Topological transition and helicity conversion of vortex torus knots and links are studied using direct numerical simulations of the incompressible Navier–Stokes equations. We find three topological transitional routes (viz. merging, reconnection and transition to turbulence) in the evolution of vortex knots and links over a range of torus aspect ratios and winding numbers. The topological transition depends not only on the initial topology but also on the initial geometry of knots/links. For small torus aspect ratios, the initially knotted or linked vortex tube rapidly merges into a vortex ring with a complete helicity conversion from the writhe and link components to the twist. For large torus aspect ratios, the vortex knot or link is untied into upper and lower coiled loops via the first vortex reconnection, with a helicity fluctuation including loss of writhe and link, and generation of twist. Then, the relatively unstable lower loop can undergo a secondary reconnection to split into multiple small vortices with a similar helicity fluctuation. Surprisingly, for moderate torus aspect ratios, the incomplete reconnection of tangled vortex loops together with strong vortex interactions triggers transition to turbulence, in which the topological helicity decomposition fails due to the breakdown of vortex core lines.

The dynamics of two slender Hopf-linked vortex rings at vortex Reynolds numbers ( $Re \equiv \varGamma /\nu, \mathrm {circulation/viscosity}$ ) $2000$ , $3000$ and $4000$ is studied using direct numerical simulations of the incompressible Navier–Stokes equations. Under self-induction, the initially perpendicularly placed vortex rings approach each other and reconnect to form two separate vortex rings. The leading ring is closely cuddled and further undergoes secondary reconnection to form two even smaller rings. At high $Re$ , the leading ring and the subsequent smaller rings are unstable and break up into turbulent clouds consisting of numerous even smaller-scale structures. Although the global helicity $H$ remains constant before reconnection, it increases and then rapidly decays during reconnection – both the growth and decay rates increase with $Re$ . In the two higher $Re$ (i.e. 3000 and 4000) cases, $H$ further rises after the first reconnection and reaches a quasi-plateau with the asymptotic value continuously increasing with $Re$ – suggesting that $H$ for viscous flows is not conserved at very high $Re$ . Further flow analysis demonstrates that significant numbers of positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates is the main reason for the complex evolution of $H$ . By examining the topological aspects of the helicity dynamics, we find that, different from $H$ , the sum of link and writhe ( $L_k+W_r$ ) continuously drop during reconnection. Our results also clearly demonstrate that the twist, which increases with $Re$ , plays a significant role in the helicity dynamics, particularly at high $Re$ .

We propose the helicity-conserved Navier–Stokes (HCNS) equation by modifying the non-ideal force term in the Navier–Stokes (NS) equation. The corresponding HCNS flow has strict helicity conservation, and retains major NS dynamics with finite dissipation. Using the helical wave decomposition, we show that the pentadic interaction of Fourier helical modes in the HCNS dynamics is more complex than the triadic interaction in the NS dynamics, and enhanced variations for left- and right-handed helicity components cancel each other in the HCNS flow to keep the invariant helicity. A comparative study of HCNS and NS flow evolutions with direct numerical simulation elucidates the influence of the helicity conservation on flow structures and statistics in the vortex reconnection and isotropic turbulence. First, the HCNS flow evolves towards a Beltrami state with a $-4$ scaling law of the energy spectrum at high wavenumbers at long times. Second, large-scale flow structures are almost identical during the viscous reconnection of vortex tubes in the two flows, whereas many more small-scale helical structures are generated via the pentadic mode interaction in the HCNS flow than in the NS flow. Moreover, we demonstrate that parity breaking at small scales can trigger a notable helicity variation in the NS flow. These findings hint that the helicity may not be conserved in the inviscid limit of the NS flow.

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