8,899 research outputs found
Velocity, energy and helicity of vortex knots and unknots
In this paper we determine the velocity, the energy and estimate writhe and
twist helicity contributions of vortex filaments in the shape of torus knots
and unknots (toroidal and poloidal coils) in a perfect fluid. Calculations are
performed by numerical integration of the Biot-Savart law. Vortex complexity is
parametrized by the winding number , given by the ratio of the number of
meridian wraps to that of the longitudinal wraps. We find that for vortex
knots and toroidal coils move faster and carry more energy than a reference
vortex ring of same size and circulation, whereas for knots and poloidal
coils have approximately same speed and energy of the reference vortex ring.
Helicity is dominated by the writhe contribution. Finally, we confirm the
stabilizing effect of the Biot-Savart law for all knots and unknots tested,
that are found to be structurally stable over a distance of several diameters.
Our results also apply to quantized vortices in superfluid He.Comment: 17 pages, 8 figures, 2 table
On the groundstate energy of tight knots
New results on the groundstate energy of tight, magnetic knots are presented.
Magnetic knots are defined as tubular embeddings of the magnetic field in an
ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear
coordinate system is introduced and the magnetic energy is determined by the
poloidal and toroidal components of the magnetic field. Standard minimization
of the magnetic energy is carried out under the usual assumptions of volume-
and flux-preserving flow, with the additional constraints that the tube
cross-section remains circular and that the knot length (ropelength) is
independent from internal field twist (framing). Under these constraints the
minimum energy is determined analytically by a new, exact expression, function
of ropelength and framing. Groundstate energy levels of tight knots are
determined from ropelength data obtained by the SONO tightening algorithm
developed by Pieranski (Pieranski, 1998) and collaborators. Results for torus
knots are compared with previous work done by Chui & Moffatt (1995), and the
groundstate energy spectrum of the first prime knots (up to 10 crossings) is
presented and analyzed in detail. These results demonstrate that ropelength and
framing determine the spectrum of magnetic knots in tight configuration.Comment: 26 pages, 9 figure
Impulse of Vortex Knots from Diagram Projections
AbstractIn this paper I extend the area interpretation of linear and angular momenta of ideal vortex filaments to complex tangles of filaments in space. A method based on the extraction of area information from diagram projections is presented to evaluate the impulse of vortex knots and links. The method relies on the estimate of the signed areas of sub-regions of the graph resulting from the pro- jection of the vortex axes on the plane of the graph. Some examples based on vortex rings interaction, vortex knots and links are considered for illustration. This method provides a complementary tool to estimate dynamical properties of complex fluid flows and it can be easily implemented in real-time diagnostics to investigate fluid dynamical properties of complex vortex flows
New developments in topological fluid mechanics
In this paper we present and discuss ideas and new results in three different research areas of topological fluid mechanics. First, we propose a conjectured experiment to produce and observe, for the first time, vortex knotting in real fluids. Next we provide a new ropelength bound for tight, magnetic knots in ideal magnetohydrodynamics. Finally, we present a novel interpretation of eigenvalue
analysis of tensor fields in terms of integral geometry by using the information on form factors provided by structural complexity analysis
Tackling Fluid Structures Complexity by the Jones Polynomial
AbstractBy making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this heuristic approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it gives also rise to new interesting problems in mathematical physics and it offers new tools to perform real-time numerical diagnostics of complex flows
Editorial to selected papers from the 2022 IMEKO International Conference on 'Metrology for Archaeology and Cultural Heritage'
Kelvin Wave Cascade and Decay of Superfluid Turbulence
Kelvin waves (kelvons)--the distortion waves on vortex lines--play a key part
in the relaxation of superfluid turbulence at low temperatures. We present a
weak-turbulence theory of kelvons. We show that non-trivial kinetics arises
only beyond the local-induction approximation and is governed by three-kelvon
collisions; corresponding kinetic equation is derived. On the basis of the
kinetic equation, we prove the existence of Kolmogorov cascade and find its
spectrum. The qualitative analysis is corroborated by numeric study of the
kinetic equation. The application of the results to the theory of superfluid
turbulence is discussed.Comment: 4 pages, RevTe
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