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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

Published online by Cambridge University Press:  18 April 2002

ANDREW G. WALTON
Affiliation:
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 → ∞, 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.

Type
Research Article
Copyright
© 2002 Cambridge University Press

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