In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate[1]), as would be seen by an observer located at that point and traveling along with the flow.
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.
could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow.
would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule.
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by
[5] In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
-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.
For example, in the laminar flow within a pipe with constant cross section, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls.
Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories.
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.
In general, the presence of viscosity causes a diffusion 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.
Viscous effects introduce frictional losses and time dependence.
[13] This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents.
A rotating-vane vorticity meter was invented by Russian hydraulic engineer A.
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.
[14] Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity"[15] and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic Research[16]).
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.
According to the Kutta–Joukowski theorem, lift per unit of span is the product of circulation, airspeed, and air density.
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.
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 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 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 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 general circulation models (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation.
In atmospheric science, helicity of the air motion is important in forecasting supercells and the potential for tornadic activity.