A theoretical model on vortex ring formation based on the roll-up of a vortex sheet with finite thickness

Abstract

The formation process of a vortex ring can be delineated with the roll-up of a vortex sheet, and the latter idealizes the vortex layer into a sheet with zero-thickness in inviscid fluids. In view of the mass transport from the nozzle into the rolling spiral in vortex ring generation experiments, the effects of the finite thickness of the detaching shear layer on the formation model of vortex rings are investigated in the present study. In this paper, we introduced a theoretical model termed a vortex-source sheet, which is able to describe the induced motion of a vortex sheet with nonzero-thickness. A relationship between the flow and the circulation of a fluid element is developed in two-dimensional, inviscid, conservative flows, making the motion equation of the vortex-source sheet discretized into a closed equation set. The numerical solutions show that the predictions of the vortex-source sheet model on the geometric structure of the vortex ring are much closer to the experimental results than the conventional similarity theory. The reasons for this result are also discussed. Moreover, the consideration of the mass transport also gives a deeper insight into the picture of vortex ring generation. However, there still exists a discrepancy between the theoretical prediction and the experimental result on the circulation of the formed vortex rings. We speculate that it may be attributed to the convective entrainment of the environmental fluids during the formation of vortex rings, and this assumption needs to be studied further.