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.