Compact Navier–Stokes trefoils in large domains with finite dissipation
Year: 2025
Type: article
Abstract: For a perturbed trefoil vortex knot evolving under the Navier–Stokes equations, a sequence of $\nu$ -independent times $t_m$ are identified that correspond to a set of scaled, volume-integrated vorticity moments $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}$ , with this hierarchy $t_\infty \leqslant \ldots \leqslant t_m\ldots t_1=t_x\approx 40$ and $\mathcal{O}_{\textit{Vm}}=(\int _{V\ell }|\omega |^{2m}\,{\rm d}V)^{1/2m}$ . For the volume-integrated enstrophy $Z(t)$ , convergence of $\sqrt {\nu }Z(t)=\bigl (\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}}(t)\bigr )^2$ at $t_x=t_1$ marks the end of reconnection scaling. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. Meanwhile $Z$ accelerates, leading to approximate finite-time $\nu$ -independent convergence of the energy dissipation rate $\epsilon (t)=\nu Z(t)$ at $t_\epsilon \sim 2t_x$ . This is sustained over a finite temporal span of at least $\Delta T_\epsilon \searrow 0.5 t_\epsilon$ , giving Reynolds number independent finite-time, temporally integrated dissipation, $\Delta E_\epsilon =\int _{\Delta T_\epsilon }\epsilon \,{\rm d}t$ , and thus satisfies one definition for a dissipation anomaly, with enstrophy spectra that are consistent with transient $k^{1/3}$ Lundgren-like inertial scaling over some of the $\Delta T_\epsilon$ time. A critical factor in achieving these temporal convergences is how the computational domain $V_\ell =(2\ell \pi )^3$ is increased as $\ell \sim \nu ^{-1/4}$ , for $\ell =2$ to 6, then to $\ell =12$ , as $\nu$ decreases. Appendix A shows compatibility with e
Source: Journal of Fluid Mechanics
Author: Robert M. Kerr
Institution: University of Warwick
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7114910900