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**Turbulence. The legacy of A. N. Kolmogorov.**
*(English)*
Zbl 0832.76001

Cambridge: Cambridge Univ. Press. xiii, 296 p., £45.00; $ 80.00/hbk (1995).

This textbook presents a modern account of turbulence put into historical perspective. After a short introductory chapter 1, the qualitative material presented in chapter 2 naturally leads to a presentation of the basic symmetries of the incompressible Navier-Stokes equations. We also discuss here the basic conservation laws (energy, helicity, etc.) This includes a ‘scale-by-scale energy budget equation’ which allows us to make sense of the idea of transfer of energy among different scales of motion without requiring at this stage a probabilistic formalism.

In chapter 3 we address the question of why a probabilistic description of turbulence is appropriate. The presentation is made in the spirit of modern (but elementary) dynamical systems theory. In chapter 4 we present some of the basic tools of probability theory, which are frequently used in turbulence.

Then, we turn to two of the most basic experimental laws of fully developed turbulence (chapter 5) which provide direct motivation for Kolmogorov’s 1941 theory. This theory is presented in chapter 6 in an unusual way: instead of beginning with the first 1941 turbulence paper with its (now) somewhat questionable hypotheses, we start with a different set of hypotheses linked to the basic symmetries of the Navier- Stokes equation and to the experimental laws reported in the previous chapter. This includes the assumption that the solutions are (statistically) scale-invariant, but no assumption about the value of the scaling exponent. We then discuss Landau’s objection to one aspect of the 1941 theory.

Phenomenology of turbulence is presented in chapter 7. To avoid a ‘black magic’ impression, we present phenomenology only after more systematic theory for which it is essentially a shorthand system. Intermittency, a particular case of the break-down of the 1941 theory, is discussed at length in chapter 8. Kolmogorov’s (1961, 1962) work is presented only after the discussion of various more recent intermittency models and in the light of exact results about admissible deviations to his 1941 theory. Chapter 9 is a guided tour of further reading on turbulence; it also contains additional historical material.

In chapter 3 we address the question of why a probabilistic description of turbulence is appropriate. The presentation is made in the spirit of modern (but elementary) dynamical systems theory. In chapter 4 we present some of the basic tools of probability theory, which are frequently used in turbulence.

Then, we turn to two of the most basic experimental laws of fully developed turbulence (chapter 5) which provide direct motivation for Kolmogorov’s 1941 theory. This theory is presented in chapter 6 in an unusual way: instead of beginning with the first 1941 turbulence paper with its (now) somewhat questionable hypotheses, we start with a different set of hypotheses linked to the basic symmetries of the Navier- Stokes equation and to the experimental laws reported in the previous chapter. This includes the assumption that the solutions are (statistically) scale-invariant, but no assumption about the value of the scaling exponent. We then discuss Landau’s objection to one aspect of the 1941 theory.

Phenomenology of turbulence is presented in chapter 7. To avoid a ‘black magic’ impression, we present phenomenology only after more systematic theory for which it is essentially a shorthand system. Intermittency, a particular case of the break-down of the 1941 theory, is discussed at length in chapter 8. Kolmogorov’s (1961, 1962) work is presented only after the discussion of various more recent intermittency models and in the light of exact results about admissible deviations to his 1941 theory. Chapter 9 is a guided tour of further reading on turbulence; it also contains additional historical material.

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Fxx | Turbulence |

01A70 | Biographies, obituaries, personalia, bibliographies |