The Many Faces of Vortices

Perhaps the best known vortex is the vortex in the outlet of a bath tub. Almost every day we can see the remnants of vortices in the sky, when airplanes are passing by. Vortices are formed when the air flows round the airplane wing. Besides, the jet streams of the engines — the fuselage vortices (formed in the junction between the fuselage and wing) — together with the unstable shear layer are rolled into rather strong tip vortices, downstream from the wing. The latter can clearly be recognized in the wake of an agricultural plane flying at a low altitude over a field and emitting insecticides through a spreading system mounted under the wing.

Wakes of vortices in the sky

In general, the near wake of the flow just downstream from the wing is characterized by rather irregular vortex structures, containing many smaller vortices with their axes orientated in the flight direction. These vortices can be seen in the laser-light-sheet picture on the previous page.

At cruising speed at a high altitude the tip vortices are visualized by the exhaust gases of jet engines. In the combustion process of kerosene in the engine, carbon dioxide, water vapor, nitrous oxides, and soot are generated. At the altitudes at which planes fly, the temperatures are low, so water vapor condenses and freezes or sublimes, and clouds of ice are formed. The clouds are captured by the tip vortices generating those long vapor trails which are often visible in the clear sky.

How long the vapor trails can exist depends on many influence factors, mainly on temperature, prevailing winds, the humidity of air and on other factors. Sometimes they last several minutes; on other occasions, several hours. Besides, it has been noticed that under certain conditions vapor trails form vortex-ring-like structures.

This phenomenon is usually referred to as the Crow Instability, named after the American scientist S. C. Crow, who, in 1970, was the first to show, by theoretical considerations, that the mutual interaction of two tip vortices could lead to the amplification of displacement perturbations, the axial wavelength of which is typically several times the initial distance separating the tip vortices. Later, in 1997, this problem was investigated by T. Leweke and C. H. K. Williamson in a laboratory experiment in France. The researchers entirely succeeded in confirming the Crow theory.

Von Kármán Vortex Street

There are other vortices to be found in the atmosphere. With the aid of the satellite Landsat 7, a von Kármán Vortex Street was detected on the lee side of the Alejandro-Selkirk Isle of the Juan Fernandez Archipelago in the Pacific Ocean, about 800 km west of Chili.

The conditions for the formation of such a sequence of vortices were first formulated by the Hungarian scientist Th. von Kármán in 1911. He found this array of vortices in the wake of a circular cylinder, with its axis positioned at a right angle to the oncoming flow.

Two things are of particular interest about the formation of a whole sequence of counter-rotating vortices. First, the vortex street of the Alejandro Selkirk Isle would have never been discovered without satellite technology; the other point is that such a small rocky island (its area is about 44 sq km, its highest mountain is 1319 m high, well reaching into the clouds) was able to generate a huge von Kármán Street.

Von Kármán Vortex Streets are still being studied since periodic vortex shedding can be dangerous in many instances: for example, in 1940 the Tacoma Narrows Bridge in the Sate of Washington, USA, was destroyed by a von Kármán Street.

Vortex streets offer an unbelievably large variety of arrangements. To give just one example, we can mention a study conducted by G. Erhardt of the Aerodynamisches Institut of the RWTH, Aachen, Germany. In 1979, Erhardt studied the flow through and around a ring placed at a right angle to the flow.

The vortex structures shed from the inner and outer edges of the ring are vortex rings which form pairs similar in shape to the clouds that were observed on the leeward side of the Alejandro-Selkirk Isle. Obviously, the small size of the ring, measured in cm, is unimportant for the initiation of the vortex street. It functions in the same way as the island, whose coastline is several kilometers long.

Killer Hurricanes

Although whirlwinds, cyclones, hurricanes, and tornadoes do not exactly belong to the topic under consideration, at a certain time of their life they can be categorized as slender vortices, before they grow into monster storms, twisters, or killer hurricanes as they are often called in the USA.

Occasionally, small funnel clouds may appear in Europe: they can be seen in the photographs of the German Meteorological Service. The funnels rise from the ground to the upper cloud layer. When they mature to monster storms they can acquire more than a trillion watts of power in their winds. They have recently occurred more frequently than in former times and have devastated large areas, as in 2005, when the hurricane Katrina flooded New Orleans and the surrounding region.

Small slender vortices can also be simulated in laboratory experiments, similarly to von Kármán Vortex Street discussed earlier. In 1990, T. Sawada and T. Leweke of the Aerodynamisches Institut of the RWTH, Aachen, generated a slender vortex in the form of a starting vortex. The experiment was carried out in a glass container with a quadratic cross-section, equipped with a flap extending from one side wall to the other on the opposite side, and hinged to the third. The container was filled with water, and a starting vortex was generated by rotating the flap by a certain angle. The flow was visualized with the help of a light-sheet technique, by injecting dyes of different colors at the trailing edge of the flap at six axial positions. Photographs of the flow were taken in two selected illuminated planes, one parallel to and the other normal to the axis of the vortex.

In the series of photos taken “half face”, the dyes make it possible for everyone to see the formation, development and disintegration of the originally slender vortex due to its self-induced axial flow motion. The irregular structure of the flow in the core of the vortex is clearly seen in the pictures taken “en face” — in the panel parallel to the axis of the vortex. These pictures bear a resemblance to the pictures of hurricanes taken from satellites or from the International Space Station. In the second phase of experiments the container was turned by ninety degrees, so that the axis of the flap was vertical. The upper wall was removed and quartz sand was spread over the bottom. Then the container was filled with water again, and the investigation of vortex formation in the fluid with a free surface over the solid bottom was carried out, the sand playing the role of a dye, “marker” of vertical motion.

When the flap was turned, a starting vortex was formed just like in the experiment described before. Due to the drop of pressure in the vortex center the sand was shoveled from the bottom. If the speed of rotation of the flap was large enough, the core remained almost straight over a certain distance just above the bottom; higher up, it took on the shape of a spiral. Although the curved part of the core was almost horizontal and parallel to the bottom, the sand grains did not sink but remained in the core.

Similar core deformations were also observed in tornadoes in North America as, for example, reported by A. B. C. Whipple in his book Storm. There the author showed a series of photographs describing the time-development of the tornado observed on July 6, 1978, in North Dakota (USA). The core of the tornado with a funnel cloud around it was visualized by water vapor, yielding almost the same shape as generated in the miniature tornado in the experiment described above.

Bubble and Spiral

When the vortex core begins to deviate from a straight line and forms a spiral, it is called a spiral-type vortex breakdown. It also occurs in other types of flows, for example, in flows in turbo machines. One example of the internal vortical flow is the swirling flow in the model of a water turbine’s diffuser investigated by Swiss scientists. The vortex core which is formed when the flow goes through the diffuser changes and takes on the shape of a spiral.

The swirling flow in a pipe with a variable cross-section is another example. A spiral-type breakdown is often preceded by another form of vortex breakdown, the bubble-type. A breakdown process is initiated when the pressure in the pipe increases in the axial direction. At the onset of the process a double-ring vortex structure is formed, with one ring being located on the downstream side of the bubble, visible in the picture, and the other inside the bubble, upstream of the downstream ring. The pressure rises until a stagnation point is formed, downstream from which the fluid starts flowing in the opposite direction.

Before breakdown the bubble is almost axially symmetric; yet, later the flow loses its symmetry and changes into a spiral-type breakdown. Before the spiral is formed, the downstream vortex of the bubble is shed and moves downstream. The symmetry is lost and the vortex ring disintegrates, leaving a region of high pressure downstream from the stagnation point, around which the deformed stream-tube of the core is spiraling. Although the swirling pipe flow has been studied extensively in the past two decades, the conditions necessary for the transition from a bubble- to spiral-type breakdown are not known as yet.

In 1978, J. H. Faler and S. Leibowich, in the USA, were able to carry out an experiment in which the bubble and spiral remained in a stable position in the flow. Another twenty years passed before this double configuration of vortex breakdown was successfully simulated on a high-performance computer with a numerical solution of the Navier-Stokes equations. In 1997, M. Weimer of the Aerodynamisches Institut of the RWTH, Aachen, was able to produce the same flow patterns as were observed in the Faler-Leibovich experiment. The computation revealed that after the stagnation point was formed on the axis of the vortex, the bubble moved somewhat upstream. It, then, did not migrate any further but remained in the same position.

Airplanes and spaceplanes

Vortex breakdown can also occur on wings of supersonic airplanes and space transportation systems. Their wings often have a triangular shape. They generate a vortex system on their leeward side, enhancing the lift at high angles of attack. The vortex system consists of a large primary vortex, two or three smaller vortices, a secondary, tertiary, and sometimes even a quaternary vortex, as well as a shear layer vortex. The low pressure in the core of the primary vortex increases the lift in a non-linear fashion.

Since at high angles of attack the pressure on the upper part of the wing increases in the main flow in the direction towards the trailing edge, the pressure rise affects the vortex structures. If it is high enough, the primary vortex tends to break down.

Using flow visualization techniques, W. Limberg and A. Stromberg of the Aerodynamisches Institut of the RWTH, Aachen, showed that the breakdown modes observed in the swirling pipe flow also occurred on the leeward side of the model of a space transportation system.

Bifurcated pipes

After G. Hagen, in 1839, and J. Poiseulle, in 1840, published their papers on pipe flows, most of what happened in these flows could be described with the formulas these authors had derived, and as far as vortical structures were concerned pipe flows did not seem to offer any excitement. The situation is, however, completely different if one considers curved pipes or bifurcated pipes.

Although the former add only curvature to the problem, it makes all the difference in comparison to flows in straight pipes. Bifurcated pipe flows have an additional complexity in the sense that several flow modes are possible, depending on the direction of the flow motion and their magnitude. This problem was investigated by R. Neikes of the Aerodynamisches Institut of the RWTH, Aachen, in 1990.

The curved pipe generates a secondary flow varying from cross-section to cross-section, which together with the main flow leads to a pigtail-like twisting of the streamlines. The twisting of the streamlines seems to hint that vortices are being formed owing to the bending of the pipe. This is evidenced by the pictures of a dyed flow. The twisting of the streamlines also takes place if a second pipe is joined to the mother pipe at a right angle. Strong unsteady vortex formation is also observed if the inflow is through both sides of the mother branch. The ring- and horseshoe-like vortex structures, which are periodically formed in the bifurcated pipe flow travel downstream with the main flow. The frequency of formation strongly depends on the volume fluxes and on the Reynolds number of the flow (ratio of characteristic inertia and viscous forces).

Vortices in automotive engine

In recent years the formation of vortex structures has also been investigated with the aim to see whether the combustion process in automotive engines could be improved by designing a vortex ring with whose aid fuel in the cylinder could be spread out more efficiently than by fuel injection alone.

At the Aerodynamisches Institut of the RWTH, Aachen, the first results of studies of flows in cylinders of piston engines were obtained by H. Weiss in 1988. He constructed a test stand with a transparent cylinder into which water could be sucked by the motion of the piston. A fluorescent dye was injected through the open valve slot. The experiments revealed that two vortex rings were generated during the suction stroke.

In later studies, A. Abdelfattah of the Aerodynamisches Institut was able to simulate vortex rings with numerical techniques. The results of the studies raised the question of whether vortex structures could be designed in such a way that they could be used to enhance the fuel-air mixing in the cylinder and reduce fuel consumption. A few years later, in 2003, this goal was achieved by A. Abdelfattah et al. in an industrial application at BMW in Munich.

In conclusion, it should be emphasized again that we can encounter many vortex structures in different situations. Our knowledge about vortices is incomplete, so investigation will continue for many years to come. However, hopefully the data given here can help to understand these beautiful, not always predictable, physical phenomena. Like other unique natural phenomena, vortices can excite imagination and urge people to look for answers to crucial questions.

The author and publishers are grateful to doctor of physics and mathematics V. N. Vetlutsky (the Institute of Theoretical and Applied Mechanics of the SBRAS, Novosibirsk) for assisting with the article.