Motion and Decay of a Vortex Ring

Abstract

In the classical inviscid theory of a vortex ring, the velocity at a point near the vortex ring becomes singular due to terms of r⁻¹ and ln r where r is the shortest distance from the point to the vortex ring. Also the velocity of the vortex ring depends on the logarithm of the effective radius of the cross section of the vortex ring and is infinite for zero radius. The effect of the viscosity in the inner core of the vortex ring is now included and the inner viscous solution is matched with the classical inviscid solution of the outer region by the boundary layer technique. By means of the systematic matching, the singularities of r⁻¹ and ln r in the classical inviscid theory is removed. By the requirement that the velocity at the center of the viscous core is finite, a unique and finite value is obtained for the velocity of the translation of the vortex ring which is decreasing with respect to time as − ln (ντ), where ν is the kinematic viscosity. From this analysis, the effective radius of the cross section of the vortex ring can be identified as 2(ντ)^½. The variable τ is transformed from the time variable t by the relationship τ = ∫₀^t R(t′)dt′/R(t), where R(t) is the radius of the ring.