Journal of Fluid Mechanics

Determining modes and fractal dimension of turbulent flows

P.  Constantin a1, C.  Foias a1, O. P.  Manley a2 and R.  Temam a3
a1 Indiana University, Bloomington, IN 47405
a2 U.S. Department of Energy, Washington DC 20545
a3 Laboratoire d'Analyse Numérique, CNRS and Université Paris-Sud, 91405 Orsay

Article author query
constantin p   [Google Scholar] 
foias c   [Google Scholar] 
manley op   [Google Scholar] 
temam r   [Google Scholar] 


Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.

(Published Online April 20 2006)
(Received January 17 1984)
(Revised July 31 1984)