《湍流》是根据弗里希多年来授课的教材整理而成的，主要内容是以Kolmogorov学派的湍流理论为主，系统介绍了半个世纪以来这一理论的发展及今后的研究方向，《湍流》内容新，重点突出，阐述深入浅出。最后一章还专门向读者介绍了如何阅读参考资料。

《湍流》(作者弗里希) 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.

《湍流》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.

Preface

CHAPTER 1

Introduction

1.1 Turbulence and symmetries

1.2 Outline of the book

CHAPTER 2

Symmetries and conservation laws

2.1 Periodic boundary conditions

2.2 Symmetries

2.3 Conservation laws

2.4 Energy budget scale-by-scale

CHAPTER 3 Why a probabilistic description of turbulence?

3.1 There is something predictable in a turbulent signal

3.2 A model for deterministic chaos

3.3 Dynamical systems

3.4 The Navier-Stokes equation as a dynamical system

CHAPTER 4

Probabilistic tools: a survey

4.1 Random variables

4.2 Random functions

4.3 Statistical symmetries

4.4 Ergodic results

4.5 The spectrum of stationary random functions

CHAPTER 5

Two experimental laws of fully developed turbulence

5.1 The two-thirds law

5.2 The energy dissipation law

CHAPTER 6

The Kolmogorov 1941 theory

6.1 Kolmogorov 1941 and symmetries

6.2 Kolmogorov's four-fifths law

6.2.1 The Karman-Howarth-Monin relation for anisotropic turbulence

6.2.2 The energy flux for homogeneous turbulence

6.2.3 The energy flux for homogeneous isotropic turbulence.

6.2.4 From the energy flux relation to the four-fifths law

6.2.5 Remarks on Kolmogorov's four-fifths law

6.3 Main results of the Kolmogorov 1941 theory

6.3.1 The Kolmogorov-Obukhov law and the structure functions

6.3.2 Effect of a finite viscosity: the dissipation range

6.4 Kolmogorov and Landau: the lack of universality

6.4.1 The original formulation of Landau's objection

6.4.2 A modern reformulation of Landau's objection

6.4.3 Kolmogorov and Landau reconciled?

6.5 Historical remarks on the Kolmogorov 1941 theory

CHAPTER 7

Phenomenology of turbulence in the sense of Kolmogorov 1941

7.1 Introduction

7.2 Basic tools of phenomenology

7.3 The Richardson cascade and the localness of interactions

7.4 Reynolds numbers and degrees of freedom

7.5 Microscopic and macroscopic degrees of freedom

7.6 The distribution of velocity gradients

7.7 The law of decay of the energy

7.8 Beyond phenomenology: finite-time blow-up of ideal flow

CHAPTER 8

Intermittency

8.1 Introduction

8.2 Self-similar and intermittent random functions

8.3 Experimental results on intermittency

8.4 Exact results on intermittency

8.5 Intermittency models based on the velocity

8.5.1 The//-model

8.5.2 The bifractal model

8.5.3 The muitifractal model

8.5.4 A probabilistic reformulation of the multifractal model

8.5.5 The intermediate dissipation range and multifractal universality

8.5.6 The skewness and the flatness of velocity derivatives according to the muitifractal model

8.6 Intermittency models based on the dissipation

8.6.1 Multifractal dissipation

8.6.2 Bridging multifractality based on the velocity and multifractality based on the dissipation

8.6.3 Random cascade models

8.6.4 Large deviations and multifractality

8.6.5 The lognormal model and its shortcomings

8.7 Shell models

8.8 Historical remarks on fractal intermittency models

8.9 Trends in intermittency research

8.9.1 Vortex filaments: the sinews of turbulence?

8.9.2 Statistical signature of vortex filaments: dog or tail?

8.9.3 The distribution of velocity increments

CHAPTER 9

Further reading: a guided tour

9.1 Introduction

9.2 Books on turbulence and fluid mechanics .

9.3 Mathematical aspects of fully developed turbulence

9.4 Dynamical systems, fraetals and turbulence

9.5 Closure, functional and diagrammatic methods

9.5.1 The Hopfequation

9.5.2 Functional and diagrammatic methods

9.5.3 The direct interaction approximation

9.5.4 Closures and their shortcomings

9.6 Eddy viscosity, multiscale methods and renormalization

9.6.1 Eddy viscosity: a very old idea

9.6.2 Multiscale methods

9.6.3 Applications of multiscale methods in turbulence

9.6.4 Renormalization group (RG) methods

9.7 Two-dimensional turbulence

9.7.1 Cascades and vortices

9.7.2 Two-dimensional turbulence and statistical mechanics.

9.7.3 Conservative dynamics punctuated by dissipative events

9.7.4 From Flatland to three-dimensional turbulence

References

Author index

Subject index