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 . [1] 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 .[2] 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.