Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "circulation" or "rotation" (or more strictly, the local angular rate of rotation) in a fluid. Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ...

Fluid dynamics

In fluid dynamics, vorticity is the curl of the fluid velocity. It can also be considered as the circulation per unit area at a point in a fluid flow field. It is a vector quantity, whose direction is along the axis of the fluid's rotation. For a two-dimensional flow, the vorticity vector is perpendicular to the plane. Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... This article is about the cURL command line tool. ... The velocity of an object is simply its speed in a particular direction. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

For a fluid having locally a "rigid rotation" around an axis (i.e., moving like a rotating cylinder), vorticity is twice the angular velocity of a fluid element. An irrotational fluid is one whose vorticity=0. Somewhat counter-intuitively, an irrotational fluid can have a non-zero angular velocity (e.g. a fluid rotating around an axis with its tangential velocity inversely proportional to the distance to the axis has a zero vorticity) (see also forced and free vortex) The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex is a spinning, often turbulent, flow (or any spiral motion) with closed streamlines. ...

One way to visualize vorticity is this: consider a fluid flowing. Imagine that some tiny part of the fluid is instantaneously rendered solid, and the rest of the flow removed. If that tiny new solid particle would be rotating, rather than just translating, then there is vorticity in the flow.

Diagram illustrating vorticity in a fluid: (from article vorticity) Diagram created by Tarquin in Adobe Illustrator. ...

In general, vorticity is a specially powerful concept in the case that the viscosity is low (i.e. high Reynolds number). In such cases, even when the velocity field is relatively complicated, the vorticity field can be well approximated as zero nearly everywhere except in a small region in space. This is clearly true in the case of 2-D potential flow (i.e. 2-D zero viscosity flow), in which case the flowfield can be identified with the complex plane, and questions about those sorts of flows can be posed as questions in complex analysis which can often be solved (or approximated very well) analytically. The Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and is used for determining whether a flow will be laminar or turbulent. ... In fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations Note that ∇ · v is something different than ∇ v The equations above imply , or Laplaces equation, holds. ...

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity (may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously). In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i.e. low mach number) fluids: Mach number (Ma) (pronounced: mæk, mɑːk) is defined as a ratio of the speed of an object or flow relative to the speed of sound in the medium through which it is travelling: Vo/Vs where Vo is the speed of the object and Vs is the speed...

Even for real flows (3-dimensional and finite Re), the idea of viewing things in terms of vorticity is still very powerful. It provides the most useful way to understand how the potential flow solutions can be perturbed for "real flows." In particular, one restricts attention to the vortex dynamics, which presumes that the vorticity field can be modeled well in terms of discrete vortices (which encompasses a large number of interesting and relevant flows). In general, the presence of viscosity causes a diffusion of vorticity away from these small regions (e.g. discrete vortices) into the general flow field. This can be seen by the diffusion term in the vorticity transport equation. Thus, in cases of very viscous flows (e.g. Couette Flow), the vorticity will be diffused throughout the flow field and it is probably simpler to look at the velocity field (i.e. vectors of fluid motion) rather than look at the vorticity field (i.e. vectors of curl of fluid motion) which is less intuitive. This article or section does not cite its references or sources. ... The term Couette flow refers to the laminar flow of a viscous liquid in the space between two surfaces, one of which is moving relative to the other. ...

Related concepts are the vortex-line, which is a line which is everywhere tangent to the local vorticity; and a vortex tube which is the surface in the fluid formed by all vortex-lines passing through a given (reducible) closed curve in the fluid. The 'strength' of a vortex-tube is the integral of the vorticity across a cross-section of the tube, and is the same at everywhere along the tube (because vorticity has zero divergence). It is a consequence of Helmholtz's theorems (or equivalently, of Kelvin's Circulation Theorem) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. The vortex tube, also known as the Ranque-Hilsch vortex tube, is a heat pump with no moving parts. ... In fluid mechanics, Helmholtzs theorems describe the behaviour of vortex lines in a fluid. ...

Note however that in a three dimensional flow, vorticity (as measured by the volume integral of its square) can be intensified when a vortex-line is extended (see say Batchelor, section 5.2). Mechanisms such as these operate in such well known examples as the formation of a bath-tub vortex in out-flowing water, and the build-up of a tornado by rising air-currents.

Atmospheric sciences

In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since atmospheric circulations important to meteorology are largely horizontal, the vertical vorticity vector is virtually negligible. It is therefore common to use only the horizontal component of the vorticity vector for meteorological applications. In the Northern Hemisphere, vorticity is positive for counter-clockwise (i.e., cyclonic) rotation, and negative for clockwise (i.e, anti-cyclonic) rotation. It is the same in the Southern Hemisphere although the rotational direction differs to that in the Northern Hemisphere. Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. ... AIR is a three-letter abbreviation with multiple meanings, as described below: The Annals of Improbable Research, a monthly magazine devoted to scientific humour All India Radio - Indias Government Radio service AIR, a popular electronica band from France. ...

Relative and absolute vorticity are defined as the z-components of the curls of relative (i.e., in relation to Earth's surface) and absolute wind velocity, respectively. This article is about the cURL command line tool. ... Earth (often referred to as the Earth, or the earth), is the third planet in the solar system in terms of distance from the Sun, and the fifth largest. ... Wind is the roughly horizontal movement of air (as opposed to an air current) caused by uneven heating of the Earths surface. ... The velocity of an object is simply its speed in a particular direction. ...

This gives

for relative vorticity and

for absolute vorticity, where u and v are the zonal (x direction) and meridional (y direction) components of wind velocity. The absolute vorticity at a point can also be expressed as the sum of the relative vorticity at that point and the Coriolis parameter at that latitude (i.e., it is the sum of the Earth's vorticity and the vorticity of the air relative to the Earth). In geography, geophysics, and meteorology, zonal usually means along a latitude circle, i. ... Meridional is a geographic term that means along a north-south direction, or relative to a meridian (opposite: zonal, east-west). ...

A useful related quantity is potential vorticity. The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the z direction. But if the absolute vorticity is divided by the vertical spacing between levels of constant entropy (or potential temperature), the result is a conserved quantity of adiabatic flow, termed potential vorticity (PV). Because diabatic process which can change PV and entropy occur relatively slowly in the atmosphere, PV is useful as an approximate tracer of air masses over the timescale of a few days, particularly when viewed on levels of constant entropy. In thermodynamics, entropy, symbolized by S, is a state function of a thermodynamic system defined by the differential quantity , where dQ is the amount of heat absorbed in a reversible process in which the system goes from the one state to another, and T is the absolute temperature. ... The potential temperature of a parcel of air at pressure is the temperature that the parcel would aquire if adiabatically brought to a standard reference pressure , usually 1 bar. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... This article covers adiabatic processes in thermodynamics. ... In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born-Oppenheimer approximation. ... There are several uses for Tracer: Tracer ammunition Tracer isotope or radioactive tracers as used in positron emission tomography, SPECT and autoradiography. ...

The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 500 hPa geopotential) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting utilized that equation. A simplified form of the vorticity equation for an inviscid, divergence-free flow, the barotropic vorticity equation can simply be stated as where is the material derivative and is absolute vorticity, with being relative vorticity, defined as the vertical component of the curl of the fluid velocity and f is... Rossby (or planetary) waves are large-scale motions in the ocean or atmosphere whose restoring force is the variation in Coriolis effect with latitude. ... A trough is an elongated region of relatively low atmospheric pressure, often associated with fronts. ... A ridge is a geological feature that features a continuous elevational crest for some distance. ... The pascal (symbol: Pa) is the SI unit of pressure. ... Geopotential is the potential of the Earths gravity field. ...

In modern numerical weather forecasting models and GCMs, vorticity may be one of the prognostic variables. GCM can refer to: General circulation model Owen Roberts International Airport in Grand Cayman, Cayman Islands. ... A variable that a GCM predicts by integration of a physical equation, typically vorticity, divergence, temperature, surface pressure, and water vapor concentration. ...

Other fields

Vorticity is important in many other areas of fluid dynamics. For instance, the lift distribution over a finite wing may be approximated by assuming that each segment of the wing has a semi-infinite trailing vortex behind it. It is then possible to solve for the strength of the vortices using the criterion that there be no flow induced through the surface of the wing. This procedure is called the vortex panel method of computational fluid dynamics. The strengths of the vortices are then summed to find the total approximate circulation about the wing. Lift is the product of circulation, airspeed, and air density. Lift consists of the sum of all the fluid dynamic forces on a body perpendicular to the direction of the external flow approaching that body. ... Computational fluid dynamics (CFD) is the use of computers to analyze problems in fluid dynamics. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ...

See also

• Main : Application of tensor theory in engineering science, Barotropic vorticity equation, D'Alembert's paradox, Vortex, Vortical, Vorticity equation, Vortical motion, Vortex tube
• Atmospheric sciences : Prognostic variable, Carl-Gustaf Rossby, Hans Ertel
• Fluid dynamics: Biot-Savart law, Circulation, Curl, Navier-Stokes equations

Tensor theory is extremely useful in advanced engineering theory. ... A simplified form of the vorticity equation for an inviscid, divergence-free flow, the barotropic vorticity equation can simply be stated as where is the material derivative and is absolute vorticity, with being relative vorticity, defined as the vertical component of the curl of the fluid velocity and f is... DAlemberts paradox states that an inviscid (non-viscous), incompressible flow produces no drag on an object surrounded by such fluid, yet it does produce lift. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex is a spinning, often turbulent, flow (or any spiral motion) with closed streamlines. ... Vortical means pertaining to a vortex or to vortices. ... The vorticity equation is an important prognostic equation in the atmospheric sciences. ... The vortex tube, also known as the Ranque-Hilsch vortex tube, is a heat pump with no moving parts. ... A variable that a GCM predicts by integration of a physical equation, typically vorticity, divergence, temperature, surface pressure, and water vapor concentration. ... Carl-Gustav Arvid Rossby (December 28, 1898 – August 19, 1957) was a Swedish-US meteorologist who first explained the large_scale motions of the atmosphere in terms of fluid mechanics. ... The Biot-Savart law is a physical law with applications in both electromagnetics and fluid dynamics. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ... This article is about the cURL command line tool. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances like liquids and gases. ...

Further reading

• Batchelor, G. K., (1967, reprinted 2000) An Introduction to Fluid Dynamics, Cambridge Univ. Press
• Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. ISBN 0-521-81984-9
• Chorin, Alexandre J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
• Majda, Andrew J., Andrea L. Bertozzi, and D. G. Crighton, "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. ISBN 0-521-63948-4
• Tritton, D. J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0-19-854493-6
• Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0-12-059820-5

George Keith Batchelor (March 8, 1920 - March 30, 2000) was an Australian applied mathematician and fluid dynamicist. ...

References

1. "Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.
2. "Vorticity". Integrated Publishing.

External links

• Weisstein, Eric W., "Vorticity". Scienceworld.wolfram.com.
• Doswell III, Charles A., "A Primer on Vorticity for Application in Supercells and Tornadoes". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma.
• Cramer, M. S., "Navier-Stokes Equations -- Vorticity Transport Theorems: Introduction". Foundations of Fluid Mechanics.
• Parker, Douglas, "ENVI 2210 - Atmosphere and Ocean Dynamics, 9: Vorticity". School of the Environment, University of Leeds. September 2001.
• Graham, James R., "Astronomy 202: Astrophysical Gas Dynamics". Astronomy Department, UC, Berkeley.
  • "The vorticity equation: incompressible and barotropic fluids".
  • "Interpretation of the vorticity equation".
  • "Kelvin's vorticity theorem for incompressible or barotropic flow".
• "Spherepack 3.1". (includes a collection of FORTRAN vorticity program)
• "Mesoscale Compressible Community (MC2) Real-Time Model Predictions". (Potential vorticity analysis)
• "Vorticity" images via Google.