The life of a vortex knot

Abstract

The idea that the knottedness (hydrodynamic Helicity) of a fluid flow is conserved has a long history in fluid mechanics. The quintessential example of a knotted flow is a knotted vortex filament, however, owing to experimental difficulties, it has not been possible until recently to directly generate knotted vortices in real fluids. Using 3D printed hydrofoils and high-speed laser scanning tomography, we generate vortex knots and links and measure their subsequent evolution. In both cases, we find that the vortices deform and stretch until a series of vortex reconnections occurs, eventually resulting several disjoint vortex rings.This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion at the 66th Annual Meeting of the APS Division of Fluid Dynamics.Smoke rings are an everyday example of filamental vortex loops that are found in many complex flows. Vortex loops with non-trivial topologylinked or knotted together -should be conserved in ideal fluids, as first noted by Lord Kelvin [1]. In real fluids, even superfluids, the situation is more complicated, and linking can change through reconnection events, whose detailed dynamics are challenging to resolve in both theory and experiment. As a result, the dynamics of knots in non-ideal fluids is largely unresolved.Previous methods for generating vortex loops are difficult or impossible to generalize to creating vortex links and knots, so we developed a new technique based on accelerating specially shaped hydrofoils [2]. Using this method, we were able to generate linked and knotted vortices for the first 1 arXiv:1310.3321v1 [physics.flu-dyn]

On the Schrödinger map for regular helical polygons in the hyperbolic space

“…This behavior which is reminiscent of the Talbot effect in optics [21,22], has also been noticed previously for polygonal curves in R 3 [17,18]. Let us recall that a similar dynamics was observed in real fluids [23], see also [24, Figure 1] where axis-switching phenomenon is clearly visible when the evolution of vortex filaments generated in a lab experiment (https://vimeo.com/83946699) are compared with that of an equilateral triangle [25]. This complex dynamics is also present in the evolution of a single point located on the polygonal curve in both Euclidean and hyperbolic cases.…”

The Talbot effect as the fundamental solution to the free Schrödinger equation

“…The Dirac comb (2) as the datum arises thinking of the curvature of the polygon as a periodic sum of Dirac deltas. These results are supported by experiments like the one performed in [14] with a clover-like smoke ring. The corresponding mathematical simulation with a triangular vortex filament is available in [15].…”

On the Hausdorff dimension of Riemann's non-differentiable function

“…The coefficients in the numerator of γ, a and b, are coprime, so either a = q and b = −p or a = −q and b = p must hold. But then the parity condition in (19) is not kept, hence γ does not exist. These points are precisely corresponding to p/q with q ≡ 2 (mod 4), because then p is odd and p/q = p/(q/2), where q/2 is odd.…”

“…To determine the coefficients a, b, c, d of γ ∈ Γ θ as in (19), we use the theory of continued fractions. In the case of p/q, the continued fraction is finite because it is a rational number, so there exists N ∈ N such that p/q = [a 0 ; a 1 , .…”

“…A situation where this model can be applied concerns jets, such as smoke jets travelling in the air, or bubble jets travelling in water, passing through nozzles of different shapes (as can be watched in [19]). A very visual example is a smoker producing a smoke-ring with his mouth.…”