vorticity
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. It is often, but not always, defined as the curl of the velocity:
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)
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.
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.
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, with conservative body forces:
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.
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.
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.
The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around. In fluid mechanics this equation can be expressed in vector form as follows,
where, is the velocity vector, is the density, is the pressure, is the viscous stress tensor and is the body force term. The equation is valid for compressible fluid in the absence of any concentrated torques and line forces. No assumption is made regarding the relationship between the stress and the rate of strain tensors (c.f. Newtonian fluid).
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 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).
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 processes, 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.
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 height) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting utilized that equation.
In modern numerical weather forecasting models and GCMs, vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation.
Diffusion is the net movement of particles from an area of high concentration to an area of low concentration.
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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.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)
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.
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.
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, with conservative body forces:
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.
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.
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.
Vorticity Equation
- Main article: Vorticity equation
The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around. In fluid mechanics this equation can be expressed in vector form as follows,
where, is the velocity vector, is the density, is the pressure, is the viscous stress tensor and is the body force term. The equation is valid for compressible fluid in the absence of any concentrated torques and line forces. No assumption is made regarding the relationship between the stress and the rate of strain tensors (c.f. Newtonian fluid).
Atmospheric sciences
In the atmospheric sciences, vorticity is the rotation of air around a vertical axis. 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.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 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).
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 processes, 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.
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 height) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting utilized that equation.
In modern numerical weather forecasting models and GCMs, vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation.
Meteorological data and variables |
---|
Atmospheric pressure Baroclinity Cloud Convection CAPE CIN Dew point Heat index Humidex Humidity Lifted index Lightning Pot T Precipitation Sea surface temperature Surface solar radiation Surface weather analysis Temperature Theta-e Visibility Vorticity Wind chill Water vapor Wind |
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.See also
Atmospheric sciences
Fluid dynamics
- Biot-Savart law
- Circulation
- Navier-Stokes equations
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, "Vorticity and Incompressible Flow". Cambridge University Press; 2002. 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
References
- "Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.
- "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)
Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion).
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In fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted . If is the fluid velocity and is a unit vector along the closed curve :
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Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion).
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cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, TFTP, SCP, SFTP, Telnet, DICT, and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password
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velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed.
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In fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted . If is the fluid velocity and is a unit vector along the closed curve :
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FLUID (Fast Light User Interface Designer) is a graphical editor that is used to produce FLTK source code. FLUID edits and saves its state in text .fl files, which can be edited in a text editor for finer control over display and behavior.
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed at which an object is rotating along with the direction in which it is rotating.
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In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields.
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In fluid mechanics, the Reynolds number is the ratio of inertial forces (v_{s}ρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of forces for given flow conditions.
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potential flow obeys the following equations
(zero rotation)
Equivalently,
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(zero rotation)
- (zero divergence = volume conservation)
Equivalently,
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The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
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The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
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The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the total derivative (that is, the local
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Mach number (Ma) (pronounced: [mɑːk], [mɑx], [mæk], see IPA) is a dimensionless measure of relative speed.
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vortex dynamics matured as a subfield of fluid mechanics, always commanding at least a major chapter in treatises on the subject. Thus, H. Lamb's well known Hydrodynamics (6th ed., 1932) devotes a full chapter to vorticity and vortex dynamics as does G. K.
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- This article is about the physical mechanism of diffusion. For alternative meanings, see diffusion (disambiguation).
Diffusion is the net movement of particles from an area of high concentration to an area of low concentration.
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In fluid dynamics, 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. The flow is driven by virtue of viscous drag force acting on the fluid.
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vortex tube, also known as the Ranque-Hilsch vortex tube, is a mechanical device that separates a compressed gas into hot and cold streams. It has no moving parts.
Pressurized gas is injected tangentially into a swirl chamber and accelerates to a high rate of rotation.
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Pressurized gas is injected tangentially into a swirl chamber and accelerates to a high rate of rotation.
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For other uses, see Helmholtz theorem.
In fluid mechanics, Helmholtz's theorems (pub. 1858) describe the motion of vortex lines and tubes in a fluid...... Click the link for more information.
In fluid mechanics, Kelvin's Circulation Theorem states "In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time"^{[1]}.
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The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the total derivative (that is, the local
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A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its stress versus rate of strain curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.
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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.
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Air or Earth's atmosphere is a layer of gases surrounding the planet Earth.
Air may also refer to:
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Air may also refer to:
- Air (1977 video game), an air combat based mainframe computer game
- Air (band), a French electronic music duo
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cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, TFTP, SCP, SFTP, Telnet, DICT, and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password
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EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001. Their greatest hit, their debut single "time after time", peaked at #13 in the Oricon singles chart.
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WIND (SOLARWIND) was a NASA spacecraft launched on November 1, 1994. It was deployed to study radio and plasma that occur in solar wind, in the Earth's magnetosphere. The spacecraft's original mission was to orbit the Sun at the L_{1}
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velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed.
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