Kármán vortex street
The term Kármán vortex street (or a von Kármán vortex street) is used in fluid dynamics to describe a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid over bluff 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.
Contents
Analysis
A vortex street will only be observed over a given 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:
- \(d\) = the diameter of the cylinder (or some other suitable measure of width of non-circular bodies) about which the fluid is flowing.
- \(V\) = the steady velocity of the flow upstream of the cylinder.
- \(\nu\,\) = the kinematic viscosity of the fluid.
or:
\[\mathrm{Re}=\frac{\rho _\infty V _\infty d }{\mu _\infty} \]
where:
- \(\rho _\infty\) = the free stream fluid density.
- \(V _\infty\) = the steady free stream velocity of the flow upstream of the cylinder.
- \(d\) = the diameter of the cylinder (or some other suitable measure of width of non-circular bodies) about which the fluid is flowing.
- \(\mu _\infty\) = the free stream dynamic viscosity of the fluid.
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 which, 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.
Engineering problems
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 nearby structures the turbulence produced by these prevents the formation of coherent vortices.[2] 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 which 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 in 1965 during high winds.
The failure of the Tacoma Narrows Bridge (1940) 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:
- f = vortex shedding frequency.
- d = diameter of the cylinder
- V = flow velocity.
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 bees 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.
See also
- Vortex shedding
- Vortex-induced vibration
- Strouhal number
- Aeroelastic flutter - where the self-resonance of the solid object dominates, e.g. flapping to give +/- Lift (force) resulting from angle of attack
References
- ↑ Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0-07-067602-2. Dover (1994): ISBN 978-0-486-43485-8.
- ↑ Script error
External links
| 40x40px | Wikimedia Commons has media related to: Von Kármán vortex street |
- 3D animation of the Vortex Flow Measuring Principle
- Vortex streets and Strouhal instability
- How Insects Fly
- YouTube — Flow visualisation of the vortex shedding mechanism on circular cylinder using hydrogen bubbles illuminated by a laser sheet in a water channel
- Various Views of von Karman Vortices, NASA pagede:Kármánsche Wirbelstraße
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