Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes

Classical linearized stability analysis predicts (neutral) stability of supersonic vortex sheets for compressible flow with normalized Mach numbers, M > √2, while recent detailed numerical simulations by Woodward indicate the nonlinear development of instabilities for M > √2 through the development and interaction of propagating kink modes in the slip-stream. These kink modes are discontinuities in the slip-stream bracked by shock waves and rarefaction waves which grow self-similarly in time. In this paper, the apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on M > √2 where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed by Woodward.