Subcritical transition in channel flows

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as R-gamma say, with gamma < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact gamma < -1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations gamma = -1 and -7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations gamma = -5/4 and -7/4. Here, through a formal asymptotic analysis of the Navier-Stokes equations, it is found that for streamwise initial perturbations gamma = -1 and -3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations gamma = -1 and -5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.