From Leonardo da Vinci to Andrei Kolmogorov to Giorgio Parisi: Five Centuries of Temporal Decay of Turbulence.

21.09.2022 15:55 - 17:15

Uriel Frisch (CNRS)

Abstract: Leonardo da Vinci (1452-1519) had a strong interest in hydrodynamics, particularly in the last 15 years of his life. Around1505, in one of his mostly famous notebooks, Leonardo got interested in "turbulence" (he was the first to propose 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) [1].
In spite of Leonardo's strong interested in mathematics, at that time, it consisted basically of geometry and simple polynomial equations. There were no tools able to describe the very slowly temporal relaxation of turbulence.
This topic would remain dormant for about 430 years, until in 1938 Kármán [2], 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 [3] found a algebraic mistake in Kármán's calculation; 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 [4]), proved later wrong by Proudman and Reid [5]. The main change in the last few decades is that fully developed turbulence is definitely not self-similar, not only is it fractal (as proposed by Mandelbrot), but it can have infinitely many fractal scalings (multifractality), as proposed by Parisi and Frisch in the eighties [6].
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 interesiing possiibility, which is being explored by Frisch, Khanin, Pandit and Roy. A brief exploration of what happens to the energy decay-law will be presented.
Will there soon be an IR-multifractal theory of the energy decay of turbulence?

[1] Frisch, U. 1995 "Turbulence. The legacy of A.N. Kolmogorov", CUP (p. 112).
[2] Karman, T. and Howarth, L. 1938. On the statistical theory of isotropic turbulence. Proc. Roy. Soc. London, Series A, vol. 164, 192-215.
[3] Kolmogorov, A.N. 1941. On degeneration of isotropic turbulence in an incompressible viscous liquid. Doklady Akad. Nauk SSSR, vol. 31, 838.
[4] Loitsiansky, L.G. 1939. Some basic laws for isotropic turbulent flow. C.A.H.I. (Moscow), Rept. 440.
[5] Proudman, I. and Reid, W.H. 1954. On the decay of a normally distributed and homogeneous turbulent velocity field, Philosoph. Trans. Royal Soc. London, Series A. Mathematical and Physical Sciences, vol. 247, pp. 163-189.
[6] Parisi G. and Frisch U. 1985 "On the singularity structure of fully developed turbulence", in Proceed. Turbulence and predictability in geophysical fluid dynamics and climate

Norbert J. Mauser

HS 13, 2. OG., OMP 1