Singular Vortex Research Articles

Critical velocity of continuous vortex formation in rotating3He-A

Vortex lines with continuous (singularity-free) core structure are usually formed when superfluid3He-A is accelerated to rotation. The critical counterflow velocityvc(T), at which vortex formation starts, has been measured with cw NMR. The onset of vortex formation is found to depend on the order parameter texture and differs from that of singular core vortices in4He-II or3He-B due to much larger length scale.

Finite-time vortex singularity and Kolmogorov spectrum in a symmetric three-dimensional spiral model.

A recent analytical model of three-dimensional Euler flows [Phys. Rev. Lett. 69, 2196 (1992)] which exhibits a finite-time vortex singularity is developed further. The initial state is symmetric and contains a velocity null (stagnation point) which is collinear with two vorticity nulls. Under some assumptions, it is shown by asymptotic analysis of the Euler equation that the vorticity blows up at the stagnation point as inverse time in a locally self-similar manner. The spatial structure of the inviscid flow in the vicinity of the singularity involves disparate small scales. The effect of a small but finite viscosity is shown to arrest the formation of the singularity. The presence of spiral structure in the initial conditions leads naturally to the model developed by Lundgren [Phys. Fluids 25, 2193 (1982)] in which the gradual tightening of spirals by differential rotation provides a mechanism for transfer of energy to small spatial scales. It is shown by asymptotic analysis of the Navier-Stokes equation, that a time-average over the lifetime of the spiral vortex in the present model yields the Kolmogorov spectrum.

Chern-Simons vortices in an open system.

A gauge invariant quantum field theory with a spacetime dependent Chern-Simons coefficient is studied. Using a constraint formalism together with the Schwinger action principle it is shown that non-zero gradients in the coefficient induce magnetic-moment corrections to the Hall current and transform vortex singularities into non-local objects. The fundamental commutator for the density fluctuations is obtained from the action principle and the Hamiltonian of the Chern-Simons field is shown to vanish only under the restricted class of variations which satisfy the gauge invariance constraint.

Dissociation of Flux Line in Unconventional Superconductor

Besides usual flux lines the topology of the “real” vector phase in a multicomponent superconductor admits an existence of half-quantum vortices in the bulk. These vortices have a divergent energy near H c 1 . However, the possibility of their existence reveals itself in an instability of a usual singular one-quantum vortex towards dissociation into the nonsingular bound pair of half-quantum vortices. Considering an example of the multicomponent Ginzburg-Landau functional we found analytically critical values of phenomenological parameters corresponding to this instability, as well as the symmetry group of the vortex pair.

Vortex dynamics of viscous fluid flows. Part 1. Two-dimensional flows

The method of product integration is applied to the vortex dynamics of two-dimensional incompressible viscous media. In the cases of both unbounded and bounded flows under the no-slip boundary condition, the analytic solutions of the Cauchy problem are obtained for the Helmholtz equation in the form of linear and nonlinear product integrals. The application of product integrals allows the generalization in a natural way of the vortex dynamics concept to the case of viscous flows. However, this new approach requires the reconsideration of some traditional notions of vortex dynamics. Two lengthscales are introduced in the form of a micro- and a macro-scale. Elementary ‘vortex objects’ are defined as two types of singular vortex filaments with equal but opposite intensities. The vorticity is considered as the macro-value proportional to the concentration of elementary vortex filaments inhabiting the micro-level. The vortex motion of a viscous medium is represented as the stochastic motion of an infinite set of elementary vortex filaments on the micro-level governed by the stochastic differential equations, where the stochastic velocity component of every filament simulates the viscous diffusion of vorticity, and the regular component is the macro-value induced according to the Biot–Savart law and simulates the convective transfer of vorticity.In flows with boundaries, the production of elementary vortex filaments at the boundary is introduced to satisfy the no-slip condition. This phenomenon is described by the application of the generalized Markov processes theory. The integral equation for the production intensity of elementary vortex filaments is derived and solved using the no-slip condition reformulated in terms of vorticity. Additional conditions on this intensity are determined to avoid the many-valuedness of the pressure in a multi-connected flow domain. This intensity depends on the vorticity in the flow and the boundary velocity at every time instant, together with boundary acceleration.As a result, the successive and accurate application of the product-integral method allows the study of vortex dynamics in a viscous fluid according to the concepts of Helmholtz and Kelvin.

High-Speed Observations of Highly Erosive Vortex Cavitation around Butterfly Valve.

In order to clarify the mechanism of the severe erosion mainly resulting from the singular vortex cavitation, we carefully observe the vortex cavitation associated with the erosion around a butterfly valve by means of a high-speed video camera, whose framing rate is 40500 per second. The velocity of cavitating flow is measured by the digital-image processing technique. It is worth noting that the vortex cavitation develops in the boundary region between the main flow region, i. e., the orifice-jet, and the recirculating region. The cavitating vortex core consists of several tiny vortex cavitations. The rotating velocity of the highly erosive vortex cavitation can also be determined by this technique.

特集 復元と可視化 画像処理による渦キャビテーション解析

In order to clarify the mechanism of the severe erosion mainly resulted by the singular vortex cavitation, we carefully try to observe the vortex cavitation associated with the erosion around a butterfly valve by means of a high-speed video-camera, whose maximum framing rate is 40, 500 per seconds. The velocity of cavitating flow are measured with the correlation method, one of the digital-image processing-technique. It is worth noticing that the vortex cavitation develops in the boundary region between the main flow region, i. e., the orifice-jet, and the recirculating region. The cavitating vortex core consists of several tiny vortex cavitation. And the rotating velocity of the highly erosive vortex cavitation can be also detected by the technique.

The collapse of an axi-symmetric, swirling vortex sheet

An axi-symmetric and swirling vortex sheet is investigated as the simplest flow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stability of a single sheet but instability for other configurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and growth rates (or frequencies) can be explicitly expressed. Subsequent nonlinear evolution is numerically simulated using a vortex method. The numerical results for an axi-symmetric swirling sheet with a vortex line along the axis of symmetry show detachment of a vortex ring from the sheet into the outer fluid, and collapse of the sheet onto the vortex line at some points. Vortex cancellation, which in the presence of viscosity would likely lead to vortex line reconnection, seems to occur in both of these phenomena. The evolution of two co-axial, axi-symmetric, swirling vortex sheets is similar.

The vorton method

After a general introduction to the vorton method, which is a vortex method resembling the 2-D point-vortex method, a set of equations describing dynamics of 3-D vortex singularities (vortons) is derived, avoiding the inconsistency in the derivation of other vorton equations which have been applied. Though inviscid, numerical simulations show reconnection phenomena.

103 画像による渦キャビテーション解析

In order to clarify the mechanism of the severe erosion mainly resulted by the singular vortex cavitation, we carefully try to observe the vortex cavitation associated with the erosion around a butterfly valve by means of a high-speed video-camera, whose framing rate is 40500 per seconds. The velocity of cavitating flow are measured by using the digital-image processing-technique. It is worth noticing that the vortex cavitation develops in the boundary region between the main flow region, i.e., the orifice-jet, and the recircurating region. The cavitating vortex core consists of several tiny vortex cavitation. And the rotating velocity of the highly erosive vortex cavitation can be also detected by the technique.

A class of unsteady self-similar flows without an ?essential? singularity and a mechanism of whirlwind formation

This article is a continuation of reference [1], in which the author represented the known self-similar incompressible flows by means of a single table. The table incorporated more than 50 problems of practical interest. Although such a representation can never be complete, the method of representation itself also proved useful, since it drew attention to the existence of interesting uninvestigated self-similar flows corresponding to the “empty” boxes of the table. For example, it turned out that the problem of unconfined flow resulting from the instantaneous generation of finite momentum at a point had been considered in the case of turbulent viscosity [2], but the analogous formulation for an ideal fluid was not known, although on the basis of dimensional considerations such a self-similar flow can exist and, in fact, was subsequently found [3]. A class of self-similar flows comprising more than 20 problems which can be combined in five different formulations is considered. These flows can have an “essential” singularity only at the initial instant of time (for example, a powerful explosion) and subsequently do not contain singularities of the energy or mass source type. Flows caused by the motion of bodies are excluded from consideration, i.e., cases in which the flow is bounded by fixed or free boundaries with zero pressure on the latter are considered. In the case of an ideal fluid vortex singularities at which no energy is released are admitted.

Close interaction between a vortex filament and a rigid sphere

The evolution of a linear vortex filament close to a rigid sphere is investigated at high Reynolds number. The limiting evolution in an ideal flow, is analysed using a cutoff method and the results are compared with those of a singular vortex approach able to account for a viscous effect on the vortex structure evolution. The computed results show the creation of a closed vortex structure in ideal flow and also, at low Reynolds number, an unrealistic reattachment of the vortex to the surface of the body. The nature of the boundary-layer development, when the no-slip condition is satisfied, is calculated near the symmetry plane. The solutions show the development of an unsteady, vortex-driven, separating boundary layer with the three-dimensional separations dependent on the initial distance of the filament from the wall. All the solutions ultimately show a rapid growth of the secondary vorticity field near the surface and suggest an ejection from the boundary layer, followed by a strong viscous-inviscid interaction.

Finite-time vortex singularity in a model of three-dimensional Euler flows.

An analytical model of three-dimensional Euler flows which exhibits a finite-time singularity is given. The singularity in vorticity occurs at a velocity field null (stagnation point) which lies on the line joining two vorticity field nulls. It is shown that the vorticity diverges inversely with time.

Topology in superfluid3He: Recent results

We discuss: the topological phase transition in nonsingular vortices in3He-A; vortices in3He-B and solitons terminating on strings; topological defects of the A-B interface: the interaction of continuous A-phase vortices with singular B-phase vortices across the interface; extended degeneracy and topology of Larmor precession; and internal topology in thin3He-A film, responsible for chiral edge states of fermions and QHE.

Insight into singular vortex flows

The method of three-dimensional vortex singularities is analyzed. The fact that it does not represent a weak solution of the Euler equations has little bearing on its validity as an operative model. Other evolution inconsistencies are a major problem. The non-solenoidality of the vortex field is analyzed and a linear filtering feedback procedure, requiring no additional computations, is introduced to allow alignment with the reconstructed solenoidal vorticity field. A rough estimation of the dissipative effect during vortex reconnection has shown a finite viscous effect that is implicitly present in the model, growing with stretching, representing the mechanism of shifting to reconnected topologies without observing strong velocity gradients. The dividing vorton method is rearranged in order to allow the quantized reproduction of a predefined core evolution law, an error estimation, due to the corpuscularity of the field, is given. Two different numerical computations are performed corresponding to a weak and a strong vortex interaction. The improvements are tested, and confrontations with existing numerical and experimental results are performed showing good agreement. The possibility of the singular vortex flow serving as a rough, but simple, model for very high Reynolds number complex flows is discussed.

Dynamics of singular vortices on a beta-plane

A new singular-vortex theory is presented for geostrophic, beta-plane dynamics. The stream function of each vortex is proportional to the modified Bessel function Ko(pr), where p can be an arbitrary positive constant. If p−1 is equal to the Rossby deformation scale Rd, then the vortex is a point vortex; for p−1 ≠ Rd the relative vorticity of the vortex contains an additional logarithmic singularity. Owing to the β-effect, the redistribution of the background potential vorticity produced by the vortices generates a regular field in addition to the velocity field induced by the vortices themselves. Equations governing the joint evolution of singular vortices and the regular field are derived. A new invariant of the motion is found for this system. If the vortex amplitudes and coordinates are set in a particular way then the regular field is zero, and the vortices form a system moving along latitude circles at a constant speed lying outside the range of the phase velocity of linear Rossby waves. Each of the systems is a discrete two-dimensional Rossby soliton and, vice versa, any distributed Rossby soliton is a superposition of the singular vortices concentrated in the interior region of the soliton. An individual singular vortex is studied for times when Rossby wave radiation can be neglected. Such a vortex produces a complicated spiral-form regular flow which consists of two dipoles with mutually perpendicular axes. The dipoles push the vortex westward and along the meridian (cyclones move northward, and anticyclones move southward). The vortex velocity and trajectory are calculated and applications to oceanic and atmospheric eddies are given.

Gap anisotropy, vortex cluster, and nucleation of vortices in rotating 3He-B

Abstract In rotating 3He-B the critical flow velocity for the nucleation of singular vortex lines is high and allows the existence of different rotating states: vortex-free counterflow, the equilibrium vortex state, and metastable states with a well-defined vortex cluster containing less than the equilibrium number of vortices. These states can be distinguished and characterized using CW NMR techniques. We derive an estimate for the magnetic field induced anistropy of the superfluid energy gap from an analysis of the spatial distribution of the order parameter texture in the vortex-free state. When vortices are introduced, their number in the vortex cluster can be counted using a simple experimental procedure. This allows us to measure the critical rotation velocity for the nucleation of vortices, which proves to correspond to the maximum flow velocity in the cylindrical NMR cell.

Mutual friction in superfluid 3He: Effects of bound states in the vortex core.

The motion of singular quantized vortex lines in superfluid $^{3}\mathrm{He}$ is considered for the A and B phases. Mutual friction is calculated within a microscopic quantum-mechanical Green's-function formalism, valid for dynamical processes. This enables us to include all the different physical phenomena in a unified approach. We consider axisymmetric vortices for temperatures considerably lower than ${\mathit{T}}_{\mathit{c}}$. In this regime, the main contribution to the force exerted on a moving vortex originates from the localized Fermi excitations occupying quantized energy eigenstates in the vortex core. These $^{3}\mathrm{He}$ quasiparticle states are similar to the quantized motion of charge in a magnetic field; thus vortex motion in $^{3}\mathrm{He}$ resembles the Hall phenomenon in metals. The outcome is that the viscous drag cannot simply be expressed through the cross sections for $^{3}\mathrm{He}$ quasiparticles scattering off the vortex, but is rather due to the mutual interactions between the localized quasiparticles and the normal excitations. Our calculations conform with the experimental values for the mutual-friction parameters. We also discuss vortex oscillations, and predict that strong dissipation should be observed at a resonant frequency of about 10 kHz, owing to transitions between the bound-state energy levels. This effect could be used for detecting and measuring the quantization of the bound-state spectrum for superfluid $^{3}\mathrm{He}$ in the vortex-core matter.

Vortex clusters in superfluid 3He-B

A unique property of 3He-B is vortex-free rotation, due to a high nucleation threshold of the singular vortices. In contrast to other superfluids, rotational states consisting of a vortex cluster surrounded by vortex-free counterflow form spontaneously and can also be prepared with a fixed number of vortices. The clusters can be analyzed by means of NMR measurements with high precision. These techniques have been exploited to study rotational states in the presence of different phase boundaries.

Dynamics of two-dimensional turbulence

Mixing of Lagrangian particles by a vortex singularity is shown to correspond to a continuous curdling process with the Lipschitz–Hölder exponent equal to (4)/(3) , demonstrating that Richardson’s l4/3 scaling is derivable from the two-dimensional kinematics of the vortex singularity. Scaling relationships describing the Lagrangian geometry of two-dimensional turbulence are linked to the rate of enhanced diffusion. The value of the two-dimensional intermittency exponent μ= (2)/(3) , derived from the author’s Lagrangian structure function theory [Phys. Fluids A 1, 1836 (1989)], when used in the Lévy flight theory of enhanced diffusion developed by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] predicts 〈R2〉∝t2.6 for the time rate of separation of labeled pairs of vortices in two dimensions. This is found to be in good agreement with the rate of dispersion measured in numerical simulations of inviscid two-dimensional turbulence. Kinetic theory arguments, supplemented by the previously derived Lagrangian structure relationships, yield an asymptotic P∝s−5/3 probability distribution function for path lengths s between vortex pair collisions. This result is consistent with path length distributions obtained from the two-dimensional simulations.