A Localized Approximation Method for Vortical Flows

Abstract

Abstract. 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.

Global existence, singular solutions, and ill‐posedness for the Muskat problem

“…However, as discussed in [4], it is useful to analytically extend the governing equations to complex values of ξ by extending the complex conjugate via Schwarz reflection. More precisely, we define…”

“…The interface between the fluids is a "vortex sheet" since the tangential velocity may be discontinuous there. An integrodifferential equation governing the evolution of the sheet is derived from the governing differential equations and boundary conditions in [4,16]. We use here a form of the equation that employs complex variable notation, following the presentation of [4].…”

“…An integrodifferential equation governing the evolution of the sheet is derived from the governing differential equations and boundary conditions in [4,16]. We use here a form of the equation that employs complex variable notation, following the presentation of [4]. Let z(ξ, t) = x(ξ, t) + iy (ξ, t) denote the location of the interface in the complex x + iy plane as a function of the parameter ξ and time t. Define also the complex interface velocity w(ξ, t) = u − iv.…”

Singularity formation for complex solutions of the 3D incompressible Euler equations

“…Derived by Moore for the Kelvin-Helmholtz problem [27,28], this method has been generalized to additional fuid interface problems by Caflisch, Orellana and Siegel [15], and in section 2 it is formulated as a general approximation for singularity formation. When applied to the Euler equations for axi-symmetric flow with swirl, the velocity u is split into two complex velocities u+ and u = t/+, in which u+ consists of the nonnegative wavenumber components of u.…”

Existence and uniqueness of analytic solution for Rayleigh–Taylor problem

“…Model equations for the location of the interface have been derived (see Baker et al [1,2], Caflisch et al [5], Moore [10], Sharp [15] and references therein). These studies are numerical and asymptotic, but important to furthering physical understanding of the flow dynamics.…”