Nearly two-dimensional solutions of Euler's equations

Abstract

Motivated by a stretched version of the Taylor-Green initial-value problem for the Euler and Navier-Stokes equations, nearly two-dimensional incompressible flows are considered involving a single direction of slow spatial variation. A multiple-scale formulation of the problem leads by contour averaging to a system that determines the slow development in space and time. It is shown that the latter system is equivalent to an axisymmetric problem with nonstandard connection between circulation, cylindrical radius, and angular velocity component. Special solutions of the system in the exact axisymmetric case suggest that near-two-dimensionality is lost in finite time.