Notches

It must be becoming evident to you that the use of the energy equation is in some way tied up with finding rational expressions on which to base coefficients to be determined experimentally in order to store data relating to the performance of the various devices that we want to use. I think that the method is very well illustrated by the use of notches for measuring flows of water that is flowing with a free surface. A notch is a specially-shaped gap in a plate. The basic shapes in use are the triangle, the rectangle and the trapezoid which can be thought of as a combination of a rectangle and a triangle. All of them have an edge having the profile shown in figure 5-17. They are not often seen in places that are used by the public but there are many in use in places like sewage treatment plants and water treatment work. I did come across one high in some hills that had been fitted for the remote measurement of the flow in a small stream. I wondered what possible purpose it could serve except feeding bureaucratic paperwork. Anyway they are useful to us in this context.

 

The sharp edged notch

One device that is frequently used is the triangular notch. I choose this because it shows us how the energy equation has been used to define a calibration coefficient when the flow is very complex and quite beyond our powers of analysis. The plate might form part of a vertical side of a tank or simply be set up vertically across a stream. Liquid then backs up behind the plate and flows over the notch. The shapes that are taken up by the liquid vary but they are all attractive to look at and interesting to study. They are called nappes, a word that describes the shape of the liquid in the same way as jet. I shall attempt to describe the nappe for a triangular notch so that we can see why it is beyond analysis.

 

Pictures 5-2 and 5-3 show water flowing over a triangular notch fitted into a channel with its axis of symmetry vertical. In truth the notch is in a piece of commercial educational equipment that is nothing like as good as the college-built rig that it replaced. In the college-built equipment the notch was in the side of a big tank that was fitted with screens to reduce the eddying. The commercial equipment placed the notch at the end of a very short channel and fed it from a deeper tank through smooth curves. You can see that the approach flow is not smooth and the result is evident in the ruffled shape of the nappe. The approach channel is also too narrow and the width affects the nappe more and more as the width of the nappe increases with depth. The accountants who forced the change did not know this.

 

The shape of this size of nappe is partly dependent on the effect of surface tension. Engineers do not pay much attention to surface tension but it acts on freely falling streams of water. Look at picture 5-1. (I repeat it for your convenience here.) You can see the air-entraining vortex through the Perspex tank. The water is rotating and, when it comes through the orifice it has a hole in it. For the want of a radially inward force the hollow jet tries to spread but then surface tension starts to provide a force to stop the spreading and is indeed sufficient to reverse it. The inward flow superimposed on the jet leads to a second expansion and sometimes a third. It was great fun to play with this tank and students enjoyed doing so. The important thing is that the surface tension of water in not to be totally ignored. It is quite capable of imposing shape on moving water if the size of the system is not too big. Full sized tugs produce a white foaming moustache of a bow wave but a scale model of say 40² in length running at the “correct” speed never produces foam because surface tension holds the eddies together.

 

In figure 5-16 I have drawn the essential feature of the flow and we need some explanation of the shape.

 

As the triangular notch has no other function than as a measuring instrument and we know that the edge of the notch is carefully made we end up with a nappe that has its shape as a result of the strength of the gravitational field, the physical properties of water and two dimensions the angle of the notch and the vertical distance between the general level of the free surface and the apex of the notch.

 

The flow pattern in the nappe is symmetrical about a plane that is normal to the plane of the notch and through its axis of symmetry. Figure 5-17 shows the flow pattern in this plane. The pressure all over the free surface of the nappe is the same, ie atmospheric. The liquid in the free surface falls as it approaches the plane of the notch and exchanges some potential energy directly for kinetic energy. The liquid that emerges at the apex will be exchanging pressure energy for kinetic energy and potential energy as it flows towards the apex. Once it has cleared the apex the liquid will be moving more quickly than that in the free surface because it has given up more potential energy to kinetic energy. At the highest point it reaches, it has still exchanged much more potential energy for kinetic energy than that at the free surface. It moves more quickly than the liquid above it.[1]

 

Figure 5-18a shows the paths of two objects that have been projected from point O and allowed to move freely. Path OA is for a slow moving object projected horizontally and OB for a faster one projected outwards and upwards. (You could run these out using Mathcad and see the shapes.) The liquid on the free surface of the nappe is trying to follow a path like OA whilst the liquid emerging at the apex is trying to go upwards and outwards to follow a path like OB. They are shown in their relative positions in figure 5-18b. Not surprisingly the liquid at the top of the nappe falls sideways, the centre comes up and the nappe spreads. This spreading would continue if it were not for surface tension which holds the nappe together and which, for water and other liquids with a high enough surface tension, brings the sides back in. The cross-section of the nappe is also shown in figure 5-16.

 

Add to this the side contractions that are evident in picture 5-3 and we have a very complex pattern of flow that might yield to analysis by finite element methods if only we could include the surface tension but that looks to be especially difficult and certainly not worth the cost and effort involved. We must expect to experiment as we did for the orifice.

 

For the triangular notch the only measurable quantities are , the head  above the apex and the angle  of the notch and once more we must test with the sides and bottom of the tank far enough away to have no effect on the flow pattern.

 

The obvious liquid to use in testing is water and data is available for the commonly used angles of 90° and 60°. This data can be stored in terms of a coefficient of discharge but for the triangular notch there is no obvious expression, like that for an orifice, on which to base the coefficient. We have to see how others have contrived a suitable expression.

 

In the case of the orifice we found an expression for the maximum conceivable flow on which to base a coefficient. The maximum flow that we can imagine for the Vee notch would be when the water emerged horizontally through a vertical triangle with the each particle of water moving with velocity  where  is the depth of the particle below the free surface. In unit time a volume having the shape shown in figure 5-20 would flow through the notch. We need to find this volume. If the vertical section at the notch plate is divided into elements as shown in figure 5-19 we can find an expression for the flow through the element and then it is only a matter of integration to find an expression having the dimensions of rate of flow and in terms of  and  both of which are measurable.

 

 

The flow through an element =  and

                                               = 

                                               = 

Therefore the notional volume passing the plane of the notch per second :-

                                               = 

                                               = 

                                               = 

                                               = 

Then, if  is the measured flow, a coefficient of discharge, , can be defined by :-

                                            = .

 

The most commonly-used triangular notch is the 90° notch (  = 45°) and it is normally used for values of  from about 25 mm to 200 mm. Below 25 mm the nappe clings to the plate and above about 200 mm the nappe becomes too wide at the surface and the flow varies too quickly with head to be useful. It is then better to change to a suitable trapezoidal notch or a weir.

 

If a triangular notch is used in an approach channel that is more than  wide and more than  deep below the apex, the shape of the nappe is not affected by the walls of the channel. Then, if  is measured at a point in the free surface that is close to the notch but far enough away for the surface not to have fallen measurably, the value of  is 0.62 and is effectively independent of  for the normal range. Then the calibration expression becomes  cubic metres/ second where  is in metres. We could not have wished for a better outcome to our search for a useful expression.

 

Let me repeat my original statement. Here we have defined a coefficient of discharge for the triangular notch using the same idea as was used for the sharp-edged orifice and if these two devices had no practical use, they would still be valuable to show, in a simple way, how a science is constructed. It is important to realise that the acceptability of the process by which these coefficients have been set up does not lie in the validity of the process but in the usefulness of the result. We shall meet this time and time again and it is much easier to understand if you can recognise what is being done.

 

The rectangular notch

I have said that there are other shaped notches and the rectangular notch has interest for us. It is just a rectangular shape in a plate in place of the triangle. It is shown in figure 5-21. I have shown the nappe and it is clear that the water flows upwards and outwards from the lower edge, (the sill) and inwards and outwards from the sides. This is another flow that we cannot analyse. However we can use the same method as before to find an expression using the two measurable quantities  the head over the sill and  the width of the sill. The flow through the element is given by  and integrating between  and  gives . Then, if the real flow is  we can define a coefficient of discharge  using:-

                                                           

The figure above shows that the nappe has two “side contractions”. These side contractions vary in size with . Figure 5-23 shows the difference in the shape of the nappe for a large flow and a small flow. We cannot expect  to be independent of . It has a value that varies around 0.6 depending on the net effect of the side contractions.

 

Inevitably there have been experiments to find out how  varies with  and  these are readily available in several empirical expressions.[2]

 

One quite practical device is the Cippoletti notch which has sides inclined at 14°. This widening with increasing head above the sill offsets the increasing contraction and gives the notch a fairly constant value of .

 

The weir.

If a plate is fitted across a channel of rectangular section to form a level sill the side contractions do not occur and the device is called a weir. I have already drawn a weir in figure 3-2. It is not a weir that might be used to measure flow because the wooden boards can be removed to control the river. However weirs of similar construction are used to measure flow and the important feature is that the nappe is ventilated to let it break free pf the weir wall. Needless to say these devices are sometimes overwhelmed by the excessive flow and river gauging is a study in its own right. Nevertheless it is still dependent on the steady flow energy equation.