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.[1] 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.