Point Vortex Model Research Articles

Vortex Characteristics of Heliostats’ Surface Wind Pressure under Resting Condition

Flat rectangular heliostats’ surface wind pressure distribution under resting condition was analyzed via wind tunnel experiment results. The flow field characteristics of the mirror in parallel with the ground were combined to reveal the distribution cause of fluctuating pressure. Phenomena of columnar vortex and conical vortex formed from mirror’s flow field under wind angles 0° and 45° were aimed to explain their internal structure using theory of point vortex model. With power spectrum of peculiar points’ fluctuating wind pressure analyzed, the surface vortex characteristics were further elaborated. The above work revealed heliostats’ surface wind pressure characteristics under resting conditions, and established the theoretical basis of its design study.

Interaction of two equal co-rotating viscous vortices in the presence of background shear

The interactions of two equal co-rotating vortices under the influence of both viscosity and uniform background shear are investigated using two-dimensional numerical simulations. A range of values of the shear strength parameter, , and initial aspect ratio, , are considered for two values of circulation Reynolds number . The primary effect of viscosity is to increase the core size a(t) in time while the primary effect of shear is to vary the separation distance b(t) in time. For sufficiently separated vortices, the motion of the vortices is well-described by a point vortex model with linear shear. The present simulations show that for a viscous symmetric vortex pair there are two distinct flow regimes, merger and separation, with the boundaries separating these regimes well-predicted by the point vortex model and largely independent of over the range tested. Results also indicate that the onset of merging occurs when attains the critical value found for vortex pairs without shear.

Onsager-Kraichnan condensation in decaying two-dimensional quantum turbulence.

Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling MonteCarlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Damped GPE simulations reveal that such OKC states can emerge dynamically, via aggregation of small-scale clusters into giant OKC clusters, as the end states of decaying 2D quantum turbulence in a compressible, finite-temperature superfluid. These statistical equilibrium states should be accessible in atomic Bose-Einstein condensate experiments.

Generalized vortex model for the inverse cascade of two-dimensional turbulence

We generalize Kirchhoff's point vortex model of two-dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with like-signed circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynamical systems theory.

The vortex sheet model for a turbulent mixing layer

The primary aim of this work is to study a temporal mixing layer using the model of instability-induced roll-up of a slightly perturbed vortex sheet in an inviscid and an incompressible fluid. A point vortex model evolves into a chaotic cloud of point vortices instead of smooth double-branched spirals. The vortex sheet model is derived from basic equations of vortex dynamics. The present model uses linear segments to interpolate the sheet. Computer simulation of the vortex sheet model is complicated as compared to point vortices. However, closed-form equations of motion do exist for vortex sheets. A prominent vortex core flanged by regular spiral windings and an irrotational fluid in between the layers is obtained. The sheet develops a highly peaked distribution of circulation density and curvature at a finite time. The problem of finite time singularity can be handled by a technique that invokes longitudinal circulation density diffusion along the sheet at singular points or by restriction of the smallest scales of motion. The accuracy of such simulations can be independently verified by using the laws of vortex dynamics and conserved physical quantities. We observe the growth of the two-dimensional shear layer with time and the merger of vortex-like structures. The dependence of the mixing layer on the initial conditions is studied and the mixing layer properties thus obtained are examined. It remains to be seen whether the vortex sheet model yields a truly turbulent mixing layer.

Point vortex model for prediction of sound generated by a wing with flap interacting with a passing vortex

Acoustic signature of a rigid wing, equipped with a movable downstream flap and interacting with a line vortex, is studied in a two-dimensional low-Mach number flow. The flap is attached to the airfoil via a torsion spring, and the coupled fluid-structure interaction problem is analyzed using thin-airfoil methodology and application of the emended Brown and Michael equation. It is found that incident vortex passage above the airfoil excites flap motion at the system natural frequency, amplified above all other frequencies contained in the forcing vortex. Far-field radiation is analyzed using Powell-Howe analogy, yielding the leading order dipole-type signature of the system. It is shown that direct flap motion has a negligible effect on total sound radiation. The characteristic acoustic signature of the system is dominated by vortex sound, consisting of relatively strong leading and trailing edge interactions of the airfoil with the incident vortex, together with late-time wake sound resulting from induced flap motion. In comparison with the counterpart rigid (non-flapped) configuration, it is found that the flap may act as sound amplifier or absorber, depending on the value of flap-fluid natural frequency. The study complements existing analyses examining sound radiation in static- and detached-flap configurations.

Collective dynamics of vortices in trapped Bose-Einstein condensates

We have calculated collective mode spectra for three-dimensional, rotating Bose-Einstein condensates in oblate harmonic traps using the microscopic Bogoliubov-deGennes field theory. For condensates with $N_v$ vortices, $N_v$ Kelvin-Tkachenko mode branches are obtained. The features of these modes are compared with those predicted by a classical point vortex model. We have created movies to visualize the motion of the vortices corresponding to the Kelvin-Tkachenko waves.

Low-order phenomenological modeling of leading-edge vortex formation

A low-order point vortex model for the two-dimensional unsteady aerodynamics of a flat plate wing section is developed. A vortex is released from both the trailing and leading edges of the flat plate, and the strength of each is determined by enforcing the Kutta condition at the edges. The strength of a vortex is frozen when it reaches an extremum, and a new vortex is released from the corresponding edge. The motion of variable-strength vortices is computed in one of two ways. In the first approach, the Brown–Michael equation is used in order to ensure that no spurious force is generated by the branch cut associated with each vortex. In the second approach, we propose a new evolution equation for a vortex by equating the rate of change of its impulse with that of an equivalent surrogate vortex with identical properties but constant strength. This impulse matching approach leads to a model that admits more general criteria for shedding, since the variable-strength vortex can be exchanged for its constant-strength surrogate at any instant. We show that the results of the new model, when applied to a pitching or perching plate, agree better with experiments and high-fidelity simulations than the Brown–Michael model, using fewer than ten degrees of freedom. We also assess the model performance on the impulsive start of a flat plate at various angles of attack. Current limitations of the model and extensions to more general unsteady aerodynamic problems are discussed.

Trapped instability and vortex formation by an unstable coastal current

This paper addresses the instability of a two-layer coastal current in a quasigeostrophic model; the potential vorticity (PV) structure of this current consists in two uniform cores, located at different depths, with finite width and horizontally shifted. This shift allows both barotropic and baroclinic instability for this current. The PV cores can be like-signed or opposite-signed, leading to their vertical alignment or to their hetonic coupling. These two aspects are novel compared to previous studies. For narrow vorticity cores, short waves dominate, associated with barotropic instability; for wider cores, longer waves are more unstable and are associated with baroclinic processes. Numerical experiments were performed on the f-plane with a finite-difference model. When both cores have like-signed PV, trapped instability develops during the nonlinear evolution: vertical alignment of the structures is observed. For narrow cores, short wave breaking occurs close to the coast; for wider cores, substantial turbulence results from the entrainment of ambient fluid into the coastal jet. When the two cores have opposite-signed PV, the nonlinear regimes range from short wave breaking to the ejection of dipoles or tripoles, via a regime of dipole oscillation near the wall. The Fourier analysis of the perturbed flow is appropriate to distinguish the regimes of short wave breaking, of dipole formation, and of turbulence, but not the differences between regimes involving only vortex pairs. To explain more precisely the regimes where two vortices (and their wall images) interact, a point vortex model is appropriate.

Inviscid and viscous vortex models for Richtmyer–Meshkov instability

We present simple inviscid and viscous vortex models for the Richtmyer–Meshkov (RM) instability, in the limit of fluids of the same densities. The evolution of the interface is driven by a row of alternating point vortices. A regularization method for the singular vortices is applied to the inviscid model for stable calculations. The regularized vortex model provides fine resolutions for the interface, improving previous results on the evolution of the RM instability. The point vortex model, for the inviscid fluid, is validated by comparing with the results of the full vortex sheet model. The viscous vortex model for the unstable interface is based on the multi-Lamb–Oseen vortices. We show that the viscous vortex model gives the main qualitative features of the evolution of the viscous RM instability.

Interactions of surface and deep anticyclonic eddies in the Bay of Biscay

In 1990, satellite observations revealed that an anticyclonic surface eddy (a SWODDY, for Slope Water Oceanic eDDY) followed a cycloidal trajectory north of the Iberian coast in the Bay of Biscay. To understand the mechanisms underlying such a trajectory, we study the evolution of an idealized surface eddy in a two-layer flat-bottom quasi-geostrophic model. The effect of several processes is studied, notably the presence of deep anticyclonic vorticity. This deep vorticity may result either from the tilting of the swoddy itself, or from the presence of an anticyclonic eddy of different origins, such as a meddy (Mediterranean Water EDDY). We also study the influence of a zonal coast south of the swoddy, via the “mirror effect”.Firstly, a point-vortex model is used on the f-plane. When the surface and deep vortices lie much farther away from the coast than from each other, their motion is close to the addition of a mutually induced rotation and of a quasi-uniform zonal drift induced by the mirror vortices. The sensitivity of the rotation and translation characteristics to vortex position, strength, thickness and to a surrounding flow, are investigated. Such a surface-deep vortex interaction can reasonably well represent the observed motions of the swoddy, if the deep vortex is far enough from the surface one, and if they have comparable strengths.Then, a numerical code of the two-layer quasi-geostrophic equations is used to model finite-area vortices, again on the f-plane. Vertical alignment of the surface and deep vortices or vortex pairing with the mirror image, are not observed. Vortex splitting due to mutual shearing effects occurs only for vortices with very different strengths. Vortex trajectories similar to the ones observed are reproduced by the model for equal strength vortices.Complementary data, from the CONGAS experiments in 2004–2007, and from the ARGO profiling float database, are used to show that meddies (or at least coherent salinity anomalies at 1000m depth) can originate from the continental slope near 45°N, 8°W and move regularly northeastward toward the region of swoddy generation and drift. These data also show that meddy–swoddy (“deep vortex–surface vortex”) encounters may occur in the southeastern Bay of Biscay.

Simple kinematic models for the environmental interaction of tropical cyclones in vertical wind shear

Abstract. A major impediment to the intensity forecast of tropical cyclones (TCs) is believed to be associated with the interaction of TCs with dry environmental air. However, the conditions under which pronounced TC-environment interaction takes place are not well understood. As a step towards improving our understanding of this problem, we analyze here the flow topology of a TC immersed in an environment of vertical wind shear in an idealized, three-dimensional, convection-permitting numerical experiment. A set of distinct streamlines, the so-called manifolds, can be identified under the assumptions of steady and layer-wise horizontal flow. The manifolds are shown to divide the flow around the TC into distinct regions. The manifold structure in our numerical experiment is more complex than the well-known manifold structure of a non-divergent point vortex in uniform background flow. In particular, one manifold spirals inwards and ends in a limit cycle, a meso-scale dividing streamline encompassing the eyewall above the layer of strong inflow associated with surface friction and below the outflow layer in the upper troposphere. From the perspective of a steady and layer-wise horizontal flow model, the eyewall is well protected from the intrusion of environmental air. In order for the environmental air to intrude into the inner-core convection, time-dependent and/or vertical motions, which are prevalent in the TC inner-core, are necessary. Air with the highest values of moist-entropy resides within the limit cycle. This "moist envelope" is distorted considerably by the imposed vertical wind shear, and the shape of the moist envelope is closely related to the shape of the limit cycle. In a first approximation, the distribution of high- and low-θe air around the TC at low to mid-levels is governed by the stirring of convectively modified air by the steady, horizontal flow. Motivated by the results from the idealized numerical experiment, an analogue model based on a weakly divergent point vortex in background flow is formulated. The simple kinematic model captures the essence of many salient features of the manifold structure in the numerical experiment. A regime diagram representing realistic values of TC intensity and vertical wind shear can be constructed for the point-vortex model. The results indicate distinct scenarios of environmental interaction depending on the ratio of storm intensity and vertical-shear magnitude. Further implications of the new results derived from the manifold analysis for TCs in the real atmosphere are discussed.

Wake structure of a deformable Joukowski airfoil

We examine the vortical wake structure shed from a deformable Joukowski airfoil in an unbounded volume of inviscid and incompressible fluid. The deformable airfoil is considered to model a flapping fish. The vortex shedding is accounted for using an unsteady point vortex model commonly referred to as the Brown–Michael model. The airfoil’s deformations and rotations are prescribed in terms of a Jacobi elliptic function which exhibits, depending on a dimensionless parameter m , a range of periodic behaviors from sinusoidal to a more impulsive type flapping. Depending on the parameter m and the Strouhal number, one can identify five distinct wake structures, ranging from arrays of isolated point vortices to vortex dipoles and tripoles shed into the wake with every half-cycle of the airfoil flapping motion. We describe these regimes in the context of other published works which categorize wake topologies, and speculate on the importance of these wake structures in terms of periodic swimming and transient maneuvers of fish.

Stabilization of vortices in the wake of a circular cylinder using harmonic forcing

We explore whether vortex flows in the wake of a fixed circular cylinder can be stabilized using harmonic forcing. We use Föppl's point vortex model augmented with a harmonic point source-sink mechanism which preserves conservation of mass and leaves the system Hamiltonian. We discover a region of Lyapunov-stable vortex motion for an appropriate selection of parameters. We identify four unique parameters that affect the stability of the vortices: the uniform flow velocity, vortex equilibrium positions, forcing amplitude, and forcing frequency. We assess the robustness of the controller using a Poincaré section.

Robust numerical method for integration of point-vortex trajectories in two dimensions

The venerable two-dimensional (2D) point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ordinary differential equations which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between a pair of vortices relative to all other intervortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remains constant to high accuracy.

Vortex-induced chaotic mixing in wavy channels

Mixing is studied in open-flow channels with conformally mapped wavy-wall profiles, using a point-vortex model in two-dimensional irrotational, incompressible mean flow. Unsteady dynamics of the separation bubble induced by oscillatory motion of point vortices located in the trough region produces chaotic mixing in the Lagrangian sense. Significant mass exchange between passive tracer particles inside and outside of the separation bubble forms an efficient mixing region which evolves in size as the vortex moves in the unsteady potential flow. The dynamics closely resembles that obtained by previous authors from numerical solutions of the unsteady Navier–Stokes equations for oscillatory unidirectional flow in a wavy channel. Of the wavy channels considered, the skew-symmetric form is most efficient at promoting passive mixing. Diffusion via gridless random walks increases lateral particle dispersion significantly at the expense of longitudinal particle dispersion due to the opposing effect of mass exchange at the front and rear of the particle ensemble. Active mixing in the wavy channel reveals that the fractal nature of the unstable manifold plays a crucial role in singular enhancement of productivity. Hyperbolic dynamics dominate over non-hyperbolicity which is restricted to the vortex core region. The model is simple yet qualitatively accurate, making it a potential candidate for the study of a wide range of vortex-induced transport and mixing problems.

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

We generalize the concepts of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Mobius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.

Dynamics of quantum vortices in a toroidal trap

The dynamics of quantum vortices in a two-dimensional annular condensate held under a toroidal trapping potential composed of a harmonic term plus Gaussian term are considered by numerically simulating the Gross-Pitaevskii equation. Families of solitary wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometry the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previously. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with different velocities. The stability of the solitary wave sequences for the annular condensate without a persistent flow is also investigated by numerically evolving the solitary wave solutions in time. In addition, the interaction of vortex-vortex pairs and vortex-antivortex pairs is considered and it is demonstrated that the collisions are either elastic or inelastic depending on the magnitude of the angular velocity. The similarities and differences between numerically simulating the Gross-Pitaevskii equation and using a point vortex model for these collisions are elucidated.

Point-vortex model for Lagrangian intermittency in turbulence

Experiments have shown that Lagrangian statistics in turbulent flows display Gaussianly distributed velocity values and non-Gaussianly distributed velocity differences or accelerations. Coherent flow structures in the form of vortices have often been proposed to play an important role in this behavior. Here we examine the origin of these statistics using both continuously stirred n -body point-vortex simulations and analytic random variable transformation in a simplified model of randomly distributed vortices. We conclude that Lagrangian velocity distributions can be understood in terms of dominant nearest vortex neighbor contributions. Accelerations likewise reflect vortical contributions, but at smallest temporal increment are dominated, not by the motion of the Lagrangian tracers, but by vorticity reconfiguration within the domain.

Simultaneous particle-image velocimetry–planar laser-induced fluorescence measurements of Richtmyer–Meshkov instability growth in a gas curtain with and without reshock

The structure of the concentration and velocity fields in a light-heavy-light fluid layer subjected to an impulsive acceleration by a shock wave (Richtmyer–Meshkov instability) is studied using simultaneous particle-image velocimetry and planar laser-induced fluorescence (PLIF) measurements (performed in such flows for the first time). The initial condition prior to shock impact is accurately characterized using calibrated PLIF measurements to enable comparisons of the evolving structure to numerical simulations. Experiments performed on a SF6 curtain in air (Atwood number, At=0.67), after single shock by a Mach 1.2 shock wave and reshock by the reflected wave, show that the reshock wave has a dramatic impact on the evolution of the unstable structure. After first shock and in the absence of reshock(s), the structure widths agree well with an analytical extension to the nonlinear point vortex model [J. W. Jacobs et al., “Nonlinear growth of the shock-accelerated instability of a thin fluid layer,” J. Fluid Mech. 295, 23 (1995)] that accounts for the nonuniform spacing of the row of counter-rotating vortices that drive the flow. However, upon reshock, the width deviates significantly from the singly shocked case, and a substantial rise in the growth rate is observed. Enhanced mixing, destruction of the ordered velocity field, and an increase in both the positive and negative circulations ensue. Large velocity fluctuations relative to the mean flow, and the advection of a wide spectrum of vortex scales combine to mix the flow well and create turbulent conditions in the reshocked structure.