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The Quarterly Journal of Mechanics and Applied Mathematics 2006 59(4):651-673; doi:10.1093/qjmam/hbl021
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Q. Jl Mech. Appl. Math, Vol. 59. No. 4 © The author 2006. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

The instability of the flow in a suddenly blocked pipe

Nathaniel Jewell and James P. Denier{dagger}

( School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia )

{dagger} jim.denier{at}adelaide.edu.au


  Abstract

This paper considers the decay of Poiseuille flow within a suddenly blocked pipe. For small to moderate times the flow is shown to consist of an inviscid core flow coupled with a boundary layer at the pipe wall. A small-time asymptotic solution is developed and it is shown that this solution is valid for times up to the point at which the boundary layer fills the whole pipe. A small-time composite solution is used to initiate a numerical marching procedure which overcomes the small-time singularity that arises in the flow and so allows us to describe the ultimate decay of the flow within a blocked pipe. The stability of this flow is then considered using both a quasi-steady approximation and a transient-growth analysis based upon marching solutions of the linearized Navier–Stokes equations. Our transient stability analysis predicts a critical Reynolds number, for transition to turbulence, in the range 970 < Re < 1370.


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