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By Tsinober, A. (1937-)

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8 The common view of turbulence dynamics involves the Richardson-Kolmogorov cascade of energy (the famous poem by Richardson). g. in all spatially developing turbulent flows. All the examples given in the previous section on the partly turbulent flows are such, both free and wall-bounded. The important point is that the so-called ‘cascade’ takes place not in the physical space. We will return to this point and to the notion of scale later in chapter 5 (see also appendix C). 18 CHAPTER 1 - S TRONGLY DIFFUSIVE (random waves are not).

However, Kolmogorov’s ideas on the experimentalist’s difficulties in distinguishing between quasi-periodic systems with many basic frequencies and genuinely chaotic systems have not yet been formalized (Arnold, 1991). In other words it is very difficult if not impossible to make such a distinction in practice. Here is the right place to note that there is an important difference between the number of degrees of freedom rougly proportional to the number of ordinary differential equations necessary to adequately represent a system described by partial differential equations (NSE) and the dimension of the attractor of the system (if such exists).

We emphasize that statistical description is not synonymous to statistical theorization. g. solitons in the systems described by the Korteveg de Vries and Shrödinger equations, and shocks in the Burgers equation and . Again most of the examples given in the previous section clearly show the multi-scale nature of turbulent flows in space. Turbulent flows exhibit also quite complex behaviour in time as well, see plate 6. - LOSS OF PREDICTABILITY. Two initially nearly (but not precisely) identical turbulent flows become unrecognizably different on the time scale of dynamical interest.

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An informal introduction to turbulence by Tsinober, A. (1937-)

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