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.
The average vorticity in a small region of fluid flow is equal to the circulation [math]\displaystyle{ \Gamma }[/math] around the boundary of the small region, divided by the area A of the small region.
- [math]\displaystyle{ \omega_{av} = \frac {\Gamma}{A} }[/math]
Notionally, the vorticity at a point in a fluid is the limit as the area of the small region of fluid approaches zero at the point:
- [math]\displaystyle{ \omega = \frac {d \Gamma}{dA} }[/math]
Mathematically, the vorticity at a point is a vector and is defined as the curl of the velocity:
- [math]\displaystyle{ \vec \omega = \vec \nabla \times \vec v . }[/math]
One of the base assumptions of the potential flow assumption is that the vorticity [math]\displaystyle{ \omega }[/math] is zero almost everywhere, except in a boundary layer or a stream-surface immediately bounding a boundary layer.
Because a vortex is a region of concentrated vorticity, the non-zero vorticity in these specific regions can be modelled with vortices.
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
Vorticity Media
Animation of a rigid-body-like 2D fluid flow. Highlighted area shown in companion images
Animation of a parallel 2D flow with shear. Highlighted area shown in companion image
Animation of an irrotational 2D vortex flow. Highlighted area shown in companion image
Absolute velocities around a point in
Absolute velocities around a point in
Absolute velocities around a point in
Relative velocities around a point in
Relative velocities around a point in
Relative velocities around a point in
Diagram illustrating vorticity in a fluid: 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.
Other websites
- 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 Archived 2004-05-04 at the Wayback Machine". 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 Archived 2004-07-02 at the Wayback Machine".
- "Interpretation of the vorticity equation Archived 2004-07-02 at the Wayback Machine".
- "Kelvin's vorticity theorem for incompressible or barotropic flow Archived 2004-07-02 at the Wayback Machine".
- "Spherepack 3.1". (includes a collection of FORTRAN vorticity program)
- "Mesoscale Compressible Community (MC2)[dead link] Real-Time Model Predictions". (Potential vorticity analysis)