......Page 63
3.5 The shell spacing and energy conservation......Page 64
3.6 Parameter space for the GOY model......Page 65
3.7 2D and 3D shell models......Page 68
3.8 Other quadratic invariants......Page 69
3.9 Triad interactions and nonlinear fluxes......Page 70
3.10 The special case … = 1......Page 72
3.11 The Sabra shell model......Page 73
3.12 Problems......Page 75
4.1 The nonlinear fluxes......Page 77
4.2 3D GOY and Sabra models......Page 79
4.3 Phase symmetries......Page 83
4.4 Equivalent of the four-fifth law......Page 84
4.5 Problems......Page 85
5 Chaotic dynamics......Page 86
5.1 The Lyapunov exponent......Page 87
5.2 The attractor dimension......Page 91
5.3 The attractor for the shell model......Page 95
5.4 Predictability......Page 97
5.5 Large scale predictability......Page 98
5.6 The finite size Lyapunov exponent......Page 99
5.7 Limited predictability of the small scales......Page 101
5.8 Problems......Page 105
6.1 The helicity spectrum......Page 107
6.3 The helicity dissipation scale......Page 109
6.4 Helicity in shell models......Page 112
6.5 A generalized helical GOY model......Page 117
6.6 Problems......Page 120
7.1 Kolmogorov's lognormal correction......Page 122
7.2 The Beta-model......Page 125
7.3 The multi-fractal model......Page 126
7.4 Intermittency in shell models......Page 127
7.5 Probability densities and intermittency......Page 128
7.6 Problems......Page 132
8.1 The statistical ensemble......Page 133
8.2 The partition function......Page 135
8.3 Phase space geometry......Page 139
8.4 Statistical equilibrium and turbulence......Page 140
8.5 Cascade or equilibrium......Page 142
8.6 Problems......Page 145
A.1 Velocity Fourier transforms in 3D......Page 147
A.2 Rotation of the velocity field......Page 149
A.3 Product rules for Levi–Civita symbols......Page 150
A.4 Divergence theorem (integration by parts)......Page 151
A.5 The vorticity equation......Page 152
A.6 Helicity......Page 153
A.8 The 2D case......Page 154
A.9 Scaling consequence of lognormal assumption......Page 156
References......Page 159
Index......Page 163
