A linear model of a finite amplitude Helmholtz instability

Abstract

Abstract The Helmholtz instability of a vortex sheet separating two fluids in relative motion is un­bounded in a simple linear model of the interaction of sound with the sheet. This paper presents a model which limits the amplitude of a harmonic wave in a physically realistic way but remains mathematically tractable. It is based on the idea that growth is limited by the onset of turbulence between the fluids when the Helmholtz wave reaches a critical size. An important consequence of the theory is a strong enhancement of the sound scattered up­stream, which is significant both in the context of forward noise produced by a jet and possibly also of jet screech. The requirement of causality is of central importance in determining the correct solution, and detailed general results on the theory of zero ultradistribu­tions are presented to establish an analytic definition of causality for the class of solutions encountered.