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Incontinuum mechanics,vorticity is apseudovector (or axial vector)field that describes the localspinning 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 theflow. It is an important quantity inthe dynamical theory offluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the generation oflift on wings.[2][3]
Mathematically, the vorticity is thecurl of theflow velocity:[4][3]
where is thenabla operator. Conceptually, could be determined by marking parts of a continuum in a smallneighborhood of the point in question, and watching theirrelativedisplacements as they move along the flow. The vorticity would be twice the meanangular velocity vector of those particles relative to theircenter of mass, oriented according to theright-hand rule. By its own definition, the vorticity vector is asolenoidal field since
In atwo-dimensional flow, is always perpendicular to the plane of the flow, and can therefore be considered ascalar field.
The dynamics of vorticity are fundamentally linked to drag through the Josephson-Anderson relation.[5][6]
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by, defined as thecurl of the velocity field describing the continuum motion. InCartesian coordinates:
We may also express this in index notation as.[7] In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
In a two-dimensional flow where the velocity is independent of the-coordinate and has no-component, the vorticity vector is always parallel to the-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector:
The vorticity is also related to the flow'scirculation (line integral of the velocity) along a closed path by the (classical)Stokes' theorem. Namely, for anyinfinitesimalsurface elementC withnormal direction and area, the circulation along theperimeter of is thedot product where is the vorticity at the center of.[8]
Since vorticity is an axial vector, it can be associated with a second-order antisymmetric tensor (the so-called vorticity or rotation tensor), which is said to be the dual of. The relation between the two quantities, in index notation, are given by
where is the three-dimensionalLevi-Civita tensor. The vorticity tensor is simply the antisymmetric part of the tensor, i.e.,
In a mass of continuum that is rotating like a rigid body, the vorticity is twice theangular velocity vector of that rotation. This is the case, for example, in the central core of aRankine vortex.[9]
The vorticity may be nonzero even when all particles are flowing along straight and parallelpathlines, if there isshear (that is, if the flow speed varies acrossstreamlines). For example, in thelaminar flow within a pipe with constantcross section, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the idealirrotational vortex, where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of continuum that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocityabout their center of mass is zero.
| Example flows: | ||
 |  |  |
| Rigid-body-like vortex v ∝r | Parallel flow with shear | Irrotational vortex v ∝1/r |
| wherev is the velocity of the flow,r is the distance to the center of the vortex and ∝ indicatesproportionality. Absolute velocities around the highlighted point: | ||
 |  |  |
| Relative velocities (magnified) around the highlighted point | ||
 |  |  |
| Vorticity ≠ 0 | Vorticity ≠ 0 | Vorticity = 0 |
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. If that tiny new solid particle is rotating, rather than just moving with the flow, then there is vorticity in the flow. In the figure below, the left subfigure demonstrates no vorticity, and the right subfigure demonstrates existence of vorticity.
The evolution of the vorticity field in time is described by thevorticity equation, which can be derived from theNavier–Stokes equations.[10]
In many real flows where the viscosity can be neglected (more precisely, in flows with highReynolds number), the vorticity field can be modeled by a collection of discrete vortices, the vorticity being negligible everywhere except in small regions of space surrounding the axes of the vortices. This is true in the case of two-dimensionalpotential flow (i.e. two-dimensional zero viscosity flow), in which case the flowfield can be modeled as acomplex-valued field on thecomplex plane.
Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes adiffusion 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.[11]
Avortex line orvorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation[12]
where is the vorticity vector inCartesian coordinates.
Avortex tube is the surface in the continuum formed by all vortex lines passing through a given (reducible) closed curve in the continuum. The 'strength' of a vortex tube (also calledvortex flux)[13] is the integral of the vorticity across a cross-section of the tube, and is the same everywhere along the tube (because vorticity has zero divergence). It is a consequence ofHelmholtz's theorems (or equivalently, ofKelvin's circulation theorem) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence.[14]
In a three-dimensional flow, vorticity (as measured by thevolume integral of the square of its magnitude) can be intensified when a vortex line is extended — a phenomenon known asvortex stretching.[15] 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. Ya. Milovich (1874–1958). 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.[16]
Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity"[17] and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic Research[18]).
Inaerodynamics, thelift distribution over afinite wing may be approximated by assuming that each spanwise 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 ofcomputational fluid dynamics. The strengths of the vortices are then summed to find the total approximatecirculation about the wing. According to theKutta–Joukowski theorem, lift per unit of span is the product of circulation, airspeed, and air density.
Therelative vorticity is the vorticity relative to the Earth induced by the air velocity field. 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. In the northern hemisphere, positive vorticity is calledcyclonic rotation, and negative vorticity isanticyclonic rotation; the nomenclature is reversed in the Southern Hemisphere.
Theabsolute vorticity is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, theCoriolis parameter.
Thepotential vorticity is absolute vorticity divided by the vertical spacing between levels of constant(potential) temperature (orentropy). 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 isconserved in anadiabatic flow. Asadiabatic flow predominates in the atmosphere, the potential vorticity is useful as an approximatetracer of air masses in the atmosphere over the timescale of a few days, particularly when viewed on levels of constant entropy.
Thebarotropic vorticity equation is the simplest way for forecasting the movement ofRossby waves (that is, thetroughs andridges of 500 hPageopotential height) over a limited amount of time (a few days). In the 1950s, the first successful programs fornumerical weather forecasting utilized that equation.
In modern numerical weather forecasting models andgeneral circulation models (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is aprognostic equation.
Related to the concept of vorticity is thehelicity, defined as
where the integral is over a given volume. In atmospheric science, helicity of the air motion is important in forecastingsupercells and the potential fortornadic activity.[19]
{{cite journal}}: CS1 maint: DOI inactive as of September 2025 (link){{cite book}}: CS1 maint: location missing publisher (link) "Professor Milovich's float", as Joukovsky refers this vorticity meter to, is schematically shown in figure on page 196 of Collected works.