On the topology of vortex lines and tubes

Abstract

This paper examines the widespread idea that vortex lines and tubes must either close on themselves or extend to the boundary of the fluid. A survey of the origins of this misconception, and of earlier attempts to set it right, is followed by an analysis of simple flows exhibiting vortex lines and tubes which do not fit those shapes. Two types of vortex lines are discussed:dense, which comprise open lines of infinite length but confined in a finite region, andseparatrix, which comprise lines that begin or finish within the fluid, at points where the vorticity is null. The presence of these vortex lines in a vortex tube affects its topology in the following ways. Vortex tubes formed by dense vortex lines have infinite length; they self-intersect an infinite number of times but do not close on themselves. Vortex tubes formed by separatrix vortex lines (and either closed or open vortex lines) are torn apart at the points where the vorticity is null. Vortex tubes exclusively composed of separatrix vortex lines begin or finish at points or surfaces within the fluid; in this particular situation the vortex tube has zero strength.