![Karmansche Wirbelstr kleine Re](https://upload.wikimedia.org/wikipedia/commons/f/fb/Karmansche_Wirbelstr_kleine_Re.JPG)
In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, a Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of swirling
vortices
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
, caused by a process known as
vortex shedding
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, v ...
, which is responsible for the unsteady
separation of flow of a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
around blunt bodies.
It is named after the engineer and fluid dynamicist
Theodore von Kármán
Theodore von Kármán ( hu, ( szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronaut ...
, and is responsible for such phenomena as the "
singing
Singing is the act of creating musical sounds with the voice. A person who sings is called a singer, artist or vocalist (in jazz and/or popular music). Singers perform music (arias, recitatives, songs, etc.) that can be sung with or without ...
" of suspended telephone or power lines and the vibration of a car antenna at certain speeds. Mathematical modeling of von Kármán vortex street can be performed using different techniques including but not limited to solving the full Navier-Stokes equations with k-epsilon, SST, k-omega and Reynolds stress, and large eddy simulation (LES) turbulence models,
by numerically solving some dynamic equations such as the
Ginzburg-Landau equation, or by use of a
bicomplex variable.
Analysis
![Karman vortex street](https://upload.wikimedia.org/wikipedia/commons/7/7b/Karman_vortex_street.gif)
A vortex street will form only at a certain range of flow velocities, specified by a range of
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
s (''Re''), typically above a limiting ''Re'' value of about 90. The (''global'') Reynolds number for a flow is a measure of the ratio of
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l to
viscous forces
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
in the flow of a fluid around a body or in a channel, and may be defined as a
nondimensional parameter of the global speed of the whole fluid flow:
where:
*
= the free stream
flow speed (i.e. the flow speed far from the fluid boundaries
like the body speed relative to the fluid at rest, or an inviscid flow speed, computed through the Bernoulli equation), which is the original global flow parameter, i.e. the target to be non-dimensionalised.
*
= a characteristic length parameter of the body or channel
*
= the free stream
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
parameter of the fluid, which in turn is the ratio:
between:
*
= the reference fluid density.
*
= the free stream fluid
dynamic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
For common flows (the ones which can usually be considered as incompressible or isothermal), the kinematic viscosity is everywhere uniform over all the flow field and constant in time, so there is no choice on the viscosity parameter, which becomes naturally the kinematic viscosity of the fluid being considered at the temperature being considered. On the other hand, the reference length is always an arbitrary parameter, so particular attention should be put when comparing flows around different obstacles or in channels of different shapes: the global Reynolds numbers should be referred to the same reference length. This is actually the reason for which the most precise sources for airfoil and channel flow data specify the reference length at the Reynolds number. The reference length can vary depending on the analysis to be performed: for a body with circle sections such as circular cylinders or spheres, one usually chooses the diameter; for an airfoil, a generic non-circular cylinder or a
bluff body
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equ ...
or a revolution body like a fuselage or a submarine, it is usually the profile
chord or the profile thickness, or some other given widths that are in fact stable design inputs; for flow channels usually the
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channel ...
about which the fluid is flowing.
For an aerodynamic profile the reference length depends on the analysis. In fact, the profile chord is usually chosen as the reference length also for aerodynamic coefficient for wing sections and thin profiles in which the primary target is to maximize the lift coefficient or the lift/drag ratio (i.e. as usual in thin airfoil theory, one would employ the ''chord Reynolds'' as the flow speed parameter for comparing different profiles). On the other hand, for fairings and struts the given parameter is usually the dimension of internal structure to be streamlined (let us think for simplicity it is a beam with circular section), and the main target is to minimize the drag coefficient or the drag/lift ratio. The main design parameter which becomes naturally also a reference length is therefore the profile thickness (the profile dimension or area perpendicular to the flow direction), rather than the profile chord.
The range of ''Re'' values varies with the size and shape of the body from which the
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid beh ...
are
shed
A shed is typically a simple, single-story roofed structure that is used for hobbies, or as a workshop in a back garden or on an allotment. Sheds vary considerably in their size and complexity of construction, from simple open-sided ones de ...
, as well as with the
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
of the fluid. For the wake of a circular cylinder, for which the reference length is conventionally the diameter ''d'' of the circular cylinder, the lower limit of this range is ''Re'' ≈ 47. Eddies are shed continuously from each side of the circle boundary, 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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the vortices is consumed by viscosity as they move further down stream, and the regular pattern disappears. Above the ''Re '' value of 188.5, the flow becomes three-dimensional, with periodic variation along the cylinder. Above ''Re'' on the order of 10
5 at the
drag crisis, vortex shedding becomes irregular and turbulence sets in.
When a single vortex is shed, an
asymmetrical
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in pre ...
flow pattern forms around the body and changes the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
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
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
is similar to the
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
of a body or structure, it causes
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
. It is this forced vibration that, at the correct frequency, causes suspended
telephone
A telephone is a telecommunications device that permits two or more users to conduct a conversation when they are too far apart to be easily heard directly. A telephone converts sound, typically and most efficiently the human voice, into e ...
or
power line
An overhead power line is a structure used in electric power transmission and distribution to transmit electrical energy across large distances. It consists of one or more uninsulated electrical cables (commonly multiples of three for three-p ...
s to "sing" and the
antenna
Antenna ( antennas or antennae) may refer to:
Science and engineering
* Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves
* Antennae Galaxies, the name of two collid ...
on a car to vibrate more strongly at certain speeds.
In meteorology
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. The vortex street can reach over 400 km from the obstacle and the diameter of the vortices are normally 20–40 km.
Engineering problems
![Proserpine Mill - January 2012](https://upload.wikimedia.org/wikipedia/commons/4/4f/Proserpine_Mill_-_January_2012.jpg)
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. Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, due to the
vortex-induced vibrations caused, which can damage the structure, hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from
submarine
A submarine (or sub) is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability. The term is also sometimes used historically or colloquially to refer to remotely op ...
periscope
A periscope is an instrument for observation over, around or through an object, obstacle or condition that prevents direct line-of-sight observation from an observer's current position.
In its simplest form, it consists of an outer case with ...
s to industrial
chimney
A chimney is an architectural ventilation structure made of masonry, clay or metal that isolates hot toxic exhaust gases or smoke produced by a boiler, stove, furnace, incinerator, or fireplace from human living areas. Chimneys are typic ...
s and
skyscraper
A skyscraper is a tall continuously habitable building having multiple floors. Modern sources currently define skyscrapers as being at least or in height, though there is no universally accepted definition. Skyscrapers are very tall high-ris ...
s. For monitoring such engineering structures, the efficient measurements of von Kármán streets can be performed using smart sensing algorithms such as compressive sensing.
In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid beh ...
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
Helical may refer to:
* Helix, the mathematical concept for the shape
* Helical engine, a proposed spacecraft propulsion drive
* Helical spring, a coilspring
* Helical plc, a British property company, once a maker of steel bar stock
* Helicoil
A t ...
projections resembling 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
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
can be created in concrete
cooling tower
A cooling tower is a device that rejects waste heat to the atmosphere through the cooling of a coolant stream, usually a water stream to a lower temperature. Cooling towers may either use the evaporation of water to remove process heat an ...
s, 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
Aeroelasticity is the branch of physics and engineering studying the interactions between the inertial, elastic, and aerodynamic forces occurring while an elastic body is exposed to a fluid flow. The study of aeroelasticity may be broadly classif ...
.
Kármán turbulence is also a problem for airplanes, especially when landing.
Formula
This formula will generally hold true for the range 250 < Re
''d'' < 200000:
where:
*''f'' = vortex shedding frequency.
*''d'' = diameter of the cylinder
*''U'' = flow velocity.
This dimensionless parameter St is known as the
Strouhal number
In dimensional analysis, the Strouhal number (St, or sometimes Sr to avoid the conflict with the Stanton number) is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who ...
and is named after the Czech physicist,
Vincenc Strouhal
Vincenc Strouhal (Čeněk Strouhal) (10 April 1850 in Seč – 26 January 1922 in Prague) was a Czech physicist specializing in experimental physics. He was one of the founders of the Institute of Physics of the Czech part of Charles Universit ...
(1850–1922) who first investigated the steady humming or singing of telegraph wires in 1878.
History
Although named after
Theodore von Kármán
Theodore von Kármán ( hu, ( szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronaut ...
, he acknowledged that the vortex street had been studied earlier by
Arnulph Mallock
Arnulph Henry Reginald Mallock, FRS (12 March 1851 – 26 June 1933) was a British scientific instrument designer and experimentalist.
He was born in Cheriton Bishop, Devon the son of the Revd. William and Margaret (née Froude) Mallock. His fat ...
and
Henri Bénard. Kármán tells the story in his book Aerodynamics:
In his autobiography, von Kármán described how his discovery was inspired by an Italian painting of
St Christopher carrying the child
Jesus
Jesus, likely from he, יֵשׁוּעַ, translit=Yēšūaʿ, label=Hebrew/Aramaic ( AD 30 or 33), also referred to as Jesus Christ or Jesus of Nazareth (among other names and titles), was a first-century Jewish preacher and religious ...
whilst wading through water. Vortices could be seen in the water, and von Kármán noted that "The problem for historians may have been why Christopher was carrying Jesus through the water. For me it was why the vortices". It has been suggested by researchers that the painting is one from the 14th century that can be found in the museum of the
San Domenico church in Bologna.
See also
*
*
*
*
*
*
References
External links
*
*
*
*
{{DEFAULTSORT:Karman Vortex Street
Vortices
Aerodynamics
Articles containing video clips