Chapter 4     Flow lines and stable flow patterns.

 

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.

 

Organised flow and flow lines

When I started racing model yachts I thought that it would be a good idea to find out how sails work. I looked in books and found nothing that made sense. So I decided to draw the paths of the air as it flowed over the sails. I knew how to draw flow patterns, as these diagrams are called, and knew that it would involve a process of trial. At home we have square, smooth-faced slabs under one upstairs bedroom window and they look like a grid for plotting graphs. I drew the lines in chalk on these slabs and went upstairs to view them. I then erased and replaced parts of the lines over and over again until I thought that they were consistent with physics. I learnt a great deal about the fundamentals of sailing in two hours and have not had any reason to alter the flow pattern in any significant way since. You could do the same for problems that you meet but you will first need to understand flow patterns.

 

We have all seen pictures of smoke trails round cars and aerofoils used in advertisements.

They are often fiddled to carry a message that is questionable to say the least. Nevertheless they are often real if not for the purpose claimed. I think that picture 4-5 is real and it carries a valid message. We can see the angle that the aerofoil makes with the approach flow and the blips that started in line across the flow show how the smoke lines get out of synchronisation as they flow over the aerofoil.

 

A goal could be to generate these lines in some way without having to own a wind tunnel or make models to test.

 

What we are talking about is the way that an extensive mass of water flows steadily under the influence of gravity and/or pressure and some rigid boundaries. So what “rules” govern the flow?  I drew some lines in the river in figure 4-2 and what I was doing was drawing lines that you would interpret intuitively to indicate the paths of the water through space. I could draw the lines because I know and you probably know that the approach flow to a weir appears to be quite orderly. We can see in picture 4-3 that the water is flowing just like it did in the Perspex tank but this time it is going to form a sort of two-dimensional (prismatic) jet called a nappe. It seems to me to be reasonable to think that the water flowing in the river follows an infinite number of paths through space and we draw a few of these paths in order to get a mental picture of the way that the water behaves. The first thing that is special is that these paths through space cannot cross nor can a path divide into two or three new paths. So we shall represent an orderly flow by a pattern of lines that can only get closer or further apart and mostly we draw for just one plane. The next thing is that all the paths will be smooth curves that may just terminate even though the flow continues. These are points where the flow ceases to be capable of representation by a line as a result of mixing or the like.

 

There seems to be some uncertainty as the what to call these diagrammatic lines, some call them stream lines and limit them to the mathematical concept of frictionless flow and others call them flow lines because flow with friction can appear to be orderly and be usefully depicted by a flow pattern. I shall use flow lines.

 

Flow nets

Now in a real flow there will be steady changes in pressure along the lines and usually surfaces of uniform potential can be drawn that cross the flow lines and everywhere that the flow lines cross the surfaces they do so at right angles to the surface. The result is a network of lines that is called a flow net. The surfaces are called equipotential surfaces and they give a way of refining a flow pattern to comply with the energy equation and the continuity equation.

Look at frame 3-2. It is a graph drawn to equal scales using Mathcad and is just a series of rectangular hyperbolae. They could be the flow lines for a wind approaching a vertical cliff. (I am not saying that they are correct.)  Now I can try to draw the equipotential lines to complete a flow net. I have done so in frame 4-4 and I did it by eye.

 

The lines representing the equipotential surfaces are in black and even though I thought that the lines marked with a cross looked to be correct when I drew them they were clearly in error when I viewed them from another direction and I drew the better ones. The point that I want to make is that, if these nets are drawn by trial, my eye, and probably yours as well, is capable of spotting places where the lines do not cross at right angles. If, as I did for the sailing rig, you use chalk or, as I do now, use office correcting fluid ,flow nets that give a fair representation of the real flow are possible.

Let me give an example. Most people will have stood on the top of a cliff on the coast and experienced an on-shore wind. What is the flow pattern of the air flowing over the cliff really like? What can we find out just by using the continuity equation, the energy equation, Newton’s laws of motion and flow nets?

Figure 4-3 is the starting point. It represents the airflow over the sea and the cliff and on to the land. Note that these diagrams must be drawn to equal vertical and horizontal scales if the flow lines and the equipotential lines are to meet at right angles. I cannot draw the whole flow pattern but it must be evident that a cliff cannot disturb the wind to extreme heights. There must be some level at which the effect of the cliff is not detectable. The top line represents this level. The cause of the disturbance is the cliff and I have drawn a circle in which the disturbance occurs. We might reasonably expect the effect to be evident above and in front of the cliff and I have drawn two typical lines with bumps in them to show this. The three lines on the right above the sea indicate the approach flow at low level and that is almost all that is possible without a closer look at the flow at the face of the cliff. We can draw a flow net.

 

I have drawn the flow net in figure 4-4. I drew it with the sharp edge at the top of the cliff in mind because the flow must be affected by the sudden absence of the vertical surface. I supposed that the straight equipotential line would be skewed. It did not take very long to draw and I did not tidy it up as you can see. It is rather like my set of hyperbolae above.

 

The immediate and obvious effect of the cliff is to divert the flow upwards. This means that air that started with momentum in the horizontal direction acquired momentum in the vertical direction as is flowed over the cliff face. This requires two forces, one to slow it down horizontally and another to speed it up vertically. We must find them.

 

This flow net is a cross-section of a three dimensional flow but the flow will be the same for the whole length of the cliff. These flow lines really represent surfaces and air flowing between two surfaces must remain between the surfaces. The continuity equation tells us that where the flow lines diverge pressure will rise and where they converge pressure will fall. The flow lines diverge as they approach the cliff and converge again as they flow up it. The pressure in the approach flow is the same as the pressure just above the edge of the cliff so the pressure above the sea at the foot of the cliff and on the cliff face must be above atmospheric. The sea is exerting an upward force at the foot of the cliff and the cliff is exerting a horizontal force all over its face. We have our two forces.

 

The air will possess momentum as it flows upwards over the cliff edge and that momentum will not be destroyed so the air must go on upwards but I do not think that flow nets can help us any more. However there are two observations that we can make. The first concerns the flow in the region at the immediate foot of the cliff. The air in my lowest flow line is going faster that the air below it and this tends to drag the slower air along. At some point the air may produce an eddy in the anti-clockwise direction to fill that square corner. When you are next in a position to check you could find out whether this happens. In the same way there is a problem at the cliff edge because the upward flow cannot just stop so it must drag air along the land towards the sea and then upwards. Again if you are physically active you could find out what happens by just standing near the edge. You could try using a children’s bubble blower to find out or a hand held windsock as I do. You can watch gulls just flying in the rising air on the cliff race and glider pilots soaring along a ridge and at some sites where there are big ridges gliders go so high on the wave from the ridge that the pilots need oxygen. We know enough now to look with understanding and all we did was draw an approximate flow net and use continuity, energy and Newton’s laws.

 

 

 

Convergent, divergent and curving flow within a flowing fluid.

Figure 4-5 is a representation of a tank like the Perspex tank but fitted with a nozzle and not an orifice. (This makes it easier for me to draw.) I have shown the tank to be fitted with a sprinkler system so that water can be fed into the tank without seriously disturbing the water as a jet would do. If water is supplied to the tank at a steady rate, a jet, flowing vertically downwards, will emerge from the nozzle. The level in the tank will rise until the outflow from the nozzle is just equal to the inflow.

 

We have seen that the main body of the water will flow downwards quite slowly in an orderly manner and then turn towards the nozzle and eventually make a rapid exchange of pressure energy and potential energy for kinetic energy in the vicinity of the nozzle. Figure 3-5 shows one of the paths as water flows from the free surface to the exit plane of the nozzle. I have called such a path a flow line. This flow line is one of an infinite number of such lines. The flow pattern for this tank is shown in figure 4-6.

 

This flow pattern is, of course, a cross-section of a three-dimensional flow pattern and figure 4-7 shows two surfaces that contain the flow lines which start in concentric circles at the surface. The water that starts at the surface between these concentric circles must subsequently flow between the two surfaces.

 

It is useful to try to decide what factors determine a flow pattern. Clearly the flow is caused by the gravitational attraction on the water. There are two special flow lines, the one that follows the long path down the wall and across the bottom, and the central one that coincides with the axis. A more typical flow line is that shown in figure 4-5. There must be a reason for the shape of this path.

 

A flow pattern can be thought of as being made up of many tubes of more or less square section that all fit together to fill the space occupied by the flow. Every tube is bounded by an infinite number of flow lines. I have drawn one of these “square” sectioned tubes in figure 4-7.  It is evident that the water changes its direction of flow as it flows through the tube. It also changes its velocity in the direction of flow. Now we can look at what goes on around a short length of this tube to find out what forces are acting on it and what effect these forces produce.

 

 

 

 

 

 

 

 

 

Figure 4-10 shows water flowing through a short length of one of these imaginary tubes, which is of small cross-sectional area, and through which the rate of flow is . Figure 4-10a shows the front view of the elemental volume and shows water flowing into the elemental volume at section 1 where the area is  with a velocity  and out at section 2 at  and .

 

The water in the elemental volume is subjected to a pressures  and  on sections 1 and 2 shown in Figure 4-10b and these pressures produce forces of  and  acting on the water. All four sides of the element are subjected to non-uniform pressures and, because the elemental volume is tapered, this forces produced by all of these pressures produce a resultant force on the water.

 

The four side surfaces are also surrounded by water that will be moving at different velocities to that of the element. We have already acknowledged the fact that fluids resist forces producing a change in shape and so we must expect there to be tangential shearing forces, which could be in either direction, exerted on the water. The forces on two of the sides are shown in figure 4-10c.

 

Finally the mass,  of water in the elemental volume has a gravitational force exerted on it equal to  shown in figure 4-10d.

 

All these forces combine to produce an acceleration acting along the path and a centripetal acceleration to give the change in the direction of flow of the water as it flows through the elemental volume. The whole flow is made up of many elemental volumes between which normal pressures and tangential forces are exerted. The flow will become steady when, in every elemental volume the water moves at the right speed, and the volume has the right shape and is subject to the right system of forces to move in equilibrium with all the other elemental volumes. In the limit these elements tend to become just the path line and differential equations can be set up to describe the flow.

 

In the past all this was recognised and the possibility of setting up mathematical equations that might be solved to describe any flow must have been evident. Navier and Stokes chose to model the shear forces by treating the flow as laminar (See later) and both contributed to the derivation of the partial differential equations, now called the Navier-stokes equations, that might produce the desired result if they could be solved but that must have been a far-off dream for them. Now they can be solved to various levels of complexity by powerful, fast computers but it is not easy and very expensive. Engineers will mostly have to soldier on with ordinary methods. They should recognise that Computational Fluid Dynamics, the name for the application of the Navier-Stokes equations, is still just a tool albeit an expensive one, for exploring the way in which some fluid systems work. It is only as good as the input to it. I watched a television programme on the building of the cruise liner Queen Mary 2 and it reported that the model testing showed that the hull would not meet the contracted speed. We had a sequence of the head-man being towed along beside the very large model of the ship in a tiny pram and studying the flow over the bulbous bow. He came back and ordered the length of the bulb to be increased by one metre. The fact that the change was a nominal amount tells us that he had backed his experience and intellect against whatever design method had been used and, it seems, the necessary improvement was achieved. So, as an engineer, keep the initiative over the CFD and remember, “slaves do arithmetic”. [1]

 

The problem of divergent flow

We have the energy equation which has not been constructed for an idealised flow but is for one-dimensional flow and we plan to apply it to organised flow like the flow through a pipe or over a weir or through an orifice. Its success depends not on the validity of its application but on whether it serves our purpose as engineers. We shall see that it is very successful but we shall also run into limitations on its use. Unfortunately liquids cannot always flow in an orderly manner. In figures 4-11a ,b, c and d the elemental volume was shown as a converging tube and as a consequence the liquid accelerates as it flows through the elemental volume. The forces needed to produce this acceleration are caused by the pressures exerted on the liquid in the elemental volume and the inward acting side forces serve to ensure that the flow does converge. Clearly liquid can also flow through elemental volumes that diverge and then, if the flow is to diverge to fill the elemental volume, forces acting outwards must come into existence. For most of the mobile liquids this does not occur, the necessary forces do not come into existence and the liquid cannot find a condition of flowing in equilibrium in the same way as it does in convergent flow. The flow pattern breaks down into a mass of eddies and I need to offer some explanation and examples. I will do so in a later chapter.