Introduction

I think that flow patterns are so important that I devote a chapter to them. But first I want to give two examples of the need to describe flows.

 

Picture 4-1 is of a Perspex tank that was 690 mm in diameter and about 1.5 metres high. I had it made for student use. It replaced a steel tank that was used to test the expression for the time taken for such a tank to empty through a hole of 0.5² in diameter. I wanted to see what happened in the water not just look at an opaque tank. When the tank was delivered we fitted it with a brass plate with a clean round sharp-edged hole of 12 mm diameter centrally in the base. I had pipes fitted as you can see to feed dense dye into the water at sensible positions. I used to have the tank filled 24 hours before it was to be used by students so that the water could come to rest and the students knew what they were testing.

 

This photograph is of the tank when it was draining. The dye is just dribbling in from the injectors on the left and mainly falling through the water because its density is higher than that of water. Note the way that it wanders about. On the right dye is being injected much more quickly and in picture 4-2, which is a blow-up of picture 4-1, you can see that the thick streams of dye get thinner and just disappear as the water flows towards the hole.  The cloud of bits falling and leaving trails are particles of dye falling and dissolving as they go.

 

We are looking at the region in which the water is accelerating towards the hole and giving up its pressure and potential energy to kinetic energy.

 

These days it is easy to explore this process using the continuity and energy equations to set up some relationships and Mathcad or the like to process them.

 

The diameter of the jet from this tank was about 10 mm so its area  was about 0.0000785 . The velocity of the jet depends on the height  of the free surface above the base and if the height is say 1 metre the velocity from the energy equation is given by:-

                      and the flow is  

This flow takes place as if it is converging radially through a series of hemispherical surfaces centred in the hole. I have shown this in figure 3-1

 

Now we can find out how the radial velocity varies with radius. The steps in the calculation are shown in frame 4-1 together with the graph of velocity against radius.

 

The graph shows quite clearly that most of the conversion of pressure and potential energy takes place in the final 25 mm with the start of the transition being inside the 100 mm hemisphere.

 

This is our first use of the continuity and energy equations and, for those like me who cannot instantly interpret a mathematical equation, the mental image is clear and would not be altered if a much more rigorous method were to be used. It works and if you go back to the photograph we can understand how the dye gets stretched out so much that it disappears.

 

My second example is from a river. Pictures 4-3 and 4-4 are of the water flowing over a weir on the River Cray at Hall Place in Bexley UK. I think that it is an impressive example of exchange of energy in a flow of water. In order to understand it we need to look at figure 4-2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In my diagram the River Cray flows from the left towards the weir wall that is constructed of substantial timbers. All of the water including the surface flows towards the wall as I have shown using the curved lines with arrowheads. The one at the bottom moves very slowly but it turns upward as it approaches the wall. The rest moves at differing speeds and the surface level drops to give velocity to the top layer. As the flow approaches the crest of the weir the acceleration increases just as it does in the tank with the hole with the major part of the change to kinetic energy being in the last foot or so. But water flows in all directions towards the top of the wall as I have indicated. The momentum of this water cannot be destroyed and it lifts the stream of water clear of the top sill to flow freely through the air in what is called a nappe. This stream of water falls nearly vertically on to a sloping apron and there it is suddenly stopped and it splits into two flows one onward down the apron and the other back towards the weir wall. This split is inevitable because the collision produces a rise in pressure on the apron that acts in all directions and if it pushes water onwards it must also push it backwards and the two forces must be equal. The reverse flow hits the weir wall and is now diverted upwards behind the falling nappe. The diverted water goes upwards and fills the gap between the nappe and the weir wall until it stops rising when it produces a flat free surface behind the nappe. I thought it was an exhilarating thing to stumble upon.

 

If you look at picture 4-4 you can see the nappe, (The breaks in it are caused by detritus) and the free surface appears as a horizontal straight line about half way up the wall. You can also see the foam as the flow hits the rocks and the standing wave that is caused by the rocks. You can see the free surface more clearly in picture 4-3

 

So what about the energy exchanges? In the approach flow to the weir only a very small part of the flow near to the surface falls, the rest goes forward and upwards and exchanges pressure energy for potential energy and some kinetic energy. Just before it passes over the wall, in a process like that in the tank, all the energy is converted to kinetic energy to give a nappe that has atmospheric pressure on both sides. Now the nappe falls giving up potential energy for kinetic energy with no pressure energy. When it hits the apron some of the kinetic energy is exchanged for pressure energy and this causes the split and is sustained by the centripetal acceleration imposed on the flow and the flow towards the wall goes up until all the energy is in potential energy. Of course this pile of water is not capable of existence without the presence of the nappe which drags it down to produce an eddy in the column of water. 

 

I walked that part of the river many times and this flow pattern did not appear very often because sometimes the flows were too great and at others too small. At Teston in Kent UK there is a weir about the same size and small trees grow in that space under the nappe. I do not know whether the trees are robust or whether someone adjusts the weir so that the nappe never clings to the wall.