A Localized Approximation Method for Vortical Flows

An approximation method of Moore for Kelvin-Helmholtz instability is formulated as a general method for two-dimensional, incompressible, inviscid flows generated by a vortex sheet. In this method the nonlocal equations describing evolution of the sheet are approximated by a system of (local) differential equations. These equations are useful for predicting singularity formation on the sheet and for analyzing the initial value problem before singularity formation. The general method is applied to a number of problems: Kelvin-Helmholtz instability for periodic vortex sheets, motion of an interface in Hele-Shaw flow, Rayleigh-Taylor instability for stratified flow, and Krasny's desingularized vortex sheet equation. A new physically desingularized vortex sheet equation is proposed, which agrees with the finite thickness vortex layer equations in the localized approximation. AMS(MOS) subject classification. 76C05 1. Introduction. For many two-dimensional incompressible, inviscid flows, the vorticity is confined to lie on a vortex sheet, which is a free surface in the flow. The fluid velocity is then determined by the location and strength of the vortex sheet. Such flows include the Kelvin-Helmholtz flow in which the vorticity is present initially, Rayleigh-Taylor flow in which the vorticity is generated baroclinically along a curve of density discontinuity, and motion of a fluid interface in a Hele-Shaw cell for which vorticity is produced by a jump in viscosity across the interface. These problems are described in detail in subsequent sections. One of the most interesting phenomenon for such free surface flows is singularity formation on the vortex sheet. The singularity may be of weak type, such as infinite