Scaling of Navier-Stokes trefoil reconnection

Abstract

Perturbed, helical trefoil vortex knots and a set of anti-parallel vortices are examined numerically to identify the scaling of their helicity and vorticity norms during reconnection. For the volume-integrated enstrophy $Z=\int\omega^2 dV$, a new scaling regime is identified for both configurations where as the viscosity $\nu$ changes, all $\sqrt{\nu}Z(t)$ cross at $\nu$-independent times $t_x$, identified as when the first reconnection events end. Self-similar linear collapse of $B_\nu(t)=(\sqrt{\nu}Z)^{-1/2}$ can be found for $t\lesssim t_x$ by linearly extrapolating $B_\nu(t)$ to zero at critical times $T_c(\nu)$, then plotting $(T_c(\nu)-t_x)(B_\nu(t)-B_x)$ where $B_x=B_\nu(t_x)$. The size $\ell^3$ of the periodic domains must be increased as $\nu$ is decreased to maintain this scaling as implied by known Sobolev space bounds. The anti-parallel calculations show that the linear collapse of $B_\nu(t)$ begins with a quick, viscosity-independent exchange of the circulation $\Gamma$ between the original vortices and the new vortices. Up to and after the trefoil knots' first reconnection at time $t_x$, their helicity ${\cal H}$ is preserved, validating the experimental centreline helicity observation of Scheeler et al (2014a). Because the cubic Navier-Stokes velocity norm $L_3$ barely changes and the Navier-Stokes $\|\omega\|_\infty$ are bounded by the Euler values, these flows are never singular. Despite this, the Navier-Stokes $Z$ can, for a brief period, grow faster than the Euler $Z$ and the following increase in the viscous energy dissipation rate $\epsilon=\nu Z$ shows $\nu$-independent convergence at $t\approx 2t_x$. Taken together, these results could be a new paradigm whereby smooth solutions without singularities or roughness could generate a $\nu\to0$ {\it dissipation anomaly} (finite dissipation in a finite time) as $\ell\to\infty$, as seen in physical turbulent flows.