Point Vortices Research Articles

Statistical analysis of vortex condensate motion in two-dimensional turbulence

An inverse turbulent cascade in a periodic square box produces a coherent system-sized vortex dipole. We study the statistics of its motion by carrying out direct numerical simulations performed for various bottom friction α, pumping intensity ε, and fluid hyperviscosity ν. In the main approximation, coherent vortices can be considered as point vortices, and within this model, they drift at the same dipole velocity, which is determined by their circulation and mutual arrangement. The characteristic value of the dipole velocity is more than an order of magnitude smaller than the polar velocity inside coherent vortices. Turbulent fluctuations give rise to a relative velocity between the vortices, which changes the distance between them. We found that for a strong condensate, the probability density function of the vector ρ, describing the difference in the mutual arrangement of coherent vortices from half the diagonal of the computational domain, has the form of a ring. The radius of the ring weakly depends on control parameters and its width is proportional to parameter δ=ϵ−1/3L2/3α, where ϵ is the inverse energy flux and L is the system size. The random walk around the ring, caused by turbulent fluctuations, has superdiffusion behavior at intermediate times. It results in a finite correlation time of the dipole velocity, which is of the order of turnover time τK=L2/3ϵ−1/3 of system-size eddies produced by an inverse turbulent cascade. The results obtained deepen the understanding of the processes governing the motion of coherent vortices.

Random splitting of point vortex flows

Random splitting of point vortex flows

Mean-field limit of point vortices for the lake equations

Mean-field limit of point vortices for the lake equations

A lightweight vortex model for unsteady motion of airfoils

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.

Dynamics of two vortices on a finite flat cylinder

In this work, a model that describes the motion of point vortices in an ideal incompressible fluid on a finite flat cylinder is obtained. The case of two vortices is considered in detail. It is shown that the equations of motion of vortices can be represented in Hamiltonian form and have an additional first integral. A procedure of reduction to a fixed level of the first integral is proposed. For the reduced system, phase portraits are constructed, fixed points and singularities of the system are indicated.

The Vortex Surface in a Three-Body Quantum System

Vortices are structures known in our daily lives and observed in a wide variety of systems, from cosmic to microscopic scales. Relatively recent studies showed that vortices could also appear in simple quantum systems. For instance, they were observed experimentally and theoretically as isolated zeros in the differential cross section in atomic ionization processes by the impact of charged particles. In this work, we show that the appearance of these quantum vortices as point structures was not due to any intrinsic property of them, but to the use of restrictive geometries in their visualization. In particular, we show that by studying the fully differential cross section for hydrogen ionization by positron impact, these vortex points are actually a manifestation of a more complex and hitherto unexplored structure, a 3D “vortex surface”.

On the Transition of the Rayleigh-Taylor Instability in 2d Water Waves with Point Vortices

On the Transition of the Rayleigh-Taylor Instability in 2d Water Waves with Point Vortices

Zero-noise dynamics after collapse for three point vortices

The continuation of point vortex dynamics after a vortex collapse is investigated by means of a regularization procedure consisting in introducing a small stochastic diffusive term, that corresponds to a vanishing viscosity. In contrast with deterministic regularization, in which a cutoff interaction selects in the limit a single trajectory of the system after collapse, the zero-noise method produces a probability distribution supported by trajectories satisfying relevant conservation laws of the point vortex system.

Dynamics of several point vortices for the lake equations

The global asymptotic dynamics of point vortices for the lake equations is rigorously derived. Vorticity that is initially sharply concentrated around N N distinct vortex centers is proven to remain concentrated for all times. Specifically, we prove weak concentration of the vorticity and in addition strong concentration in the direction of the steepest ascent of the depth function. As a consequence, we obtain the motion law of point vortices following at leading order the level lines of the depth function. The lack of strong localization in the second direction is linked to the vortex filamentation phenomena. The main result allows for any fixed number of vortices and general assumptions on the concentration property of the initial data to be considered. No further properties such as a specific profile or symmetry of the data are required. Vanishing topographies on the boundary are included in our analysis. Our method is inspired by recent results on the evolution of vortex rings in 3D axisymmetric incompressible fluids.

Gaussian Fluctuations for Interacting Particle Systems with Singular Kernels

We consider the asymptotic behaviour of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein–Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot–Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier–Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein–Uhlenbeck process. The method relies on the martingale approach and the Donsker–Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of the large deviation principle.

Active intensity vortex and stagnation point singularities in a shallow underwater waveguide.

Vector acoustic properties of a narrowband acoustic field are observed as a function of range from a source towed in waters of depth 77 m on the New England Mud Patch. At the source frequency (43 Hz), the waveguide supported three trapped modes, with mode 2 weakly excited owing to the towed source depth. The receiving sensor was positioned 1.45 m above the seafloor with a sampling range aperture of 2500 m. The vector acoustics observations enabled study of vortex regions that encompass two singular points for active acoustic intensity: the vortex point, which is co-located with a dislocation, and stagnation point. Interpretative modeling, based on the normal modes and using a geoacoustic model consistent with those emerging from studies conducted at this location, is in agreement with these measurements. Model-data comparisons were based on the first-order variables of acoustic pressure and velocity along with inverse Hankel transforms, which yield normalized horizontal wavenumber spectra, and second-order variables in the form of horizontal and vertical intensity as well as non-dimensional intensity-based ratios. These measures provide a degree of observational confirmation of some vortex region properties. Both observations and modeling point to a gradual deepening of such regions with increasing range owing to sediment attenuation.

Weak-strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation

A correspondence is established between the dynamics of the two-vortex system and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super- (chiral) and critical phases. As a result, a trivial permutation symmetry of the point vortices induces a weak-strong coupling duality in the NCLP. We show that quantum two-vortex systems with non-zero total vorticity can be generated by applying conformal bridge transformation to a two-dimensional quantum free particle or to a quantum vortex-antivortex system of zero total vorticity. The sub- and super-critical phases of the quantum NCLP are generated in a similar way from the 2D quantum free particle in a commutative or non-commutative plane. The composition of the inverse and direct transformations of the conformal bridge also makes it possible to link the non-chiral and chiral phases in each of these two systems.

Linear Stability of Self-Similar Motions of Three Point Vortices in a Generalized Two-Dimensional Fluid System

Linear Stability of Self-Similar Motions of Three Point Vortices in a Generalized Two-Dimensional Fluid System

Axis-symmetric Onsager clustered states of point vortices in a bounded domain

We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to a two-dimensional disc. Our analysis is based on the mean field of bounded point vortices in the microcanonical ensemble. The clustered vortex states are specified by the inverse temperature β and the rotation frequency ω, which are the conjugate variables of energy E and angular momentum L, respectively. The formation of the axis-symmetric clustered vortex states (azimuthal angle independent) involves the separating of vortices with opposite circulation and the clustering of vortices with the same circulation around the origin and edge. The state preserves SO(2) symmetry while breaking symmetry. We find that, near the uniform state, the rotation-free clustered state (ω = 0) emerges at particular values of L 2/E and β. At large energies, we obtain asymptotically exact vortex density distributions, whose validity condition gives rise to the lower bound of β for the rotation-free states. Noticeably, the obtained vortex density distribution near the edge at large energies provides a novel exact vortex density distribution for the corresponding chiral vortex system.

Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Polynomial sequences related to point vortex equilibria with three strengths

In this paper a family of sequences of polynomials is studied that generalize the Adler–Moser and Loutsenko polynomial sequences related to point vortex equilibria. The polynomials come in pairs with roots that correspond to point vortex configurations in equilibrium, and they have continuous parameters so that the equilibria form positive-dimensional sets in configuration space. The polynomial pairs satisfy a bilinear differential equation and consequently have special properties. The differential recursion relations used to generate the sequences are found by exploiting a symmetry of this equation. Raising and lowering operators are constructed which map between the different levels of the lattice of solutions. In this way the new polynomials are shown to have some of the same special algebraic properties as the Adler–Moser polynomials, such as the Wronskian structure and Stieltjes root sums.

Point vortex dynamics on Kähler twistor spaces

In this paper, we provide tools to study the dynamics of point vortex dynamics on CPn and the flag manifold F1,2(C3). These are the only Kähler twistor spaces arising from four-manifolds. We give an explicit expression for Green’s function on CPn, which enables us to determine the Hamiltonian H and the equations of motions for the point vortex problem on CPn. Moreover, we determine the momentum map μ:F1,2(C3)→su*(3) on the flag manifold.

ACCURACY OF CALCULATIONS OF THE VELOCITY FIELD FROM THE SYSTEM OF VORTEX SEGMENTS USING GRAPHIC PROCESSORS

The paper explored the possibility of using graphics processors to calculate the velocity field from a system of vortex segments in the ideal fluid three-dimensional simulation. This study is important in view of the large number of methods that use discrete vortex structures in modeling of three-dimensional flows. When simulating numerically urban buildings by vortex methods, a huge number of vortex segments are generated. At the same time, the algorithm complexity for calculating such systems is proportional to the square of the number of elements. This requires significant computing resources, especially over a long time or with a large number of fluid-affected objects. Graphics processors have long been actively used to speed up this kind of calculations, in particular, for vortex structures modeling. However, it should be noted that the maximum simulation speed is achieved when using float type variables and this leads to an increase in calculation errors. Therefore, when using the GPU in methods where it is necessary to calculate the velocity field from a set of vortex elements, for example, point vortices in the method of discrete vortices or vortex segments in the method of discrete vortex frames, we must be sure that we do not lose the accuracy of calculations.
 Two systems of vortex segments were considered: segments located along a straight line and segments leaving the plate during flow separation. The main attention is paid to the study of the accuracy of calculations, the formation of data structures and their transfer to the graphics processor. The results indicate that the correct organization of data transfer from the central processor to the graphic processor and vice versa allows to significantly improve the speed of calculations without substantial loss of calculations accuracy.

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Structure dominated two-dimensional turbulence: formation, dynamics and interactions of dipole vortices

Two dimensional turbulence in geophysical fluids and plasma physics tends to be spotty, intermittent and rich in large scale structures such as coherent vortices or zonal flows, due to various mechanisms of self organization. Nonlinear solutions that rely on the vanishing of nonlinearity, especially the dipole vortex solution, stand out as key aspects of this structure dominated turbulence state. Using numerical simulations, it is demonstrated that an initial condition with a small number of high intensity turbulent patches, evolves towards a state dominated by coherent structures, and in particular dipole vortices, as each patch is organized into a finite number of dipole vortices that are ejected from this initially active region. In order to study the details of this process, an initial condition of two Gaussian peaks of the stream function is considered, and it was shown to result in a Chaplygin–Lamb dipole if the peaks have the same amplitude, or a Flierl–Stern–Whitehead dipole that rotates in the direction implied by the excess of vorticity if they do not. Analytical estimates for the velocity, the radius and the radius of curvature of the resulting dipole vortex is given in terms of the peaks and widths of the initial conditions. These are then verified by a detailed comparison of the analytical form of the vorticity of the dipole vortex and its numerical realization. It is argued that since these coherent structures are spared from the strong shear forces normally exerted by the nonlinearities, and can coexist with other localized solutions, or large scale flow patterns, they provide the backbone of the structure dominated or ‘sporadic’ turbulent state in two dimensions, on top of which other structures, waves and instabilities can develop. In order to elucidate these, a number of collision scenarios are considered. It is also shown that a simple two point vortex approximation to a dipole vortex seems to be appropriate for describing their evolution far from each-other, or for computing head on collisions between two or more dipole vortices, but not in the case of close or grazing collisions or their interaction with a nontrivial large scale flow.