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 $\nu^{1/4}{\it O}_{V1}$ with this hierarchy $t_\infty\le\dots\le t_m\dots t_1=t_x\approx40$ and ${\it O}_{Vm}=(\int_{V\ell}|\omega|^{2m}dV)^{1/2m}$. For $Z(t)={\it O}^2_{V1}(t)$ 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 $\Delta T_\epsilon\searrow 0.5 t_\epsilon$, 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 $\nu^{1/4}{\it O}_{Vm}(t)$ 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 $t\sim t_\epsilon$.