In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, vorticity is a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
that describes the local
spinning
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psych ...
. It is an important quantity in
the dynamical theory of
fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of
vortex rings.
Mathematically, the vorticity
is the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the
flow velocity :
:
where
is the
nabla operator. Conceptually,
could be determined by marking parts of a continuum in a small
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the point in question, and watching their ''relative''
displacements as they move along the flow. The vorticity
would be twice the mean
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector of those particles relative to their
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, oriented according to the
right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
.
In a
two-dimensional flow
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
,
is always perpendicular to the plane of the flow, and can therefore be considered a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
.
Examples
In a mass of continuum that is rotating like a rigid body, the vorticity is twice the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector of that rotation. This is the case, for example, in the central core of a
Rankine vortex
The Rankine vortex is a simple mathematical model of a vortex in a Viscosity, viscous fluid. It is named after its discoverer, William John Macquorn Rankine.
The vortices observed in nature are usually modelled with an Potential flow#Examples of ...
.
The vorticity may be nonzero even when all particles are flowing along straight and parallel
pathline
Streamlines, streaklines and pathlines are field lines in a fluid flow.
They differ only when the flow changes with time, that is, when the flow is not steady.
Considering a velocity vector field in three-dimensional space in the framework of ...
s, if there is
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
(that is, if the flow speed varies across
streamlines). For example, in the
laminar flow
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
within a pipe with constant
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal
irrotational vortex
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 ...
, where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of continuum that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocity ''about their center of mass'' is zero.
:
Another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the continuum becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, then there is vorticity in the flow. In the figure below, the left subfigure demonstrates no vorticity, and the right subfigure demonstrates existence of vorticity.
:
Mathematical definition
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by
, defined as the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the velocity field
describing the continuum motion. In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
:
:
In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
In a two-dimensional flow where the velocity is independent of the
-coordinate and has no
-component, the vorticity vector is always parallel to the
-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector
:
:
The vorticity is also related to the flow's
circulation (line integral of the velocity) along a closed path by the (classical)
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
. Namely, for any
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
surface element with
normal direction and area
, the circulation
along the
perimeter of
is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
where
is the vorticity at the center of
.
[Clancy, L.J., ''Aerodynamics'', Section 7.11]
Evolution
The evolution of the vorticity field in time is described by the
vorticity equation, which can be derived from the
Navier–Stokes equations.
In many real flows where the viscosity can be neglected (more precisely, in flows with high
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 ...
), the vorticity field can be modeled by a collection of discrete vortices, the vorticity being negligible everywhere except in small regions of space surrounding the axes of the vortices. This is true in the case of two-dimensional
potential flow (i.e. two-dimensional zero viscosity flow), in which case the flowfield can be modeled as a
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
field on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of vorticity away from the vortex cores into the general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation.
Vortex lines and vortex tubes
A vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation
:
where
is the vorticity vector in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
A vortex tube is the surface in the continuum formed by all vortex lines passing through a given (reducible) closed curve in the continuum. The 'strength' of a vortex tube (also called vortex flux) is the integral of the vorticity across a cross-section of the tube, and is the same everywhere along the tube (because vorticity has zero divergence). It is a consequence of
Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are ...
(or equivalently, of
Kelvin's circulation theorem) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence.
In a three-dimensional flow, vorticity (as measured by the
volume integral of the square of its magnitude) can be intensified when a vortex line is extended — a phenomenon known as
vortex stretching
In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. ...
. This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents.
Vorticity meters
Rotating-vane vorticity meter
A rotating-vane vorticity meter was invented by Russian hydraulic engineer A. Ya. Milovich (1874–1958). In 1913 he proposed a cork with four blades attached as a device qualitatively showing the magnitude of the vertical projection of the vorticity and demonstrated a motion-picture photography of the float's motion on the water surface in a model of a river bend.
Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity" and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic Research
Films by Hunter Rouse — IIHR — Hydroscience & Engineering
).
Specific sciences
Aeronautics
In aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
, the lift distribution over a finite wing
A finite wing is an aerodynamic wing with tips that result in trailing vortices.http://www.aerospaceweb.org/question/aerodynamics/q0167.shtml This is in contrast to an infinite wing
Infinite may refer to:
Mathematics
*Infinite set, a set th ...
may be approximated by assuming that each spanwise 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
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
. The strengths of the vortices are then summed to find the total approximate circulation about the wing. According to the Kutta–Joukowski theorem, lift is the product of circulation, airspeed, and air density.
Atmospheric sciences
The relative vorticity is the vorticity relative to the Earth induced by the air velocity field. This air velocity field is often modeled as a two-dimensional flow parallel to the ground, so that the relative vorticity vector is generally scalar rotation quantity perpendicular to the ground. Vorticity is positive when – looking down onto the earth's surface – the wind turns counterclockwise. In the northern hemisphere, positive vorticity is called cyclonic rotation, and negative vorticity is anticyclonic rotation Anticyclonic rotation, or anticyclonic circulation, is atmospheric motion in the direction opposite to a planet's rotation. For Earth, this motion is in a clockwise direction in the Northern Hemisphere and counterclockwise in the Southern Hemispher ...
; the nomenclature is reversed in the Southern Hemisphere.
The absolute vorticity is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, the Coriolis parameter.
The potential vorticity
In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It i ...
is absolute vorticity divided by the vertical spacing between levels of constant (potential) temperature (or entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
). The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the vertical direction, but the potential vorticity is conserved in an adiabatic flow. As adiabatic flow predominates in the atmosphere, the potential vorticity is useful as an approximate tracer
Tracer may refer to:
Science
* Flow tracer, any fluid property used to track fluid motion
* Fluorescent tracer, a substance such as 2-NBDG containing a fluorophore that is used for tracking purposes
* Histochemical tracer, a substance used for tr ...
of air masses in the atmosphere over the timescale of a few days, particularly when viewed on levels of constant entropy.
The barotropic vorticity equation The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. It ...
is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridge
A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...
s of 500 hPa HPA may refer to:
Organizations
* Harry Potter Alliance, a charity
* Halifax Port Authority, Canada
* Hamburg Port Authority, Germany
* Hawaii Preparatory Academy, a school in Hawaii, US
* Health Protection Agency, UK
* Heerespersonalamt, the Ger ...
geopotential height
Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height (altitude above mean sea level) that accounts for the variation of gravity with latitude and altitude. Thu ...
) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting
Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in ...
utilized that equation.
In modern numerical weather forecasting models and general circulation model
A general circulation model (GCM) is a type of climate model. It employs a mathematical model of the general circulation of a planetary atmosphere or ocean. It uses the Navier–Stokes equations on a rotating sphere with thermodynamic terms f ...
s (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation Prognostic equation - in the context of physical (and especially geophysical) simulation, a prognostic equation predicts the value of variables for some time in the future on the basis of the values at the current or previous times.
For instance, ...
.
Related to the concept of vorticity is the helicity , defined as
:
where the integral is over a given volume . In atmospheric science, helicity of the air motion is important in forecasting supercells and the potential for tornadic activity.
See also
* Barotropic vorticity equation The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. It ...
* D'Alembert's paradox
* Enstrophy In fluid dynamics, the enstrophy can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particu ...
* Velocity potential
* Vortex
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 ...
* Vortex tube
The vortex tube, also known as the Ranque-Hilsch vortex tube, is a mechanical device that separates a compressed gas into hot and cold streams. The gas emerging from the hot end can reach temperatures of , and the gas emerging from the cold end ...
* Vortex stretching
In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. ...
* Horseshoe vortex
The horseshoe vortex model is a simplified representation of the vortex system present in the flow of air around a wing. This vortex system is modelled by the ''bound vortex'' (bound to the wing) and two '' trailing vortices'', therefore having ...
* Wingtip vortices
Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift.Clancy, L.J., ''Aerodynamics'', section 5.14 One wingtip vortex trails from the tip of each wing. Wingtip vortices are sometimes named ''trailing'' ...
Fluid dynamics
* Biot–Savart law
* Circulation
* Vorticity equations
* Kutta–Joukowski theorem
Atmospheric sciences
* Prognostic equation Prognostic equation - in the context of physical (and especially geophysical) simulation, a prognostic equation predicts the value of variables for some time in the future on the basis of the values at the current or previous times.
For instance, ...
* Carl-Gustaf Rossby
* Hans Ertel Hans Ertel (March 24, 1904 in Berlin – July 2, 1971 in Berlin) was a German natural scientist and a pioneer in geophysics, meteorology and hydrodynamics.
Life and work
Hans Ertel began his scientific career at the former ''Preußischen Meteorolog ...
References
Bibliography
*
*
*
*
*
* Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London
* "
Weather Glossary
'"' The Weather Channel Interactive, Inc.. 2004.
* "
'". Integrated Publishing.
Further reading
* Ohkitani, K., "''Elementary Account Of Vorticity And Related Equations''". Cambridge University Press. January 30, 2005.
* Chorin, Alexandre J., "''Vorticity and Turbulence''". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994.
* Majda, Andrew J., Andrea L. Bertozzi, "''Vorticity and Incompressible Flow''". Cambridge University Press; 2002.
* Tritton, D. J., "''Physical Fluid Dynamics''". Van Nostrand Reinhold, New York. 1977.
* Arfken, G., "''Mathematical Methods for Physicists''", 3rd ed. Academic Press, Orlando, Florida. 1985.
External links
* Weisstein, Eric W., "
Vorticity
'". Scienceworld.wolfram.com.
* Doswell III, Charles A., "
'". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma.
* Cramer, M. S., "''Navier–Stokes Equations -
Introduction''". Foundations of Fluid Mechanics.
* Parker, Douglas, "''ENVI 2210 – Atmosphere and Ocean Dynamics
'". School of the Environment, University of Leeds. September 2001.
* Graham, James R., "''Astronomy 202: Astrophysical Gas Dynamics''". Astronomy Department, UC Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public university, public land-grant university, land-grant research university in Berkeley, California. Established in 1868 as the University of Californi ...
.
** "
The vorticity equation: incompressible and barotropic fluids
'".
** "
'".
** "
'".
* "
Spherepack 3.1
'". (includes a collection of FORTRAN vorticity program)
* "
Real-Time Model Predictions''". (Potential vorticity analysis)
{{Meteorological variables
Continuum mechanics
Fluid dynamics
fr:Tourbillon (physique)