## Subcritical transition in channel flows

###
S. J. Chapman

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, Trefethen, Reddy and Driscoll 1993
conjectured that in fact
gamma<-1. Subsequent numerical experiments by
Lundbladh, Henningson and Reddy 1993 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.