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Oxford Centre for Industrial and Applied Mathematics
Oxford University
Abstract-
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. [1] conjectured that in fact
gamma<-1. Subsequent numerical experiments [2] indicated that
for streamwise initial perturbations gamma = -1 and -7/4 for plane
Couette and plane Poiseuille flow respectively (using subcritical
Reynolds numbers for Poiseuille flow), while for oblique
initial perturbations gamma = -5/4 and -7/4.
We show, through a formal asymptotic analysis of the
Navier-Stokes equations, that for streamwise initial
perturbations in fact gamma = -1 and -3/2 for plane
Couette and plane Poiseuille flow respectively (factoring out the
unstable modes for Poiseuille flow), while for oblique
initial perturbations gamma = -7/6 and -17/12. Furthermore we
show why the numerically determined threshold exponents are not the true
asymptotic values.
[1] L.N. Trefethen, A.E. Trefethen, S.C. Reddy, T.A. Driscoll. Hydrodynamic Stability Without Eigenvalues, Science 261, 578-584 (1993).
[2] A. Lundbladh, D.S. Henningson, S.C. Reddy. Threshold Amplitudes for Transition in Channel Flows, Proceedings of the ICASE Workshop on Transition, Turbulence and Combustion (1993).
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