Vortex ring

A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid or gas; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant flow in a vortex ring is said to be toroidal, more precisely poloidal.

Spark photography image of a vortex ring in flight.

Vortex rings are plentiful in turbulent flows of liquids and gases, but are rarely noticed unless the motion of the fluid is revealed by suspended particles—as in the smoke rings which are often produced intentionally or accidentally by smokers. Fiery vortex rings are also a commonly produced trick by fire eaters. Visible vortex rings can also be formed by the firing of certain artillery, in mushroom clouds, and in microbursts.[1][2]

A vortex ring usually tends to move in a direction that is perpendicular to the plane of the ring and such that the inner edge of the ring moves faster forward than the outer edge. Within a stationary body of fluid, a vortex ring can travel for relatively long distance, carrying the spinning fluid with it.

Structure
Flow around an idealized vortex ring

In a typical vortex ring, the fluid particles move in roughly circular paths around an imaginary circle (the core) that is perpendicular to those paths. As in any vortex, the velocity of the fluid is roughly constant except near the core, so that the angular velocity increases towards the core, and most of the vorticity (and hence most of the energy dissipation) is concentrated near it.

Unlike a sea wave, whose motion is only apparent, a moving vortex ring actually carries the spinning fluid along. Just as a rotating wheel lessens friction between a car and the ground, the poloidal flow of the vortex lessens the friction between the core and the surrounding stationary fluid, allowing it to travel a long distance with relatively little loss of mass and kinetic energy, and little change in size or shape. Thus, a vortex ring can carry mass much further and with less dispersion than a jet of fluid. That explains, for instance, why a smoke ring keeps traveling long after any extra smoke blown out with it has stopped and dispersed.[3] These properties of vortex rings are exploited in the vortex ring gun for riot control and vortex ring toys such as the air vortex cannons.[4]

Formation

One way a vortex ring may be formed is by injecting a compact mass of fast moving fluid (A) into a mass of stationary fluid (B) (which may be the same fluid). Viscous friction at the interface between the two fluids slows down the outer layers of A relative to its core. Those outer layers then slip around the mass A and collect at the rear, where they re-enter the mass in the wake of the faster-moving inner part. The net result is a poloidal flow in A that evolves into a vortex ring.

This mechanism is commonly seen, for example, when a drop of colored liquid falls into a cup of water. It is also often seen at the leading edge of a plume or jet of fluid as it enters a stationary mass; the mushroom-like head ("starting plume") that develops at the tip of the jet has a vortex-ring structure.

Vortex ring of a microburst

A variant of this process may occur when a jet within a fluid hits a flat surface, as in a microburst. In this case the poloidal spinning of the vortex ring is due to viscous friction between the layer of fast outward flow near the surface and the slower-moving fluid above it.

A vortex ring is also formed when a mass of fluid is impulsively pushed from an enclosed space through a narrow opening. In this case the poloidal flow is set in motion, at least in part, by interaction between the outer parts of the fluid mass and the edges of the opening. This is how a smoker expels smoke rings from the mouth, and how most vortex ring toys work.

Vortex rings may also be formed in the wake of a solid object that falls or moves through a fluid at sufficient speed. They may form also ahead of an object that abruptly reverses its motion with the fluid, as when producing smoke rings by shaking an incense stick. A vortex ring can also be created by a spinning propeller, as in a blender.

Other examples

Vortex ring state in helicopters
Main article: Vortex ring state
The curved arrows indicate airflow circulation about the rotor disc. The helicopter shown is the RAH-66 Comanche.

Air vortices can form around the main rotor of a helicopter, causing a dangerous condition known as vortex ring state (VRS) or "settling with power". In this condition, air that moves down through the rotor turns outward, then up, inward, and then down through the rotor again. This re-circulation of flow can negate much of the lifting force and cause a catastrophic loss of altitude. Applying more power (increasing collective pitch) serves to further accelerate the downwash through which the main-rotor is descending, exacerbating the condition.

In the human heart

A vortex ring is formed in the left ventricle of the human heart during cardiac relaxation (diastole), as a jet of blood enters through the mitral valve. This phenomenon was initially observed in vitro[5][6] and subsequently strengthened by analyses based on color Doppler mapping[7][8] and magnetic resonance imaging.[9][10] Some recent studies[11][12] have also confirmed the presence of a vortex ring during rapid filling phase of diastole and implied that the process of vortex ring formation can influence mitral annulus dynamics.

Bubble rings

Releasing air underwater forms bubble rings, which are vortex rings of water with bubbles (or even a single donut-shaped bubble) trapped along its axis line. Such rings are often produced by scuba divers and dolphins.[13]

Separated vortex rings
Pappus of the dandelion which produces a separated vortex ring in order to stabilize flight.

There has been research and experiments on the existence of separated vortex rings (SVR) such as those formed in the wake of the pappus of a dandelion. This special type of vortex ring effectively stabilizes the seed as it travels through the air and increases the lift generated by the seed.[14][15] Compared to a standard vortex ring, which is propelled downstream, the axially symettric SVR remains attached to the pappus for the duration of its flight and uses drag to enhance the travel.[15][16]

Theory

Historical studies

Vortex rings must have been known for as long as people have been smoking, but a scientific understanding of their nature had to wait for the development of mathematical models of fluid dynamics, such as the Navier-Stokes equations.

Vortex rings were first mathematically analyzed by the German physicist Hermann von Helmholtz, in his 1858 paper On Integrals of the Hydrodynamical Equations which Express Vortex-motion.[17][18][19] The formation, motion and interaction of vortex rings have been extensively studied.[20]

Spherical vortices

For many purposes a ring vortex may be approximated as having a vortex-core of small cross-section. However a simple theoretical solution, called Hill's spherical vortex[21] after the English mathematician Micaiah John Muller Hill (1856–1929), is known in which the vorticity is distributed within a sphere (the internal symmetry of the flow is however still annular). Such a structure or an electromagnetic equivalent has been suggested as an explanation for the internal structure of ball lightning. For example, Shafranov used a magnetohydrodynamic (MHD) analogy to Hill's stationary fluid mechanical vortex to consider the equilibrium conditions of axially symmetric MHD configurations, reducing the problem to the theory of stationary flow of an incompressible fluid. In axial symmetry, he considered general equilibrium for distributed currents and concluded under the Virial Theorem that if there were no gravitation, a bounded equilibrium configuration could exist only in the presence of an azimuthal current.

Instabilities

A kind of azimuthal radiant-symmetric structure was observed by Maxworthy[22] when the vortex ring traveled around a critical velocity, which is between the turbulence and laminar states. Later Huang and Chan[23] reported that if the initial state of the vortex ring is not perfectly circular, another kind of instability would occur. An elliptical vortex ring undergoes an oscillation in which it is first stretched in the vertical direction and squeezed in the horizontal direction, then passes through an intermediate state where it is circular, then is deformed in the opposite way (stretched in the horizontal direction and squeezed in the vertical) before reversing the process and returning to the original state.

See also

References
  1. "The Microburst as a Vortex Ring". Forecast Research Branch. NASA. Archived from the original on 2011-07-18. Retrieved 2010-01-10.
  2. Chambers, Joseph R. (Jan 1, 2003). "Wind Shear". Concept to Reality: Contributions of the Langley Research Center to US Civil Aircraft of the 1990s (PDF). NASA. pp. 185–198. hdl:2060/20030059513. Archived from the original on 2007-10-09. Retrieved 2007-10-09.
  3. Batchelor, G.K. (1967), An introduction to fluid dynamics, Cambridge University Press, pp. 521–526, ISBN 978-0-521-09817-5
  4. Physics in a Toroidal Vortex: Air Cannon Physics Central, American Physical Society . Accessed January 2011.
  5. Bellhouse, B.J., 1972, Fluid mechanics of a model mitral valve and left ventricle, Cardiovascular Research 6, 199–210.
  6. Reul, H., Talukder, N., Muller, W., 1981, Fluid mechanics of the natural mitral valve, Journal of Biomechanics 14, 361–372.
  7. Kim, W.Y., Bisgaard, T., Nielsen, S.L., Poulsen, J.K., Pedersen, E.M., Hasenkam, J.M., Yoganathan, A.P., 1994, Two-dimensional mitral flow velocity profiles in pig models using epicardial echo Doppler Cardiography, J Am Coll Cardiol 24, 532–545.
  8. Vierendeels, J. A., E. Dick, and P. R. Verdonck, Hydrodynamics of color M-mode Doppler flow wave propagation velocity V(p): A computer study, J. Am. Soc. Echocardiogr. 15:219–224, 2002.
  9. Kim, W.Y., Walker, P.G., Pedersen, E.M., Poulsen, J.K., Oyre, S., Houlind, K., Yoganathan, A.P., 1995, Left ventricular blood flow patterns in normal subjects: a quantitative analysis by three dimensional magnetic resonance velocity mapping, J Am Coll Cardiol 26, 224–238.
  10. Kilner, P.J., Yang, G.Z., Wilkes, A.J., Mohiaddin, R.H., Firmin, D.N., Yacoub, M.H., 2000, Asymmetric redirection of flow through the heart, Nature 404, 759–761.
  11. Kheradvar, A., Milano, M., Gharib, M. Correlation between vortex ring formation and mitral annulus dynamics during ventricular rapid filling, ASAIO Journal, Jan–Feb 2007 53(1): 8–16.
  12. Kheradvar, A., Gharib, M. Influence of ventricular pressure-drop on mitral annulus dynamics through the process of vortex ring formation, Ann Biomed Eng. 2007 Dec;35(12):2050–64.
  13. Don White. "Mystery of the Silver Rings". Archived from the original on 2007-10-26. Retrieved 2007-10-25.
  14. Ledda, P. G.; Siconolfi, L.; Viola, F.; Camarri, S.; Gallaire, F. (2019-07-02). "Flow dynamics of a dandelion pappus: A linear stability approach". Physical Review Fluids. 4 (7): 071901. Bibcode:2019PhRvF...4g1901L. doi:10.1103/physrevfluids.4.071901. ISSN 2469-990X.
  15. Cummins, Cathal; Seale, Madeleine; Macente, Alice; Certini, Daniele; Mastropaolo, Enrico; Viola, Ignazio Maria; Nakayama, Naomi (2018). "A separated vortex ring underlies the flight of the dandelion" (PDF). Nature. 562 (7727): 414–418. Bibcode:2018Natur.562..414C. doi:10.1038/s41586-018-0604-2. ISSN 0028-0836. PMID 30333579. S2CID 52988814.
  16. Yamamoto, Kyoji (November 1971). "Flow of Viscous Fluid at Small Reynolds Numbers Past a Porous Sphere". Journal of the Physical Society of Japan. 31 (5): 1572. Bibcode:1971JPSJ...31.1572Y. doi:10.1143/JPSJ.31.1572.
  17. von Helmholtz, H. (1858), "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" [On Integrals of the Hydrodynamical Equations which Express Vortex-motion], Journal für die reine und angewandte Mathematik (in German), 56: 25–55
  18. von Helmholtz, H. (1867). "On Integrals of the hydrodynamical equations, which express vortex-motion" (PDF). Philosophical Magazine. Series 4. 33 (226). doi:10.1080/14786446708639824. (1867 translation of 1858 journal article)
  19. Moffatt, Keith (2008). "Vortex Dynamics: The Legacy of Helmholtz and Kelvin". IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence. IUTAM Bookseries. 6: 1–10. doi:10.1007/978-1-4020-6744-0_1. ISBN 978-1-4020-6743-3.
  20. An Introduction to Fluid Dynamics, Batchelor, G. K., 1967, Cambridge UP
  21. Hill, M.J.M. (1894). "On a spherical vortex". Philosophical Transactions of the Royal Society of London A. 185: 213–245. Bibcode:1894RSPTA.185..213H. doi:10.1098/rsta.1894.0006.
  22. Maxworthy, T. J. (1972) The structure and stability of vortex ring, Fluid Mech. Vol. 51, p. 15
  23. Huang, J., Chan, K.T. (2007) Dual-Wavelike Instability in Vortex Rings, Proc. 5th IASME/WSEAS Int. Conf. Fluid Mech. & Aerodyn., Greece
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