Influid dynamics, avortex (pl.:vortices orvortexes)[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 insmoke rings,whirlpools in thewake of a boat, and the winds surrounding atropical cyclone,tornado ordust devil.
Vortices are a major component ofturbulent flow. The distribution of velocity,vorticity (thecurl of the flow velocity), as well as the concept ofcirculation 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.
In the dynamics of fluid, a vortex is fluid that revolves around the line of flow. This flow of fluid might be curved or straight. Vortices form from stirred fluids: they might be observed insmoke rings,whirlpools, in the wake of a boat or the winds around atornado ordust devil.
Vortices are an important part ofturbulent 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 anddrain 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.
A key concept in the dynamics of vortices is thevorticity, avector that describes thelocal 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 theright-hand rule) while its length is twice the ball'sangular velocity. Mathematically, the vorticity is defined as the curl (or rotational) of thevelocity field of the fluid, usually denoted by and expressed by thevector analysis formula, where is thenabla operator and is the local flow velocity.[5]
The local rotation measured by the vorticity 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, may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis.
In theory, the speedu of the particles (and, therefore, the vorticity) in a vortex may vary with the distancer from the axis in many ways. There are two important special cases, however:
In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern[citation needed], where the flow velocityu is inversely proportional to the distancer. Irrotational vortices are also calledfree vortices.
For an irrotational vortex, thecirculation 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. The angular momentum per unit mass relative to the vortex axis is therefore constant,.
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 asr goes to zero. Within that region, the flow is no longer irrotational: the vorticity becomes non-zero, with direction roughly parallel to the vortex axis. TheRankine vortex is a model that assumes a rigid-body rotational flow wherer is less than a fixed distancer0, 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 viscousNavier–Stokes equations, known as aLamb–Oseen vortex.
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 speedw about its vertical axis, the water will eventually rotate in rigid-body fashion. The particles will then move along circles, with velocityu equal towr.[6] In that case, the free surface of the water will assume aparabolic shape.
In this situation, the rigid rotating enclosure provides an extra force, namely an extra pressuregradient in the water, directed inwards, that prevents transition of the rigid-body flow to the irrotational state.
Vortex structures are defined by theirvorticity, the local rotation rate of fluid particles. They can be formed via the phenomenon known asboundary 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 theno-slip condition. This rapid negative acceleration creates aboundary 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 (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 combattingpressure 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 thetrailing edge of abluff 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 asplash effect. The velocity streamlines are immediately deflected and decelerated so that the boundary layer separates and forms atoroidal vortex ring.[8]
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 eachvortex 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 avortex 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 toHelmholtz'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 closedtorus-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 eachpropeller, or theturbofan of eachjet 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.
The fluid motion in a vortex creates a dynamicpressure (in addition to anyhydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it, in accordance withBernoulli'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 constantdensity, the dynamic pressure is proportional to the square of the distancer from the axis. In a constantgravity field, thefree surface of the liquid, if present, is a concaveparaboloid.
In an irrotational vortex flow with constant fluid density and cylindrical symmetry, the dynamic pressure varies asP∞ −K/r2, whereP∞ 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 tor2. The shape formed by the free surface is called ahyperboloid, or "Gabriel's Horn" (byEvangelista Torricelli).
The core of a vortex in air is sometimes visible because water vaporcondenses as the low pressure of the core causesadiabatic 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.
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 likehelices andcycloids. 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 besolenoidal.
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 ofHelmholtz's second theorem. Thus vortices (unlikesurface waves andpressure 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 ringtoys andguns.
Two or more vortices that are approximately parallel and circulating in the same direction will attract and eventually merge to form a single vortex, whosecirculation will equal the sum of the circulations of the constituent vortices. For example, anairplane wing that is developinglift will create a sheet of small vortices at its trailing edge. These small vortices merge to form a singlewingtip vortex, less than onewing chord downstream of that edge. This phenomenon also occurs with other activeairfoils, such aspropeller 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 exhibitviscosity 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.
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