Linking of vortex rings

Abstract

THE topology of vortex lines, which trace the local vorticity in a fluid flow just as streamlines trace the velocity, is important in attempts to understand, describe and control flows in various applications. Changes in this topology may, for example, affect mixing in flows, and may be significant for the dynamics of turbulence. In particular, the behaviour of closed loops of vortex lines (vortex rings) has been studied ever since Kelvin's 'vortex atom' theory, and contributed to Tait's development of the topological theory of knots. Several recent experimental and computational studies^l–9 have explored the 'reconnection' of initially distinct vortex rings; particularly elegant are Schatzle's¹⁰ experiments in which two vortex rings, inclined towards one another, go through two reconnections, after which two new rings, comprising half of each of the originals, emerge. Here we describe three-dimensional numerical simulations which establish a simple mechanism by which the linking of two vortex rings may be achieved starting from an unlinked initial state. Appropriate initial states were identified by simulating the unlinking of two initially linked vortex rings and then reversing the vorticity of the final state and running the simulation backwards. This computational procedure sheds light on why, both in experiment and simulation, linking is not always achieved from an arbitrary initial configuration of unlinked vortex rings set on a collision course.