Steady Vortex Rings Research Articles

A spectral method for unbounded flow in a cylindrical coordinate system

Fourier expansions in the radial direction for unbounded flows expressed in a cylindrical coordinate system are proposed. By appropriate coordinate mapping and periodic extension in the r direction, periodic boundary conditions required by Fourier expansions and infinite differentiability demanded by spectral convergence are established. Appropriate zero factors for the general Fourier expansions are given at the axis and at infinity in order to remove the numerical singularity at r=0 and to satisfy all the boundary conditions. The effectiveness of these expansions are demonstrated by the simulation of steady axisymmetrical vortex rings in ideal fluid and the numerical simulation of the head-on collision of two coaxial, equal and opposite vortex rings.

On the bifurcation of steady vortex rings from a Green function

This paper is a supplement to [2]. There, solutions ψ., (μ) of certain free-boundary problems, depending on a small parameter μ, are related to the Green function G., (a(μ)) of a second-order, elliptic, partial differential equation defined on a bounded set Ω ⊂ ℝ2. The free boundary is the boundary of a set A(μ) ⊂ Ω that is unknown a priori and corresponds to the cross-section of a steady vortex ring or of a confined plasma in equilibrium, for example.

Existence of steady vortex rings in an ideal fluid

Abstract We prove the existence of global steady vortex rings in an ideal fluid with given propagation speed W > 0 , flux constant k ⩾ 0 and any bounded, positive, nondecreasing vorticity function.

Existence of steady vortex rings in an ideal fluid

We prove the existence of global steady vortex rings in an ideal fluid with given propagation speed W> 0, flux constant k⩾ 0 and any bounded, positive, nondecreasing vorticity function.

The uniqueness of a family of steady vortex rings

The uniqueness of a family of steady vortex rings

A global branch of steady vortex rings.

Abstract : A steady vortex ring of prescribed strength and propagation speed can be described in terms of a Strokes stream function psi. A flux constant k measures the flow through the center of the axisymmetric vortex ring. For k = 0, Hill in 1984 found an explicit solution for the semi-linear elliptic equation satisfied by psi. In this paper it is shown that there is an unbounded, closed, connected branch of solutions emanating from Hill's vortex in the space of pairs (k, psi). (Author)

The nonlinear instability of Hill's vortex

The nonlinear instability of Hill's spherical vortex, subject to axisymmetric perturbations is considered. The problem is formulated as a nonlinear integrodifferential equation for the motion of the vortex boundary. This equation is solved employing a numerical procedure which involves a piecewise representation of the vortex contour with discrete elements. This formulation offers an efficient method for studying a variety of vortex flows in axisymmetric geometry.Our results indicate that if Hill's vortex becomes a prolate spheroid, a certain amount of rotational fluid is detrained from the rear stagnation point of the vortex, leaving behind a reduced vortex of approximately spherical shape. The amount of detrained fluid is a function of the initial deformation. If the vortex becomes an oblate spheroid, irrotational fluid is entrained into the vortex from the rear stagnation point, reaches the front vortex boundary, and circulates along the vortex boundary in a spiral pattern. In this fashion, the vortex reduces to a nearly steady vortex ring whose asymptotic structure is a function of the initial deformation. The structure of the asymptotic rings arising from oblate vortices is similar to that of steady rings described by Norbury (1973). The vortex speed is shown to tend to a constant value for prolate perturbations, and to fluctuate around a mean value for oblate perturbations.

The uniqueness of Hill's spherical vortex

The mathematical description of steady vortex rings, in an ideal fluid occupying the whole space R3, can be approached in various ways. The physical basis of the problem, its history up to 1973, and several formulations are outlined in [12], pp. 14–21. Another, quite different formulation and the plan for a corresponding existence theory are presented in [6]. Further existence theorems, variational principles and results are to be found in [3], [7], [11], [13] and [20]. Here we state only definitions and equations that seem relevant to our immediate purpose.

Vortex Rings: Existence and Asymptotic Estimates

The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter $\lambda$; as $\lambda \to \infty$ the vortex ring tends to a torus whose cross-section is an infinitesimal disc.

Vortex rings: existence and asymptotic estimates

The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter $\lambda$; as $\lambda \to \infty$ the vortex ring tends to a torus whose cross-section is an infinitesimal disc.

Nonlinear desingularization in certain free-boundary problems

We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown setA that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter ofA tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call “nonlinear desingularization”, is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem.

A global theory of steady vortex rings in an ideal fluid

A global theory of steady vortex rings in an ideal fluid

A family of steady vortex rings

Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.

A global theory of steady vortex rings in an ideal fluid

A global theory of steady vortex rings in an ideal fluid

A steady vortex ring close to Hill's spherical vortex

AbstractThe existence of a steady vortex ring close to Hill's spherical vortex is established, and an approximate description of its boundary is given. The vorticity in the ring is proportional to the distance from the axis of symmetry. The core propagates steadily in an unbounded fluid at rest at infinity. The boundary of the vortex ring is close to an interior stream surface of Hill's vortex.

Examples of steady vortex rings of small cross-section in an ideal fluid

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r(of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/ris considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.

On steady vortex rings of small cross-section in an ideal fluid

This paper is concerned with vortex rings, in an unbounded inviscid fluid of uniform density, that move without change of form and with constant velocity when the fluid at infinity is at rest. The work is restricted to rings whose cross-sectional area is small relative to the square of a mean ring radius. An existence theorem is proved for distributions of vorticity in the core that are arbitrary, apart from the condition imposed by the equation of motion and certain smoothness requirements. The method of proof relies on the nearly plane, or two-dimensional, nature of the flow in the neighbourhood of a small cross-section, and leads to approximate but explicit formulae for the propagation speed and shape of the vortex rings in question.