Vorticity

In continuum mechanics, the vorticity is a pseudovector 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.

Conceptually, vorticity could be determined by marking the part of continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. This quantity must not be confused with the angular velocity of the particles relative to some other point.

More precisely, the vorticity is a pseudovector field ω→, defined as the curl (rotational) of the flow velocity u→ vector. The definition can be expressed by the vector analysis formula:

${\vec {\omega }}\equiv \nabla \times {\vec {u}}\,,$

where ∇ is the del operator. The vorticity of a two-dimensional flow is always perpendicular to the plane of the flow, and therefore can be considered a scalar field.

The vorticity is related to the flow's circulation (line integral of the velocity) along a closed path by the (classical) Stokes' theorem.^[2] Namely, for any infinitesimal surface element C with normal direction n→ and area dA, the circulation dΓ along the perimeter of C is the dot product ω→ ∙ (dA n→) where ω→ is the vorticity at the center of C.^[2]

Many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity rather than the basic concepts of pressure and velocity. This applies, in particular, to the formation and motion of vortex rings.

Examples

In a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, of water in a tank that has been spinning for a while around its vertical axis, at a constant rate.

The vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines, if there is shear (that is, if the flow speed varies across streamlines). 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. 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 ideal irrotational 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 velocity about their center of mass is zero.

Example flows:
Rigid-body-like vortex v ∝ r	Parallel flow with shear	Irrotational vortex v ∝ 1/r
where v is the velocity of the flow, r is the distance to the center of the vortex and ∝ indicates proportionality. 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.

Mathematical definition

Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ω→, defined as the curl or rotational of the velocity field v→ describing the continuum motion. In Cartesian coordinates:

${\begin{array}{rcl}{\vec {\omega }}&=&\nabla \times {\vec {v}}\;=\;\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\times (v_{x},v_{y},v_{z})\\&=&\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}},\;{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}},\;{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\end{array}}$

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 z coordinate and has no z component, the vorticity vector is always parallel to the z axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector z→:

${\vec {\omega }}\;=\;\nabla \times {\vec {v}}\;=\;\left({\tfrac {\partial }{\partial x}},{\tfrac {\partial }{\partial y}},{\tfrac {\partial }{\partial z}}\right)\times (v_{x},v_{y},0)\;=\;\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right){\vec {z}}\,.$

Evolution

The evolution of the vorticity field in time is described by the vorticity equation, which can be derived from the Navier–Stokes equations.

In many real flows where the viscosity can be neglected (more precisely, in flows with high Reynolds number), the vorticity field can be modeled well 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 clearly true in the case of 2-D potential flow (i.e. 2-D zero viscosity flow), in which case the flowfield can be modeled as a complex-valued field on the complex plane.

Vorticity is a useful tool to understand how the ideal potential flow solutions can be perturbed to model real flows. 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 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 than at the vorticity.

Vortex lines and vortex tubes

A vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation ^[3]

${dx \over \omega _{x}}={dy \over \omega _{y}}={dz \over \omega _{z}},$

where ${\vec {\omega }}=(\omega _{x},\omega _{y},\omega _{z}),$ is the vorticity vector in cartesian coordinates.

A vortex 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 called vortex flux)^[4] 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 of Helmholtz's theorems (or equivalently, of Kelvin'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.

In a three dimensional flow, vorticity (as measured by the volume integral of its squared magnitude) can be intensified when a vortex-line is extended — a phenomenon known as vortex stretching.^[5] This phenomenon occurs in the formation of a bath-tub vortex in out-flowing water, and the build-up of a tornado by rising air-currents.

Helicity is vorticity in motion along a third axis in a corkscrew fashion.

Vorticity meters

Rotating-vane vorticity meter

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 float's motion on the water surface in a model of river bend.^[6]

Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity"^[7] and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic Research^[8]).

Specific sciences

Aeronautics

In aerodynamics, 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. According to the Kutta–Joukowski theorem, lift is the product of circulation, airspeed, and air density.

Atmospheric sciences

The relative vorticity is the vorticity of the air velocity field relative to the Earth. This is often modeled as a two-dimensional flow parallel to the ground, so that the relative vorticity vector is generally perpendicular to the ground, and can then be viewed as a scalar quantity, positive when the vector points upward, negative when it points downwards. Therefore, vorticity is positive when the wind turns counter-clockwise (looking down onto the Earth's surface). In the Northern Hemisphere, positive vorticity is called cyclonic rotation, and negative vorticity is anticyclonic rotation; the nomenclature is reversed in the Southern Hemisphere.

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 potential vorticity is absolute vorticity divided by the vertical spacing between levels of constant entropy (or potential temperature). The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the z direction, but the potential vorticity is conserved in an adiabatic flow, which predominates in the atmosphere. The potential vorticity is therefore 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 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 (GCM's), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation.

Helicity of the air motion is important in forecasting supercells and the potential for tornadic activity.

See also

• Barotropic vorticity equation
• D'Alembert's paradox
• Enstrophy
• Velocity potential
• Vortex
• Vortex tube
• Vortex stretching
• Vortical
• Horseshoe vortex
• Wingtip vortices

Fluid dynamics

• Biot–Savart law
• Circulation
• Vorticity equations
• Kutta–Joukowski theorem

Atmospheric sciences

• Prognostic equation
• Carl-Gustaf Rossby
• Hans Ertel

References

1. Lecture Notes from University of Washington Archived October 16, 2015, at the Wayback Machine.
2. Clancy, L.J., Aerodynamics, Section 7.11
3. Kundu P and Cohen I. Fluid Mechanics.
4. Introduction to Astrophysical Gas Dynamics Archived June 14, 2011, at the Wayback Machine.
5. Batchelor, section 5.2
6. Joukovsky N.E. (1914). "On the motion of water at a turn of a river". Matematicheskii Sbornik. 28.. Reprinted in: Collected works. 4. Moscow; Leningrad. 1937. pp. 193–216; 231–233 (abstract in English). "Professor Milovich's float", as Joukovsky refers this vorticity meter to, is schematically shown in figure on page 196 of Collected works.
7. National Committee for Fluid Mechanics Films Archived October 21, 2016, at the Wayback Machine.
8. Films by Hunter Rouse — IIHR — Hydroscience & Engineering Archived April 21, 2016, at the Wayback Machine.

Bibliography

• Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London ISBN 0-273-01120-0
• "Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.
• "Vorticity". Integrated Publishing.

Further reading

• Batchelor, G. K. (2000) [1967], An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-66396-2
• 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, Florida. 1985. ISBN 0-12-059820-5

External links

• 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)