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Bounds for threshold amplitudes in subcritical shear flows

Published online by Cambridge University Press:  26 April 2006

Gunilla Kreiss ,
Anders Lundbladh  and
Dan S. Henningson
Gunilla Kreiss
Affiliation:
Department of Numerical Analysis and Computer Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Anders Lundbladh
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden and The Aeronautical Research Institute of Sweden (FFA), Box 11021, S-161 11 Bromma, Sweden
Dan S. Henningson
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden and The Aeronautical Research Institute of Sweden (FFA), Box 11021, S-161 11 Bromma, Sweden
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Abstract

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R−21/4. Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R−1. The discrepancy is discussed in the light of a model problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 270 , 10 July 1994 , pp. 175 - 198
Copyright
© 1994 Cambridge University Press

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