We study the large time behavior of boundary and pressure-gradient driven incompressible fluid flows in elongated two-dimensional channels with emphasis on estimates for their degrees of freedom, i.e., the dimension of the attractor for the solutions of the Navier-Stokes equations. for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form ca Re3/2 where c is a universal constant, a denotes the aspect ratio of the channel (length/width), and Re is the Reynolds number based on the channel width and the imposed "outer" velocity scale. for fixed pressure gradient driven channel flows we obtain an upper bound of the form c1 α Re2, where c1 is another universal positive constant and the Reynolds number is based on a velocity defined by the infimum, 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 © 1998 Elsevier Science B.V.
C. R. Doering and X. Wang, "Attractor Dimension Estimates for Two-Dimensional Shear Flows," Physica D: Nonlinear Phenomena, vol. 123, no. 1 thru 4, pp. 206 - 222, Elsevier, Jan 1998.
The definitive version is available at https://doi.org/10.1016/S0167-2789(98)00121-3
Mathematics and Statistics
Keywords and Phrases
Background Flows; Channel Flows; Energy Dissipation Rate; Global Attractor; Hausdorff and Fractal Dimensions; Lieb-Thirring Inequality; Navier-Stokes Equations; Reynolds Number; Small Length Scales
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 1998