Mathematics Institute, University of Warwick, UK

Simulated Navier-Stokes trefoil reconnection

The evolution and self-reconnection of a perturbed helical trefoil vortex knot is simulated, then compared to recent experimental measurements (Scheeler et al., PNAS 2014). The results might hold the clue to resolving the inconsistency between the current mathematics restricting singularities of the Navier-Stokes and experiments, for which there is always finite energy dissipation in a finite time, even if they do not solve the Clay Prize problem for Navier-Stokes. The analysis to get there began with qualitative comparisons between the simulations and experiments using three-dimensional vorticity diagnostics. Before reconnection begins, there is a long period with deformations similar to the experiment. During this phase the helicity and self-linking number of the trefoil are preserved. After reconnection has clearly begun, a Reynolds number independent fraction of the initial helicity is dissipated in a finite time th ? 72, in contrast to the experiment that claims that the helicity is preserved during reconnection. Two timescales are proposed. One based up two large-scale properties: Circulation of the filament and the dimensions of the trefoil weave ta = 9. The times that reconnection begins and ends, tc ? 31 and th respectively, are multiples of this time. The other timescale is viscous and based upon ta and a dimensionless function of the circulation, viscosity, trefoil size and radius of the vortex filament. This describes the time over which the first reconnection step completes and the scaling of the final period of helicity decay ?th. The helicity decay of all of the simulations collapse under this scaling. These results are compared to the experiments, with tc ? 600ms, ?th ? 200ms and th ? 1200. However, the experiment ends at only t=800ms, so what do the simulations find for t= th-2?th? The highest Reynolds number case finds a 2% depletion in the helicity at its equivalent time, consistent with a slight dip in the experimental helicity. So there are no inconsistencies between the experiment and simulations. Which is the basis for concluding that the very small viscosity, ? ? 0 mathematical restrictions upon finite-time dissipative behaviour do not apply to the trends observed over this range of modest viscosities so long as the radii of the vortices also decrease.