WHEN A STREAM OF WATER FALLS from a faucet onto a smooth sink with an open drain, a circle forms around the point of impact. The radius of the circle depends on the amount of water in the stream. The smaller the flow, the smaller the circle. Inside the circle the water is shallower than it is outside. The transition between the two heights is called a hydraulic jump. It is a stationary shock wave that is the hydraulic analogue of the atmospheric shock wave created by a supersonic airplane.

Stationary hydraulic jumps develop in many common situations but probably go unnoticed. They may arise in water flowing down a driveway or along a curb. Some can be seen in irrigation canals or small streams. The most dramatic hydraulic jumps occur in certain rivers when the tide comes in from the sea. These large-scale hydraulic jumps, called bores, move upstream at speeds of as much as 12 knots, are as much as 20 feet in height and extend across the full width of the river. The sudden and unexpected appearance of a bore can cause trouble for vessels on the river.

Of the three general types of surface waves on water, only shallow-water gravity waves matter in the explanation of hydraulic jumps. In these waves the motion is controlled by the gravitational pull on the water after the water is displaced at first. The speed of such a wave depends primarily on the depth of the water Deep-water gravity waves, which develop on ocean surfaces, do not depend on the depth and play no role in hydraulic jumps, since the jumps occur in relatively shallow water. Ripples of the kind caused by small insects on the surface of a pond are controlled by the surface tension of the water rather than by gravity. Such ripples, which are sometimes called capillary waves, have relatively small wavelengths (a few centimeters or less) and play no role in hydraulic jumps.

The speed of flowing water is often compared with the speed at which shallow-water gravity waves move over still water of the same depth If the water is moving faster than the waves, the flow is said to be supercritical. It is subcritical if the flow is slower than the waves. A critical flow is seen when the speeds match. A hydraulic jump develops when a supercritical flow switches to a subcritical one. The transition is sudden and possibly chaotic because during the change the flow becomes quite unstable.

Shallow-water gravity waves are generated when a barrier interferes with the normal flow of the water. If the flow is subcritical, the waves can travel both upstream and downstream. In some situations one wave might remain stationary just downstream of the barrier. A1though the speed of a wave depends primarily on the depth of the water, it also depends on the wavelength of the wave. In the range of wavelengths that can be generated by a barrier one upstream wave may have a speed matching the speed of flow of the water. That wave remains in place because it travels upstream at the same speed as the water travels downstream. The wave is constantly reinforced by the interference of the barrier with the flow of water. The other waves generated by the barrier travel away from it and die out as their energy is dissipated.

When the flow is supercritical, none of the waves move faster than the water and thus none can move upstream. They are all swept downstream by the flow. If the flow is changing from supercritical to subcritical, a stationary wave can be formed when the flow passes through the critical state. Then a match is possible between the speed of an upstream wave and the speed of the flow. From such a stationary wave a hydraulic jump can develop.

A ridge across a streambed provides an example of how a barrier can create supercritical flow, a standing wave and a hydraulic jump in an initially subcritical flow. If the ridge is small and provides only a slight resistance to the flow, the movement remains subcritical, but a small stationary wave may arise behind the ridge. With more resistance from a larger ridge the wave is larger.

A further increase in the ridge and its resistance might create a wave with a relatively shallow region behind the ridge and in front of the first crest. Because the speed of shallow-water gravity waves depends on the depth of the water, the speed of a wave would be relatively small in such a shallow region, forcing the flow there to be supercritical. The first crest downstream of the shallow region would restore subcritical flow because of the increase in depth. If the shape of the water surface is maintained, the transition from the shallow region (supercritical flow) to the deeper one (subcritical flow) is a hydraulic jump. If the ridge does not itself create such a stationary wave, one might still develop in this way if a second ridge farther downstream prevents a wave from the first ridge from being carried downstream.

The shape of a hydraulic jump is normally classified according to a condition of flow called the Froude number. It is the ratio of the square of the speed of flow to the square of the speed of shallow-water gravity waves over still water of the same depth. The Froude number is greater than 1 for supercritical flow, equal to 1 for critical flow and less than 1 for subcritical flow.

The shape of a jump depends on the Froude number of the supercritical flow ahead of the jump. (In principle the flow is unstable only at a Froude number of 1, but the instability is so chaotic that there is no precise location for this event. Hence the Froude number ahead of the jump is employed for classification.) If the number is between 1 and 3, the jump is said to be undular; it consists of a large first crest and smaller crests trailing downstream. After them the water surface is relatively smooth. At an initial Froude number between 3 and 6 the transition between the two water levels is smoother and the trailing waves are absent. The jump is said to be weak.

If the initial Froude number lies between 6 and 21, the transition gives rise to large unstable waves that may travel for considerable distances downstream from the jump. This kind of jump is called an oscillating jump because of the irregular waves. A steady jump, which lacks the destructive waves, is created when the initial Froude number is between 21 and 80. Roughly half of the kinetic energy of the inflowing water is dissipated in the turbulence of the jump. At higher initial Froude numbers the jump is again rough and irregular, dissipating as much as 85 percent of the kinetic energy of the water and sending potentially destructive waves downstream.

The amount of dissipation of energy can be crucial in the design of a sluice for a dam and of other systems for draining water. It is sometimes necessary to decrease the kinetic energy of a flow in order to avoid damaging the water channel. Placing a barrier across the bottom or a sluice gate across the top might be beneficial. The barrier either must be designed carefully or must be adjustable so that the Froude number of the supercritical flow can be controlled and the hydraulic jump does not send potentially destructive waves downstream.

Whether or not a gate on a channel creates a hydraulic jump depends on the steepness of the channel. For a given volume

of water flowing down a channel each second the slope determines the speed of flow and the depth of the water. The two are related in a simple way. When the channel is steep, the flow is rapid and the depth is shallow. With a more gradual slope the flow is slower and the water is deeper. A steep channel is defined as one that develops a supercritical flow of water. A critical channel generates a critical flow and a gradual channel a subcritical flow.

If a sluice gate is lowered into the water of a gradual channel in such a way that the flow emerging from below the gate is supercritical, a hydraulic jump will form to restore the subcritical flow the water would normally have in the channel. If the supercritical flow from the gate builds up in a critical channel, there may not be a distinct jump. The flow changes from supercritical to critical and remains there, making the surface unstable. If the sluice gate causes a supercritical flow in a steep channel, the flow remains supercritical. It also remains relatively stable down the entire channel.

A hydraulic jump can also turn up when a steep channel joins a gradual one, forcing the water to make the transition from a supercritical to a subcritical flow. In the steep channel the water has a rather large Froude number and thus remains supercritical and relatively 11 steady because it is fairly stable against small perturbations from any obstacles. When the water flows into the gradual channel, it must go slower and become deeper in keeping with the normal flow down such a slope. The Froude number is reduced to approximately the value where the flow becomes unstable. against perturbations from obstacles. Surface waves are formed, and the hydraulic jump develops to complete the transition to subcritical flow. The result is the stationary wave that brings a sudden and dramatic change in the depth of the flowing water.

This jump may not form near the place where the steep channel joins the gradual one. Its location depends in part on the slope of the gradual channel. The greater the slope of that channel is, the more gradually the Froude number of the flow is reduced. The critical value of 1 is reached farther downstream. Only then will the jump form. If the gradual channel is only slightly angled, the Froude number is reduced sooner. Then the jump forms nearer the junction of the two channels or even up in the steep channel.

When I began investigating hydraulic jumps recently, I struggled with several 11 fundamental questions. Why do jumps occur? As I have mentioned, a jump serves as a transition from a supercritical to a subcritical flow, but why is the transition necessary and why must it be made suddenly in a jump? If jumps suddenly increase the depth of flow, why do they not form in subcritical flows? And why are there no jumps at transitions from one supercritical flow to another with a lower Froude number?

Let me tackle the questions by considering a supercritical flow from a sluice gate or a steep channel that joins a gradual channel. The speed at which water can run uniformly down the gradual channel is governed by the slope and roughness of the channel. The problem is that the water entering the gradual channel is moving faster than the water in it. Forces from the water already in the channel begin to- slow the entering water. In order for all the entering water to be moved downstream at the slower speed the depth must increase. That is why supercritical water flowing onto a gradual slope begins to move slower and get deeper.

During this rather gradual transition the Froude number of the water decreases to nearly 1 and the flow becomes sensitive to perturbations it encounters in the channel. By sensitive I mean that if the flow meets even a modest resistance from an obstacle, the height of the surface increases considerably. Suppose the flow (at a Froude number near 1) meets a small ridge in the channel. The resistance of the ridge to the flow may be small, but it forces the surface of the water upward by a relatively large amount. If the flow had encountered the ridge when it had a different Froude number, the resulting rise would not have been as dramatic.

When the surface of water is suddenly pushed up, waves are created. (You can achieve the same result in a sink or a tub

by pushing suddenly down on the surface of the water or by raising your submerged hand to the surface.) The uplifting rapidly completes the transition to the greater depth required by the gradual channel, and some of the waves it creates remain as part of the jump.

Such a rapid transition is not made between two subcritical or two supercritical flows for several reasons. Suppose a subcritical flow moves from a gradual channel to a more gradual one. In the second channel the water would have to slow down and become deeper so that all of it could flow at the speed dictated by the forces on it. The transition between the two depths, however, would not form a jump. As the entering water began to move slower and get deeper its Froude number would decrease, moving farther from the critical value of 1 where the flow is sensitive. If the flow meets a small obstacle during the transition, the resistance of the obstacle to the flow would not force the surface of the water to rise much. The flow would be disturbed by the obstacle but would quickly regain its stability.

Something similar happens when water flows from one supercritical channel into another one with a smaller slope. The Froude number is reduced, but it remains above 1 and the flow is not sensitive to small obstacles. Resistance to the flow might cause a negligible increase in height, but the change would not be dramatic and any wave generated would be swept downstream by the supercritical flow.

Some of the properties of hydraulic jumps can be observed by simply positioning two flat planes in the water flowing from a faucet. I held a glass windowpane in the stream from my kitchen faucet, tilting the glass so that the water poured onto a flat piece of wood I held snugly against the lower end of the glass. T he water ran down the glass onto the wood and then off the edge of the wood into the sink. This arrangement of two tilted surfaces differs from the flow of water out of a steep channel into a gradual one because the water is not confined to a channel by side walls. Nevertheless, the dependence of the jump on the angle of tilt can be observed.

By adjusting the angle of the glass and the wood I could create a small hydraulic jump almost anywhere on the wood. The glass had to be tilted at a large angle to ensure that the water moving along it was supercritical. If I made the tilt of the wood fairly steep but still less steep than that of the glass, the jump appeared farther from the intersection of the glass and the wood. With a negligible tilt (the wood may even have been horizontal) the jump was at the intersection.

I also experimented with hydraulic jumps in tilted channels by means of a simple rig consisting of a rubber washbasin, two aluminum channels and some plastic tape. The metal channels were made out of spare aluminum strips that were three or four feet long and several inches wide. I bent both strips into a rectangular channel and taped one of them to the side of the washbasin where I had cut out part of the wall. The channels could also be made out of spare eaves-trough material. I installed a hose so that the basin would continually overflow. Thus a constant amount of water flowed down the channel.

At the lower end of the first channel I taped the second channel. With blocks of wood under the channels I could adjust the channel angles. The upper channel was steep, the lower one gradual. Water ran from the basin down the steep channel, down the gradual channel and then into a drain in the basement floor. I had fastened the channel to the basin with plastic tape. It was not a satisfactory arrangement because the tape eventually became loose, but I was able to make most of my observations before the structure collapsed.

Although creating a jump was easy, I could not tell what kind of jump it was. A much better arrangement would be to make the channels out of transparent plastic so that the experimenter could observe the jump from the side. Then it would be easier to determine whether the surface of the water was smooth or had relatively large waves.

To investigate the effects of a ridge or a miniature sluice gate I removed the lower channel and placed obstacles in the upper one. My gate was a metal plate almost as wide as the channel. By adjusting the slope of the channel and the depth of the plate I could generate the various sluice-gate effects. My ridge was a narrow mound of Silly Putty across the width of the channel. By adjusting the height of the ridge and the angle of the slope I again could control whether a hydraulic jump was created.

As an alternative to a ridge you can build a smooth and gradual bump in the channel bed. The theoretical analysis of a subcritical flow encountering such a bump on a gradual slope is not easy. One of two possible results is that the surface of the water may rise over the bump if the flow becomes supercritical. The other is that the surface may dip above the bump if the flow remains subcritical.

The hydraulic jump surrounding a stream of water from a faucet was apparently first reported by Lord Rayleigh. In his 1914 paper "On the Theory of Long Waves and Bores" Rayleigh organized the equations governing the kinetic energy and momentum in a bore. As a postscript he offered his simple observations of the hydraulic jump in a sink, noting that it was governed by the same principles.

Since Rayleigh's work the hydraulic jump in a kitchen sink has received little attention until quite recently. One of the most interesting discussions of the jump was published by R. G. Olsson and E. T. Turkdogan of the United States Steel Corporation. They made a stream of water fall onto a flat circular plate held perpendicular to the stream. The water formed a jump on the plate.

The water came from a tank in which the water level was kept constant. The diameter of the stream was controlled by an aperture. When a hydraulic jump formed on the plate, Olsson and Turkdogan measured the depth inside and outside the jump by inserting into the water a vernier height gauge with a needle pointer. (Anything wider would have disturbed the flow too much, destroying the circular jump and increasing the depth of the water close to the gauge.) The average depth inside the jump was between .1 and .9 millimeter. In the region beyond the jump the depth was between one millimeter and three millimeters.

Olsson and Turkdogan also estimated the speed of flow inside the jump by making high-speed motion pictures (2,000 frames per second) of bits of cork moving on the surface of the water. The speed was almost constant until the water reached the jump, where it began to slow down. In other experiments they replaced the water with more viscous fluids. In general the higher the viscosity, the smaller the radius of the jump. I generated hydraulic jumps with a variety of objects placed in the path of the stream from my faucet. A plate, a frying pan and even the flat side of a table knife created the full circular jump or at least sections of it. Make sure to provide drainage from a sink or a container. If the water gets too deep, the jump will be destroyed.

When I made the stream fall onto a glass windowpane, I could pour a small amount of soap powder into the area of supercritical flow. The soap was immediately whisked out to the jump, where it turned to foam in the turbulence. The undissolved soap collected just beyond the jump, left there by the sudden slowing of the water as it changed to a subcritical flow.

If you create a hydraulic jump in a dark sink or an iron skillet illuminated with fluorescent light, you might see colors in the supercritical area. I find circular bands of blue and yellow (or orange) around the point of impact of the falling stream. In incandescent light or sunlight the colors are missing. A fluorescent lamp gives rise to the colors because its light is not constantly white, notwithstanding its appearance to the eye. The light consists of three main components. One component is the long-lived phosphorescence of a phosphor on the inside of the tube. The second is the short-lived phosphorescence of another phosphor. The third is the emission spectrum of the mercury excited by the current discharged through the tube. The steady emission from the long-lived phosphorescence is most intense in the yellow region of the spectrum. The periodic emission from the short-lived phosphorescence is in the blue region. Thus, although the averaged output of a fluorescent lamp looks white, it is actually a flickering blue on a constant background of yellow.

When a falling stream of water hits an object, ripples flow out to the hydraulic jump. Since the ripples change the tilt of the water surface, they also change the reflection of light from the water. As the ripples sweep outward from the impact area to the jump they reflect yellow light at times and a mixture of yellow and blue at other times. If the train of ripples is continuous, you will see concentric circles of yellow and yellow-blue around the impact point.

Many fluids found in a kitchen can be employed to generate hydraulic jumps when they are poured onto a flat, well-drained obstacle. I held a glass pane in streams of corn oil, vinegar, beer, syrup, honey and a solution of cornstarch. The fluids with relatively low viscosity showed hydraulic jumps similar to the ones formed with water.

The honey, which was at room temperature, was too viscous to create a hydraulic jump. Instead of colliding with the pane and then spreading out to the sides in a supercritical flow the stream of honey merged slowly with the thin layer of honey already on the pane. At the point of impact a cylindrical, thin stream wrapped itself around in a rope coil and a broad stream moved back and forth to form a ribbon.

The cornstarch solution belongs to the class of fluids called non-Newtonian. They are characterized by viscosities that can be varied not only by a change in temperature but also by stress. I described these strange fluids in this department for November, 1978. The viscosity of a cornstarch solution immediately becomes high when the solution is stressed. When the stress is removed the viscosity returns immediately to the lower value.

I prepared a moderately thick solution of cornstarch and water. It was thicker than water but not too thick to flow. When I poured it into a large frying pan, the stream showed a hydraulic jump resembling the type seen in water. As the level of fluid in the pan rose, the hydraulic jump shrank in radius until it looked as though it would disappear. As the radius decreased, the wall of the jump became steeper but never displayed any turbulence. At the smallest radius the wall appeared to be concave. The falling stream thickened near the impact point, collided with the solution already in the pan and then spread to the sides to form the jump. Farther out, at a radius of a few centimeters, a slight increase in height was visible.

The strange appearance of the fluid around the impact area was probably due to the non-Newtonian nature of my cornstarch solution. When the solution collides with the fluid in the pan, the viscosity increases in the stream just above the impact point and in the solution just below it. As a result the stream hits a fairly rigid surface even though the surface is only a solution of cornstarch. The fluid in the stream is sent out horizontally in a flow that is supercritical.

After the collision the fluid is still under stress, and so it maintains its relatively high viscosity. Therefore the hydraulic jump has a small radius. The fluid flows horizontally into the jump and then a backwash carries part of the solution to the top of the jump. The overhang of the rim is probably due to a relaxation of the stress and viscosity of the fluid there. As soon as the fluid is pulled back down along the wall and into the stresses of the outward horizontal flow the viscosity increases again. The result is a concave wall around the hydraulic jump.

When I ran water onto the pane, it formed a circular hydraulic jump. When I tilted the pane, the jump was distorted. The uphill region moved closer to the stream and the downhill region moved farther away. When water broke through the uphill region, it did not spread out much but instead flowed downhill in the ridge of the jump.

Hydraulic jumps can develop in gases as well as in liquids. One of the most dramatic examples of an atmospheric hydraulic jump was photographed by Peter H. Hildebrand of the Illinois State Water Survey and published in the Bulletin of the American Meteorological Society in June, 1977. A dense cloud from a summer thunderstorm was moving to the north across Chicago toward Lake Michigan. As the high-speed cloud, which was 200 to 300 meters high, encountered less dense air it apparently broke into a hydraulic jump, with a sharp front wall followed by several waves. The jump was visible because the dense cloud was noticeably darker than the less dense air.

Bibliography

ON THE THEORY OF LONG WAVES AND BORES. John William Strutt, Baron Rayleigh, in Scientific Papers: Vol. 6, 1911-1919. Cambridge University Press, 1920.

RADIAL SPREAD OF A LIQUID STREAM ON A HORIZONTAL PLATE. R. G. Olsson and E. T. Turkdogan in Nature. Vol. 211, NO. 5051, pages-813-816; August 20, 1966.

OPEN CHANNEL FLOW. Stephen Whitaker in Introduction to Fluid Mechanics. Prentice-Hall, Inc., 1968.

THE TIDES AS WE SEE THEM. Edward P. Clancy in The Tides: Pulse of the Earth. Doubleday & Company, Inc., 1968.

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