Generation and Decay of Viscous Vortex Rings

Abstract

Decay of a vortex ring in a viscous fluid is discussed by using a solution in the form of an asymptotic expansion for large time t. It is found that the velocity of the vortex ring varies as t^−1.5 in the final state of low Reynolds number. The asymptotic expansion is not uniformly valid, and an improvement is made by using the method of matched asymptotic expansions.
Generation and development of vortex rings are simulated by numerical integration of the Navier-Stokes equation as an initial and boundary value problem. Time variations of physical quantities such as total energy, impulse and velocity of the vortex rings, etc. are obtained, and have been shown to approach asymptotically to those obtained from the asymptotic expansion. Comparison with experimentally produced vortex rings is also given briefly.