Numéro |
J. Phys. I France
Volume 7, Numéro 6, June 1997
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Page(s) | 759 - 765 | |
DOI | https://doi.org/10.1051/jp1:1997190 |
J. Phys. I France 7 (1997) 759-765
A Renormalization Group Study of Three-Dimensional Turbulence
Ph. BraxDAMTP, University of Cambridge, Silver Street, Cambridge CB39EW, UK
(Received 24 January 1997, revised 12 March 1997, accepted 4 April 1997)
Abstract
We study the three-dimensional Navier-Stokes equation with a random Gaussian force acting on large wavelengths. Our work has
been inspired by Polyakov's analysis of steady states of two-dimensional turbulence. We investigate the time evolution of
the probability law of the velocity potential. Assuming that this probability law is initially defined by a statistical field
theory in the basin of attraction of a renormalization fixed point, we show that its time evolution is obtained by averaging
over small scale features of the velocity potential. The probability law of the velocity potential converges to the fixed
point in the long time regime. At the fixed point, the scaling dimension of the velocity potential is determined to be
-4/3. We give conditions for the existence of such a fixed point of the renormalization group describing the long time behaviour
of the velocity potential. At this fixed point, the energy spectrum of three-dimensional turbulence coincides with a Kolmogorov
spectrum.
© Les Editions de Physique 1997