Vorticity

From Academic Kids

Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "circulation" or "rotation" in a fluid.

Contents

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 angular velocity inversely proportional to the distance to the axis has a zero vorticity).

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.

Missing image
Vorticity_visualized_diagram.png
Illustration of the example

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:

<math> {D\omega \over Dt} = \omega \cdot \nabla u + \nu \nabla^2 \omega <math>

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, sort of presumes that the vorticity field can be modeled well in terms of discrete vorticies (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 vorticies) 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.

Atmospheric sciences

In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since the atmospheric circulation is nearly horizontal, the (3 dimensional) vorticity is nearly vertical, and it is common to talk use the vertical component as the scalar vorticity. The scalar vorticity is positive when the parcel has a counterclockwise rotation for the Northern Hemisphere. It is negative when the parcel has clockwise rotation for the Northern Hemisphere. [1]

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

<math>\zeta=\frac{\partial v_r}{\partial x} - \frac{\partial u_r}{\partial y}<math>

for relative vorticity and

<math>\eta=\frac{\partial v_a}{\partial x} - \frac{\partial u_a}{\partial y}<math>

for absolute vorticity, where u and v are the zonal (x direction) and meridional (y direction) components of wind velocity.

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.

The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 500 mb geopotential) 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 prognostic variables.

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

Further reading

  • Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge Univ Pr. January 30, 2005. ISBN 0521819849
  • Chorin, Alexandre J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0387941975
  • Majda, Andrew J., Andrea L. Bertozzi, and D. G. Crighton, "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. ISBN 0521639484
  • Tritton, D. J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0198544936
  • Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0120598205

References

  1. "Weather Glossary (http://oap2.weather.com/glossary/v.html)"' The Weather Channel Interactive, Inc.. 2004.
  2. "Vorticity (http://www.tpub.com/content/aerographer/14010/css/14010_18.htm)". Integrated Publishing.

External links

de:Vorticity cs:Vorticita ja:渦度

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