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

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". 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)