Leonardo da Vinci, Andrei Kolmogorov and Giorgio Parisi. The energy decay of turbulence from Leonardo to multifractal theory
ABSTRACT :
Leonardo da Vinci had a strong interest in hydrodynamics. Around
1505 he got interested in "turbulence" (he was the first to use this
name). Examining the "turbulences" (eddies) in the river Arno of Florence,
he found that the amplitude of the turbulence was decreasing very slowly
in time, until it would come to rest (within the surrounding river). In
spite of Leonardo's strong interest in mathematics, at that time, it
consisted basically of geometry and simple polynomial equations. There
were no tools available to describe the very slow temporal relaxation of
turbulence. This topic would remain dormant for about 430 years, until in
1938 Karman, triggered by Taylor, established that the mean energy of the
turbulence should decrease very slowly, indeed like an inverse power of
the time elapsed. Three years later, Kolmogorov himself found another
inverse power
(10/7) of the time elapsed. This, likewise was wrong, because he was
assuming a certain invariance property (Loitsiansky , proved later
wrong by Proudman and Reid). The main change in the last few decades is
that fully developed turbulence is definitely not self-similar, not only
is it fractal, but it can have infinitely many fractal scalings
(multifractality), as proposed by Parisi and Frisch in the eighties.
Furthermore, multifractality can manifest itself either at small scales or
at large scales. The latter might change the law of energy decay. Not
enough is understood for the 3D Euler equations, but large-scale
multifractality for the Burgers is an interesting possibility, which is
being explored by Frisch, Khanin, Pandit and Roy. A brief exploration of
what happens to the energy decay-law will be presented.