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Dynamics of a trefoil knotted vortex

Published online by Cambridge University Press:  27 July 2021

Jie Yao Open the ORCID record for Jie Yao [Opens in a new window] ,
Yue Yang Open the ORCID record for Yue Yang [Opens in a new window]  and
Fazle Hussain Open the ORCID record for Fazle Hussain [Opens in a new window]
Jie Yao
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
Yue Yang
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing100871, PR China
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
*
†Email address for correspondence: [email protected]
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Abstract

A slender trefoil knotted vortex is studied using direct numerical simulation of the Navier–Stokes equations for vortex Reynolds numbers ($Re \equiv \varGamma /\nu$, circulation/viscosity) up to 12 000. For initially zero twist ($T_{w,0}=0$), neither the writhe $W_r$ nor the global helicity $H$ is conserved. Initially $W_r$ slowly decreases, then suddenly drops during reconnection and becomes almost constant thence; its evolution is almost $Re$ independent. Before reconnection, $H$ also gradually decreases but sharply increases during reconnection. The evolution of $H$ after reconnection strongly depends on $Re$. While steadily decreasing at low $Re$, $H$ significantly increases before eventually decaying at high $Re$. Flow visualization, helicity decomposition and helical wave decomposition reveal that significant amounts of positive and negative twist helicities are simultaneously generated before and during reconnection. Also, the small leading and large trailing rings resulting from asymmetric reconnection have respectively negative and positive twists, which then decay differently due to different initial values, geometries and mutual induction. In particular, at high $Re$, the twist in the small ring, under stretching by the large trailing ring, decays much faster and even switches sign to become positive by the writhe-to-twist conversion – the main reason for the ‘transient growth’ of $H$. Simulations with non-zero initial twists ($T_{w,0}=7.48$ and $22.48$) reveal that the overall dynamics is similar to the $T_{w,0}=0$ case. Hence, the evolution of the trefoil knotted vortex is mainly governed by $W_r$, not $T_w$, although the latter is found to play an essential role in enstrophy growth as well as energy cascade.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A19
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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