Variational necessary and sufficient stability conditions for inviscid shear flow

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 470; no. 2172; p. 20140322
• Main Authors: Hirota, M., Morrison, P. J., Hattori, Y.
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 08.12.2014
• Subjects: Continuum Hamiltonian Hopf Bifurcation; Flow Stability; Kreĭn Signature; Variational Method; Kreĭn signature; continuum Hamiltonian Hopf bifurcation; variational method; flow stability
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2014.0322

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.