# Vortex

Last updated

In fluid dynamics, a vortex (plural vortices/vortexes) [1] [2] is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. [3] [4] Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil.

## Contents

Vortices are a major component of turbulent flow. The distribution of velocity, vorticity (the curl of the flow velocity), as well as the concept of circulation are used to characterise vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis.

In the absence of external forces, viscous friction within the fluid tends to organise the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries some angular and linear momentum, energy, and mass, with it.

## Properties

### Vorticity

A key concept in the dynamics of vortices is the vorticity, a vector that describes the local rotary motion at a point in the fluid, as would be perceived by an observer that moves along with it. Conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, and observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball (according to the right-hand rule) while its length is twice the ball's angular velocity. Mathematically, the vorticity is defined as the curl (or rotational) of the velocity field of the fluid, usually denoted by ${\displaystyle {\vec {\omega }}}$ and expressed by the vector analysis formula ${\displaystyle \nabla \times {\vec {\mathit {u}}}}$, where ${\displaystyle \nabla }$ is the nabla operator and ${\displaystyle {\vec {\mathit {u}}}}$ is the local flow velocity. [5]

The local rotation measured by the vorticity ${\displaystyle {\vec {\omega }}}$ must not be confused with the angular velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ${\displaystyle {\vec {\omega }}}$ may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis.

### Vortex types

In theory, the speed u of the particles (and, therefore, the vorticity) in a vortex may vary with the distance r from the axis in many ways. There are two important special cases, however:

• If the fluid rotates like a rigid body – that is, if the angular rotational velocity Ω is uniform, so that u increases proportionally to the distance r from the axis – a tiny ball carried by the flow would also rotate about its center as if it were part of that rigid body. In such a flow, the vorticity is the same everywhere: its direction is parallel to the rotation axis, and its magnitude is equal to twice the uniform angular velocity Ω of the fluid around the center of rotation.
${\displaystyle {\vec {\Omega }}=(0,0,\Omega ),\quad {\vec {r}}=(x,y,0),}$
${\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\Omega y,\Omega x,0),}$
${\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=(0,0,2\Omega )=2{\vec {\Omega }}.}$
• If the particle speed u is inversely proportional to the distance r from the axis, then the imaginary test ball would not rotate over itself; it would maintain the same orientation while moving in a circle around the vortex axis. In this case the vorticity ${\displaystyle {\vec {\omega }}}$ is zero at any point not on that axis, and the flow is said to be irrotational.
${\displaystyle {\vec {\Omega }}=(0,0,\alpha r^{-2}),\quad {\vec {r}}=(x,y,0),}$
${\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\alpha yr^{-2},\alpha xr^{-2},0),}$
${\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=0.}$

#### Irrotational vortices

In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern[ citation needed ], where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are also called free vortices.

For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, Γ, for any contour that does enclose the axis once. [6] The tangential component of the particle velocity is then ${\displaystyle u_{\theta }={\tfrac {\Gamma }{2\pi r}}}$. The angular momentum per unit mass relative to the vortex axis is therefore constant, ${\displaystyle ru_{\theta }={\tfrac {\Gamma }{2\pi }}}$.

The ideal irrotational vortex flow in free space is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational: the vorticity ${\displaystyle {\vec {\omega }}}$ becomes non-zero, with direction roughly parallel to the vortex axis. The Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r0, and irrotational flow outside that core regions.

In a viscous fluid, irrotational flow contains viscous dissipation everywhere, yet there are no net viscous forces, only viscous stresses. [7] Due to the dissipation, this means that sustaining an irrotational viscous vortex requires continuous input of work at the core (for example, by steadily turning a cylinder at the core). In free space there is no energy input at the core, and thus the compact vorticity held in the core will naturally diffuse outwards, converting the core to a gradually-slowing and gradually-growing rigid-body flow, surrounded by the original irrotational flow. Such a decaying irrotational vortex has an exact solution of the viscous Navier–Stokes equations, known as a Lamb–Oseen vortex.

#### Rotational vortices

A rotational vortex – a vortex that rotates in the same way as a rigid body – cannot exist indefinitely in that state except through the application of some extra force, that is not generated by the fluid motion itself. It has non-zero vorticity everywhere outside the core. Rotational vortices are also called rigid-body vortices or forced vortices.

For example, if a water bucket is spun at constant angular speed w about its vertical axis, the water will eventually rotate in rigid-body fashion. The particles will then move along circles, with velocity u equal to wr. [6] In that case, the free surface of the water will assume a parabolic shape.

In this situation, the rigid rotating enclosure provides an extra force, namely an extra pressure gradient in the water, directed inwards, that prevents transition of the rigid-body flow to the irrotational state.

### Vortex formation on boundaries

Vortex structures are defined by their vorticity , the local rotation rate of fluid particles. They can be formed via the phenomenon known as boundary layer separation which can occur when a fluid moves over a surface and experiences a rapid acceleration from the fluid velocity to zero due to the no-slip condition. This rapid negative acceleration creates a boundary layer which causes a local rotation of fluid at the wall (i.e. vorticity) which is referred to as the wall shear rate. The thickness of this boundary layer is proportional to ${\displaystyle \surd (vt)}$ (where v is the free stream fluid velocity and t is time).

If the diameter or thickness of the vessel or fluid is less than the boundary layer thickness then the boundary layer will not separate and vortices will not form. However, when the boundary layer does grow beyond this critical boundary layer thickness then separation will occur which will generate vortices.

This boundary layer separation can also occur in the presence of combatting pressure gradients (i.e. a pressure that develops downstream). This is present in curved surfaces and general geometry changes like a convex surface. A unique example of severe geometric changes is at the trailing edge of a bluff body where the fluid flow deceleration, and therefore boundary layer and vortex formation, is located.

Another form of vortex formation on a boundary is when fluid flows perpendicularly into a wall and creates a splash effect. The velocity streamlines are immediately deflected and decelerated so that the boundary layer separates and forms a toroidal vortex ring. [8]

### Vortex geometry

In a stationary vortex, the typical streamline (a line that is everywhere tangent to the flow velocity vector) is a closed loop surrounding the axis; and each vortex line (a line that is everywhere tangent to the vorticity vector) is roughly parallel to the axis. A surface that is everywhere tangent to both flow velocity and vorticity is called a vortex tube. In general, vortex tubes are nested around the axis of rotation. The axis itself is one of the vortex lines, a limiting case of a vortex tube with zero diameter.

According to Helmholtz's theorems, a vortex line cannot start or end in the fluid – except momentarily, in non-steady flow, while the vortex is forming or dissipating. In general, vortex lines (in particular, the axis line) are either closed loops or end at the boundary of the fluid. A whirlpool is an example of the latter, namely a vortex in a body of water whose axis ends at the free surface. A vortex tube whose vortex lines are all closed will be a closed torus-like surface.

A newly created vortex will promptly extend and bend so as to eliminate any open-ended vortex lines. For example, when an airplane engine is started, a vortex usually forms ahead of each propeller, or the turbofan of each jet engine. One end of the vortex line is attached to the engine, while the other end usually stretches out and bends until it reaches the ground.

When vortices are made visible by smoke or ink trails, they may seem to have spiral pathlines or streamlines. However, this appearance is often an illusion and the fluid particles are moving in closed paths. The spiral streaks that are taken to be streamlines are in fact clouds of the marker fluid that originally spanned several vortex tubes and were stretched into spiral shapes by the non-uniform flow velocity distribution.

### Pressure in a vortex

The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it, in accordance with Bernoulli's principle. One can say that it is the gradient of this pressure that forces the fluid to follow a curved path around the axis.

In a rigid-body vortex flow of a fluid with constant density, the dynamic pressure is proportional to the square of the distance r from the axis. In a constant gravity field, the free surface of the liquid, if present, is a concave paraboloid.

In an irrotational vortex flow with constant fluid density and cylindrical symmetry, the dynamic pressure varies as PK/r2, where P is the limiting pressure infinitely far from the axis. This formula provides another constraint for the extent of the core, since the pressure cannot be negative. The free surface (if present) dips sharply near the axis line, with depth inversely proportional to r2. The shape formed by the free surface is called a hyperboloid, or "Gabriel's Horn" (by Evangelista Torricelli).

The core of a vortex in air is sometimes visible because water vapor condenses as the low pressure of the core causes adiabatic cooling; the funnel of a tornado is an example. When a vortex line ends at a boundary surface, the reduced pressure may also draw matter from that surface into the core. For example, a dust devil is a column of dust picked up by the core of an air vortex attached to the ground. A vortex that ends at the free surface of a body of water (like the whirlpool that often forms over a bathtub drain) may draw a column of air down the core. The forward vortex extending from a jet engine of a parked airplane can suck water and small stones into the core and then into the engine.

### Evolution

Vortices need not be steady-state features; they can move and change shape. In a moving vortex, the particle paths are not closed, but are open, loopy curves like helices and cycloids. A vortex flow might also be combined with a radial or axial flow pattern. In that case the streamlines and pathlines are not closed curves but spirals or helices, respectively. This is the case in tornadoes and in drain whirlpools. A vortex with helical streamlines is said to be solenoidal.

As long as the effects of viscosity and diffusion are negligible, the fluid in a moving vortex is carried along with it. In particular, the fluid in the core (and matter trapped by it) tends to remain in the core as the vortex moves about. This is a consequence of Helmholtz's second theorem. Thus vortices (unlike surface waves and pressure waves) can transport mass, energy and momentum over considerable distances compared to their size, with surprisingly little dispersion. This effect is demonstrated by smoke rings and exploited in vortex ring toys and guns.

Two or more vortices that are approximately parallel and circulating in the same direction will attract and eventually merge to form a single vortex, whose circulation will equal the sum of the circulations of the constituent vortices. For example, an airplane wing that is developing lift will create a sheet of small vortices at its trailing edge. These small vortices merge to form a single wingtip vortex, less than one wing chord downstream of that edge. This phenomenon also occurs with other active airfoils, such as propeller blades. On the other hand, two parallel vortices with opposite circulations (such as the two wingtip vortices of an airplane) tend to remain separate.

Vortices contain substantial energy in the circular motion of the fluid. In an ideal fluid this energy can never be dissipated and the vortex would persist forever. However, real fluids exhibit viscosity and this dissipates energy very slowly from the core of the vortex. It is only through dissipation of a vortex due to viscosity that a vortex line can end in the fluid, rather than at the boundary of the fluid.

## Further examples

### Summary

In the dynamics of fluid, a vortex is fluid that revolves around the axis line. This fluid might be curved or straight. Vortices form from stirred fluids: they might be observed in smoke rings, whirlpools, in the wake of a boat or the winds around a tornado or dust devil.

Vortices are an important part of turbulent flow. Vortices can otherwise be known as a circular motion of a liquid. In the cases of the absence of forces, the liquid settles. This makes the water stay still instead of moving.

When they are created, vortices can move, stretch, twist and interact in complicated ways. When a vortex is moving, sometimes, it can affect an angular position.

For an example, if a water bucket is rotated or spun constantly, it will rotate around an invisible line called the axis line. The rotation moves around in circles. In this example the rotation of the bucket creates extra force.

The reason that the vortices can change shape is the fact that they have open particle paths. This can create a moving vortex. Examples of this fact are the shapes of tornadoes and drain whirlpools.

When two or more vortices are close together they can merge to make a vortex. Vortices also hold energy in its rotation of the fluid. If the energy is never removed, it would consist of circular motion forever.

## Related Research Articles

In physics, the Coriolis force is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

In physics, angular velocity, also known as angular frequency vector, is a vector measure of rotation rate, that 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.

In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point, 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.

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.

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 the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

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.

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.

In fluid dynamics, the Taylor number (Ta) is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about an axis, relative to viscous forces.

In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal. This basic state is known as circular Couette flow, after Maurice Marie Alfred Couette, who used this experimental device as a means to measure viscosity. Sir Geoffrey Ingram Taylor investigated the stability of Couette flow in a ground-breaking paper. Taylor's paper became a cornerstone in the development of hydrodynamic stability theory and demonstrated that the no-slip condition, which was in dispute by the scientific community at the time, was the correct boundary condition for viscous flows at a solid boundary.

Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

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 first predicted by Lars Onsager in 1949 in connection with superfluid helium. Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction. 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.

Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength and frequency. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.

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.

In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids, for example liquid water and the air in the Earth's atmosphere. Unlike liquids, gases cannot form a free surface on their own. Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface.

In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum.

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 fluid dynamics, the Craik–Leibovich (CL) vortex force describes a forcing of the mean flow through wave–current interaction, specifically between the Stokes drift velocity and the mean-flow vorticity. The CL vortex force is used to explain the generation of Langmuir circulations by an instability mechanism. The CL vortex-force mechanism was derived and studied by Sidney Leibovich and Alex D.D. Craik in the 1970s and 80s, in their studies of Langmuir circulations.

## References

### Notes

1. "vortex". Oxford Dictionaries Online (ODO). Oxford University Press. Retrieved 2015-08-29.
2. "vortex". Merriam-Webster Online. Merriam-Webster, Inc. Retrieved 2015-08-29.
3. Ting, L. (1991). Viscous Vortical Flows. Lecture notes in physics. Springer-Verlag. ISBN   978-3-540-53713-7.
4. Kida, Shigeo (2001). Life, Structure, and Dynamical Role of Vortical Motion in Turbulence (PDF). IUTAMim Symposium on Tubes, Sheets and Singularities in Fluid Dynamics. Zakopane, Poland.
5. Vallis, Geoffrey (1999). Geostrophic Turbulence: The Macroturbulence of the Atmosphere and Ocean Lecture Notes (PDF). Lecture notes. Princeton University. p. 1. Archived from the original (PDF) on 2013-12-28. Retrieved 2012-09-26.
6. Clancy 1975 , sub-section 7.5
7. Sirakov, B. T.; Greitzer, E. M.; Tan, C. S. (2005). "A note on irrotational viscous flow". Physics of Fluids. 17 (10): 108102–108102–3. Bibcode:2005PhFl...17j8102S. doi:10.1063/1.2104550. ISSN   1070-6631.
8. Kheradvar, Arash; Pedrizzetti, Gianni (2012), "Vortex Dynamics", Vortex Formation in the Cardiovascular System, London: Springer London, pp. 17–44, doi:10.1007/978-1-4471-2288-3_2, ISBN   978-1-4471-2287-6 , retrieved 2021-03-16