Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane Couette flow
SPEC, CEA Saclay, 91191 Gif-sur-Yvette, France
2 École Nationale des Ponts et Chaussées, 77455 Marne-la-Vallée, France
Revised: 6 July 1999
Published online: 15 March 2000
Various experiments have outlined generic properties of the subcritical transition to turbulence in plane Couette flow. A low order model of a self-sustaining process has been derived by Waleffe  from the Navier-Stokes equations for a sinusoidal shear flow. This paper investigates the weakly non-linear properties and the phase space analysis of this model, including the dependence on the model parameters. It is shown that the asymptotic dynamics essentially reduces to a bidimensional manifold, that many trajectories exhibit long transients, and that a statistical description of the nonlinear response to finite amplitude perturbations is needed in order to recover the bifurcation diagram from an experimental point of view. Comparison with recent experimental results obtained in the plane Couette flow finally outlines the relevance of this kind of approach.
PACS: 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems / 47.20.Ft – Instability of shear flows / 47.20.Ky – Non linearity (including bifurcation theory)
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000