# Vorticity

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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.

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true vector, also known, in this context, as a polar vector, which on reflection matches its mirror image.

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

## Contents

Conceptually, vorticity 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. 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.

In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a rigid body's centre of rotation revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. In general, angular velocity is measured in angle per unit time, radians per second in SI units, and is usually represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space.

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:

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector characterize the rotation at that point.

In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile.

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

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.

Fluid motion can be said to be a two-dimensional flow when the flow velocity at every point is parallel to a fixed plane. The velocity at any point on a given normal to that fixed plane should be constant.

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

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 along the perimeter of C is the dot product ω ∙ (dAn) where ω is the vorticity at the center of C. [2]

In fluid dynamics, circulation is the line integral around a closed curve of the velocity field. Circulation is normally denoted Γ. Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolai Zhukovsky.

In vector calculus, and more generally differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative over the whole of Ω, i.e.,

In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

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.

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid or gas; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant flow in a vortex ring is said to be toroidal, more precisely poloidal.

The name vorticity was created by Horace Lamb in 1916 [3] .

## 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, in the central core of a Rankine vortex.

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 vortexv ∝ r Parallel flow with shear Irrotational vortexv ∝ 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. In the figure below, the left subfigure demonstrates no vorticity, and the right subfigure demonstrates existence of vorticity.

## 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:

{\displaystyle {\begin{aligned}{\vec {\omega }}=\nabla \times {\vec {v}}&={\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}v_{x}&v_{y}&v_{z}\end{pmatrix}}\\[6px]&={\begin{pmatrix}{\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}}\end{pmatrix}}\,.\end{aligned}}}

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:

{\displaystyle {\begin{aligned}{\vec {\omega }}=\nabla \times {\vec {v}}&={\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}\times \left(v_{x},v_{y},0\right)\\[6px]&=\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right){\vec {z}}\,.\end{aligned}}}

## 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 two-dimensional potential flow (i.e. two-dimensional 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 [4]

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

where ω = (ωx, ωy, ωz) 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) [5] 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 the square of its magnitude) can be intensified when a vortex line is extended — a phenomenon known as vortex stretching. [6] This phenomenon occurs in the formation of a bathtub vortex in outflowing 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. [7]

Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity" [8] and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic Research [9] ).

## 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 counterclockwise (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 (GCMs), 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.

## Related Research Articles

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

The vorticity equation of fluid dynamics describes evolution of the vorticity ω of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid . The equation is:

In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic atmosphere is one for which the density depends on both the temperature and the pressure; contrast this with a barotropic atmosphere, for which the density depends only on the pressure. In atmospheric terms, the barotropic zones of the Earth are generally found in the central latitudes, or tropics, whereas the baroclinic areas are generally found in the mid-latitude/polar regions.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

A shear stress, often denoted by τ, is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent, i.e., the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.

In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex filaments. These theorems apply to inviscid flows and flows where the influence of viscous forces are small and can be ignored.

In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

In fluid dynamics, helicity is, under appropriate conditions, an invariant of the Euler equations of fluid flow, having a topological interpretation as a measure of linkage and/or knottedness of vortex lines in the flow. This was first proved by Jean-Jacques Moreau in 1961 and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity.

The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant. It is also known as the Gödel solution or Gödel universe.

Potential vorticity (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers.

In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was predicted by Lars Onsager in 1947 in connection with superfluid helium. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil and any two-dimensional bodies including circular cylinders translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid and the circulation around the airfoil. The circulation is defined as the line integral around a closed loop enclosing the airfoil of the component of the velocity of the fluid tangent to the loop. It is named after Martin Kutta and Nikolai Zhukovsky who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory, but it is a good approximation for real viscous flow in typical aerodynamic applications.

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

The Lambda2 method, or Lambda2 vortex criterion, is a vortex core line detection algorithm that can adequately identify vortices from a three-dimensional fluid velocity field. The Lambda2 method is Galilean invariant, which means it produces the same results when a uniform velocity field is added to the existing velocity field or when the field is translated.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

## 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. Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Press.
4. Kundu P and Cohen I. Fluid Mechanics.
5. Introduction to Astrophysical Gas Dynamics Archived June 14, 2011, at the Wayback Machine .
6. Batchelor, section 5.2
7. 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.
8. National Committee for Fluid Mechanics Films Archived October 21, 2016, at the Wayback Machine .
9. Films by Hunter Rouse — IIHR — Hydroscience & Engineering Archived April 21, 2016, at the Wayback Machine .

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