Nonlinear Development of Instabilities in Supersonic Vortex Sheets II: Resonant Interaction Among Kink Modes

Abstract

In contrast with the predictions of classical linearized stability analysis, recent large-scale numerical simulations of supersonic vortex sheets at high Mach numbers predict the nonlinear development of instabilities. These numerical experiments document that the mechanism of nonlinear instability is through the generation and interaction of propagating kink modes along the vortex sheet. These kink modes are discontinuities that travel along the slip stream at various velocities bracketed by shock waves and rarefaction waves that grow self-similarly in time. In earlier work, the authors explained the generation at small amplitude of three distinct quantitative families of kink modes with a structure like that observed in the numerical simulations. In this paper the authors develop simplified asymptotic equations for the further generation and interaction of these kink modes at small amplitudes. Exact and numerical solutions of the simplified asymptotic equations are developed in this paper, and thesesolutions predict amplification of the kink modes as time evolves through resonant interaction among kink modes; through the asymptotics this amplification of the kink modes automatically predicts simultaneous rapid growth of the vorticity in the vortex sheet to enhance instability and strong growth of the bracketed shock and rarefaction waves. These predictions of the simplified asymptotic equations provide a detailed explanation of supersonic vortex sheet instability in qualitative agreement with the results documented in the earlier large scale numerical simulations.