Numerical simulations of circular vortex sheets by using two regularization models

Abstract

We perform numerical simulations for circular vortex sheets by using two regularization models and present the late-time evolution of unstable interfaces. We also give a linear stability analysis for the circular vortex sheet and find that the sheet is always susceptible to the Kelvin-Helmholtz instability. The numerical results show that the motion of the circular sheet is similar to that of the periodic vortex sheet at an early time, but the secondary instability at a late time depends on the wavenumber of the initial perturbation. The interface of the initial two-modes evolves two primary and two secondary roll-ups while the interface of the initial four-modes has four symmetric roll-ups until late times and does not form a secondary instability. The non-symmetric interface of the single-mode develops to a structure more complex than that of the symmetric interfaces, having continuously evolved roll-ups and a strong interaction of spirals. We also give quantitative comparisons of the Krasny model and the Beale-Majda model. The solutions of the two models are similar on a large scale, but are different on a small scale. The solution of the Beale-Majda model exhibits somewhat irregular features for a small regularization parameter while the Krasny model gives well-regularized solutions with fine resolution.