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 $\mathcal{H}$ 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 $h\lesssim 0$ sheets form during reconnection. For the Gaussian/Lamb-Oseen profile helicity $\mathcal{H}$ 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 $\mathcal{H}$ 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.

arXiv comment35 pages, 31 figures