## The Theory of Homogeneous TurbulenceThis is a reissue of Professor Batchelor's text on the theory of turbulent motion, which was first published by Cambridge Unviersity Press in 1953. It continues to be widely referred to in the professional literature of fluid mechanics, but has not been available for several years. This classic account includes an introduction to the study of homogeneous turbulence, including its mathematic representation and kinematics. Linear problems, such as the randomly-perturbed harmonic oscillator and turbulent flow through a wire gauze, are then treated. The author also presents the general dynamics of decay, universal equilibrium theory, and the decay of energy-containing eddies. There is a renewed interest in turbulent motion, which finds applications in atmospheric physics, fluid mechanics, astrophysics, and planetary science. |

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### Contents

INTRODUCTION | 1 |

12 Mathematical formulation of the problem | 3 |

13 Brief history of the subject | 7 |

MATHEMATICAL REPRESENTATION OF THE FIELD OK TURBULENCE | 14 |

22 The complete statistical specification of the field of turbulence | 17 |

23 Mean values of velocity products | 19 |

24 General properties of the velocity correlation and spectrum tensors | 23 |

25 Fourier analysis of the velocity field | 28 |

55 Dynamical equations for isotropic turbulence | 99 |

THE UNIVERSAL EQUILIBRIUM THEORY | 103 |

62 Turbulent motion at large Reynolds number | 106 |

63 The hypothesis of independence of Fourier components for distant wavenumbers | 109 |

64 The universal equilibrium | 114 |

65 The inertial subrange | 121 |

66 The energy spectrum in the equilibrium range | 125 |

DECAY OF THE ENERGYCONTAINING EDDIES | 133 |

THE KINEMATICS OF HOMOGENEOUS TURBULENCE | 34 |

32 The vorticity correlation and spectrum tensors | 38 |

33 Symmetry conditions | 40 |

34 Isotropic turbulence | 45 |

SOME LINEAR PROBLEMS | 55 |

42 Passage of a turbulent stream through wire gauze | 58 |

43 Effect of sudden distortion of a turbulent stream | 68 |

THE GENERAL DYNAMICS OF DECAY | 76 |

52 The flow of energy | 82 |

53 The permanence of big eddies | 88 |

54 The final period of decay | 92 |

72 Evidence for the existence of a unique statistical state of the energycontaining eddies | 139 |

73 The quasiequilibrium hypothesis | 148 |

74 The equilibrium at large wavenumbers for moderate Reynolds numbers | 155 |

75 Heisenbergs form of the energy spectrum in the quasi equilibrium range | 161 |

THE PROBABILITY DISTRIBUTION OF ux | 169 |

82 The hypothesis of a normal distribution of the velocity field associated with the energycontaining eddies | 174 |

83 Determination of the pressure covariance | 177 |

84 The smallscale properties of the motion | 183 |

BIBLIOGRAPHY OF RESEARCH ON HOMOGENEOUS TURBULENCE | 188 |

196 | |

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### Common terms and phrases

anemometer approximately asymptotic average Batchelor and Townsend Central Limit Theorem correlation functions curves described determined different Reynolds numbers dynamical equation effect energy spectrum energy transfer energy-containing eddies energy-containing range equilibrium range exist experimental field of turbulence final period flatness factor fluid Fourier analysis Fourier components Fourier transform G. I. Taylor gauze homogeneous turbulence hypothesis inertia forces inertial subrange initial conditions integral isotropic turbulence joint-probability distribution kinetic energy large Reynolds numbers large wave-numbers linear measurements moderate Reynolds numbers motion associated Navier-Stokes equation normal distribution obtained parameters period of decay points pressure probability density function probability distribution problem Proc product mean values quantities quasi-equilibrium range of wave-numbers relation scalar function solution spectrum function spectrum tensor statistically independent Stewart and Townsend stream theoretical transfer of energy turbulent motion universal equilibrium variation vector velocity components velocity correlation velocity field velocity-product mean values viscous forces vorticity wave-number space wind tunnel zero