Drop-formed vortex rings—The generation of vorticity

Abstract

Vortex rings are seen to form when dyed water drops strike a water surface and their formation and structure depend on height of fall and surface tension. The assumption that a vortex sheet envelopes the penetrating drop, frequently stated without explanation in the literature, does not explain these factors and this paper shows why it is incorrect. Alternative mechanisms have been proposed in the literature but none explains adequately the vorticity generation or the restriction of vortex ring formation to low Weber numbers. This paper proposes a mechanism based on the generation of vorticity on relaxation of surface stresses at coalescence. The condition that the surface viscous stress be continuous across the water-air interface leads to a boundary condition on vorticity and the total amount of vorticity generated depends on the quantity which can be diffused into the fluid interior from the boundary during coalescence. At low values of Weber number this condition appears to be sufficient to generate enough vorticity to allow flow separation at the surface, such separation being a necessary condition for vortex sheet roll-up and ring production. The existence of a critical Weber number above which vortex rings do not form is the result of a balance between the rate at which the ring of contact moves outward associated with on the one hand, the action of surface tension forces and, on the other, the rate of surface destruction due to the coming together of surfaces. If surface destruction dominates then the fluid elements to which the surface viscous stress boundary condition applies will become part of the fluid interior before diffusion has carried significant vorticity away from the surface.