APPROXIMATE THEORY FOR THE NONLINEAR EVOLUTION OF VORTEX SHEETS.

摘要

We study the nonlinear evolution of vortex sheet problems and formation of singularities, mainly: the flat 2-d vortex sheet perturbed initially by a small analytic disturbance, the circular vortex sheet, and the ring wing problem or nacelle. For the first, by a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet and used it to formally show that a sinusoidal perturbation of the flat vortex sheet develops a singularity at time t(,c) = 2(VBAR)log (epsilon)(VBAR) + 0(log(VBAR)log (epsilon)(VBAR)). We present a simplified derivation of Moore's approximate equation and the equivalent conservation law and show short time existence using Lax's a priori estimate theory for 2 x 2 systems of nonlinear conservation law. We also present a valid asymptotic expansion for a singular solution of Moore's equation and evaluate Birkhoff's singular integral in some interesting cases that give information about the validity of Moore's approximate equation, type of singularity and motion of singularities. For the circular vortex sheet we present its linear theory and propose an approximate system of equations to study its nonlinear evolution. Finally, for the ring wing or nacelle problem we present a Moore's type of equation to study its nonlinear evolution.