Dynamics of a trefoil knotted vortex

A slender trefoil knotted vortex is studied using direct numerical simulation of the Navier-Stokes equations for vortex Reynolds numbers (Re Gamma/nu, circulation/viscosity) up to 12000. For initially zero twist (T-w,T-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,T-0 = 7.48 and 22.48) reveal that the overall dynamics is similar to the T-w,T-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.

Automated summary

The study of a slender trefoil knotted vortex reveals that the evolution of the vortex is mainly governed by the change in writhe, rather than twist, although twist plays a significant role in enstrophy growth and energy cascade. Reconnection causes changes in twist helicity and global helicity, strongly dependent on Reynolds number. Flow visualization and helical wave decomposition show the generation of both positive and negative twist helicities before and during reconnection.