Axisymmetric vortex sheet roll-up.

Abstract

This thesis presents a numerical study of axisymmetric vortex sheet roll-up applied to vortex ring formation. The calculations are performed using a vortex blob method which resolves spiral vortex sheet roll-up. A convergence study with respect to the smoothing parameter is performed and detailed information about the vortex sheet is obtained. The first problem studied is the evolution of an initially flat circular vortex sheet obtained by giving a disk immersed in ideal fluid an impulse and then removing it from the fluid. For a short time, the computed axisymmetric vortex sheet rolls up and obeys 2-d similarity theory in a moving frame of reference. For longer times, the computed solution approximates a steady propagating vortex ring with vorticity concentrated in a core, in good agreement with a symmetric core model proposed by Saffman (1975). A slight asymmetry in the computed vorticity distribution is responsible for a small deviation of the vortex ring radius and translation velocity from the theoretical results of Taylor (1953) and Saffman (1975). The second problem studied is vortex ring formation at the edge of a circular tube due to the motion of a piston which ejects fluid from the tube. An experiment performed by Didden (1979) is simulated numerically using a vortex blob model which incorporates vortex sheet separation at a sharp edge. Good agreement between computation and experiment is found for the sheet shape and the vortex ring trajectory. Except in an initial time period, during which viscous effects, are dominant in the experiment, the computed and experimental circulation shedding rates agree reasonably well. Experiments show that the vortex ring trajectory does not obey 2-d similarity theory predictions over long times (Didden 1979, Auerbach 1987a). The present computations confirm this experimental observation and clarify previous explanations proposed for the absence of self-similarity.