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Energy growth in viscous channel flows

Published online by Cambridge University Press:  26 April 2006

Satish C. Reddy  and
Dan S. Henningson
Satish C. Reddy
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA
Dan S. Henningson
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USAandAeronautical Research Institute of Sweden (FFA), Box 11021, S-16111 Bromma, Sweden
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Abstract

In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille–Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 209 - 238
Copyright
© 1993 Cambridge University Press

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