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Stable manifolds and the transition to turbulence in pipe flow

Published online by Cambridge University Press:  25 May 2009

D. VISWANATH  and
P. CVITANOVIĆ
D. VISWANATH*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
P. CVITANOVIĆ
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: divakar@umich.edu
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Abstract

Lower branch travelling waves and equilibria computed in pipe flow and other shear flows appear intermediate between turbulent and laminar motions. We take a step towards connecting these lower branch solutions to transition by deriving a numerical method for finding certain special disturbances of the laminar flow in a short pipe. These special disturbances cause the disturbed velocity field to approach the lower branch solution by evolving along its stable manifold. If the disturbance were slightly smaller, the flow would relaminarize, and if slightly larger, it would transition to a turbulent state.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 627 , 25 May 2009 , pp. 215 - 233
Copyright
Copyright © Cambridge University Press 2009

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