Simulated Navier-Stokes trefoil reconnection

Abstract

The evolution and self-reconnection of a perturbed trefoil vortex knot is simulated, then compared to recent experimental measurements (Scheeler et al. 2014a). Qualitative comparisons using three-dimensional vorticity isosurfaces and lines, then quantitative comparisons using the helicity. To have a single initial reconnection, as in the experiments, the trefoil is perturbed by 4 weak vortex rings. Initially there is a long period with deformations similar to the experiment during which the energy, continuum helicity and topological self-linking number are all preserved. In the next period, once reconnection has clearly begun, a Reynolds number independent fraction of the initial helicity is dissipated in a finite time. In contrast, the experimental analysis finds that the helicity inferred from the trajectories of hydrogen bubbles is preserved during reconnection. Since vortices reconnect gradually in a classical fluid, it is suggested that the essential difference is in the interpretation of the reconnection timescales associated with the observed events. Both the time when reconnection begins, and when it ends. Supporting evidence for the strong numerical helicity depletion is provided by spectra, a profile and visualisations of the helicity that show the formation of negative helicity on the periphery of the trefoil. A single case with the same trajectory and circulation, but a thinner core, replicates this helicity depletion despite larger Sobolev norms, showing that the reconnection timescale is determined by the initial trajectory and circulation of the trefoil, not the initial vorticity. This case also shows that the very small viscosity, $\nu\rightarrow 0$ mathematical restrictions upon finite-time dissipative behavior do not apply to this range of modest viscosities.