Journal of Fluid Mechanics



The temporal evolution of neutral modes in the impulsively started flow through a circular pipe and their connection to the nonlinear stability of Hagen–Poiseuille flow


ANDREW G. WALTON a1
a1 Mathematics Department, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ, UK

Abstract

The linear stability of the impulsively started flow through a pipe of circular cross-section is studied at high Reynolds number R. A crucial non-dimensional time of O(R7/9) is identified at which the disturbance acquires internal flow characteristics. It is shown that even if the disturbance amplitude at this time is as small as O(R−22/27) the subsequent evolution of the perturbation is nonlinear, although it can still be followed analytically using a multiple-scales approach. The amplitude and wave speed of the nonlinear disturbance are calculated as functions of time and we show that as t [rightward arrow] [infty infinity], the disturbance evolves into the long-wave limit of the neutral mode structure found by Smith & Bodonyi in the fully developed Hagen–Poiseuille flow, into which our basic flow ultimately evolves. It is proposed that the critical amplitude found here forms a stability boundary between the decay of linear disturbances and ‘bypass’ transition, in which the fully developed state is never attained.

(Received January 2 2001)
(Revised October 23 2001)



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