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Optimal energy density growth in Hagen–Poiseuille flow

Published online by Cambridge University Press:  26 April 2006

Peter J. Schmid  and
Dan S. Henningson
Peter J. Schmid
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195. USA.
Dan S. Henningson
Affiliation:
Aeronautical Research Institute of Sweden (FFA), Box 11021, S-16111 Bromma, Swedenand Department of Mechanics, Royal Institute of Technology, S-10044 Stockholm, Sweden
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Abstract

Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and ε-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 277 , 25 October 1994 , pp. 197 - 225
Copyright
© 1994 Cambridge University Press

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