Construction of knotted vortex tubes with the writhe-dependent helicity

Abstract

We propose a feasible method for constructing knotted vortex tubes with the finite thickness and arbitrary complexity and develop an accurate algorithm to implement this method in numerical simulations. The central axis of the knotted vortex tube is determined by the parametric equation of a given smooth and non-degenerate closed curve. The helicity of the vortex tube is only proportional to the writhe of the vortex axis, a geometric measure for coiling of vortex tubes. This vortex construction can facilitate the investigation of the conversion of writhe to twist in the helicity evolution of knotted vortex tubes. As examples, we construct velocity–vorticity fields of trefoil, cinquefoil, and septafoil vortex knots. These vortex knots are used as initial conditions in the direct numerical simulation of viscous incompressible flows in a periodic box. In the evolution of vortex knots from simple flows to turbulent-like flows, all the knots are first untied. Then the vortex topology is invariant and the helicity is almost conserved for the trefoil knot, whereas the helicity decays rapidly during the breakdown and coaxial interactions of pinch-off vortex rings for cinquefoil and septafoil knots.