Title: The computation of 2D unstable manifolds in models of turbulent shear flow

Abstract:
The publication of a landmark paper by Kawahara and Kida (2001) on the
relevance of unstable periodic orbits to the dynamics of shear flow has
initiated intense efforts to explain such phenomena as bursting and
subcritical transition in terms of dynamical systems theory. Results
similar to and stronger then those of Kawahara and Kida were since found
in a variety of geometries, such as flow through pipes, ducts and
channels. In all these geometries there exists a laminar flow profile
which remains asymptotically stable up to high, sometimes infinite,
Reynolds number. Thus, the onset of turbulence cannot be explained in
terms of a straightforward bifurcation scenario as is often found in
rotating or differentially heated flows. Instead, the relevant dynamical
structures seem to be periodic orbits which live on the boundary of the
domain of attraction of the laminar flow profile in phase space. We can
study the geometry of the basin boundary through the stable and unstable
manifolds of such orbits. However, the computation of manifolds embedded
in a high-dimensional phase space is a hard task. In this presentation I
will show some recent results obtained with a low-order model of shear
flow and with a full-fledged simulation of plane Couette flow.