Navier-Stokes bounds and scaling for compact trefoils in $(2\ellπ)^3$ domains
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arXiv
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2024
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| _version_ | 1866909509749833728 |
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| author | Kerr, Robert M. |
| author_facet | Kerr, Robert M. |
| contents | For a perturbed trefoil vortex knot evolving under the Navier-Stokes equations, a sequence of $ν$-independent times $t_m$ are identified corresponding to a set of scaled, volume-integrated vorticity moments $ν^{1/4}{\it O}_{V1}$ with this hierarchy $t_\infty\le\dots\le t_m\dots t_1=t_x\approx40$ and ${\it O}_{Vm}=(\int_{V\ell}|ω|^{2m}dV)^{1/2m}$. For $Z(t)={\it O}^2_{V1}(t)$ the volume-integrated enstrophy, convergence of $\sqrtνZ(t)$ 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 $ν$-independent convergence of the energy dissipation rate $ε(t)=νZ(t)$ at $t_ε\sim 2t_x$ and sustained over a finite span $ΔT_ε\searrow 0.5 t_ε$, giving Reynolds number independent finite-time, dissipation, $ΔE_ε=\int_{ΔT_ε}εdt$, and thus satisfying one definition for a {\it dissipation anomaly}. Evidence for transient Kolmogorov-like enstrophy spectra is found over ${ΔT_ε}$. A critical factor in achieving these temporal convergence laws is how the domain $V_\ell=(2\ellπ)^3$ is increased as $\ell\simν^{-1/4}$, for $\ell=2$ to 6, then to $\ell=12$, as $ν$ decreases. $(2\ellπ)^3$ domain compatibility with established $(2π)^3$ mathematics in appendix allows small $ν$ Navier-Stokes solutions. Two spans of $ν$ are considered. Over the first factor of 25 decrease in $ν$, all of the $ν^{1/4}{\it O}_{Vm}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease in $ν$, $\ell$ is increased to $\ell=12$, there is only convergence of $ν^{1/4}Ω_{V\infty}(t)$ to $t_\infty$ and later $\sqrtνZ(t)$ convergence at $t_1=t_x$ and $ε(t)$ over $t\sim t_ε$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Navier-Stokes bounds and scaling for compact trefoils in $(2\ellπ)^3$ domains Kerr, Robert M. Fluid Dynamics For a perturbed trefoil vortex knot evolving under the Navier-Stokes equations, a sequence of $ν$-independent times $t_m$ are identified corresponding to a set of scaled, volume-integrated vorticity moments $ν^{1/4}{\it O}_{V1}$ with this hierarchy $t_\infty\le\dots\le t_m\dots t_1=t_x\approx40$ and ${\it O}_{Vm}=(\int_{V\ell}|ω|^{2m}dV)^{1/2m}$. For $Z(t)={\it O}^2_{V1}(t)$ the volume-integrated enstrophy, convergence of $\sqrtνZ(t)$ 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 $ν$-independent convergence of the energy dissipation rate $ε(t)=νZ(t)$ at $t_ε\sim 2t_x$ and sustained over a finite span $ΔT_ε\searrow 0.5 t_ε$, giving Reynolds number independent finite-time, dissipation, $ΔE_ε=\int_{ΔT_ε}εdt$, and thus satisfying one definition for a {\it dissipation anomaly}. Evidence for transient Kolmogorov-like enstrophy spectra is found over ${ΔT_ε}$. A critical factor in achieving these temporal convergence laws is how the domain $V_\ell=(2\ellπ)^3$ is increased as $\ell\simν^{-1/4}$, for $\ell=2$ to 6, then to $\ell=12$, as $ν$ decreases. $(2\ellπ)^3$ domain compatibility with established $(2π)^3$ mathematics in appendix allows small $ν$ Navier-Stokes solutions. Two spans of $ν$ are considered. Over the first factor of 25 decrease in $ν$, all of the $ν^{1/4}{\it O}_{Vm}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease in $ν$, $\ell$ is increased to $\ell=12$, there is only convergence of $ν^{1/4}Ω_{V\infty}(t)$ to $t_\infty$ and later $\sqrtνZ(t)$ convergence at $t_1=t_x$ and $ε(t)$ over $t\sim t_ε$. |
| title | Navier-Stokes bounds and scaling for compact trefoils in $(2\ellπ)^3$ domains |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2401.03578 |