Transition between tranquil and rapid flow and vice versa.

Anyone who has ever followed the course of a river, especially in its upper reaches where river control works are not so common, will have observed different modes of flow. In some places the river will flow with a smooth glass-like surface, and that is tranquil flow[1], exceptionally, at other places where there are no stones or rocks and where the flow is close to the critical depth, it will move with clearly defined undulations that change their positions randomly, and in other places the flow will move quite quickly with only small disturbance of the surface and the flow is probably rapid. I think that it is difficult to be certain about rapid flow in a natural water-course but, at fittings, it may well be obvious. Look at figure 10-4 of water flowing over a weir. On the right the approach flow is smooth and is clearly above the critical depth. On the steeply-sloping apron of the weir the flow is clearly rapid and below the critical depth. Now look at the swan sliding down the same apron in 10-5 when the flow is much less. Just look at the surface bobbing up and down.[2] It must be flow that is close to the critical depth. If we look at the ways that the transitions between tranquil and rapid flow take place it may be possible to be find other evidence in the rest of the flow pattern to be certain.

So how can we be certain to change tranquil flow to rapid flow? The certain way is to let the water flow under a gate. I have shown a sluice gate in a rectangular channel in figure 10-6. The water approaches at depth  and leaves at . There will be very little loss of energy in the flow under the gate and the depths  and  are alternate depths. The proportions shown could be for a flow per unit width of about 1 . (This is our first use of the model restricted to horizontal channels.)

 

There are other ways of changing from tranquil flow to rapid flow but this sluice gate will be sufficient for the time being. It is more important to look at the ways in which rapid flow can change to tranquil flow.

 

Graph 10-3 shows the depth, the kinetic energy and the specific energy for a flow of 1  in a rectangular channel. Suppose that the flow is at the depth corresponding the flow from the sluice gate in figure 10-6 and that a change must take place to tranquil flow. This change can take place in several ways but, if the flow from the sluice gate is into a channel of uniform section, the most simple way is to fit a line of blocks across the channel, as in figure 10-7, to obstruct the rapid flow and just slow it down with a very significant loss of specific energy. This is a method used very extensively in natural watercourses to stop rapid flow and avoid scouring caused by rapid flow over weirs and under gates.[3]

 

However our interest must be in the transition from rapid to tranquil flow in an unobstructed channel. So let me suppose that we had a laboratory channel, like the one shown diagrammatically in figure 10-8, of about 6² width and 12² deep (150mm by 300mm) that could be set up with a sluice gate halfway along it. Now suppose that we also had lots and lots of extensions to the channel so that it could be made just as long as we like. (We could imagine operating on the level floor of one aisle of a supermarket.) Suppose that the flow is adjusted to give the condition shown in figure 10-5. The depth in the header , and the depth after the sluice gate  are alternate depths and the flow downstream of the gate is rapid. This rapid flow will now discharge along a level channel and inevitably be affected by friction between the water and the channel. What we need to know is how this water would behave as the length of the channel is increased.

The effect of friction will be to reduce the specific energy and this means that the velocity will decrease and the depth increase to follow the line from A in graph 10-4 towards the critical depth. If the channel is short, as in figure 10-9, the depth of the water will increase steadily in rapid flow and discharge over the end of the channel in a fall without reaching the critical depth. If the channel is extended with the bed remaining horizontal the level at the end will become greater but the flow remain rapid. This cannot go on, a length must be reached where the depth approaches the critical depth. When it does the flow as it approaches the fall will become very confused with the undulations and obvious swirls that are typical of flow near to the critical depth. This is shown in figure 10-10. The point has been reached where the flow cannot suffer any further drop in specific energy. There must be a major change in the flow pattern when the length of the channel is increased still more.

 

The water “solves” this problem by using a hydraulic jump. In figure 10-11 the water flowing with rapid flow from under the gate loses energy and the depth increases, probably imperceptibly, and then, in a hydraulic jump, increases its depth to something somewhat lower than the depth in the header. Then the water flows much more slowly in tranquil flow and will continue along the channel losing energy. This time the loss of energy leads to a decrease in depth not to an increase in depth as it did in rapid flow. The final steady state is when the flow settles down to pass through the critical depth in the fall at the end of the channel.  The coming into existence of the jump has no effect on the flow under the gate but a further increase in the length of the channel must ultimately affect the flow upstream of the gate. We need to see how this happens.

I glibly chose a point 1 on the graph of specific energy versus depth for my diagram but, if this is science, that point must be repeatable for a given length of channel and position of the sluice gate. Its position is not accidental. For every length of channel the flow between the gate and the end of the channel will become steady when the loss in the rapid flow before the jump plus the loss in the jump plus the loss to friction in tranquil flow just brings the water to the critical depth at the fall. It follows that, for a given flow per unit width, the position and the height of the jump varies with the length of the channel. In order to understand this it is necessary to look a little more closely at the jump.

 

In graph 10-4 I have shown alternate depths A and B on a graph of depth versus specific energy. I have supposed that a hydraulic jump will occur from point A. The jump cannot be from A to B because there will be a loss in the jump. I have drawn the jump as taking place between A and B¢. Then, if this were to be the channel shown in figure 10-10, the level would follow the line from B¢ to  and reach  in the fall.

Unlike the flow below the critical depth further extension of the channel will alter the depths throughout the channel and cause the hydraulic jump to move towards the gate and ultimately it will submerge the flow from under the gate and affect the conditions in the header. The jump will change in character as it approaches the gate and we need to see why and how.

 

In graph 10-5 I have drawn four positions A,B,C and D on the rapid flow curve between A and the critical depth and the final depths A¢,B¢,C¢and D¢ on the tranquil flow curve. These jumps could take place as the length of the channel is increased causing the loss of energy to friction to increase and the jump to take place between two points that get closer to the critical depth. It is a matter of observation that these jumps are very different in appearance and I must explain why this is the case.

 

Figure 10-12 shows water flowing steadily through a hydraulic jump. The rapid flow before the jump may be quite orderly but in the jump and for some way beyond it the flow will be disorderly. My diagram is intended to depict the flow in no special way as it can be very different. In the earlier parts of this text I have pointed out the value of treating the pressure distribution both before and after an event such as this jump as if it is the same as that for stationary water. I have drawn the static pressure distribution for sections before and after the jump. Clearly, as it flows through the channel between the two sections, the water is acted on by two horizontal forces, one larger than the other, and not in line. These two forces can be replaced by one force and a couple. As the water flows through the channel between the two sections the net force will continuously reduce the linear momentum and the couple will continuously increase the angular momentum. As the height of the jump increases the difference in magnitude of the forces increases as does the magnitude of the couple. This results in jumps of different appearance as the shape of the jump changes from a simple undulation in which there must be backward rotation even if it is not discernable to a wave with a very evident back-roll[4].

Others have been able to discern five forms of hydraulic jump that presumably have different combinations of the backwards rolling of the water on the surface and the divergent flow under the surface. They have been given names and are associated with ranges of the Froude number, , before the jump. For Fr =1 to 1.7 the jump takes the form of two or more clearly defined smooth waves and is called an undular jump. For Fr =1.7 to 2.5 the surface at the jump is characterised by a series of small backwards rolling waves on top of a smooth divergent flow followed by a relatively smooth surface down stream and this is called a weak jump.  The troublesome jump is one that has values of Fr before the jump of 2.5 to 4.5. In this one the divergence is unstable and forms an oscillating stream with no set frequency under the rolling waves and the combination produces large irregular waves that can be quite destructive in natural channels. It is called an oscillating jump. For Fr between 4.5 and 9 a clearly defined back rolling wave forms on top of smooth diverging flow. This is the most steady form of jump and is called a steady jump. At higher values of Fr the back roll is carried away intermittently by the diverging stream underneath it. This produces a surface that is disturbed by intermittent, untidy waves. It is called a strong jump.  Whatever form the jump may take the eddying will cause a loss of energy. As a result the water does not reach the alternate depth after the jump.

 

A jump is about 5 times as long as it is high.

 

What is interesting about this is that the character of the jump can be categorised by the use of the Froude number. For a channel the number is the ratio  where  is the velocity at a given section of the flow and evaluated from  and  which has the dimensions of velocity. Channels come in all sizes and in the forgoing text it is clear that a valuable parameter is the flow per unit width, q, but it is also clear that a given value of q is typically associated with a size of channel. In the case of the jumps we want a parameter that is independent of the size of the channel. The Froude number appears to serve this purpose.

 

It is very easy to produce a hydraulic jump by running water vertically from a tap on to a more or less horizontal surface. This produces the familiar ring that is in fact a hydraulic jump. It is much too small to examine in detail but ink can be dropped into the water to show the reduction in velocity that occurs at the jump. The shape of the jump is determined by the effect of surface tension and this masks any eddying that may occur in the jump. You can go one step further and run the water on to the middle of a dinner plate. The raised edges make the water deeper and bubbles stream towards the jet to show the back roll. We would have to examine hydraulic jumps in large channels to observe the details of the flow pattern in the transition.