Chapter 10.   The flow of liquids in open ducts.

 

Introduction.

At one time the study of the flow of liquids in channels was a regular topic in degree courses for students of mechanical engineering. Then it was omitted because of the need to make time for the studies in other fields of fluid flow. I think that there was a case for omitting the flow in long channels because that is properly the job of civil engineers. However water flows through and over all sorts of fittings that are employed in connection with flow in long channels and the very fact that the behaviour of the water in these fittings can be observed directly is of significant value to mechanical engineers. The direct value lies in the similarity between flow in these fittings and the flow of gas in convergent-divergent nozzles and in supersonic flow in general. We cannot see gas but we can see water and this can give a very useful mental picture.

 

I spent a very long time on this chapter concentrating on the mechanical aspects and gradually concluded that this topic is a valuable exercise in the application of science to a very practical problem and, as this text is going on the internet, there is no reason to omit anything on the grounds of space as might be the case for a book. There is a more general value to the student in that the “theory” and “practice” or, as I prefer the physics and its application can be separated.

 

For many years I could not see any line of demarcation between physics as taught in a physics department in a college or university and mechanical engineering. Indeed the student projects in physics were often engineering projects. Then a tele-professor, adjudicating on a design competition, said that the group sponsoring a machine for capturing energy from the wind had clearly understood the physics of the aerofoils that had been used and had also solved the practical problems of their application in the machine. He had no doubt where the line between physics and its application lay. The study of flow in channels is tailor made for a clear division. So I plan to introduce the topic in general, present the physics in the form that it is used and then apply it to some practical applications.

 

The study of open channels

Water flows in a channel when the channel connects a source of water at one level to a discharge point at some lower level and it is the gravitational field that sustains the flow. The flow is resisted by “friction” forces that are of the same character as those in pipes. The most common channels are natural water-courses, that is rivers and brooks, and man-made ducts such as canals and drainage channels. These channels can vary from smooth-surfaced, straight, unobstructed channels to rough, winding channels full of weed. Man-made channels usually have geometrical shapes, typically trapezoidal, rectangular, circular or semi-circular and perhaps parabolic. The problem of quantifying the friction loss in such channels is even more formidable than that for pipes.

 

The downward slope in normal channels is very small. I cannot detect the slope when I walk along the banks of the River Medway in Kent in England yet many rivers have much smaller slopes. It would be difficult to detect the slope of the surface and even more difficult to decide whether the slope of the surface was the same as that of the river-bed. Yet civil engineers need to understand the mechanics of such rivers and mechanical engineers are no worse off if, when they see water flowing in a river, they can answer the question “ why does it do that?”

 

Channels are generally long with fittings occurring at intervals for various reasons. On rivers it is common to have fixed weirs that are associated with locks, to make the river navigable. Rivers have seasonal variations of flow and the weirs usually have adjustable gates that normally cope with these changes but may well be overwhelmed in time of flood. There are other man-made channels that are often straight and made with a uniform section in ways that depend on the nature of the ground through which the channel has been cut. Some have sections with sloping sides to suit the use of digging machines, others are lined with very smooth concrete with vertical sides. Treatment works use channels in various ways. These man-made channels also employ fittings for various purposes.

 

Just walking alongside such channels leads the engineer to doubt whether any analysis is possible. Certainly the flow is going on in the open and subject to all sorts of natural activity like weed growth, algae growth, organic and non-organic debris that may not be visible and the effects of the wind.[1]  An engineer might ponder how the speed of the flowing water might be measured and conclude that it cannot be easy. I do not think that he would immediately think of a double float, a hunter watch and a man on horseback. I am pointing out that the study of channel flow is never going to be as amenable to analysis as say the use of the aerofoil in the wings of an airliner. It is a difficult topic to study and it will always be subject to random and unpredictable effects. The best that we can do is to try to understand the behaviour of water in channels and exercise our engineering skill to make the best shot we can at design. My impression is that over the years this study has become very successful even through the use of empirical methods.

 

The study of channels can be in two sections, the study of the long runs of channel and the study of the fittings but both depend on the same physics and so that is where I must start and it is the Bernoulli equation, adapted to channels, momentum and continuity on which it all depends.

 

The concept of specific energy.

Those who have gone before us have settled on the following as the best way to proceed. They used the idea of adapting the Bernoulli equation to a channel and, by reckoning energy terms from the bed of the channel, created a new form of the energy equation. They chose to treat the flow as one-dimensional, ignore friction, treat the channels as horizontal and deal only with rectangular channels. These decisions cleared away many of the complications the subsequent analysis proved to be adequate to get a very good idea of the behaviour of water flowing in a channel and in some cases facilitate further progress to practical applications. I need to explain the consequences of restricting our model to a horizontal channel so that the decision can be reopened when we come to the problem of allowing for friction.

 

Water flows in a channel because the weight, , of the water has a component force along the channel as shown in figure 10-1 where a bed slope of 1 in 30 has been drawn because it is about as small as I can manage. This is a very large slope for a practical channel and some channels like the drainage channels on the Somerset levels in England have a slopes as small as 1 in 1000 and, in America, some rivers have slopes as small as 1 in 14,000. . The flowing water will move relative to the sides and bottom of the channel and experience a resistance to flow just like a pipe. The common mode of flow in a straight, uniform channel is one where the depth changes gradually along the length with, of course, a gradual change in velocity. It is almost impossible to observe this change because it takes place over long lengths and it is small. Nevertheless, on the large rivers of Europe, barges without sails or engine were floated down river at about 4 knots, 2 of which were the speed of the river and 2 were from the slope of the surface. They were controlled by sweeps.

 

Those concerned with flow in long channels will seek ways to relate the flow and the depth of flow to the dimensions of the channel and to the slope of its bed. We have the experience with the application of the energy equation to the flow in pipes to draw on and we can use similar simplifications notably to treat the flow as one-dimensional with the major difference that we treat the pressure distribution in the flow as being the same as if the water were to be stationary.[2] We need a starting point and this will come from the use of the momentum equation that we used in the sudden enlargement but now we must apply it to a short length of the channel. I have shown this short length in figure 10-2 and in this figure I have added the friction forces. We can say that steady flow will occur in a channel when the sum of all the forces acting on the element equal the change in momentum per second in passing through the element. There will be two forces acting in opposite directions on the two cross-sections of the element, the friction forces and, of course, the component of the weight of the element acting along the channel. If the friction force and the component of weight are equal there will be no change in velocity.

 

I have said that usually the depth changes as the water flows along the channel but there is the possible case of flow at a uniform depth. If we wanted a straight rectangular channel to carry a given flow at a given uniform depth so that there was no change in velocity we would have to be able to adjust either the friction or the slope of the bed of the channel and normally this is not possible. It follows that there is only one flow in a given channel that will take place at uniform depth. This is often called the normal depth although it is not normal at all. Uniform flow is special because when flow occurs at uniform depth the gravitational force causing flow is equal to the frictional resistance to flow. We would expect that, where there are different bed slopes, a small depth and high velocity would be associated with a large bed slope and vice versa.

 

This clears the way for our one-dimensional model of a horizontal rectangular channel in which friction is ignored.

 

If we are to apply the Bernoulli equation we have to decide how to do it. Let us suppose that a flow line passes through two points 1 and 2 in two sections distance l apart as shown in figures 10-3a and 10-3b where the depth at 2 is slightly greater than the depth at 1. In both diagrams the channel has been drawn as a rectangular channel but it could be any shape.

 

In figure 10-3b I have drawn the section of the channel on a slope and above some arbitrary datum. So point 1 is higher relative to the datum than point 2. The water at point 1 and at point 2 has kinetic energy, pressure energy because it is below the surface and potential energy because it is above the datum. These energies, that may be evaluated as energy/unit weight or as head, can be summed and, in channel flow texts, this is called the total head and given the symbol .

Then, for points 1 and 2 on the same flow line we can say that the total heads are :-

     and

     .

We can rewrite  as  and  as  and then we get :-

                       and,  .

If we now use the supposition that the pressure distribution with depth is the same as that in a stationary fluid we can write :-

 and , and this is the major difference between flow with a free surface and flow in a pipe. Whatever section of free-surface flow we care to study, the sum of the pressure energy and the potential energy equals the depth. This permits us to rewrite the total heads as :

                                    and,

What has happened is that the potential energy at any point in the flow has been split into two parts. One of these is the height of the bed above datum and the other is the height of the point above the bed. Clearly the group  is really the total head, i.e. the total energy/unit weight, measured from the bed of the channel as datum. This is a special form of the total head that applies only to free surface flow and it is called the specific energy and denoted .

 

I now leave the question of flow in a channel with a sloping bed until I deal with long channels and concentrate on some of the implications of the specific energy equation.

 

Specific energy and the critical depth.

Now we can look at just one section in a rectangular channel. This is comparable with the treatment of flow in pipes where we related flow, area, kinetic energy and power using only continuity and the energy equation. Here the specific energy is the same for all the water at a given section and we can explore the physics of this specific energy. We have  and if we suppose that the width of the channel is  we can say, for a flow of , that . If now we put , where  is the flow per unit width,  and we can say that :-

        and this is an equation in  and constants. Now there is a choice to be made. In my view it is best for me to draw graphs of the way  and kinetic energy vary with depth because I cannot convert mathematical symbols into spatial mental models. Generally I like graphs to have the independent variable plotted as the x-axis and I try never to suppress the zero but, in this case, the independent variable is the depth and plotting this horizontally is counter-intuitive. I think that, in this text, I will plot it both ways and my reader can choose which presentation suits best. I need figures in order to plot a graph and, as a velocity of 1 m/s is high enough to keep silt in suspension it must be a useful velocity to consider. For this velocity and a depth of 2 m the flow/metre width  =2m/s/m. Taking 2 m as the starting value for depth we can use Mathcad to plot graphs of kinetic head, the sum of the pressure energy and potential energy relative to the bed and of the specific energy .

Graph 10-1a gives depth plotted horizontally and Graph 10-1b gives it plotted vertically. Neither is a wholly satisfactory presentation but I get on best with 10-1a.

 

The figures are plotted in graph 10-1a or 10-1b and the graphs give us a first insight to the flow patterns that are observable in real channels. The first observation must be that for any value of the specific energy there can be two depths. It is useful to give these two depths a common name and they are often called either conjugate depths or alternative depths. I shall use alternate depths.[3] The second observation is the existence of a minimum value of the specific energy for the selected value of flow/unit width. For our value of  this minimum appears to be 1.15 J/N and it occurs at a depth of about 0.74 m. In addition, for every depth above 0.74 m there is a depth below 0.74 m that has the same specific energy. They are alternative depths. The shape of the graph at the minimum has implications. The minimum is not sharply defined and a relatively large change in depth can be associated with a small change in specific energy. This means that flow at the minimum specific energy will never be clearly defined. All this comes from a graph of quite simple quantities.

There is one important graph that must be plotted with depth horizontal and that is the graph of velocity versus depth. We shall need it later in this text. It is graph 10-2. It tells us that quite high velocities might be possible but engineers view high velocities with extreme caution. There is always a snag.

 

 

 

 

 

 

 

 

 

 

The critical depth, tranquil flow and rapid flow.

In the creation of a science the existence of some special value of a variable leads us to give it a name. The depth at which the specific energy takes its minimum is called the critical depth and denoted . There is nothing critical about it but some name has to be used to identify it and this is the one that is in common use. It becomes familiar with use. Clearly our graph indicates that we shall need to think of the flow as being able to occur in one of two regimes, that is, above and below the critical depth. As usual we want a short name to distinguish between these two modes of flow. Some call flow above the critical depth tranquil and below the critical depth rapid, others use super-critical and sub-critical probably because these names sound dramatic. In this text I will use tranquil and rapid as being much more descriptive of what we see when we go out to look at these two modes of flow. However these descriptive terms do not mean that the mode of flow is instantly recognisable, only flow at or near the critical depth is easily recognised from the disorganised undulations of the free surface but you will not see it very often.

 

It is of interest to find out how  is related to the minimum specific energy. This can be done if we differentiate to find the conditions for the minimum ;-

    and . Now if we put  at the critical depth it follows that:-                                        .

 From this it follows that the critical value of the specific energy is,  but as the value of  we get .

In other words, at the critical depth in this rectangular channel, the kinetic energy is 1/3 of the specific energy or the critical depth is 2/3 of the total head. For   =0.741 m and  =1.112 m which confirms the readings from the graph. This result gives us an insight to the behaviour of water flowing in rectangular channels but it must also be broadly the same for other practical sections.

 

It will be useful later to note that, if  is the velocity at the critical depth, . Then for tranquil flow  and for rapid flow . [4]

 

The line of d versus E in figure 12-4 is just one of a family of curves and graph 10-3 shows curves for values of flow per unit width of 3, 2.5, 2, 1.5, 1, and 0.5 m³/s/m.

 

This graph tells us that the behaviour of water flowing in a channel is the same regardless of the flow and that two regimes of flow can occur. The area on the graph between the line of critical depth and the maximum depth is the region of tranquil flow and the region between the line of critical depth and the x-axis is the region of rapid flow. I have used the presentation with the depth plotted vertically so that the depths for tranquil flow appear to be greater than the depths for rapid flow as, indeed, they are.

 

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[5], 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.[6] 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.[7]

 

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[8].

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.

 

Loss in a hydraulic jump.

It is possible to make an estimate of the loss of energy in a hydraulic jump in a horizontal rectangular channel using the same method as was used to find an expression for the loss of energy in a sudden enlargement in a pipe. I cannot imagine it having any great value for mechanical engineers but the method used is likely to be useful when trying to analyse other flows that are important in mechanical engineering. It turns out that we must first find the depth after the jump in terms of the depth before the jump and then substitute in the total head equation.

 

This can be done for a channel of rectangular section if we isolate, between two cross sections, a part of the water containing the hydraulic jump. I have shown this diagrammatically in figure 10-13 which is a redraft of 10-9. The jump is now between two sections 1 and 2. There are two forces that are not in line and, as I have said earlier, these forces reduce the speed of flow and impart angular momentum to the water as it goes through the jump. The normal decision taken in order to get a result is first to treat the flow as being one-dimensional and then to ignore the fact that the forces are out of line and equate the net force on the section of water in the direction of motion and the resulting change in momentum per second. Whether this is justified depends not on validity of the mathematics but on whether it is any value when it is used in ordinary engineering analysis. My experience is that it is useful.

 

The force in direction of motion =  = .

Now, if we regard the velocities in sections 1 and 2 as uniform, we can say that the resulting change in momentum per second is equal to .

 It follows that,  and that :-

                                      

 We can also use continuity of flow to write  and, from this,

          and then  .

This is an equation in the flow per unit width and the depths before and after the jump. It must be possible to manipulate it to plot a graph of  against  for a given value of q.[9] It is common to leave the manipulation to the reader but I will lay it out in full.

 from which,  .

This can be rearranged to give a quadratic equation in  and other quantities that are known.

 

      and this is the required quadratic equation in .

It can be manipulated to give a solution and a graph plotted from it.

                                or 

.

This gives the final form:-   which is quite elegant.

 

Mathcad can be used us to explore the physics of the hydraulic jump in a rectangular channel. The outcome is in graph 10-6. I have used a value of  of  as before and plotted everything against values of the depth before the jump. The depth after the jump comes from the equation directly above. Once the final depth is known the loss can be evaluated from  and the loss as a proportion of  by division. We have expressions for the critical velocity and for the critical depth and these are constants for a given value of flow per unit width of  equal to 4.43 m/s and 0.74 m. These can be used to plot the values of Fr before and after the jump.

 

The graphs have no meaning above the critical depth of 0.74 m.

 

In assessing this graph one should remember that nothing has been said about the depth required before the gate to create the rapid flow initially. At the small depth of 0.1 m the velocity before the jump is about 20 m/s and this would be associated with a kinetic head of 20.4 m and the loss is 17.63 m. If this comes from a transition from tranquil flow then the upstream depth would be about 20 m, a depth that is most unlikely to occur.

 

Not surprisingly this physics tells us that, as the upstream Froude number increases, the loss in the jump as a proportion of the upstream total energy also increases. The strong jump is useful as an energy dissipation device.

 

This graph and its implications result from delving into a very simple mathematical model of the flow through a hydraulic jump.  It is worth going into MathCad and plotting these graphs for other rates of flow.

 

The Froude number

The Froude number is just a device that we choose to use when it suits our purpose. It has no counterpart in reality like a depth. It is dimensionless because it is the ratio of two velocities one of which, the numerator, is normally an actual velocity. The denominator is usually derived from a length and is a notional velocity that is used as a reference value and possibly called a critical value because it might be the line of demarcation between two regimes of flow.

 

For displacement hulls a critical speed is worked out for the speed of a wave that has a wavelength equal to twice the length of the waterline when stationary. The idea behind this is shown in figure 10-14 where the model yacht is on a wave with crests at bow and stern. There is an expression for the speed of this wave and it is speed of the wave =  in some consistent units. It is often called the critical speed but it can be exceeded without the need to plane but progress at speeds in excess of this critical speed involve a pronounced bow up attitude and very high resistance. It is a good guide to the maximum practical speed that can be attained by a displacement hull. Organisers of full sized yacht racing use this speed in the form hull speed =  knots where  is the length of the hull in feet to set up handicaps according to hull size.  Obviously a Froude number can be constructed from the ratio of actual speed in knots over hull speed and, at the critical speed, Fr will be unity.

 

But Froude, who was involved with model testing, did not use this idea. He needed only to claim that if two hulls of exactly the same shape but of different sizes move at speeds that are proportional to the square roots of their lengths their bow waves would be exactly proportional to their lengths. There is the common element here that speed and length are related by a square root relationship. This is hardly surprising when one considers Bernoulli with his interchangeability of kinetic energy and potential energy. We have a ratio :-

         wave height on the model/wave height on the prototype = Ölength model/length prototype.

It has nothing to do with the critical speed used as the hull speed above.

 

In the flow of water in channels there is also a critical velocity as we have seen. It divides rapid flow and tranquil flow. It occurs where the depth is also critical. We can say that flows at velocities below the critical velocity or above the critical depth are tranquil or above the critical velocity and below the critical depth are rapid. The Froude number is then defined as Fr =  and when flow is at the critical depth the value of Fr is unity. If the Froude number is to be used in some way in connection with channel flow we must have an expression for the critical velocity in terms of a length. The obvious length is the depth that will be the critical depth and the mathematical model used in this text shows that the critical velocity is given by . This looks to be very convenient and simple and it leads to Fr =  having unit value at the critical depth and critical speed. However  and then the expression for Fr looks altogether more troublesome. I suppose that the use of the Froude number in connection with channels it would have disappeared if it had not been useful in relating the behaviour of water flowing in one channel to that of water flowing in a second channel that was much larger.

 

Some people have taken this Froude number on board to completely redraft the models for channel flow in terms of Froude number. I suppose that the outcome looks new and attractive to mathematicians but engineers go out and look at leaves moving on water to get an idea of velocity and red and white poles to find the depth but there is a great lack of gadgets giving Froude numbers. It is not surprising that they feel more at home with mathematical models in depth and velocity.

 

The oblique “hydraulic jump”

I am working towards the Venturi flume partly because it is an excellent vehicle to practice using simple mathematical models in mechanical engineering and partly because of the analogy with the flow of gases through convergent-divergent nozzles. Figure 10-15 shows a photograph of the flow pattern in rapid flow in a laboratory channel with a piece shaped like an isosceles triangle with small base angles attached to one wall. At the first change in direction at the wall an undular wave is usually seen to form on the surface at an angle to the wall of the channel. In this wave the water appears to change direction to follow the oblique wall, to increase in depth, but to continue to flow with rapid flow. At the next change in direction the surface level falls smoothly, the velocity increases, and the water flows more or less parallel to the second face. Then at the third change in direction another surface wave-forms and then the flow follows the wall at the original depth. The waves are all attenuated quite quickly and die out before reaching the other wall if the channel is of reasonable width. It seems that it can be explained in terms of a sequence of a hydraulic jump followed by a hydraulic drop and then by another hydraulic jump.

These waves can be seen in the gutters of streets when there is heavy rain.

 

I am not wholly convinced that they are hydraulic jumps

 

 

The speed of waves in channels.

Observation of waves in a channel shows that a seemingly unlimited variety of waves can occur. The wind blowing on still water in a channel may produce small ripples on one day and organised well-defined waves moving along the channel on the next depending on the speed and direction of the wind. But flowing water often has stationary criss-cross wave patterns that are generated by the flow and, at other points, a stationary pattern of stippled waves. These waves may be of intellectual interest to the observer but the interest to the mechanical engineer is in the wave that moves along the channel as the result of a disturbance.

 

We have to have a way of creating a wave to study. Let us start by imagining a long horizontal channel of rectangular section that is partly filled with water that is at rest. Now suppose that a block that fills the channel is at rest on the bed. The arrangement is shown diagrammatically in the upper diagram in figure 10-17. Suppose now that the movable block is suddenly made to move steadily along the channel at a velocity . It is a matter of observation that this causes a wave to be transmitted quite quickly along the surface and that in front of the wave the water is at rest and behind the wave moving at the speed of the moving block at a new depth. This wave will appear to roll along the surface.

 

It is possible to find a useful expression for the speed of this wave using the ideas of one-dimensional flow, continuity and momentum and, if we find this expression for a wave moving in stationary water, we can also regard it as being the speed of a wave relative to the water.

 

If we start at the instant at which the block starts to move at some velocity  we can say that in some subsequent time interval  the wave will have moved a distance  where  is the speed of the wave. During this time a force caused by the difference in levels before and after the wave will have given a mass of water occupying the space between the block and the wave a new velocity .

 

If we make the usual simplifications we can say that the net force equals :-

                                  .

This force will have acted on a mass of water equal to  to give it velocity c. Therefore, from the statement that the impulse of a force equals the change in momentum that it produces, we can write:-

                               from which :-

                                                       

We can also relate these quantities by saying that the stationary mass, , is equal to the  moving  mass, . From this .

 

This now requires some manipulation to produce an expression that relates  and the two depths in the channel . If we expand the square term in the impulse equation and divide through by  we get:-  . But  or , and from this :- 

                                                        

This is an interesting result as it says that the speed of the wave depends on the square root of the arithmetic mean of the depths before and after the wave and that we can deduce that a wave of a given height would travel more quickly in a deep channel than in a shallow one and, that in a given channel, a large wave would travel more quickly than a small one.

 

This has certain implications. Suppose that we consider our moving block to have taken a short but finite time to reach its speed. During that time the depth in front of it will have been increasing and we can consider that a series of waves of different heights have been transmitted. As the speed of the wave increases as the depth increases the later waves will catch up with the earlier ones and the wave will first become steeper and then roll. Once it has started the rolling will continue. As there is nothing in the derivation of the expression for the speed of the wave that would preclude its use for the block moving in the opposite direction we can consider the “negative” wave. The falling level will produce a series of waves each slower moving than the one before it. The negative wave will become longer and longer.

 

Had the block been made to move in the opposite direction it would not have led to a similar wave producing a drop in level and water keeping up with the moving block. Instead, the surface takes on a slope with the leading edge of the slope advancing quickly and the unbalanced pressure force acts to accelerate the water slowly. Clearly either of these two effects could be superimposed on water that is moving in a channel. Sometimes this occurs naturally especially in tidal rivers where a rapid rise in the tide will produce a bore. More often positive and negative waves are produced by rapid adjustment of sluice gates. These effects can be demonstrated in laboratory channels.

 

A further point of interest is that a very small wave will move at a speed given by  where d is the depth and then the speed of the wave will be equal to the critical speed, that is, Fr for the wave will be 1.

 

These waves have links to shock waves in gases and the intensifying of the positive-going waves and the attenuation of the negative-going waves has some part to play in producing wakes behind ships and aeroplanes and buulets.

 

Quite large waves can be produced in channels and around 1850 a service of high speed, horse drawn, passenger boats ran on some canals “surfing” along on such waves. The technique was to make the horses pull vigorously at the start so that a wave would form in front of the boat. Then, with an extra effort the boat could be lifted over the wave to get in front of it. From there on the boat would be carried forward by the wave and the effort required from the horses much reduced although the boat was moving much more quickly than it would otherwise do with the same effort. The boat had produced an isolated wave moving along the canal and, in modern parlance, surfed along on the wave. The boats covered 4 miles in about 20 minutes and ran to a timetable.[10]

I spotted the picture in 10-17 in a calendar and could think of no reason for the wave that is almost square across the canal in line with the stem of the boat. It is clearly not a part of the bow wave. When I was writing this chapter and thinking about the waves in a channel I recalled meeting a river “steamer” on the River Thames at Kew when the water level was low and I was in an eight-oared racing boat. Ahead of the steamer the water drained away from us and flowed under the steamer leaving us in danger of going aground. The propellers just pulled the water away from under our boat. The propeller of the narrow boat in 10-18 must be doing the same and the result is this hydraulic “drop” ahead of the boat. My text also required that there be a wave across the canal behind the boat in which the level rises. I had the picture in 10-19 that had been cut from a newspaper and the drop is evident again on the right of the picture and the “jump” as well. It is very intriguing. It pays to look for these odd examples of physics at work and make a scrap-book of them.

 

Even prolonged study of a particular wave pattern in the sea seems not to lead to any real understanding of what is going on. However there are two quite different forms of wave motion one of which occurs in deep water and one in shallow water.

 

In deep water the motion of particles of water as a consequence of wave motion on the surface varies with depth. The particles behave as if they form elastic strings with one end fixed to some deep point that is stationary and the other to a point in the surface that moves in a circle about another fixed point which is vertically above the first. The radius vector for each circle is progressively out of phase with its neighbour and the resultant motion is one that approximates to the observed motion in waves.

 

The vertical component of the motion at depth transfers no energy horizontally but the net effect is one of continuous transfer of momentum forwards and near to the surface a series of particles that are equally spaced horizontally in the direction of motion of the waves will have equal phase differences. Then for particles spaced out by half a wave length one particle will be at the top of its circle and travelling forwards when the other is at the bottom of its circle and travelling backwards. The combination of such motions is such that the particles when at the top carry energy forwards but underneath the particles simply fill the space under the crests and do not transmit energy.

 

As waves come ashore on a shelving beach they roll as is all too evident. The tsunami that hit South East Asia was almost undetectable as it travelled in the open sea but, when it came ashore the wavelength was very long and the backwards flow deep down under the wave first pulled the sea away from the beaches and then the tsunami came roaring inland as a great rolling wave.

 

Waves are worth watching.

 

Flow over a hump in the bed of a channel.

I want to look at the flow over a hump on the bed of a rectangular channel where the datum for specific energy changes. 

 

In figure 10-20 I have drawn a channel with a horizontal bed with a hump in the bed. The hump is an arc of a circle and is 0.5 m high in the middle and 4 m long. I have used the same flow per unit width of  by way of illustration and I intend to draw three typical surface profiles for different approach depths.

 

The presence of the hump obviously upsets the way in which we have been using specific energy until now where we used the horizontal bed as datum. The key observation is that graph 10-4 is not restricted to a particular channel but is for a particular flow per unit width of  in a rectangular channel. It relates the sum of the potential energy and the pressure energy, i.e. the depth, kinetic energy, and specific energy to depth. It can have any datum level. This means that graphs of specific energy versus depth for a given flow per unit width can be drawn for any point on the profile of the hump as the origin but the specific energy at every point must be measured from the same datum, usually the horizontal bed. Graph 10-4 also showed the critical depth to be 0.74 m and, in the channel in figure 10-20, I have shown a line representing the critical depth of 0.74 m for  in the channel with its hump. It clearly follows the profile of the channel.

 

I did not find this diagram easy to draw and it needs careful explanation. I have chosen to consider only three points on the profile of the hump. I called them A, B and C. For each point I prepared a reduced version of graph 10-1b with depth and specific energy plotted vertically and to the same vertical scale as the hump. These three graphs will have different datum levels corresponding to the elevation of points A, B and C. The graph for A has been be fitted directly on to the profile of the bed with its origin on the bed and graph C has its origin at the topmost point of the hump. There is no space for the graph for B so I have set it to the right with its horizontal axis at the level of B. and C

 

Now, as this piece of physics is dependent on ignoring friction, the specific energy is unchanged as the water flows over the hump. I have drawn three chain-dotted green lines representing lines of constant specific energy. One is arbitrarily at a level of 2 metres above the bed, a second goes through at the level of the minimum specific energy at the top of the hump and the third is part way between the other two. All three cut the specific energy lines in the three graphs. On each graph I have projected downwards from the intersections of the green lines on to the red depth line and shown the projection lines in black. Then for each graph I have projected across to the relevant vertical through A, B and C to give 3 points on each of three profiles. I have shown the projection for point C with chain dotted red lines but omitted the others because of the congestion of lines.

 

Then, in full red lines, I have drawn the three surface profiles.

 

Now we have to think about the implications of these three profiles. The upper two profiles are only possible if the channel beyond the hump is full so that the level after the highest point has been passed can rise again. Then, for this case of no friction means that the profiles are both symmetrical about the middle of the hump.

 

It is the third profile that dips down to the critical depth that is of special interest. We have to decide what happens after the water has passed the middle of the hump. If the level were to be high in the down stream channel the profile over the hump would again be symmetrical but, if the level were to be low, the flow would change to rapid flow and a hydraulic jump could appear in the channel probably on the hump.

 

But it is obvious that I could quite easily have drawn another constant specific energy line below the one that is at the minimum value of the specific energy curve for C. That line would have had no meaning because it would require the flow to have a specific energy below the minimum possible specific energy for this flow per unit width. So there is a minimum value for the upstream depth for this flow of  and that value comes when the depth over the top of the hump is the critical depth.

 

I concluded that a general exploration of the behaviour over this hump could only be made using a computer programme and that programme would require a loop and a conditional statement. I am not alone in finding that I cannot understand the programming methods used in MathCad and, if I did, I do not have sufficient use for them to be able to remember the methods. How I wish that BASIC had not been killed off. It would have been very useful to the jobbing engineer because the architecture was totally obvious and was not a memory exercise.

 

Nevertheless I think that I have done enough to come to terms with this problem.

 

Control of rivers.

Most rivers of useful size have been fitted with flow control devices to make them suitable for navigation and/or for extracting power. The basic object in each case is to increase the depth along the river either to give sufficient depth for boats or to impound water and so store potential energy in its new elevation to drive machinery.

 

In order to raise the level of a river, obstructions have to be placed in the bed at suitable intervals. The increase in depth upstream of each obstruction reduces the mean velocity for a given flow and that, in turn, means that the energy loss between the obstructions is much reduced compared with that in natural flow. The slope of the surface is then much less than the average bed slope and as a result the depth increases as the water approaches the obstruction so creating a difference in level at each obstruction. It is this difference in level which permits the operation of water wheels or water turbines to extract  the potential energy of the water before discharging the water into the  next section of the river. If the obstructions are fitted to permit navigation the water must still pass the obstruction and some means has to be found to dissipate the potential energy that has been accumulated if it is not to be turned into a high velocity flow that may damage the bed by scouring.

 

The design of the obstruction must take into account the normal seasonal variation of flow and of flood and drought conditions. During a drought the flow cannot be stopped completely at one obstruction and so deprive river users down stream of what flow may be possible. In time of flood it may be more important to protect centres of population than to reduce all the obstructions to a minimum but generally the design of the obstruction should either permit it effectively to be removed or to be bypassed.

 

Keeping in mind that many flow control devices have designs that may be 200 years old, and the difference in size of rivers, it is not surprising to find a wide variation in design for devices serving essentially the same purpose. Broadly for water-power a dam will be built and a discharge device such as a side channel with a sluice gate used to handle floods. For navigation a dam and a lock are essential although the dam may act as a weir to carry the normal flow when the flow is not carried intermittently by the locks. A side discharge and sluice may also be fitted. All the fittings have to be designed on the essentially simple physics given above. The sluice gate and the drum gate will be considered here as typical examples of such fittings.

 

The sluice gate.

The normal function of a sluice gate is to reduce the depth of tranquil flow in a channel. Just as in valves in pipes this means that the sluice gate must cause a loss of energy. The gate is merely a full width gate in a rectangular channel arranged so that it can be opened by raising it from the bed of the channel. Its function is to convert some of the potential head upstream to kinetic energy that can then be dissipated in some way, preferably in the channel immediately down stream of the gate, which will be designed to localise the loss and to resist erosion. There are several methods of dissipating the kinetic energy and these are typically, in a hydraulic jump, in the intense eddying caused by blocks set in the bed and by rapid flow down a sloping apron followed by a hydraulic jump or blocks.

 

The level downstream of the gate is not determined by the position of the gate but by the position of the next gate downstream and this may mean that the flow from the gate is submerged. This interferes with the normal function of the dissipation devices but somehow or other the surplus energy will be dissipated if only in uncontrolled eddying. This means that the depth profile of a river is determined by the positions of the gates acting together each controlling the upstream depth. Of course changes take place slowly and there is no immediate response to changes of gate positions as would occur in a pipe following valve adjustment.

 

 A section through a gate followed by blocks in the bed is shown in figure 10-21. The gate is shown in the partly open position and it could clearly operate at any position between nearly closed and fully open. Normally the functional range of gate positions is limited by the need to produce rapid flow after the gate.

 

We cannot predict the relationship between gap under the gate, the steady upstream depth and the flow but rectangular gates with no side contractions will all have the same relationship.

 

Experimental data on the flow through a sluice gate is available and it is usually stored by the use of a coefficient of discharge. The coefficient is defined through an expression like that for a sharp-edged orifice. It is :-

                    , where  is the width of the gate,  is the height under the gate,  is the depth before the gate and  is a coefficient of discharge. This is clearly a rational expression just like that for the orifice. It is the product of a velocity and an area, the area being that under the gate and the velocity being derived from a measurable height. It is not the outcome of an attempt to predict the flow; it is simply a practical basis for a coefficient. The value of  for practical sized gates depends on the ratio  and, if the discharge from the gate is submerged, on the ratio . Typical values for , for free discharge, range from about 0.4 for at low values of  to about 0.6 for high values such as 10.

 

The radial or Tainter gate.

Where a river is used for navigation there is a need to maintain the levels in the river despite variations of flow. At one time, control devices were continually supervised by lock-keepers who lived nearby, but there is an obvious advantage in devising control gates which work automatically. Such devices have evolved over the years and modern designs are based on the drum gate. This is part of a cylinder mounted on a lever that is pivoted on a horizontal axis as shown in figure 10-23. The attraction of the radial gate is that, if the pivot is also the centre of the arc of the drum, the forces exerted by the water on the gate have a resultant force that always acts through the pivot and this minimises the force needed to move the gate. The depth upstream can be sensed with floats and these may be very large and provide sufficient force to alter continuously the position of the gate in response to changes in depth. In other circumstances electrically powered control systems, that may form part of a sophisticated system the whole river, may be used to move the gate.

 

 The gate is really just a curved sluice gate and when it is half open the flow pattern is very like that of a sluice gate. When it is nearly closed it is more like a convergent nozzle of rectangular cross section producing a high velocity jet.  The radial gate is followed by energy dissipating devices like those used with a sluice gate.

 

The radial gate is a device that can be designed quite satisfactorily using the physics of this chapter and the mechanics of floats and levers. Data for the flow under a radial gate is stored in the same way as that for the sluice gate.

 

Measurement of flow.

There are plenty of reasons for wanting to measure flow in open channels. In water treatment works dosing of a stream of water with some reagent is a normal process that may not need to be carried out very accurately but needs to be carried out by methods better than guesswork. The management of a river and all the inflows to it require some record keeping if, for instance, a decision is required on the advisability of extracting water for drinking or process use. The flows from springs vary in unexpected ways and records of the flows over very long periods may be extremely useful. Here we will examine two devices for measuring flow, the broad-crested weir and the venturi flume.

 

The broad-crested weir.

If an obstruction is to be built across a river and there is a good reason to want to measure the flow, the flow measurement device and the obstruction may as well be one device. The broad-crested weir is an adaptation of a wall for the measurement of flow.

 

In order to understand the broad-crested weir let us investigate the physics of a specially shaped wall across a rectangular channel. The wall is shown in figure 10-24. The front is radiused to produce acceptably smooth flow over the top of the wall that may be flat and level or angled downwards to give a slope. The wall has a square downstream edge. There is nothing that is adjustable in a broad-crested weir and the profile of water flowing over the weir depends mainly on the upstream depth unless the whole thing is submerged by exceptional conditions downstream. In figure 10-24 I have drawn several profiles for the water for different upstream depths and it is immediately obvious that there will be very different profiles for the case when the upstream depth is so low that the water only just flows over the wall and when the upstream depth is so great that the wall only produces a fall that is submerged downstream. In the range between these extremes the broad-crested weir can be used as an effective measuring device.

 

We can get a useful expression for the flow over the weir if we ignore friction over the crest of the weir and apply our usual physics of channel flow. This broad-crested weir is just another hump in the bed of a channel and steady flow over it will occur when the critical depth occurs somewhere on the crest. I have shown one free surface profile in figure 10-25. In this profile the surface falls to the critical depth somewhere just beyond the end of the radius and then remains steady until the fall at the down stream edge.

 

We must have a datum level for the specific energy and, as we saw for the hump used previously, the datum must be the same for the flow both upstream and over the crest. We cannot use the upstream depth because it might vary over time due to silting so we must use the crest of the weir.

For critical flow :-  where  is the critical depth for a flow per unit width of . From this we get .

 

We also have for critical flow  where  is the specific energy reckoned from the crest of the weir.

From these we get  where  is the width of the weir and  the volume flow and, as the kinetic energy in the upstream flow is small  can be put equal to the upstream depth relative to the crest say .

 

Then :-                   where  are in metres     

 

This is an interesting outcome because it gives an expression for flow over the wall that can be used with only one measurement, that is, the upstream depth above the crest. This expression can be exploited in the design of a practical device for measuring flow.

 

No doubt weirs of all sorts were in use long before this physics was first derived. They would have been calibrated, perhaps by measuring the velocity using double floats at some point upstream where the area could be measured. The evolution of the physics above gave a calibration expression that was probably just as accurate without direct calibration. That accuracy comes from the facts that the loss in converging flow is small and that, in practical weirs, the velocity distribution is not far from being uniform. However the expression is only applicable if the flow on the crest of the wall is critical at some point. This is a constraint that would not apply if the weir had been directly calibrated.

 

 The actual flow is affected by friction and the profile must be one where the depth falls below the critical and then rises to the critical depth at the fall at the discharge edge. The problem then becomes that of designing a weir profile that will have the critical depth somewhere on its crest for the range of flows that are expected. We have already noted the fact that the weir can be overwhelmed by an excessive depth downstream and the weir cannot then be used with this calibration expression to measure flow. At the other extreme, at very low flow the flow on the crest of the weir will be chaotic as a result of encrustation etc. and this invalidates the calibration expression.

 

One way of extending the measuring range of the weir is to let the crest slope away from the radius. Sloping the crest extends the range of low flows that can be measured by producing critical conditions near to the end of the radius and rapid flow over the crest. Figure 10-24 shows that, as the upstream depth increases, the point at which critical depth occurs advances towards the fall and that the approach to the fall extends towards the radius. Extending the length of the crest would clearly extend the range of high flows that can be measured. Unfortunately the extension of the length of the crest, when coupled with a slope, impairs the validity of the calibration expression for high flows and the greater the slope the worse the calibration expression becomes. As usual a compromise has to be made and the designer must determine the maximum depth over the weir for which the flow is critical. Computer methods may be used to put free surface profiles to different proposed weir profiles but the outcome has to be taken in the light of experience.

 

I think that one must be careful when assessing such a device as this broad-crested weir. We all tend to look for accuracy in measuring devices and deem those that are inherently inaccurate as not really satisfactory but just suppose that the depth on the crest of a weir had been measured daily on the same weir over say 100 years. It would not much matter whether the calibration expression was accurate because information can be extracted about long-term trends in the change of flow, and, depending on what information was sought, on other things as well. Furthermore, if the shape of the weir has been unchanged or if records existed of any changes in shape, it may be possible to improve the calibration expression in the light of changes in the analytical tools available to us now and in the future. A broad crested weir is much better than nothing.

 

The Venturi flume

The broad-crested weir is a device that combines the functions of measuring a flow and dissipating the potential energy where a reduction in level of a channel is required. There are many applications where the flow must be measured with a minimum of loss of energy and the Venturi flume has evolved for this purpose. The broad-crested weir depends for its success as a measuring device on a contraction in the flow produced by raising the bed of a channel and so producing critical flow conditions on the crest. This permits the evolution of a calibration expression. A transition to critical flow can just as easily be produced in a horizontal channel by a contraction in width or by a combination of a contraction in width and a small hump in the bed. This is the basis of design of the Venturi flume.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 10-26 represents the plan and side elevation of a convergent-divergent fitting in a channel. It is likely that it could be a Venturi flume.  The best shape for a contraction would be one that produced no oblique waves on the surface whilst producing the desired change from tranquil flow to critical flow. There seems to be no shape that does this so, in the interests of simplicity I have let the contraction have straight sides and given it a contraction ratio of 2.5 to 1. I have shown a short parallel throat and then a relatively long divergence back to the original width. The flow throughout is the same but, as the channel narrows, the flow per unit width increases and in this case by 2.5 times. Inevitably the level drops to produce a profile like that shown in the side elevation of the flume in figure 10-16 but the surface will be criss-crossed with oblique waves.

 

I have re-drafted graph 10-3 to become graph 10-7 and supposed that the flume to be carrying a flow of  at the full width of the channel. This means that, at the throat, the flow per unit width will be . Now the loss in a contraction is always small so this change in flow per unit width takes place at sensibly constant specific energy. Once the throat is passed the depth goes on falling and the flow becomes rapid and, provided that the downstream flow pattern  is suitable, reaches a new depth that is the alternate depth to the upstream depth. Then there will be a hydraulic jump in the flow just down-stream of the flume.

 

This can be represented on graph 10-7 as the line 1 to 2 which represents, on the graph, the flow through the contraction to the throat where the depth must be critical and line 2 to 3 which takes place in the divergence. Then, in the hydraulic jump, the level rises to 4 and a depth lower than .

 

The new feature is that whilst the flow is unchanged in the contraction the flow per unit width changes. In addition, because the flow is in a contraction the loss will be very small and the specific energy the same throughout. It is evident that, providing that rapid flow can take place in the divergence, the flow will go though critical conditions somewhere in the throat. It makes sense to apply our specific energy equation to section 1 in the channel just before the contraction and to the section in the throat 2 where somewhere conditions are critical. We get:-

                                               .

Given critical conditions at section 2 we know that . Then we can put:-

                                              .

If the flow is  we have  where suffix  indicates throat and  is the flow in the throat. Now we can substitute and : -                 .

From this we get :-        and     .  If we simplify again we get   and multiplying through by  the final form is :-

                                              .

On examination it becomes clear that, as the ratio of the widths of the convergence is fixed for a given flume, the ratio of the depth of the approach flow to the critical depth in the throat is also fixed. It means that we can find the flow from a knowledge of the upstream depth if the water is free to flow with rapid flow in some or all of the divergence.

 

We already have                                       .         

So                                                           

And then                                               

This is the final form except for evaluating the constant when  in SI units.

 

This is an interesting outcome because it says that the flow is dependent on the upstream level and the width of the throat. The width of the channel is not relevant.

 

This is the physics of a Venturi flume and an engineer would be expected to carry on from here to design a Venturi flume for some application. No doubt the channel to which the flume is to be fitted exists or is planned and the expected range of flow known. If the flume is to be a convergent-divergent fitting the most obvious dimension that is required is the width of the throat. There must be some link between the ratio of the width of the throat and the width of the channel to the ratio of the depth upstream of the flume and the depth at the throat where the flow is critical. We can explore that link.

 

We have for a Venturi flume . This can be rearranged to give :-

 and if  this becomes .

Then

.

A graph can be drawn of r versus k and this is shown in graph 10-8.

 

Be careful with this plot because I have suppressed the zero and started at a ratio of 1 because lower ratios are invalid. Note the way that high values of the width ratio lead to quite impractical flumes and that the upper limit of the depth ratio is 1.5 as we might expect. In the range of practical ratios for the depth are really from about 0.5 to about 2.5.

 

Keep in mind that detritus of all sorts get into open channels and the wider the throat the less likelihood there is of a rubbish jam. 

 

These figures may give a ratio of depths but they do not give us any idea of the relationship between flow and upstream depth. For this we need another equation.

 

We can go back to the continuity equation and if critical conditions exist somewhere in the throat, and write         .

We know that  and this can be substituted to give, .

Once more we have this awkward critical depth term but we can accept a minor approximation and put  and then   and .  This can also be plotted to give graph 10-9. Once more I have suppressed the zero because the depths before the throat for widths less than 1 metre are not really practical. This is not a graph that can be plotted in a general way and I have used a flow per unit width of  and let the channel width vary from 2 to 5 metres. The graphs have no meaning for an upstream depth below 1.11 metres when the width of the throat equals that of the channel and the black line shows the boundary.

 

The implication of this graph is again that big flows take place in big channels.

 

In my view these graphs and the physics above are sufficient to design a simple convergent-divergent Venturi flume and the use of a mathematics package makes exploration of possible designs very easy. It is well to remember that the simplification of ignoring the kinetic energy of the approach flow when compared with the sum of the pressure and potential energies, i.e. the depth, and this introduces a small error of 1% or 2% in the calibration for practical designs and losses in the convergent were also ignored and these my give another error of about the same magnitude. We can expect a coefficient of discharge of between 0.95 and 0.97.

 

There is the question of the net loss of energy caused by the fitting of the Venturi flume. The major loss occurs in the hydraulic jump and that must be present if the flume is to be of any practical value. Then the loss depends on the size of the jump and that depends on the down stream depth. If the downstream depth is low the jump will occur as I showed it in figure 10-6 and the loss will be the maximum. As the depth increases the jump will move into the divergence becoming smaller in height as it does so. Eventually it reaches the throat and, whilst this gives the smallest loss, it also starts to interfere with our requirement that critical conditions shall exist in the throat. So it is best if the jump is as close to the throat as possible consistent with it not interfering with calibration.

 

It is possible to buy Venturi-flumes made in fibreglass up to about the physical proportions of a motor car. Beyond this size each design is unique and will probably be made in concrete. In use the upstream depth can be measured using a float gauge that might well be in a side chamber. It can be monitored with a recorder. It may well be that the Venturi meter will be too dependent on the down stream conditions and then a hump is incorporated on the flume with its maximum height at the throat. Then the calibration expression involves the difference in level between the upstream surface and the top of the hump.

 

Friction losses in channels.

One only has to look at a river to see that quantifying the friction loss in the fluid as it flows along is going to be difficult. Indeed one is tempted to think that it is virtually impossible for natural channels and only prismatic channels have sufficient regularity of cross-section and flow pattern for us to be at all optimistic. However attempts must be made to quantify the loss if we are to design channels to given specifications, for example a channel to convey water from a river to drive a water-power installation.

 

This presents me with a problem in choosing how to present this topic. I am aware of the high-powered mathematical treatment that channel flow has attracted and I am equally aware of the way that practitioners of civil engineering combine experience and comprehensible science to design channels. This text is in sympathy with the practical men not the mathematicians. For this reason I shall stay with the empirical approach that is closely identified with Antoine Chézy (1718-1798). Chézy was about 50 years old when he started collecting data on the flow in channels and it seems that he was working on the River Seine and a canal at Courpalet in France. It must have been a time consuming task. He presented the outcome in 1775 when he was 57 and then a second paper in 1776 in which the formula that was to secure his place in history first appeared. Here we are in 2008, 230 years later, still using his expression but in a form that rolls off the tongue more easily than the original. We must look to see why this is so.

 

Chézy was not first in the field. In 1749 Velsen claimed that the velocity of the water in a channel should be proportional to the square root of the bed slope and in 1757 Brahms said that what we would now call the resistance to flow would equal the “accelerative” action (due to gravity of course) for steady flow and said that it would be proportional to the square of the velocity.

 

These are the two statements that Chézy joined together to give his famous formula. I do not know how the transition was made but it must have depended on the principles inherent in the figure 10-27. There I have chosen to use a rectangular channel but it could have been of any prismatic shape. It depicts a section of uniform flow between two cross-sections distance  apart moving at constant speed in a prismatic channel. If one takes the statement due to Brahms, and supposes that the resistance is exerted on the water by a shearing action on the area of contact between the water and the channel, the area involved for some length  of prismatic channel will be the length times the wetted perimeter. In figure 10-27 the wetted perimeter is the width of the bed plus twice the height of one side of the flow. Then the resistance to motion according to Brahms would be given by:-

                                where  is the length of the section,  is the wetted perimeter,  is the mean velocity of flow and  is some coefficient to be determined by experiment.

 

This block of water will be moving in a channel that has a slope and I have shown a slope of 1°. It is as small as I can draw. It is of course 1 in 57 and is very large by the standards of real bed slopes. The section of flow will have a weight that acts at its centre of gravity and this weight will have a component that acts parallel to the bed. I have tried to show it with a force triangle but I have had to miss out part of the vector representing the weight. The weight can be found from the volume times the weight density . [11] If we call the slope  (Chézy used .) we can put:-

                                     the component of the weight =  where  is the area of cross section of the flow.

 

Then :-  =  from which .

Chézy changed this to  and set about finding values for the coefficient  

 

Now we have to ask how Chézy thought that he was going to proceed from here. Perhaps I have things in the wrong order in that test data came first and the equation later. I think that, in those far off days, experimenters were often faced with a graph giving a relationship between two sets of measurements, in this case, slope and velocity and then had to find some mathematical expression with one or more coefficients to represent the experimental curve and perhaps find some over-arching relationship. It looks as though Chézy sought to find values for his coefficient  for types of channel, e.g for small rivers, for large rivers, for ditches and so on. In fact all he had were the values for  for the Seine (44) and for the canal at Courpalet (272). This gave him numbers on an, as yet non-existent, scale of values for  and must have enabled him to get quite close to the value for the proposed canal. In my view this is a very successful strategy. Chézy had  to allow for the size of the canal,  to account for the slope and a value of  to account for the state of the channel.

 

Others, notably Darcy, Bazin, Manning and Kutter, followed this same strategy in that they experimented to give much better values for  over a much wider range of channel size and wetted surface to complete Chézy’s scale for .  They each produced formulae for  that varied in complexity.

 

On the way they called  the mean hydraulic depth and denoted it  and, in so doing, created a single dimension that more or less described the size and depth of the channel and they used the Chézy expression in the  form that we now recognise :-                             

 

We need to look more carefully at this expression.  has the dimension of length so it will vary numerically with the system of units.  is not a number, it has units and this means that its numerical value will also change with the system of units employed.  has units because it contains  and . There can be no doubt that water is by far the most likely liquid to flow in channels and for which most data has been gathered is water. If we accept  is no longer a variable only the value of  varies with the system of units. So it seems that this expression that is so attractive in its simplicity is not very suitable for the creation of systems for storing data on the value of .[12] Fortunately the work of Manning (1816-97) fits very consistently with the Chézy expression and only one number has to be changed to switch to SI units from Imperial units. 

 

The normally used expression for  is attributed to the Irish engineer Manning who gave us, in Imperial units :-

                                  where  is called the Manning coefficient and is dependent on the character of the wetted surface. Clearly Manning’s expression can be combined with Chezy to give :-

                                                             and this is the commonly used expression. If it is to be used with SI units it becomes:-

                                                            and  is independent of units.

Just as the Darcy expression became the expression used for the storage of data for pipes even though there were expressions available that appeared to be more accurate, the Manning expression is the one to have found favour for channels. It is a fact of life for an empirical science; it is the most useful in all the circumstances that predominates. The others are still there for those who fancy using them.

 

What had been done was that Darcy, Bazin, Manning and Kutter gave us a scale for  in terms of the roughness of the channel as described in words and not symbols. Then engineers can compare the construction of any proposed channel with the descriptions and select a value for . This is what is done for pipes.

 

TABLE 1
Description of surface	Manning
Smooth cement or planed timber	0.009 – 0.010
Sawn timber, rusted iron, ashlar* and well-laid brickwork	0.012 – 0.013
Masonry from uncut stones or rubble, inferior brickwork	0.017
Inferior rubble masonry or masonry in neglected condition and canals with earth beds in very good condition	0.02
Canals with earth beds in good condition, in moderate condition, in bad condition	0.022, 0.025, 0.030
Badly neglected channels	0.035
* ashlar is cut stone for face work on walls.

 

I have already said that gathering data for channels is very difficult because it must take place in the open and everything is difficult to measure. Just where is the surface on water flowing in a river? How do you measure a difference in level of 50 mm over a distance of 1 kilometre with an accuracy that justifies spending more time on the data in table 1?

 

It is worth exploring the expression for Chézy combined with Manning. If we select a value of  of say 0.02 as being typical, and give the channel a width of say 2 metres, we can find values of  to sustain a flow of, say , (the value used previously) at various depths. It will be convenient to find values of the velocity at the same time.

In graph 10-10 I have plotted the slope necessary to sustain a flow per unit width of  at any depth. On the graph I have added the critical depth of 0.74  that has been calculated before. In order to sustain this depth the slope must be 1 in 428. In graph 10-11 I have plotted the velocity for a given depth. The critical velocity is 2.7 .

 

These figures indicate the difficulties that confront the experimenter wishing to measure friction loss in channels. Flow at a depth of 1 metre corresponds to a speed of about 2 m/s in the channel that is to 7.2 km/h. This is about twice the speed needed to keep silt in suspension in a channel but would stop the growth of weed, yet the slope on the surface is only 0.001, that is, 1/1000 which is barely perceptible. Over 10m it is only 7 mm and, if the undulations in roads as a result of earth movement is any guide, it would not be long before any channel laid to this accuracy had moved. Furthermore there will be disturbances in the surface of the water to contend with in real channels and the measurement of flow is never likely to be accurate.

 

At a depth of 1 metre the velocity would be 2 m/s that is known to be too fast for weed to grow. This is quite a high velocity in practical channels yet the slope would be 1 in 248. It follows that for ordinary channels laid at slopes of less than say 1 in 250 it would be hard to say whether the flow was in fact at a uniform depth.

 

In assessing the results of the calculations of bed slopes above we must recognise that, whilst it is likely that the use of the Chézy expression becomes less and less justified as d decreases, the implications of the d versus slope figures is clear; excessive slopes are needed to sustain the small depths and high velocities and very small slopes are adequate to sustain useful velocities.

 

Gradually varying flow in channels.

If you care to walk a river that has been fitted with weirs and locks to make it navigable it is evident that the depth gradually increases between one weir and the next. There may be good reasons for wanting to be able to predict the profile of the surface. It will never be easy for a natural channel because the shape and texture of the river-bed vary in a random way. What can be achieved by using mathematics depends on one’s willingness to invest time in the calculations. This means that we must have some basic method to start with.

 

It is important to recognise that we are considering steady flow where the change in depth with distance is very gradual. If this is the case we can start by applying the total head equation to some short length  of the flow and show how things are changing. A diagram is needed and I have drawn one in figure 10-28. This figure represents a short length of the flow in a channel between two sections 1 and 2 that are a short distance apart. The diagram can never be drawn in good proportion because the slopes of the bed of the channel and of the surface are very small indeed. I have shown the length  grossly exaggerated and the slopes as small as I can draw them consistent with indicating kinetic energies and losses.

 

I will regard the channel as being rectangular in section and having a bed slope that is uniform along the whole length  of the channel. This decision can be reopened if necessary depending on the application because the outcome of this analysis is a differential equation that must be integrated and this could be replaced with lots of integrations to take account of changes in the channel in its length.

 

During the flow from 1 to 2 the water will lose energy to internal friction, the velocity of flow will fall by a small amount and the depth will increase. So let the velocity increase from  to  and the depth increase from  to . The increase in velocity is, in fact, negative and this will be evident in the outcome.

 

It is best to start with the total head equation (not the specific energy equation) and apply it to sections 1 and 2. In order to do this a datum is needed and I have used the bed of the channel at section 2.

 

Then :-

                  where  is the slope of the bed and 

These two differ by the loss to friction as shown in figure 10-28 and then:-

                               

Now we have to do something about the loss to friction in this short length. As there is no set way to deal with friction we can put the friction loss proportional to the length, that is,  and leave the decision about  until later. Then the equation becomes:-

                                        

This can be expanded to give :-

                                           and this reduces to :-

                                                             

In the limit:-                                            

This gives a very inconvenient term in  but it can be eliminated by using the continuity equation.

We have  and this can be differentiated to give  because  is constant. It follows that :-       and then that :-  

 But  where  is the width of the channel. And then :-

                                     

 This can be rearranged into its final form :-

                                     

This expression tells u that if the loss to friction is equal to the slope the depth will not change. Also if  the slope of the free surface is negative, that is, it falls but if  it rises. This may appear to be inconsistent with the behaviour of water in the laboratory channel but one must remember that this expression is for gradually varying flow.

 

Now we have to think how to make use of this expression. One must ask how rivers get the water that flows in them. There are only three ways. The first is from rain falling directly into them but this must be trivial. The next is from the direct run off of rainwater from the surface of the land in the catchment area and the last is by percolation through the ground. Sometimes you can see water seeping or even trickling from the banks of a river and it is quite easy to see run-off. So not only do we have a bed that changed randomly but flow changing along the length in a random way.

 

The expression is a differential equation and it will need to be solved many times to give a surface profile and each step could have its starting values and coefficients changed to suit the know information. This is not a daunting problem these days but it takes the whole exercise into another realm where it is only worth doing for a real application. It is a specialised topic.

 

Endnote

I have spent many hours writing this chapter and the time has been necessary because of the sheer complexity of the behaviour of water in channels. The whole exercise has been based on much simplified models of the flow and these have proved to be very taxing in the mathematical manipulation needed to produce the most convenient forms of the ultimate expressions. However it seems to me that these simple models give the best value for effort in forming a mental model of the way the water behaves. They certainly make the observable behaviour of rivers and their fittings comprehensible.

 

This text has been confined to channels of rectangular cross section. There are other sections of which the large pipe running partly full and the trapezoidal section are the most common. The pipe is used for sewers and for drainage and the trapezoidal shape for surface channels because of the need to use diggers for their construction. The behaviour of water in these channels is the same as that for the rectangular channel and much of the quantifying is readily adaptable.

 

Of course I did not start from scratch and I am indebted to others for finding their way through the maze of different mathematical routes that might be possible. Sadly most of my effort went to filling gaps in the sequence that were covered by notes such as “by simplifying it can be shown that”. I found it to be enjoyable.