Locally self-similar, finite-time collapse in a high-symmetry vortex filament model

Abstract

A locally self-similar solution is found using a vortex filament model. The solution is steady in a rescaled frame with magnification (${\mathrm{t}}_{\mathrm{crit}}$-t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}2}$ about the origin. A finite-time singularity results in which velocity, vorticity, and enstrophy scale as ${\mathrm{t}}_{\mathrm{crit}}$-t to powers -1/2, -1, and -1/2, respectively. The initial flow is six closed vortex contours symmetric around and propagating toward the origin. The self-similar inner solution consists of three orthogonal filament quadrupoles centered about the origin. The solution is attracting within a space of symmetries preserved by the incompressible Navier-Stokes and Euler equations. The numerical method consists of piecewise straight vortex segments with a standard variable core regularization model. Small core deformation is modeled with a two-length scale core function. This solution is similar to the candidate singular flow suggested by Boratav and Pelz [Phys. Fluids 6, 2757 (1994)] in their large-scale pseudospectral simulations. The steady inner solution has a set of hyperbolic critical points around which singular focusing occurs. It is conjectured that the singularity is pointwise in time as well as in space, and a smooth expanding solution exists which is symmetric with the collapsing solution about the critical time.