Helicity dynamics in viscous vortex links

Abstract

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 $Re$ .

Lagrangian dynamics and regularity of the spin Euler equation

“…2021) and Hopf link (Aref & Zawadzki 1991; Kivotides & Leonard 2021; Yao et al. 2022). Under the self-induced velocity, such vortex tubes can be gradually stretched, twisted and flattened, and form nearly singular vortical structures.…”

Vortex bursting and associated twist dynamics on helical vortex tubes and vortex rings

“…In particular, this could occur on knotted vortex loops and vortex links, which typically evolve towards simpler topologies through successive events of reconnection and merging (Kleckner & Irvine 2013;Zhao & Scalo 2021). The detailed helicity dynamics of these vortices has been analysed carefully by Zhao & Scalo (2021), Shen et al (2022) and Yao et al (2022), and demonstrates the generation of axial flow from the reconnection regions. However, since these studies focus on the reconnection and unlinking process, they do not follow the flow for sufficiently long to track the evolution of the coiled vortex lines.…”

Evolution of dissipative fluid flows with imposed helicity conservation

“…We study the helicity dynamics of the asymmetric collision of vortex rings (Yao et al. 2022) with in NS and HCNS flows. The parametric equations of the vortex centrelines of two initial unlinked vortex rings are and Here, is a rotating angle to pose asymmetry and is a separation distance.…”

“…2021) and Hopf links (Yao et al. 2022) at high Reynolds numbers found a ‘transient growth’ of helicity when vortex reconnection occurs, implying that helicity is not conserved in the inviscid limit. Since helicity and its time derivative are not positive definite, the study of helicity in limiting conditions appears to be more challenging than that of kinetic energy or the mean dissipation rate (Batchelor 1953; Sreenivasan 1984, 1998; Kaneda et al.…”