Vortex (fluid dynamics)

Fig. 1: Air vortex: wake vortex (colored)
Fig. 2: A vortex is a special case of a vortex

In fluid mechanics, a vortex or vortex is a rotating movement of fluid elements around a straight or curved axis of rotation. The term vortex is rather intuitive - see pictures - and cannot be precisely formulated mathematically.

In fluids with low viscosity (air and water) the flow velocity is greatest in the larger eddies in the center and decreases in inverse proportion to the distance from the center. The vortex strength is large in the center and almost zero in the outer area of ​​the vortex, which is why fluid elements there hardly rotate around themselves. The opposite is true for the pressure that is lowest in the center. Eddies tend to form elongated vortex tubes that can move, twist, bend and stretch with the current.

In meteorology and in the aerodynamic vortex play an important role.

phenomenology

This section is of phenomenology , i. H. dedicated to the physically given vortex phenomena.

Formation of the vortex

Fig. 3: hydrodynamic simulation of a Rayleigh-Taylor instability
Fig. 4: Rayleigh-Bénard convection cells

Eddy currents are introduced through (curved) walls, fanned by external forces, forced through conservation of angular momentum, or are a consequence of the balancing effort of fluids left to themselves ( Second Law of Thermodynamics ).

Round bezels like the glass in Fig. 2 or the step jump in a water roller guide currents in a circle. In aerodynamics and hydrodynamics, important eddies arise when a vehicle moves through a fluid, especially when planes or cars move through air or ships through water. As a result of flow breaks on the A-pillar of cars or on a cylinder (see animation in Fig. 7) permanent eddy currents arise. Air eddies are not only involved in the generation of audible sound in organ pipes .

One force that creates eddies are inertial forces in fluids that rotate as a whole, such as the earth's atmosphere . The Coriolis effect deflects currents within the body of air into a circular motion. This is the reason why high and low pressure areas form eddies in the atmosphere. The polar vortex and jet stream are also vortices created by inertial forces.

When fluids strive towards a center, fluid elements with angular momentum cannot simply plunge straight into it: The preservation of the angular momentum first forces them onto a circular orbit around the center, whereby protoplanetary disks in interstellar space, i.e. eddies of dust and gas, arise around young stars in the center.

Vortices are only indirectly guided by external influences when fluid masses with different properties meet. The fluid masses can differ in their temperature, their speed or their density, among other things. At the interfaces between the fluid masses, if there is a sufficient difference in the properties (at the tipping point), instabilities occur that lead to eddies and, in the further course, to turbulent flow in which eddies of different sizes intensively mix the masses. Such a phenomenon is the Kelvin-Helmholtz instability between two differently fast currents or the Rayleigh-Taylor instability between two differently heavy liquids (Fig. 3). The mixing leads to an equalization of the gradient, unless external influences maintain the gradient. Then permanent, circling convection cells can arise ( trade wind , Walker circulation , convection zone of the sun , Fig. 4).

Properties of rotary motion

Fig. 5: A horseshoe vortex behind a glider created by a theoretical, constant lift force on the wing

Eddy currents often do not form a steady flow, so they can change their shape and move as a whole. In this case, the paths of the fluid elements are not closed curves, but rather helical lines or cycloids . The axis of rotation of the vortex, which is analogous to the current line defined vortex line may be a curved, twisting and moving as a whole line ( Tornado ). The spinning motion can be combined with radial, center-to-center or away-center flow, creating vortices and spirals like drains. A purely circling vortex without a radial velocity component is called a source-free .

Because of the conservation of angular momentum, vortices cannot easily stop or start rotating. Once the vortex has dissolved, it disappears, which is the statement of Helmholtz's first vortex law . The circular motion ( circulation ) of a ring made of fluid elements is a conservation quantity ( Kelvin's vortex law ), which is constant over the length of a vortex tube (third Helmholtz vortex law). Therefore, eddies tend to form extended vortex tubes in the fluid, which is clear, for example, from the horseshoe vortex, see Fig. 5. However, due to friction effects and dissipation , real eddies dissolve over time and the circulation in eddies decreases.

The fluid elements are carried along by the vortex, which is a consequence of Helmholtz's second law of vortices. Eddy currents can transport mass, angular momentum and energy over considerable distances, which can be many times their size, with only minor losses ( smoke rings ).

In larger eddies, the fluid elements do not rotate around themselves, but are shifted parallel in a circle. This fact leads to shearings between the fluid elements which increase towards the center of the vortex. Viscosity reduces this shear in the center of the vortex or in small vortices, so that there is a quasi-rigid rotation there. These frictional effects dissipate the rotational energy and ultimately lead to the dissolution of the eddies (see energy cascade of turbulent flows and # Rankine eddies below).

Pressure and temperature distribution in eddies

Fig. 6: Condensation of water vapor in the center of eddies behind a Boeing B-747.

In a fluid with low viscosity, in a steady flow, neglecting external forces, the sum of the kinetic energy and the static pressure , the total pressure , is constant along a streamline. The static pressure is the pressure felt by a fluid element that is moved along with the flow. In larger eddies, the flow velocity increases towards the center, which is why the static pressure there decreases. In a real gas, in a constant volume, decreasing pressure goes hand in hand with decreasing temperature, which is why the temperature is lowest in the center of the vortex. Therefore, the center of such vortices is due to contrails sometimes visible, see Fig. 6. The vortex strength is in the edge vortices at the wingtips greatest and form is clearly defined vortex tubes. The intensity of the vertebral tubes decreases towards the trunk and the vertebral tubes are less clearly defined.

Eddies and turbulence

Eddy currents are the main component of turbulent flows, but not every eddy is part of a turbulent flow. Turbulent currents contain eddies on all size scales that seem to move in a disordered manner. A distinction that has proven itself in practice is made with the Reynolds equations : The physical quantities - here the speed is of particular interest - are divided into a mean value and a statistical fluctuation value. The fluctuation value deals with the random, fluctuating eddies, while the time-independent mean value contains the stationary eddies.

Fig. 7: Animation of the formation of a vortex street

A borderline case is the periodic separation of eddies, such as in Kármán's vortex street , which occurs behind a cylinder in which the Reynolds number is not too high , see animation in Fig. 7. In the case of the cylinder, which is initially laminar, eddy-free, periodic separation forms as the flow velocity increases: Es alternating left- and right-rotating vertebrae detach in a characteristic pattern. These vortices are neither stationary nor chaotic. As the flow rate increases, the vortex street changes into turbulent flow: more vortices occur, so that the flow resistance increases. During the transition to turbulence, the size of the eddies and the times at which they are separated vary more and more. When the turbulence is fully developed, eddies are present on all size scales. This transition from laminar, eddy-free flow to turbulent flow with a transition area that has clear structures is a widespread phenomenon.

Vortex types

Potential vortex

Fig. 8: Potential vortex with streamlines (blue) and fluid elements (turquoise)

The potential vortex or free vortex is a classic example of a rotation-free potential flow , see Fig. 8. Large vortices in fluids with low viscosity are well described with this model. Examples of a potential vortex are the bathtub drain far from the discharge, but also, to a good approximation, a tornado . The angular velocity in these eddies is greatest in their center where on the other hand the pressure is at a minimum. Because of this speed distribution, which differs from a rigid body movement, the fluid elements are deformed.

Because the rotation "red" of the velocity field according to ${\ displaystyle {\ vec {v}}}$

${\ displaystyle | \ operatorname {red} \, {\ vec {v}} | = 0}$

disappears, the fluid elements always point in the same direction despite their circular motion in the vortex. If you are mathematically accurate, the above equation only applies outside the center, i.e. for while taking the center with you, with the two-dimensional Dirac delta function and the vortex strength${\ displaystyle (x, y) \ neq \ operatorname {0},}$${\ displaystyle \ operatorname {rot} \, {\ vec {v}} = {\ vec {\ Phi}} _ {0} \ cdot \ delta (x, y)}$${\ displaystyle {\ vec {\ phi}} _ {0}.}$

Because of this total freedom of rotation for all points outside the center, locally it cannot be concluded that there is a vortex movement. Only the observation of a larger area or over longer periods of time allows these eddies to be recognized. In computational fluid mechanics , the smallest volume considered is the finite volume for which so-called vortex criteria were formulated to distinguish vortices from shear layers.

Solid body vortex

Fig. 9: Solid body vortex with streamlines (blue) and fluid elements (turquoise)

A solid vortex forms z. B. If, after a correspondingly long start-up time, a liquid rotates in a vessel on a turntable with constant angular velocity as a rigid body, see Fig. 9. Fluid particles that are moved along rotate around their own axis without being deformed. In a solid vortex is ${\ displaystyle \ omega}$

${\ displaystyle | \ operatorname {red} \, {\ vec {v}} | = 2 \ omega \ neq 0 \ ,.}$

As in a potential vortex, all fluid particles move on concentric circular paths, but the speed and pressure distribution is completely different: the speed is greatest on the outside and slowest on the inside, so that the pressure is lowest on the outside and highest on the inside.

Rankine vortex

Fig. 10: Sketch of a Rankine vortex

The Rankine vortex according to William John Macquorn Rankine is a vortex model that connects the potential vortex in the outer area with the solid-state vortex in the center, see Fig. 10. The potential vortex describes an outflow flow in the outer area well, where the circumferential speed decreases with the radius (blue curve) and there is no rotation ( red curve). As the fluid elements approach the center, unrealistically high shear rates develop in the fluid in the potential vortex. In the Rankine vortex, tenacity forces below a certain core radius r 0 prevent the shearing and a quasi-rigid rotation occurs. Within the core radius, the peripheral speed is therefore proportional to the radius and the rotation ω is constantly unequal to zero. In real fluids, the transition from the outer to the core flow will not be abrupt but rather smooth (dashed red and blue curves). The effect of the missing outside and inside rotation of the particles is indicated by matches floating with them. ${\ displaystyle v _ {\ varphi}}$${\ displaystyle \ omega = | \ operatorname {red} {\ vec {v}} | \ ,,}$

Hamel-Oseen'scher vortex

Fig. 11: Circumferential speed of the Hamel-Oseen vortex in comparison with the rigid rotation and the potential vortex

The Hamel-Oseen vortex (by Carl Wilhelm Oseen , Georg Hamel ) is a vortex model that exactly satisfies the Navier-Stokes equations , which describe the flow of real fluids well. The fluid flows in a purely circular but time-dependent, unsteady manner around the vortex center. The viscosity consumes the kinetic energy of the vortex over time and the flow velocity decreases monotonically over time. At the beginning of the movement or in the limit of vanishing viscosity, the vortex is a potential vortex. Otherwise the velocity profile of the Hamel-Oseen vortex is limited and corresponds to a Rankine vortex in the vortex core as well as in the outer area, see Fig. 11.