Influence of winding number on vortex knots dynamics

Chiara Oberti,Renzo L Ricca

doi:10.1038/s41598-019-53548-w

Chiara Oberti, Renzo L Ricca

Open Access

https://doi.org/10.1038/s41598-019-53548-w

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Abstract

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. Results have been obtained for 56 knots and unknots up to 51 crossings. In agreement with previous numerical results we prove that in general the propagation speed increases with the number of toroidal coils, but we notice that the number of poloidal coils may greatly modify the motion. Indeed we prove that for increasing aspect ratio and number of poloidal coils vortex motion can be even reversed, in agreement with previous numerical observations. These results demonstrate the importance of three-dimensional features in vortex dynamics and find useful applications to understand helicity and energy transfers across scales in vortical flows.

Highlights

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics
Work on determining the influence of geometric and topological features on the dynamics of vortex knots has been limited to approximated models, based either on the use of the so-called localized induction approximation (LIA)[16,17,18,19] or regularized Biot-Savart law by cut-off methods[8,10,19,20]
In this paper we provide a mathematical proof that the effects of winding number — a topological invariant of torus knots — are of primary importance on the motion of vortex knots in the context of classical, ideal fluid mechanics

Summary

OPEN Influence of winding number on vortex knots dynamics

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. In order to make analytical progress we assume uniform vorticity on a small, circular vortex cross-section, and vortex geometry given by parametric equations of knots standardly embedded on a mathematical torus This allows us to take advantage of the rotational symmetry of these knot types, by reducing the Biot-Savart integral to a line integral, function solely of torus aspect ratio λ and winding number w (see the following section for definitions).

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