A simplification of the vorticity equation and an extension of the vorticity persistence th.pdf

A simplification of the vorticity equation and an extension of the vorticity persistence theorem to three dimensions T. S. Morton Department of Mechanical, Aerospace Biomedical Engineering, University of Tennessee Space Institute 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA A simplified form of the vorticity equation is derived for arbitrary coordinate systems. The present work unifies and extends the previous findings that vorticity is conserved in planar Euler flow, while in axisymmetric Euler rings it is the ratio of the vorticity to the distance from the symmetry axis that is conserved. The unifying statement is that in any Euler flow, all components of the vorticity tensor of a streamline coordinate system that are normal to the streamline direction are conserved along streamlines. This is true for both two- and three-dimensional flows, whether the flow is axisymmetric or not, with or without swirl. What remains of the nonlinear convective terms in the vorticity equation, after the mathematical simplification, is the Lie derivative of the vorticity tensor with respect to fluid velocity. A temporal derivative is defined which, when set to zero, expresses either the continuity or vorticity equation (excluding the viscous term), depending upon the argument supplied to it. 1. Introduction The Navier-Stokes equation presents various difficulties to those seeking to solve it. Because of the presence of partial derivatives in all three spatial variables and the vectorial nature of the equation, the most promising solution methods involve the use of streamlined coordinate systems, which can consolidate the spatial dependence into a single variable. The solution can then be found by integrating along streamlines. An example of this is the following two-dimensional solution of Oseen (1910), given by: 1 2( ) /(4 )3 4 x tO e t Γ ω πν ?= ν , 1 2,ω ω = 0 , (1) ( )1 2( ) /(4 )2 1 2 12 ( ) x tOv e x νΓ π ?= ? , 1 3,v v = 0 . (2) He