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# Kármán vortex street

A Kármán vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow over bluff bodies. They are named after the engineer and fluid dynamicist, Theodore von Kármán.

## 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 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. The governing parameter, the Reynolds number, is essentially a measure of the ratio of inertial to viscous forces in the flow and is defined as

$Re=frac\left\{Vd\right\}\left\{nu\right\}$

Where:

• $d$ = diameter of the cylinder (or some other suitable measure of width of non-circular bodies)
• $V$ = steady velocity of the flow upstream of the cylinder
• $nu,$ = The kinematic viscosity of the fluid.

Over a large Re range (477 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 vortex is shed, an asymmetrical flow pattern forms around the body, which therefore changes the pressure distribution. This means that the alternate shedding of vortices can create periodic lateral 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, when at the correct frequency, causes suspended telephone or power lines to "sing", the antennae on your car to vibrate more strongly at certain speeds and it is also responsible for the fluttering of Venetian blinds as the wind passes through them.

## Engineering problems

Periodic forcing set up in this way 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. In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, as long as 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 unsymmetrical three-dimensional flow thereby discouraging the alternate shedding of vortices. 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 1968 during high winds.

When considering a long circular cylinder, the frequency of vortex shedding is given by the empirical formula

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

Where:

• f = vortex shedding frequency

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.

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.