Kármán vortex street

Visualisation of the vortex street behind a circular cylinder in air; the flow is made visible through release of oil vapour in the air near the cylinder

In fluid dynamics, a Kármán vortex street (or a von Kármán vortex sheet) is a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore von Kármán,[1] and is responsible for such phenomena as the "singing" of suspended telephone or power lines, and the vibration of a car antenna at certain speeds.

Analysis

Animation of vortex street created by a cylindrical object; the flow on opposite sides of the object is given different colors, showing that the vortices are shed from alternating sides of the object
A look at the Kármán vortex street effect from ground level, as air flows quickly from the Pacific ocean eastward over Mojave desert mountains.

A vortex street will only form at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90. The Reynolds number is a measure of the ratio of inertial to viscous forces in the flow of a fluid and may be defined as:

\mathrm{Re}=\frac{Vd}{\nu}\

where:

or:

\mathrm{Re}=\frac{\rho _\infty V _\infty d }{\mu _\infty}

where:

The range of Re values will vary with the size and shape of the body from which the eddies are being shed, as well as with the kinematic viscosity of the fluid. Over a large Re range (47<Re<105 for circular cylinders) eddies are shed continuously from each side of the body, forming rows of vortices in its wake. The alternation leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream, and the regular pattern disappears.

When a single vortex is shed, an asymmetrical flow pattern forms around the body and changes the pressure distribution. This means that the alternate shedding of vortices can create periodic lateral (sideways) forces on the body in question, causing it to vibrate. If the vortex shedding frequency is similar to the natural frequency of a body or structure, it causes resonance. It is this forced vibration that, at the correct frequency, causes suspended telephone or power lines to "sing" and the antenna on a car to vibrate more strongly at certain speeds.

In meteorology

Kármán vortex street caused by wind flowing around the Juan Fernández Islands off the Chilean coast

The flow of atmospheric air over obstacles such as islands or isolated mountains sometimes gives birth to von Kármán vortex streets. When a cloud layer is present at the relevant altitude, the streets become visible. Such cloud layer vortex streets have been photographed from satellites.[2]

Engineering problems

Simulated vortex street around a no-slip cylindrical obstruction
The same cylinder, now with a fin, suppressing the vortex street by reducing the region in which the side eddies can interact
Chimneys with spirals outside to break up vortices

In low turbulence, tall buildings can produce a Kármán street so long as the structure is uniform along its height. In urban areas where there are many other tall structures nearby, the turbulence produced by these prevents the formation of coherent vortices.[3] Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, and hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from submarine periscopes to industrial chimneys and skyscrapers.

In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, providing it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached. Obviously, for a tall building or mast, the relative wind could come from any direction. For this reason, helical projections that look like large screw threads are sometimes placed at the top, which effectively create asymmetric three-dimensional flow, thereby discouraging the alternate shedding of vortices; this is also found in some car antennas. Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building being driven at the same frequency.

Even more serious instability can be created in concrete cooling towers, for example, especially when built together in clusters. Vortex shedding caused the collapse of three towers at Ferrybridge Power Station C in 1965 during high winds.

The failure of the original Tacoma Narrows Bridge was originally attributed to excessive vibration due to vortex shedding, but was actually caused by aeroelastic flutter.

Formula

\frac{fd}{V}=0.198\left (1-\frac{19.7}{Re}\right )\

where:

This formula will generally hold true for the range 250 < Re < 2 × 105. The dimensionless parameter fd/V is known as the Strouhal number and is named after the Czech physicist, Vincenc Strouhal (1850–1922) who first investigated the steady humming or singing of telegraph wires in 1878.

Insect flight

Recent studies have shown that insects such as flies borrow energy from the vortices that form around their wings during flight. Vortices inherently create drag. Insects can recapture some of this energy and use it to improve speed and maneuverability: They rotate their wings before starting the return stroke, and the wings are lifted by the eddies of air created on the downstroke. The high frequency oscillation of insect wings means that many hundreds of vortices are shed every second. However, this leads to a symmetric vortex street pattern, unlike the ones shown above.

History

Although named after Theodore von Kármán,[4][5] he acknowledged[6] that the vortex street had been studied earlier by Mallock[7] and Bénard.[8]

See also

References

  1. ↑ Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0-07-067602-2. Dover (1994): ISBN 978-0-486-43485-8.
  2. ↑ "Rapid Response - LANCE - Terra/MODIS 2010/226 14:55 UTC". Rapidfire.sci.gsfc.nasa.gov. Retrieved 2013-12-20.
  3. ↑ Irwin, Peter A. (September 2010). "Vortices and tall buildings: A recipe for resonance". Physics Today (American Institute of Physics) 63 (9): 68–69. Bibcode:2010PhT....63i..68I. doi:10.1063/1.3490510. ISSN 0031-9228.
  4. ↑ T. von Kármán: Nachr. Ges. Wissenschaft. Göttingen Math. Phys. Klasse pp. 509–517 (1911) and pp. 547–556 (1912).
  5. ↑ T. von Kármán: and H. Rubach, 1912: Phys. Z.", vol. 13, pp. 49–59.
  6. ↑ T. Kármán, 1954. Aerodynamics: Selected Topics in the Light of Their Historical Development (Cornell University Press, Ithaca), pp. 68–69.
  7. ↑ A. Mallock, 1907: On the resistance of air. Proc. Royal Soc., A79, pp. 262–265.
  8. ↑ H. Bénard, 1908: Comptes rendus de l'Académie des Sciences (Paris), vol. 147, pp. 839–842, 970–972.

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