Scaling properties towards vortex reconnection under Biot–Savart evolution

Abstract

Reconnection of a vortex filament under the Biot–Savart law is investigated numerically using a vortex ring twisted in the form of a figure-of-eight. For the numerical method, the vortex ring is divided into piecewise linear segments, and the Biot-Savart integral is approximated by a summation over the segments with a cut-off method to deal with the singular terms. It is demonstrated that the centre part of the skewed vortex ‘chopsticks’, where the interaction is maximal, tends to approach and accelerate to form a singularity while making a ‘tent-like’ structure as shown by de Waele and Aarts (1994 Phys. Rev. Lett. 72 pp 482–5). The minimum separation of the chopsticks, the maximum velocity and the maximum axial strain rate show clear scaling exponents near the singularity consistent with Leray scaling for self-similar solutions of the Navier–Stokes equations.