Dependence of trefoil vortex knots upon the initial vorticity profile

Abstract

Six sets of Navier-Stokes trefoil vortex knots in $(2\pi)^3$ domains show how the shape of the initialprofile influences the evolution of the enstrophy $Z$, helicity ${\cal H}$ and dissipation-scale. Significant differences develop even when all have the same three-fold symmetric trajectory, the same initial circulation and the same range of the viscosities $\nu$. Maps of the helicity density $h=u\cdot\omega$ onto vorticity isosurfaces patches show where $h\lesssim0$ sheets form during reconnection. For the Gaussian/Lamb-Oseen profile helicity ${\cal H}$ grows significantly, with only a brief spurt of enstrophy growth as thin braids form then decay during reconnection. The remaining profiles are algebraic. For the untruncated algebraic cases,$h<0$ vortex sheets form in tandem with $\nu$-independent convergence of $\sqrt{\nu}Z(t)$ at a common $t_x$. For those with the broadest wings, enstrophy growth accelerates during reconnection, leading to approximately $\nu$-independent convergent finite-time dissipation rates $\epsilon=\nu Z$. By mapping terms from the budget equations onto centerlines, the origins of the divergent behavior are illustrated. Lamb-Oseen has six locations of centerline convergence form with local negative helicity dissipation, $\epsilon_h<0$, and small, but positive $h$. Later, the sum of these localized patches of $\epsilon_h<0$ leads to a positive increase in the global ${\cal H}$ and suppression of enstrophy production. For the algebraic profiles: There are only three locations of centerline convergence, each with spans of less localized $\epsilon_h<0$ and some $h<0$. Spans that could be the seeds for the $h<0$ vortex sheets that form in the lower half of the trefoil as the $\sqrt{\nu}Z(t)$ phase begins and can explain accelerated growth of the enstrophy and evidence for finite-time energy dissipation $\Delta E_\epsilon$. Despite the initial symmetries.