Algebraic instability caused by acoustic modes in supersonic shear flows

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Abstract

Perturbations in a shear flow exhibit rather complex behavior - waves may grow algebraically even when the spectrum of disturbances is entirely neutral (no exponential instability). A shear flow brings about non-selfadjoint property, invalidating the standard notion of dispersion relations, and it also produces a continuous spectrum that is a characteristic entity in an infinite-dimension phase space. This paper solves an initial value problem using the Laplace transform and presents a new-type of algebraic instability that is caused by resonant interaction between acoustic modes (point spectrum) and vortical continuum mode (continuous spectrum). Such a resonance is possible when variation of velocity shear is comparable to sound speed.

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