Self-similar shedding of vortex rings

The roll-up of an initially spherical vortex sheet into a vortex ring is computed using the vortex blob method. The ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a ring that sheds circulation. The process repeats itself at smaller and smaller scales in a self-similar manner. The relation between the vortex shedding and the energy of the vortices is investigated. In contrast, an initially cylindrical vortex sheet rolls up into a vortex pair that sheds essentially no circulation. Consider a sphere immersed in stagnant inviscid fluid which is impulsively set into motion. The resulting potential flow is induced by an axisymmetric vortex sheet in place of the sphere. This paper studies the evolution of the vortex sheet under its selfinduced velocity, as if the sphere were dissolved and the fluid within it allowed to move with the flow. The axisymmetric flow is compared to the planar flow generated by the impulsive motion of a cylinder. Rottman, Simpson & Stansby (1987) performed an experiment simulating the cylindrical scenario by quickly removing a hollow cylinder immersed in a crossflow. They also computed this flow using a vortex-incell method and compared numerical and experimental results. Rottman & Stansby (1993) computed the planar flow using the vortex blob method. The axisymmetric flow was computed by Winckelmans et al. (1995) using a three-dimensional vortex particle method. Here, we compute the planar and axisymmetric flow to longer times than in prior work, using the vortex blob method. The method consists of regularizing the singular governing equations by introducing an articial parameter (Chorin & Bernard 1973; Anderson 1985; Krasny 1986). Comparisons with solutions of the Navier{Stokes equations (Tryggvason, Dahm & Sbeih 1991) and with experimental measurements (Nitsche & Krasny 1994) show that the method approximates viscous flow well for suciently small values of the articial smoothing parameter and viscosity. The computed planar and axisymmetric sheets roll up into a vortex pair and a vortex ring respectively as they travel in the direction of the given impulse. However, the vortex ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a vortex ring. This observed shedding and roll-up repeats itself in a self-similar manner, forming a sequence of vortex rings of decreasing size in the tail of the leading ring. In contrast, the vortex pair does not shed any signicant amount of circulation. The results are shown to be essentially independent of the flow regularization. The relation between the observed shedding and the energy of the vortex rings is also discussed, motivated by the work of Gharib, Rambod & Shari (1998) relating the energy and the circulation of vortex rings generated in laboratory experiments.