Smooth Vorticity Research Articles

Vortex splitting in two-dimensional fluids and non-neutral electron plasmas with smooth vorticity profiles

Initially, elliptical, quasi-two-dimensional (2D) fluid vortices can split into multiple pieces if the aspect ratio is sufficiently large due to the growth and saturation of perturbations known as Love modes on the vortex edge. Presented here are experiments and numerical simulations, showing that the aspect ratio threshold for vortex splitting is significantly higher for vortices with realistic, smooth edges than that predicted by a simple “vortex patch” model, where the vorticity is treated as piecewise constant inside a deformable boundary. The experiments are conducted by exploiting the E × B drift dynamics of collisionless, pure electron plasmas in a Penning–Malmberg trap, which closely model 2D vortex dynamics due to an isomorphism between the Drift–Poisson equations describing the plasmas and the Euler equations describing ideal fluids. The simulations use a particle-in-cell method to model the evolution of a set of point vortices. The aspect ratio splitting threshold ranges up to about twice as large as the vortex patch prediction and depends on the edge vorticity gradient. This is thought to be due to spatial Landau damping, which decreases the vortex aspect ratio over time and, thus, stabilizes the Love modes. Near the threshold, asymmetric splitting events are observed in which one of the split products contains much less circulation than the other. These results are relevant to a wide range of quasi-2D fluid systems, including geophysical fluids, astrophysical disks, and drift-wave eddies in tokamak plasmas.

Precursory patterns to vortex nucleation in stirred Bose-Einstein condensates

Within the Gross-Pitaevskii theory, we study precursory pattern formations to the nucleation of vortex dipoles in a two-dimensional Bose-Einstein condensate stirred by a Gaussian potential. We introduce a smooth superfluid vorticity field and its conservative current, which capture very well the gradual process of vortex nucleation as a mechanism of topological singularities acquiring smooth cores. This is characterized by the localization of the superfluid vorticity into core regions which harbor the phase slips. For more impenetrable obstacles, we find that there are additional phase slips that do not acquire cores, thus remaining pinned as ghost vortices to the potential. We show that the vortex kinematics is slaved to the superfluid vorticity current, which determines not only the onset of nucleation but also the shedding dynamics.

The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x1,x2,x3):0<x3<x2<x1}. In this domain, we prove local well-posedness for Cα vorticities not necessarily vanishing on the boundary with any 0<α<1, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R3 with bounded and piecewise smooth vorticities.

Hamiltonian description of vortex systems

In the framework of 2D ideal Hydrodynamics a vortex system is defined as a smooth vorticity function having few positive local maxima and negative local minima separated by curves of zero vorticity. Invariants of such structures are discussed following from the vorticity conservation law and invertibility of Lagrangian motion. Hamiltonian formalism for vortex systems is developed by introducing new functional variables diagonalizing the original non-canonical Poisson bracket.

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

A steady solution of the incompressible Euler equation on a toroidal surface of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, , where and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c, d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29, 417–440. (doi:10.1017/S0022112067000941)) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498, 381–402. (doi:10.1017/S0022112003007043)) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R/r. A comparison with the Stuart vortex on the flat torus is also made.

Uniformly Rotating Smooth Solutions for the Incompressible 2D Euler Equations

In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.

MC-Smooth: A mass-conserving, smooth vorticity–velocity formulation for multi-dimensional flows

A novel vorticity–velocity formulation of the Navier–Stokes equations – the Mass-Conserving, Smooth (MC-Smooth) vorticity–velocity formulation – is developed in this work. The governing equations of the MC-Smooth formulation include a new second-order Poisson-like elliptic velocity equation, along with the vorticity transport equation, the energy conservation equation, and Nspec species mass balance equations. In this study, the MC-Smooth formulation is compared to two pre-existing vorticity–velocity formulations by applying each formulation to confined and unconfined axisymmetric laminar diffusion flame problems. For both applications, very good to excellent agreement for the simulation results of the three formulations has been obtained. The MC-Smooth formulation requires the least CPU time and can overcome the limitations of the other two pre-existing vorticity–velocity formulations by ensuring mass conservation and solution smoothness over a broader range of flow conditions. In addition to these benefits, other important features of the MC-Smooth formulation include: (1) it does not require the use of a staggered grid, and (2) it does not require excessive grid refinement to ensure mass conservation. The MC-Smooth formulation is a computationally attractive approach that can effectively extend the applicability of the vorticity–velocity formulation.

Recovery of Stokes waves from velocity measurements on an axis of symmetry

We provide a new method to recover the profile of Stokes waves, and more generally of waves with smooth vorticity, from measurements of the horizontal velocity component on a vertical axis of symmetry of the wave surface. Although we consider periodic waves only, the extension to solitary waves is straightforward.

On the viscosity influence on a helical vortex flament evolution

Helical vortices whose parameters have a strong influence on the efficiency of the apparatus is often occur in technical devices using swirling flow (cyclones, separators, etc.). To date the internal structure of such vortices is poorly understood. In [1] a model of helical vortex with uniform vorticity distribution in the core is proposed. Vortices arising in real flow always have a smooth vorticity distribution due to the viscosity action. The problem on steady moving helical vortices with the vortex core of small size in an inviscid fluid was solved in [2]. The non-orthogonal ‘helical’ coordinate system was introduced that allowed author to reduce the problem to two dimensional one. However, the velocity of the vortex motion was written only in the form of a quadratures computation of which is difficult. This paper presents first attempt for research on the diffusion and dynamics of a viscous helical vortex.

Symplectic Structures and Dynamics on Vortex Membranes

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-meancurvature flow on vortex membranes of codimension 2 in any R n , which generalizes to any dimension the classical binormal, or vortex filament, equation in R 3 . This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.

Number of degrees of freedom of two-dimensional turbulence

We derive upper bounds for the number of degrees of freedom of two-dimensional Navier-Stokes turbulence freely decaying from a smooth initial vorticity field omega(x,y,0)=omega0. This number denoted by N is defined as the minimum dimension such that for n>or=N, arbitrary n-dimensional balls in phase space centered on the solution trajectory omega(x,y,t); for t>0, contract under the dynamics of the system linearized about omega(x,y,t). In other words, N is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that N<or=C1Re when the phase space is endowed with the energy norm and N<or=C2Re(1+ln Re)1/3 when the phase space is endowed with the enstrophy norm. Here C1 and C2 are constant and Re is the Reynolds number defined in terms of omega0, the system length scale, and the viscosity nu. The linear (or nearly linear) dependence of N on Re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number. This result is in a sharp contrast to the forced case, for which well-known estimates for the Hausdorff dimension DH of the global attractor scale highly superlinearly with nu(-1). We argue that the "extra" dependence of DH on nu(-1) is not an intrinsic property of the turbulent dynamics. Rather, it is a "removable artifact" brought about by the use of a time-independent forcing as a model for energy and enstrophy injection that drives the turbulence.

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker–Planck equation satisfied by the N -body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1 / N , where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N → + ∞ . For spatially homogeneous systems, we study the two-body correlation function, in a Debye–Hückel approximation valid at the order O ( 1 / N ) . The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye–Hückel model of electrolytes.

Kinetic theory of 2D point vortices from a BBGKY-like hierarchy

Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solutions in powers of 1 / N (where N is the number of vortices) in a proper thermodynamic limit N → + ∞ , and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order O ( 1 / N ) . This equation was obtained previously [P.H. Chavanis, Phys. Rev. E 64 (2001) 026309] from a more abstract projection operator formalism. If we consider axisymmetric flows and make a Markovian approximation, we obtain a simpler kinetic equation which can be studied in great detail. We discuss the properties of these kinetic equations in regard to the H -theorem and the convergence (or not) towards the statistical equilibrium state. We also study the growth of correlations by explicitly calculating the time evolution of the two-body correlation function in the linear regime. In a second part of the paper, we consider the relaxation of a test vortex in a bath of field vortices and obtain the Fokker–Planck equation by directly calculating the second (diffusion) and first (drift) moments of the increment of position of the test vortex. A specificity of our approach is to obtain general equations, with a clear physical meaning, that are valid for flows that are not necessarily axisymmetric and that take into account non-Markovian effects. A limitation of our approach, however, is that it ignores collective effects.

Interaction of two equal vortices on a β plane

The interaction of two equal vortices under the influence of a gradient of background vorticity (β) is studied numerically and experimentally. If the initial shape and vorticity distribution of the vortices is fixed, two parameters determine the evolution: the normalized intercentroid distance d*=d/R, where R is the radius of the vortex; and the normalized gradient of background vorticity β*=βR/ω, where ω is the peak vorticity of the vortex. Alternate ways of identifying regimes of behavior in the parameter plane (d*,β*) are presented. These are applied to numerical simulations of interaction of vortices with steplike, steep and smooth vorticity profiles. It is found that the critical distance for merger decreases with increasing β* for all vortex types, and that vortices with smooth vorticity profile are the most merger-prone vortices. Laboratory experiments were done in a rotating water tank with a flat sloping bottom providing the β effect. The vortices produced have a smooth vorticity profile and show the same behavior observed in the simulations, except that, as a result of viscous effects, the critical merger distance is shifted towards larger values of d*.

Using a partial-wave method for sound-mean-flow scattering problems.

We present a semianalytical method, based on a partial-wave expansion and valid in the short wavelength limit for small Mach number flows, to analyze sound-vortical-flow interactions. It is more powerful than ray-tracing methods because it gives both amplitude and phase of the sound wave, and because it is less restrictive on the smallness of the wavelength. In contrast with the Born approximation approach, this method allows the computation of the sound field in the whole interaction domain (including the near field), and preserves energy conservation. Vortical flows with finite circulation are amenable to our analysis, which gives a satisfactory description of wave front dislocation by vorticity, in good agreement with direct numerical simulations. We extend previous versions of this method to the case of smooth vorticity profiles which are observed in aeroacoustics experiments.

String limit of vortex current algebra

The Poisson structure generating the Hamiltonian dynamics of string vortices is reconstructed within the current algebra picture as a limiting case of the standard brackets associated to fluids with a smooth vorticity field. The approach implemented bypasses the use of Dirac's procedure. The fine structure of the dynamical algebra is derived for planar fluids by implementing an appropriate spatial fragmentation of the vorticity field, and the limit to the point vortex gas is effected. The physical interpretation of the resulting local currents is provided. Nontrivial differences characterizing the canonical quantization of point vortices and the current algebra quantization are also illustrated.

Finite-time singularity in the vortex dynamics of a string

We analyze the dynamics of a perfectly flexible string with a constant length and a vanishing inner friction. The local angular velocity of line elements in this seemingly simple mechanical system is shown to have many mathematical and physical properties in common with vorticity in the three-dimensional incompressible Euler equation. It is demonstrated that initially smooth vorticity fields lose their regularity within finite time in a self-similar process, and that the peak vorticity grows as ${\ensuremath{\omega}}_{\mathrm{max}}\ensuremath{\sim}(T\ensuremath{-}{t)}^{\ensuremath{-}1}.$

Remarks on quantum vortex theory on Riemann surfaces

Quantized point vortex theories on a compact Riemann surface of arbitrary genus (in the zero total vorticity case) are investigated. By taking meromorphic functions thereon as order parameters and resorting to the Weil-Kostant, Abel, Riemann and Riemann-Roch theorems, a natural phase space and Hamiltonian for the vortex-antivortex configurations is exhibited, leading to explicit vortex-antivortex coherent states wave functions via geometric quantization. Furthermore, a relationship between point and smooth vorticities is established by means of Green functions associated to divisors on a Riemann surface and Poincaré duality, thereby yielding a natural regularization of the singular theory.

Long-wave instability of a vortex ring

The instability of barrel-shaped vibrations of a vortex ring in an ideal fluid is investigated. These vibrations, stable for a vortex ring with a piecewise-uniform vorticity profile, appear to be unstable for a vortex ring with a smooth vorticity profile. The instability growth rate is found on the basis of the energy balance equation determining the energy transport from perturbations with negative energy in the critical layer to perturbations with positive energy in the rest of the flow. The curvature of the vortex ring, by virtue of which the perturbations with energies of different signs appear to be connected, plays a prominent role in the mechanism under consideration.

Fast Triangulated Vortex Methods for the 2D Euler Equations

Vortex methods for inviscid incompressible two-dimensional fluid flow are usually based on blob approximations. This paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a Delaunay triangulation of the vortices. An efficient reconstruction of the Delaunay triangulation at each step makes the method accurate for long times. The vertices of the triangulation move with the fluid velocity, which is reconstructed from the vorticity via a simplified fast multipole method for the Biot-Savart law with a continuous source distribution. The initial distribution of vortices is constructed from the initial vorticity field by an adaptive approximation method which produces good accuracy even for discontinuous initial data. Numerical results show that the method is highly accurate over long time intervals. Experiments with single and multiple circular and elliptical rotating patches of both piecewise constant and smooth vorticity indicate that the method produces much smaller errors than blob methods with the same number of degrees of freedom, at little additional cost. Generalizations to domains with boundaries, viscous flow, and three space dimensions are discussed.