Inviscid evolution of a uniform vortex dipole in a strain field

Abstract

Competing models employing anti-parallel vortex collision in search of a finite-time singularity of Euler's equation have arisen recently. Both the vortex sheet model proposed by Brenner et al. (Phys. Rev. Fluids, vol. 1, 2016, 084503) and the ‘tent’ model proposed by Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019, pp. 930–967) consider a vortex monopole exposed to a strain flow to model the evolution of interacting anti-parallel vortices, a fundamental element in the turbulent cascade. Herein we employ contour dynamics to explore the inviscid evolution of a vortex dipole subjected to an external strain flow with and without axial stretching. We find that for any strain-to-vorticity ratio $\mathcal {E}$ , the constituent vortices compress indefinitely, with weaker strain flows causing flattening to occur more slowly. At low $\mathcal {E}$ , the vortex dipole forms the well-documented head–tail structure, whereas increasing $\mathcal {E}$ results in the dipole compressing into a pair of vortex sheets with no appreciable head structure. Axial stretching effectively lowers $\mathcal {E}$ dynamically throughout the evolution, thus delaying the transition from the head–tail regime to the vortex sheet regime to higher strain-to-vorticity ratios. Findings from this study offer a bridge between the two cascade models, with the particular mechanism arising depending on $\mathcal {E}$ . It also suggests limits for the ‘tent’ model for a finite-time singularity, wherein the curvature-induced strain flow must be very weak in comparison with the vorticity density-driven mutual attraction such that the convective time scale of the evolution exceeds the core flattening time scale.