Two-dimensional Vortex Dynamics Research Articles

SIGEST

The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, \alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

Structures of wing-tip vortices at low Reynolds numbers and their modelling

The structures of wing-tip vortices at a low Reynolds number () are studied by direct numerical simulation. After confirming the basic features of the wing-tip vortices, the distributions of axial vorticity and velocity are investigated in detail. The vorticity distribution consists of a dominant symmetric component, quadrupole components responsible for elliptical deformation of the vortex core and the other small components. The symmetric component of vorticity is in reasonable agreement with the Moore-Saffman model (Moore & Saffman 1973 Proc. R. Soc. Lond. 333, 491) for the three types of wing planform considered: the rectangular, elliptic and delta wings. The agreement between numerical results and the Moore-Saffman model is not good for the symmetric component of axial velocity since the boundary layer component is small owing to the incomplete roll-up of the vortex sheet shed from the boundary layer. However, the vortex component of the axial velocity is correctly predicted by the Moore-Saffman model. The elliptical deformation of the vortex core is characterized by the aspect ratio and the direction of the major axis. It is found that the elliptical deformation is not uniform in the streamwise direction, which implies that the wing-tip vortices cannot be always modelled by a two-dimensional vortex pair. The evolution of the elliptical deformation along the streamwise direction is shown to be described by two-dimensional vortex dynamics under the boundary-layer approximation. The equations of motion of an elliptic vortex patch in a straining field predicts correctly the rate of change of the direction, but fails to predict that of the aspect ratio, for which the equation derived for axisymmetrization of vortices gives a good prediction.

Coherent vortex dynamics in a strongly interacting superfluid on a silicon chip.

Quantized vortices are fundamental to the two-dimensional dynamics of superfluids, from quantum turbulence to phase transitions. However, surface effects have prevented direct observations of coherent two-dimensional vortex dynamics in strongly interacting systems. Here, we overcome this challenge by confining a thin film of superfluid helium at microscale on the atomically smooth surface of a silicon chip. An on-chip optical microcavity allows laser initiation of clusters of quasi-two-dimensional vortices and nondestructive observation of their decay in a single shot. Coherent dynamics dominate, with thermal vortex diffusion suppressed by five orders of magnitude. This establishes an on-chip platform with which to study emergent phenomena in strongly interacting superfluids and to develop quantum technologies such as precision inertial sensors.

Vortical flow development in round ducts across scales for engine inlet applications

Turbofan engine performance depends highly on the characteristics and conditions of the inlet flow. Swirl distortions, caused by non-uniformities in flows arising from boundary layer or ground/fuselage vortex ingestion are of concern and need to be fully understood to guarantee efficiency and safety of propulsion systems. To investigate a fundamental single-vortex distortion development in a duct at different Reynolds numbers, a StreamVane distortion generating device was designed and experimentally analyzed in a small-scale low-speed wind tunnel (ReD 500,000) and in a full-scale engine testing rig (ReD 3 million). Stereoscopic particle image velocimetry was used to measure the three-component velocity fields at discrete measurement planes downstream of the distortion device. Results show that the secondary flow is generated and develops very similarly in both scales, and is mostly driven by two-dimensional (2D) vortex dynamics. Induced velocities arising from the proximity of the vortex to the duct wall causes the vortex center to convect circumferentially around the duct, in the same sense as the vortex rotation, as it travels downstream. Small-scale turbulence results show small-scale instabilities related to the development of the vortex. This work shows that the development of this vane-generated, vortex-dominated flow is largely Reynolds number independent for the covered range, so that details of similar duct-bounded flows can be analyzed in depth in small-scale experiments, decreasing development efforts and cost.

Experimental study of the stability and dynamics of a two-dimensional ideal vortex under external strain

The dynamics of two-dimensional (2-D) ideal fluid vortices is studied experimentally in the presence of an irrotational strain flow. Laboratory experiments are conducted using strongly magnetized pure electron plasmas, a technique which is made possible by the isomorphism between the drift–Poisson equations describing plasma dynamics transverse to the field and the 2-D Euler equations describing an ideal fluid. The electron plasma system provides an excellent opportunity to study the dynamics of a 2-D Euler fluid due to weak dissipation and weak 3-D effects, simple diagnosis and precise control. The plasma confinement apparatus used here was designed specifically to study vortex dynamics under the influence of external flow by applying boundary conditions in two dimensions. Additionally, vortex-in-cell simulations are carried out to complement the experimental results and to extend the parameter range of the studies. It is shown that the global dynamics of a quasi-flat vorticity profile is in good quantitative agreement with the theory of a piecewise-constant elliptical patch of vorticity, including the equilibria, dynamical orbits and stability properties. Deviations from the elliptical patch theory are observed for non-flat vorticity profiles; they include inviscid damping of the orbits and modified stability limits. The dependence of these phenomena on the flatness of the initial profile is discussed. The relationship of these results to other theoretical, numerical and experimental studies is also discussed.

Collision dynamics of two-dimensional non-Abelian vortices

We study computationally the collision dynamics of vortices in a two-dimensional spin-2 Bose-Einstein condensate. In contrast to Abelian vortex pairs, which annihilate or pass through each other, we observe non-Abelian vortex pairs to undergo rungihilation - an event that converts the colliding vortices into a rung vortex. The resulting rung defect subsequently decays to another pair of non-Abelian vortices of different type, accompanied by a magnetization reversal.

In situimaging of vortices in Bose-Einstein condensates

Laboratory observations of vortex dynamics in Bose-Einstein condensates (BECs) are essential for determination of many aspects of superfluid dynamics in these systems. We present a novel application of dark-field imaging that enables \texttt{\it in situ} detection of two-dimensional vortex distributions in single-component BECs, a step towards real-time measurements of complex two-dimensional vortex dynamics within a single BEC. By rotating a $^{87}$Rb BEC in a magnetic trap, we generate a triangular lattice of vortex cores in the BEC, with core diameters on the order of 400 nm and cores separated by approximately 9 $\mu$m. We have experimentally confirmed that the positions of the vortex cores can be determined without the need for ballistic expansion of the BEC.

Spectral energy transport in two-dimensional quantum vortex dynamics

Spectral energy transport in quantum turbulence is studied in a two-dimensional Bose-Einstein condensate at finite temperature, showing that significant transport only occurs for large systems with weak dissipation.

Quenched dynamics of two-dimensional solitary waves and vortices in the Gross–Pitaevskii equation

We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose–Einstein condensates (BECs) (2002 Nature 417 150–3). We start by identifying the fundamental state of the 2D Gross–Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of stable self-trapping and a catastrophic collapse of 2D fundamental states are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitary waves to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a single-hump state. The same setting may also be implemented in the context of optical solitons and vortices, using photonic-crystal fibers.

Noise-Assisted Microwave Up-conversion by Vortices in Thin-Film Superconductors with a dc-Biased Washboard Pinning Potential

So far the main theoretical basis for understanding and optimization of the microwave properties of vortices in type II superconductors has been relying upon the Coffey–Clem (CC) approach for the linear impedance at nonzero temperature. However, the CC model does not account for the non-linear response and the possibility to control it by changing the value of the dc transport current in a superconducting sample. For this reason, we have exactly solved the Langevin equation for the two-dimensional non-linear vortex dynamics in a dc bias-tilted cosine pinning potential in the presence of an ac current of arbitrary amplitude and frequency ω and have, thereby, substantially generalized the CC results. In this work we analyze the behavior of the non-linear response on kω-frequency in a wide range of dc and ac current densities, ω, and temperature. The kω-response is found to depend strongly on all these parameters, as exemplified for the third-harmonic (k=3) transformation coefficient Z 3. The parametric window for the most enhanced up-conversion is presented. The predicted effects can be experimentally verified in thin-film superconductors with some pinning potential of the washboard type.

Wall-modeled large-eddy simulation of high Reynolds number flow around an airfoil near stall condition

This paper investigates the capability of large-eddy simulation (LES) with wall-modeling in predicting transitional and separated flow over an airfoil near stall condition at high Reynolds number (chord based Rec=2.1×106). By incorporating the non-equilibrium effects and transition treatment in the wall model, the wall-modeled LES well predicts the mean and turbulence statistics in the region of laminar separation, turbulent transition, turbulent reattachment, and initial-mid developments of attached turbulent boundary layer. The present LES with non-equilibrium wall-model reproduces the dynamics of two-dimensional intermittent laminar separation vortices, hairpin-like vortices, and breakdown of the laminar separation vortices to three-dimensional turbulent structures that allow the boundary layer to reattach. Comparisons between the non-equilibrium and equilibrium wall-models highlight the importance of including the non-equilibrium effects in the model.

An analytical study of the self-induced inviscid dynamics of two-dimensional uniform vortices

The self-induced dynamics of a uniform planar vortex in an isochoric, inviscid fluid is analytically investigated, by writing a new nonlinear integrodifferential problem which describes the time evolution of the Schwarz function of its boundary. In order to overcome the difficulties related to the nonlinear nature of the problem, an approximate solution of the above problem is proposed in terms of a hierarchy of linear integral equations. In particular, the solution at the first order is here investigated and applied to a class of uniform vortices, the kinematics of which is well known. A rich mathematical phenomenology has been found behind the approximate dynamics. In particular, the motion of certain branch points and the consequent changes in the algebraic structure of the Schwarz function appear to be its key features. The approximate solutions are finally compared with the contour dynamics simulations of the vortex motion. The agreement is satisfactory up to times of the order of a quarter of the eddy turnover time, while it becomes more and more qualitative at later times. Eventually, the physical meaning of the approximate solutions is lost, the corresponding vortex boundaries becoming non-simple curves.

Suppression of Kelvon-induced decay of quantized vortices in oblate Bose-Einstein condensates

We study the Kelvin mode excitations on a vortex line in a three-dimensional trapped Bose-Einstein condensate at finite temperature. Our stochastic Gross-Pitaevskii simulations show that the activation of these modes can be suppressed by tightening the confinement along the direction of the vortex line, leading to a strong suppression in the vortex decay rate as the system enters a regime of two-dimensional vortex dynamics. As the system approaches the condensation transition temperature we find that the vortex decay rate is strongly sensitive to dimensionality and temperature, observing a large enhancement for quasi-two-dimensional traps. Three-dimensional simulations of the recent vortex dipole decay experiment of Neely et al. [Phys. Rev. Lett. 104, 160401 (2010)] confirm two-dimensional vortex dynamics, and predict a dipole lifetime consistent with experimental observations and suppression of Kelvon-induced vortex decay in highly oblate condensates.

Lagrangian mixing in straight compound channels

Recently Stocchino & Brocchini (J. Fluid Mech., vol. 643, 2010, p. 425 have studied the dynamics of two-dimensional (2D) large-scale vortices with vertical axis evolving in a straight compound channel under quasi-uniform flow conditions. The mixing processes associated with such vortical structures are here analysed through the results of a dedicated experimental campaign. Time-resolved Eulerian surface velocity fields, measured using a 2D particle-image velocimetry system, form the basis for a Lagrangian analysis of the dispersive processes that occur in compound channels when the controlling physical parameters, i.e. the flow depth ratio (rh) and the Froude number (Fr) are changed. Lagrangian mixing is studied by means of various approaches based either on single-particle or multiple-particle statistics (relative and absolute statistics, probability density functions (p.d.f.s) of relative displacements and finite-scale Lyapunov exponents). Absolute statistics reveal that transitional macrovortices, typical of shallow flow conditions, strongly influence the growth in time of the total absolute dispersion, after the initial ballistic regime, leading to a non-monotonic behaviour. In deep flow conditions, on the contrary, the absolute dispersion displays a monotonic growth because the generation of transitional macrovortices does not take place. In all cases an asymptotic diffusive regime is reached.Multiple-particle dynamics is controlled by rh and Fr. Different growth regimes of the relative diffusivity have been found depending on the flow conditions. This behaviour can be associated with different energy transfer processes and it is further confirmed by the p.d.f.s of relative displacements, which show a different asymptotical shape depending on the separation scales and the Froude number. Finally, an equilibrium regime is observed for all the experiments by analysing the decay of the finite-scale Lyapunov exponents with the particle separations.

Hamiltonian–Dirac simulated annealing: Application to the calculation of vortex states

A simulated annealing method for calculating stationary states for models that describe continuous media is proposed. The method is based on the noncanonical Poisson bracket formulation of media, which is used to construct Dirac brackets with desired constraints, and symmetric brackets that cause relaxation with the desired constraints. The method is applied to two-dimensional vortex dynamics and a variety of numerical examples is given, including the calculation of monopole and dipole vortex states.

Guiding of vortices and ratchet effect in superconducting films with asymmetric pinning potential

Two-dimensional vortex dynamics in a ratchet washboard planar pinning potential in the presence of thermal fluctuations is considered on the basis of a Fokker-Planck equation. Explicit expressions for two nonlinear anisotropic voltages (longitudinal and transverse with respect to the current direction) are derived and analyzed. The physical origin of these odd (with respect to magnetic field or transport current direction reversal) voltages is caused by the interplay between the even effect of vortex guiding and the ratchet asymmetry. Both voltages are going to zero in the linear regimes of the vortex motion [i.e., in the thermally activated flux flow (TAFF) and Ohmic flux flow (FF) regimes] and have a bumplike current or temperature dependence in the vicinity of the highly nonlinear resistive transition from the TAFF to the FF.

A new calculus for two-dimensional vortex dynamics

This article provides a user’s guide to a new calculus for finding the instantaneous complex potentials associated with point vortex motion in geometrically complicated planar domains, with multiple boundaries, in the presence of background flows. The key to the generality of the approach is the use of conformal mapping theory together with a special transcendental function called the Schottky–Klein prime function. Illustrative examples are given.

A shallow-water theory for annular sections of Keplerian disks

A scaling argument is presented that leads to a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. To develop a theoretical construction that will aid in physically understanding the relationship of known two-dimensional vortex dynamics to their three-dimensional counterparts in Keplerian disks. Using asymptotic scaling arguments varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full $2\pi$ angular extent of the disk. The scalings lead to dynamics which are radially geostrophic and vertically hydrostatic. It follows that a potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain is developed and is shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and are shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. Based on the framework of this theory, arguments based on geophysical notions are presented to support the assertion that the strato-rotational instability is in a generic class of barotropic/baroclinic potential vorticity instabilities. Extensions of this formalism are also proposed. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach to studying disk dynamics.

Vortex Dynamics in a Two-Dimensional Domain with Holes and the Small Obstacle Limit

In this work we examine the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations on a domain with several holes, when one of the holes becomes small. We show that the limit flow satisfies a modified Euler system in the domain with the small hole removed. In vorticity form, the limit system is the usual equation for transport of vorticity, coupled with a modified Biot–Savart law which includes a point vortex at the point where the small hole disappears, together with the appropriate correction for the harmonic part of the flow. This work extends results by Iftimie, Lopes Filho, and Nussenzveig Lopes, obtained in the context of the exterior of a single small obstacle in the plane; see [Comm. Partial Differential Equations, 28 (2003), pp. 349–379]. The main difficulty in the present situation lies in controlling the behavior of the harmonic part of the flow, which is not an exact conserved quantity. As part of our analysis we develop a new description of two-dimensional vortex dynamics in a general domain with holes.