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For other uses, see Vortex (disambiguation).

In fluid dynamics, a vortex is a region, in a fluid medium, in which the flow is mostly rotating on an axis line, the vortical flow that occurs either on a straight-axis or a curved-axis.[1][2] Moreover, the plurals of vortex are vortices and vortexes.[3]([4]) Vortices form in stirred fluids, such as liquid, gas, and plasma, thus the vortices evidenced in smoke rings, the whirlpool of the wake of a boat and of a paddle, the winds surrounding a tropical cyclone (hurricane), a tornado, and a dust devil, and vortices in the wake of an aeroplane; elsewhere, the vortex is a notable feature of the atmosphere of Jupiter.

Vortices are a major component of turbulent flow. In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. In each vortex, the fluid's flow velocity is greatest next to its axis, and decreases, in inverse proportion, to the distance from the axis. The vorticity (the curl of the flow velocity) is very high in a core region surrounding the axis, and nearly nil in the greater vortex; and the pressure drops with proximity to the axis of the vortex.





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 \vec \omega and expressed by the vector analysis formula \nabla \times \vec{\mathit{u}}, where \nabla is the nabla operator and \vec{\mathit{u}} is the local flow velocity.[5]

The local rotation measured by the vorticity \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, \vec \omega may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis.

Vortex types[edit]


A rigid-body vortex
    \vec{\Omega} = (0, 0, \Omega) , \quad \vec{r} = (x, y, 0) ,
    \vec{u} = \vec{\Omega} \times \vec{r} = (-\Omega y, \Omega x, 0) ,
    \vec \omega = \nabla \times \vec{u} = (0, 0, 2\Omega) = 2\vec{\Omega} .
An irrotational vortex
  • 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 \vec \omega is zero at any point not on that axis, and the flow is said to be irrotational.
    \vec{\Omega} = (0, 0, \alpha r^{-2}) , \quad \vec{r} = (x, y, 0) ,
    \vec{u} = \vec{\Omega} \times \vec{r} = (-\alpha y r^{-2}, \alpha x r^{-2}, 0) ,
    \vec{\omega} = \nabla \times \vec{u} = 0 .

Irrotational vortices[edit]


In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern, where the flow velocity u is inversely proportional to the distance r. For that reason, 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, \Gamma, for any contour that does enclose the axis once.[6] The tangential component of the particle velocity is then u_{\theta} = \Gamma/(2 \pi r). The angular momentum per unit mass relative to the vortex axis is therefore constant,  r u_{\theta} = \Gamma/(2 \pi).


u_{\theta} = (1 - e^{-r^2/(4\nu t)})\Gamma/(2 \pi r).

In an irrotational vortex, fluid moves at different speed in adjacent streamlines, so there is friction and therefore energy loss throughout the vortex, especially near the core.

Rotational vortices[edit]

The cloud vortex Saturn's hexagon is at the north pole of the planet Saturn.




Vortex geometry[edit]




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. This is the case, for example, of the spiral arms of galaxies and hurricanes.

Pressure in a vortex[edit]

A Plughole vortex










Two-dimensional modeling[edit]

When the particle velocities are constrained to be parallel to a fixed plane, one can ignore the space dimension perpendicular to that plane, and model the flow as a two-dimensional flow velocity field on that plane. Then the vorticity vector \vec \omega is always perpendicular to that plane, and can be treated as a scalar. This assumption is sometimes made in meteorology, when studying large-scale phenomena like hurricanes.

The behavior of vortices in such contexts is qualitatively different in many ways; for example, it does not allow the stretching of vortices that is often seen in three dimensions.

Further examples[edit]

A visible vortex formed when a C-17 uses high engine power at slow speed on a wet runway.

See also[edit]



  1. ^ Ting, L. (1991). Viscous Vortical Flows. Lecture notes in physics. Springer-Verlag. ISBN 3-540-53713-9. 
  2. ^ Kida, Shigeo (2001). Life, Structure, and Dynamical Role of Vortical Motion in Turbulence. IUTAM Symposium on Tubes, Sheets and Singularities in Fluid Dynamics. Zakopane, Poland. 
  3. ^ The Oxford English Dictionary
  4. ^ The Merriam Webster Collegiate Dictionary
  5. ^ Vallis, Geoffrey (1999). Geostrophic Turbulence: The Macroturbulence of the Atmosphere and Ocean Lecture Notes. Lecture notes. Princeton University. p. 1. Retrieved 2012-09-26. 
  6. ^ a b Clancy 1975, sub-section 7.5


External links[edit]