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 boat, or the winds surrounding a tornado or dust devil. 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 characterize 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 organize 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 with it some angular and linear momentum, energy, and mass. Contents 1 Properties 1.1 Vorticity
1.2
1.2.1 Irrotational vortices 1.2.2 Rotational vortices 1.3
2 Further examples 2.1 Summary 3 See also 4 References 4.1 Notes 4.2 Other 5 External links Properties[edit] Vorticity[edit] The
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 ω → displaystyle vec omega and expressed by the vector analysis formula ∇ × u → displaystyle nabla times vec mathit u , where ∇ displaystyle nabla is the nabla operator and u → displaystyle vec mathit u is the local flow velocity.[5] The local rotation measured by the vorticity ω → 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, ω → displaystyle vec omega may be opposite to the mean angular velocity vector of the fluid
relative to the vortex's axis.
A rigid-body vortex 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. Ω → = ( 0 , 0 , Ω ) , r → = ( x , y , 0 ) , displaystyle vec Omega =(0,0,Omega ),quad vec r =(x,y,0), u → = Ω → × r → = ( − Ω y , Ω x , 0 ) , displaystyle vec u = vec Omega times vec r =(-Omega y,Omega x,0), ω → = ∇ × u → = ( 0 , 0 , 2 Ω ) = 2 Ω → . displaystyle vec omega =nabla times vec u =(0,0,2Omega )=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 ω → displaystyle vec omega is zero at any point not on that axis, and the flow is said to be irrotational. Ω → = ( 0 , 0 , α r − 2 ) , r → = ( x , y , 0 ) , displaystyle vec Omega =(0,0,alpha r^ -2 ),quad vec r =(x,y,0), u → = Ω → × r → = ( − α y r − 2 , α x r − 2 , 0 ) , displaystyle vec u = vec Omega times vec r =(-alpha yr^ -2 ,alpha xr^ -2 ,0), ω → = ∇ × u → = 0. displaystyle vec omega =nabla times vec u =0. Irrotational vortices[edit] Pathlines of fluid particles around the axis (dashed line) of an ideal irrotational vortex. (See animation) 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 u θ = Γ 2 π r displaystyle u_ theta = tfrac Gamma 2pi r . The angular momentum per unit mass relative to the vortex axis is therefore constant, r u θ = Γ 2 π displaystyle ru_ theta = tfrac Gamma 2pi . However, the ideal irrotational vortex flow 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 ω → displaystyle vec omega becomes non-zero, with direction roughly parallel to the vortex axis.
The
u θ = ( 1 − e − r 2 4 ν t ) Γ 2 π r . displaystyle u_ theta =left(1-e^ frac -r^ 2 4nu t right) frac Gamma 2pi r . Rotational vortices[edit] The cloud vortex
A rotational vortex – one which has non-zero vorticity away from the
core – can be maintained indefinitely in that state only through the
application of some extra force, that is not generated by the fluid
motion itself.
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 evolution of the rigid-body flow to the
irrotational state.
A Plughole 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 P∞ − K/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 of a plume of water vapor caused by condensation in the low pressure and low temperature of the core; the spout 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. Stability in a vortex[edit] The vortices that you create becomes more stable after you stop shaking the container because as you shake the forces acting on the whole fluid are uneven. When you stop shaking the cup or put it down on a surface, the vortex is able to evenly distribute force to the liquid. [7] Evolution[edit] 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 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[edit] The visible core of a vortex formed when a C-17 uses high engine power (reverse-thrust) at slow speed on a wet runway. Von Kármán vortex streets formed off the island of Tristan da Cunha In the hydrodynamic interpretation of the behaviour of electromagnetic
fields, the acceleration of electric fluid in a particular direction
creates a positive vortex of magnetic fluid. This in turn creates
around itself a corresponding negative vortex of electric fluid. Exact
solutions to classical nonlinear magnetic equations include the
Landau-Lifshitz equation, the continuum Heisenberg model, the Ishimori
equation, and the nonlinear Schrödinger equation.
Bubble rings are underwater vortex rings whose core traps a ring of
bubbles, or a single donut-shaped bubble. They are sometimes created
by dolphins and whales.
The lifting force of aircraft wings, propeller blades, sails, and
other airfoils can be explained by the creation of a vortex
superimposed on the flow of air past the wing.
Summary[edit] 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 liquids 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, is 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 together 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. See also[edit] Physics portal Artificial gravity
Batchelor vortex
Biot–Savart law
Coordinate rotation
Cyclonic separation
Eddy
Gyre
Helmholtz's theorems
History of fluid mechanics
Horseshoe vortex
Hurricane
Kelvin–Helmholtz instability
Quantum vortex
Rankine vortex
Shower-curtain effect
Strouhal number
Vile Vortices
Von Kármán vortex street
References[edit] Notes[edit] ^ "vortex". Oxford Dictionaries Online (ODO). Oxford University Press.
Retrieved 2015-08-29.
^ "vortex". Merriam-Webster Online. Merriam-Webster, Inc. Retrieved
2015-08-29.
^ Ting, L. (1991). Viscous Vortical Flows. Lecture notes in physics.
Springer-Verlag. ISBN 3-540-53713-9.
^ Kida, Shigeo (2001). Life, Structure, and Dynamical Role of Vortical
Motion in
Other[edit] Loper, David E. (November 1966). An analysis of confined magnetohydrodynamic vortex flows (PDF) (NASA contractor report NASA CR-646). Washington: National Aeronautics and Space Administration. LCCN 67060315. Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge Univ. Press. Ch. 7 et seq. ISBN 9780521098175. Falkovich, G. (2011). Fluid Mechanics, a short course for physicists. Cambridge University Press. ISBN 978-1-107-00575-4. Clancy, L.J. (1975). Aerodynamics. London: Pitman Publishing Limited. ISBN 0-273-01120-0. De La Fuente Marcos, C.; Barge, P. (2001). "The effect of long-lived vortical circulation on the dynamics of dust particles in the mid-plane of a protoplanetary disc". Monthly Notices of the Royal Astronomical Society. 323 (3): 601–614. Bibcode:2001MNRAS.323..601D. doi:10.1046/j.1365-8711.2001.04228.x. External links[edit] Wikimedia Commons has media related to Vortex. Optical Vortices Video of two water vortex rings colliding (MPEG) Chapter 3 Rotational Flows: Circulation and Turbulence Vortical Flow Research Lab (MIT) – Study of flows found in nature and part of the Department of Ocean Engineering. Authority control |