The Continuous Spectrum of the Orr-Sommerfeld Equation. Part I. The Spectrum and the Eigenfunctions.

摘要

It is shown that the Orr-Sommerfeld equation governing the stability of any mean shear flow in an unbounded domain which has finite energy under some Galilean transformation has a continuous spectrum. This result applies to both the temporal and spatial stability problems. Formulae for the location of this continuum in the complex wave speed plane are given. The continuous spectrum and corresponding eigenfunctions are calculated for two samples 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, are 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.