Variational necessary and sufficient stability conditions for inviscid shear flow

A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-s...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2014-12, Vol.470 (2172), p.20140322-20140322
Main Authors: Hirota, M., Morrison, P. J., Hattori, Y.
Format: Article
Language:English
Subjects:
Continuum Hamiltonian Hopf Bifurcation
Flow Stability
Kreĭn Signature
Variational Method
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Summary:A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-self-adjoint eigenvalue problem) can be associated with positive eigenvalues of a certain self-adjoint operator. The stability is therefore determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving Rayleigh's equation. This variational stability criterion is based on the understanding of Kreĭn signature for continuous spectra and is applicable to other stability problems of infinite-dimensional Hamiltonian systems.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2014.0322