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By Tsinober, A. (1937-)

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This state of affairs seems to be changed due to two recent developments. First, it was rigorously proved (Fursikov and Emanuilov, 1995 and references therein) that the Keller-Fridman (1925) chain of equations for the moments (and consequently the Hopf equation) has a unique solution for initial conditions in an appropriately chosen functional space6. In other words a positive answer was given to the question whether the closure problem has a solution, and an estimate of convergence of approximations for the closure of the infinite chain of equations for moments was given.

All of them have in common some ad hoc assumptions of unknown validity and obscured physical and mathematical justification. In this sense all the statistical theories are not rigorous. Hans Liepmann wrote in 1979: Turbulent modelling is still on the rise owing to rapid development of computers coupled with the industrial need for management of turbulent flows. I am convinced that much of this huge effort will be of passing interest only. much of this work is never subjected to any kind of critical or comparative judgement.

However, physical plausibility aside, it is embarrassing that such an important feature of turbulence as its statistical stability should remain mathematically unresolved, but such is the nature of the subject (Orszag, 1977). 36 CHAPTER 3 (time-dependent in the statistical sense) turbulent flows are quite similar to those of statistically stationary ones as long as the Reynolds number of the former is not too small at the particular time moment of interest. , 1997; and references therein). It should be emphasized that our concern here is not with statistical theories all of which are using various ad hoc assumptions on the nature and properties mainly of the small scale structure and its relation with the rest of the flow, and/or attempting to represent turbulence as a collection of more or less simple objects.

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