Vortical Motion

Vortical Motion

 

(turbulence), the motion of a liquid or gas in which small particles of the stream are displaced not only translationally but also rotate about some instantaneous axis.

The great majority of flow patterns of a liquid or gas that occur in nature or are created in industry are vortical motion. For example, the flow of water through a tube is always vortical motion, whether laminar flow or turbulent flow is involved. The rotation of elemental volumes here is due to the fact that the flow velocity is zero on the surface of the wall because of the adhesion of liquid to the walls; the flow velocity increases rapidly with increasing distance from the walls, so that the velocities of adjacent layers differ significantly. As a result of the retarding action of the bottom layer and the accelerating action of the top layer, the particles begin to rotate—that is, vortical motion takes place. Examples of vortical motion include air eddies in the atmosphere, which often take on enormous size and form tornadoes and cyclones, water eddies that form behind bridge abutments, and whirlpools in river water.

Vortical motion may be characterized quantitatively by the vector ω of the angular rotation velocity of the particles, which depends on the coordinates of the point in the stream and on time. The vector ω is called the curl of the medium at that point; if ω = 0 in some region of the flow, then the flow pattern in that region is irrotational. The rotating particles of the stream may form vortex tubes or separate layers. The vortex tube can have neither a beginning nor an end within the fluid; it can either be closed upon itself (a vortex ring) or must begin and terminate on the interfaces of the fluid—for example, on the surface of a streamlined body, on the surface of the vessel in which the liquid is contained, on the earth’s surface in the case of tornadoes, and on the surface of the water or on the river bottom in the case of eddies in a stream of water.

The presence of eddies in a liquid causes the appearance of additional velocities in it. When a system of eddies is present in a liquid, the eddies affect the motion of the other eddies present. Thus, for example, two eddies of equal intensity G but opposite in sign impart to each other velocities v of equal magnitude and identical direction—that is, they move translationally—but two eddies whose intensities are identical in absolute value and sign rotate about an axis that passes through the midpoint of the distance separating them.

If two vortex rings have a common axis and the same direction of rotation, then the front ring will increase in diameter and will slow down because of the velocities imparted to the rear ring; meanwhile, the rear will shrink in diameter and pass through the front ring—that is, the two rings change places, and the entire process repeats (“play” of vortex rings).

The action of the frictional forces in any viscous fluid causes the vortices to gradually decrease in intensity and eventually to die away. Since the viscosity of water—and especially air—is low, eddies may persist for a fairly long time in them; for example, tornadoes sometimes travel long distances. In an inviscid medium (an ideal fluid), vortices could neither reappear nor decay. In mediums of low viscosity (water and air), vortex motion arises in parts of the flow pattern in which the force of viscosity is manifested most intensely: in the layer near the surface of the streamlined body—the boundary layer, which is filled with intensely swirling fluid. The vortices in the boundary layer detach from the surface of the streamlined body and leave behind a wake in the form of varied formations (vortex layers or streets). Vortices appearing in response to the motion of a body through a medium cause a considerable part of the lift and drag forces acting on the body. Consequently, the study of vortical motion is of great significance in design calculations of airplane wings, propellers, and turbine blades.

REFERENCES

Prandtl, L. Gidroaeromekhanika, 2nd ed. Moscow, 1951. (Translated from German.)
Fabrikant, N. Ia. Aerodinamika. Moscow, 1964.
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Because the vortical motion is stationary, the linear velocity [[bar.
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