Navier-Stokes bounds and scaling for compact trefoils in $(2\ell \pi {)}^3$ domains

Abstract

For a perturbed trefoil vortex knot evolving under the Navier-Stokes equations, a sequence of $\nu$-independent times $t_m$ are identified corresponding to a set of scaled, volume-integrated vorticity moments ${\nu }^{1/4}{O}_{V1}$ with this hierarchy $t_{\infty }\le \cdots \le t_m\dots t_1=t_x\approx 40$ and $O_{Vm}=({\int }_{V\ell }|\omega {|}^{2m}dV{)}^{1/2m}$. For $Z(t)=O_{V1}^2(t)$ the volume-integrated enstrophy, convergence of $\sqrt{\nu }Z(t)$ at $t_x=t_1$ marks the end of the reconnection scaling phase. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. $Z$ then accelerates, leading to approximate finite-time $\nu$-independent convergence of the energy dissipation rate $\epsilon (t)=\nu Z(t)$ at $t_{\epsilon }\sim 2t_x$ and sustained over a finite span $\Delta T_{\epsilon }\searrow 0.5t_{\epsilon }$, giving Reynolds number independent finite-time, dissipation, $\Delta E_{\epsilon }={\int }_{\Delta T_{\epsilon }}\epsilon dt$, and thus satisfying one definition for a {\it dissipation anomaly}. Evidence for transient Kolmogorov-like enstrophy spectra is found over $\Delta T_{\epsilon }$. A critical factor in achieving these temporal convergence laws is how the 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. $(2\ell \pi {)}^3$ domain compatibility with established $(2\pi {)}^3$ mathematics in appendix allows small $\nu$ Navier-Stokes solutions. Two spans of $\nu$ are considered. Over the first factor of 25 decrease in $\nu$, all of the ${\nu }^{1/4}{O}_{Vm}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease in $\nu$, $\ell$ is increased to $\ell =12$, there is only convergence of ${\nu }^{1/4}{\Omega }_{V\infty }(t)$ to $t_{\infty }$ and later $\sqrt{\nu }Z(t)$ convergence at $t_1=t_x$ and $\epsilon (t)$ over $t\sim t_{\epsilon }$.

21 pages, 9 frames in 7 figures