The evolution of swirling axisymmetric vortex rings

The formation time scale of axisymmetric vortex rings is studied numerically for relatively long discharge times. Experimental findings on the existence and universality of a formation time scale, referred to as the ‘formation number’, are confirmed. The formation number is indicative of the time at which a vortex ring acquires its maximal circulation. For vortex rings generated by impulsive motion of a piston, the formation number was found to be approximately four, in very good agreement with experimental results. Numerical extensions of the experimental study to other cases, including cases with thick shear layers, show that the scaled circulation of the pinched-off vortex is relatively insensitive to the details of the formation process, such as the velocity programme, velocity profile, vortex generator geometry and the Reynolds number. This finding might also indicate that the properly scaled circulation of steady vortex rings varies very little. The formation number does depend on the velocity profile. Non-impulsive velocity programmes slightly increase the formation number, while non-uniform velocity profiles may decrease it significantly. In the case of a parabolic velocity profile of the discharged flow, for example, the formation number decreases by a factor as large as four. These findings indicate that a major source of the experimentally found small variations in the formation number is the different evolution of the velocity profile of the discharged flow.

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A family of steady vortex rings

Journal of Fluid Mechanics ◽

10.1017/s0022112073001266 ◽

1973 ◽

Vol 57 (3) ◽

pp. 417-431 ◽

Cited By ~ 179

Author(s):

J. Norbury

Keyword(s):

Theoretical Work ◽

Vortex Rings ◽

Small Cross Section ◽

The Core ◽

Axisymmetric Vortex ◽

Recent Theoretical Work ◽

The Family ◽

Core Boundary ◽

Spherical Vortex ◽

Axis Of Symmetry

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.

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Advective balance in pipe-formed vortex rings

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.814 ◽

2017 ◽

Vol 836 ◽

pp. 773-796

Author(s):

Karim Shariff ◽

Paul S. Krueger

Keyword(s):

Reynolds Numbers ◽

Vortex Rings ◽

Quadratic Term ◽

Axisymmetric Vortex ◽

Order Of Magnitude ◽

Cylindrical Diffusion ◽

Type Boundary ◽

The One ◽

Type Boundary Condition ◽

Edge Layer

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

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Steady axisymmetric vortex flows with swirl and shear

Journal of Fluid Mechanics ◽

10.1017/s002211200800342x ◽

2008 ◽

Vol 613 ◽

pp. 395-410 ◽

Cited By ~ 5

Author(s):

ALAN R. ELCRAT ◽

BENGT FORNBERG ◽

KENNETH G. MILLER

Keyword(s):

Multiple Solutions ◽

Iterative Procedure ◽

General Procedure ◽

Inviscid Flow ◽

Initial Guess ◽

Vortex Rings ◽

Vortex Flows ◽

Axisymmetric Vortex

A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moffatt, tubes of vorticity extending to infinity and Beltrami flows. When there is a spherical obstacle we have found multiple solutions for each set of parameters. Flows are found by numerically solving the Bragg–Hawthorne equation using a non-Newton-based iterative procedure which is robust in its dependence on an initial guess.

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NUMERICAL STUDY OF SOUND RADIATION BY AXISYMMETRIC VORTEX RINGS

Journal of Computational Acoustics ◽

10.1142/s0218396x03001808 ◽

2003 ◽

Vol 11 (01) ◽

pp. 11-45 ◽

Cited By ~ 1

Author(s):

ELIE RIVOALEN ◽

SERGE HUBERSON ◽

OMAR M. KNIO

Keyword(s):

Sound Production ◽

Numerical Study ◽

Sound Radiation ◽

Particle Method ◽

Turnover Time ◽

Vortex Rings ◽

Vortex Particle Method ◽

Axisymmetric Vortex ◽

Dominant Period ◽

Radiated Sound

The sound production by vortex rings is investigated by means of an axisymmetric vortex particle method. The predictions are first calibrated by analyzing the noise generated by steady vortex rings that are described by the analytical solutions of Fraenkel and Norbury. The noise produced by isolated vortex rings for both nominally steady and unsteady cores is then analyzed. For nominally steady cores, computed results indicate that the efficiency of sound radiation decreases as the slenderness parameter is reduced, and the acoustic signals reveal a dominant period that is approximately half the eddy turnover time. For unsteady cores, the amplitude of the radiated sound is substantially higher than that of similar steady rings. When the initial core vorticity distribution is nonuniform, complex internal motion may also occur within the core which is also reflected in the corresponding far-field acoustic signal. Finally, the effect of vortex stretching is analyzed based on computations of two coaxial corotating vortex rings.

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Statistical mechanics of axisymmetric vortex rings

Physical Review E ◽

10.1103/physreve.65.026402 ◽

2002 ◽

Vol 65 (2) ◽

Author(s):

R. Ganesh ◽

K. Avinash

Keyword(s):

Statistical Mechanics ◽

Vortex Rings ◽

Axisymmetric Vortex

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Nested contour dynamics models for axisymmetric vortex rings and vortex wakes

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.199 ◽

2014 ◽

Vol 748 ◽

pp. 521-548 ◽

Cited By ~ 3

Author(s):

Clara O’Farrell ◽

John O. Dabiri

Keyword(s):

Vortex Formation ◽

Computational Cost ◽

Small Region ◽

Two Dimensions ◽

Vortex Rings ◽

Contour Dynamics ◽

Piston Cylinder ◽

Axisymmetric Vortex ◽

Contour Models ◽

Perturbation Size

AbstractInviscid models for vortex rings and dipoles are constructed using nested patches of vorticity. These models constitute more realistic approximations to experimental vortex rings and dipoles than the single-contour models of Norbury and Pierrehumbert, and nested contour dynamics algorithms allow their simulation with low computational cost. In two dimensions, nested-contour models for the analytical Lamb dipole are constructed. In the axisymmetric case, a family of models for vortex rings generated by a piston–cylinder apparatus at different stroke ratios is constructed from experimental data. The perturbation response of this family is considered by the introduction of a small region of vorticity at the rear of the vortex, which mimics the addition of circulation to a growing vortex ring by a feeding shear layer. Model vortex rings are found to either accept the additional circulation or shed vorticity into a tail, depending on the perturbation size. A change in the behaviour of the model vortex rings is identified at a stroke ratio of three, when it is found that the maximum relative perturbation size vortex rings can accept becomes approximately constant. We hypothesise that this change in response is related to pinch-off, and that pinch-off might be understood and predicted based on the perturbation responses of model vortex rings. In particular, we suggest that a perturbation response-based framework can be useful in understanding vortex formation in biological flows.

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