Attractor dimension estimates for two-dimensional shear flows

We study the large time behavior of boundary and pressure-gradient driven i ncompressible fluid flows in elongated two-dimensional channels with emphas is on estimates for their degrees of freedom, i.e., the dimension of the at tractor for the solutions of the Navier-Stokes equations. For boundary driv en shear flows and flux driven channel hows we present upper bounds for the degrees of freedom of the form c alpha Re-3/2 where c is a universal const ant, alpha denotes the aspect ratio of the channel (length/width), and Re i s the Reynolds number based on the channel width and the imposed "outer" ve locity scale. For fixed pressure gradient driven channel flows we obtain an upper bound of the form c'alpha Re-2, where c' is another universal positi ve constant and the Reynolds number is based on a velocity defined by the i nfimum, over all possible trajectories, of the time averaged mass flux per unit channel width. We discuss these results in terms of physical arguments based on small length scales in turbulent flows. Copyright (C) 1998 Elsevi er Science B.V.