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Edges in models of shear flow

Published online by Cambridge University Press:  13 March 2013

Norman Lebovitz  and
Giulio Mariotti
Norman Lebovitz*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA
Giulio Mariotti
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

A characteristic feature of the onset of turbulence in shear flows is the appearance of an ‘edge’, a codimension-one invariant manifold that separates ‘lower’ orbits, which decay directly to the laminar state, from ‘upper’ orbits, which decay more slowly and less directly. The object of this paper is to elucidate the structure of the edge that makes this behaviour possible. To this end we consider a succession of low-dimensional models. In doing this we isolate geometric features that are robust under increase of dimension and are therefore candidates for explaining analogous features in higher dimension. We find that the edge, which is the stable manifold of a ‘lower-branch’ state, winds endlessly around an ‘upper-branch’ state in such a way that upper orbits are able to circumnavigate the edge and return to the laminar state.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Low-dimensional models Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 721 , 25 April 2013 , pp. 386 - 402
Copyright
©2013 Cambridge University Press

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