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The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions

Published online by Cambridge University Press:  12 April 2006

Chester E. Grosch  and
Harold Salwen
Chester E. Grosch
Affiliation:
Institute of Oceanography, Old Dominion University, Norfolk, Virginia 23508
Harold Salwen
Affiliation:
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030
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Abstract

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 1 , 12 July 1978 , pp. 33 - 54
Copyright
© 1978 Cambridge University Press

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