The concepts of hydraulically smooth and hydraulically rough flow.

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Graph 7-2
I must now offer some explanation of the results we have so far. In fact I shall have to speculate as others have done before me and you must decide whether it makes sense and fits in with known physics and how you understand science.

 

When we were considering the concept of viscosity as a property of a fluid we simply accepted that the fluid behaves as if the solid surface is just like a stationary layer of fluid. Certainly the use of this decision leads to a very successful way of measuring viscosity. However it is evident that, when the flow is turbulent, the nature of the internal surface of the pipe has an influence on the friction loss in the pipe. We need some explanation for this.

 

Picture 7-9 shows the inside surface of a piece of 15mm copper tube at a magnification of about 1,000. Such a pipe is made by extrusion through a die. For copper, at least, the process leaves the surface in a very rough condition as if it sticks and tears as it emerged from the die. The white lines are the crests of hollows that appear to intertwine. In the middle there are two white lines that are  mm[1] apart and this corresponds to about 20,000 molecular “diameters”. On a molecular scale these hollows are very large.

Picture 7-11 is of a new spectacle lens magnified 20,000 times. One would regard a lens as having a very good surface finish but even so the “lumps” are about 400 molecular “diameters” across.

 

Anyone simply handling new copper pipe would have no idea of the true nature of the surface and we had to wait for the electron microscope to see this detail. It is now quite impossible for me to think of the surface of a copper pipe behaving like a stationary layer of fluid yet that is what appears to happen. It is clear that we need to know more about the structure of a fluid and especially a liquid.

 

We are only just now able to study the molecular structure of liquids[2] and it is very difficult. However some information is emerging for water, which is the most common liquid. Water is also the liquid with the most extraordinary properties. The most important matter for us is its molecular behaviour.

 

In order to start we must consider the three phases of water. When water is solid, i.e. ice, the molecules are bound together in a lattice and only near to the temperature of melting do they start to move in this lattice. During melting the molecules become free of the lattice to change to a liquid. The molecules are then free to move but are still very close together. The molecules of water have a shape that stems from its structure. The two hydrogen atoms are bonded together and the oxygen atom is bonded to both of them to produce a triangular arrangement. These T shaped molecules pack together closely and, under the action of intermolecular forces, the molecules form clusters. These clusters are very short lived and a single molecule may change the molecules with which it forms clusters as many as 1012 times every second. The size of the clusters vary but the dominant size is five molecules.

 

The next phase is the gaseous phase and then the molecules break from the close-packed but mobile state of the liquid for the molecules to “fly” freely in space and to collide with one another within a cloud of molecules. The change takes place by boiling or evaporation depending on the circumstances. Of the whole range of temperature for which water molecules can exist the liquid phase occupies only a very small part. It appears to be a transitional phase between a solid and a gas rather than a phase in its own right. It is special yet very common because our planet has settled down to have a mean temperature in the right range for many substances to be in liquid form.

 

Now, if molecules of water form clusters however short-lived they may be, there must be an attractive force between the molecules and they must touch, whatever that may mean in this context. Presumably the clusters also touch or there would be no exchanges going on. In a quantity of liquid that is not being subjected to external forces that tend to change its shape, the motion of a given molecule would appear to be random within the seething mass of molecules surrounding it. If, however, the liquid were to be subjected to forces tending to shear it, this must introduce a new element to the motion in which molecules are pulled apart as one layer slides over another. The “stretching” of the intermolecular force requires the expenditure of work and this shows up as the property viscosity. By chance the method of measuring viscosity in laminar flow in a pipe happens to measure this work and seemingly nothing else.

 

So how would the liquid near the surface behave when it is part of a flow in a pipe? If I start with the liquid in the hollows it seems to be likely that the molecules are so small compared with the size of the hollows that they will simply fill them and go on forming their clusters. It is obvious from picture 7-8 that the crests of the whorls of copper are not all at a single level so the molecules in the hollows do not form a single surface at which shearing can start. It seems instead that a layer forms that is thick enough to include all the crests and that laminar flow occurs in this layer with all the shearing of the molecular structure. We have seen that, when oil is pumped through a pipe, the inevitable disturbance created by the pumping is damped out quickly to give laminar flow. It is likely that this layer at the surface is both stable and difficult to disrupt. If the general flow pattern is laminar this layer is the one that behaves as if it is stationary. If the flow is mainly turbulent this layer still forms and a transition occurs between laminar flow and turbulent flow at some small distance from the wall. This layer next to the wall is generally called the laminar layer and, in some circumstances, the laminar sub-layer

 

However the thickness of the laminar layer decreases as the velocity of the flow increases. At low velocities in the turbulent flow region the laminar layer will be much thicker than the depth of the irregularities in any surface that has been manufactured for engineering purposes. Under these conditions the actual texture of the surface has no effect on the nature of the flow. It is as if the turbulent flow is contained in a stationary tube of the same fluid. The flow can be seen to be directly comparable with laminar flow in that the pressure drop is a function of the properties of the fluid, the velocity and the diameter. When such a flow occurs it is said to be hydraulically smooth.[3]

 

As in the case of laminar flow the loss of energy in hydraulically smooth flow in a horizontal pipe would be sustained by a steady drop in the pressure energy of the fluid and the rate of loss is higher than it would be in laminar flow because of the mixing which goes on in turbulent flow. Nevertheless the ultimate mechanism by which the pressure energy is absorbed into the molecular structure of the fluid is still that of viscosity. It follows that the friction loss in hydraulically smooth flow is a function of density, viscosity, velocity and the diameter of the pipe and the existence of the smooth pipe curve is no longer so surprising.

 

Another important consequence of the smooth pipe curve is that, at the same Reynolds' number, all fluids flowing at speeds when the flow can be regarded as hydraulically smooth, appear to behave in the same way. Recalling Reynolds' "mass of curls" in a liquid that has only just changed to turbulent flow we must expect the size of these curls to decrease as the value of  increases and the intensity of the mixing increases. If, in this mixing, molecules move together in groups (each containing many clusters) we must expect the size of these groups to reduce as the mixing becomes more intense. The implication of the smooth pipe curve is that the mean of the random flow pattern is the same for all fluids flowing under hydraulically smooth conditions at the same value of .

 

For all practical pipes there is a range of  for which the flow is hydraulically smooth and  can be obtained from the extended version of Stanton and Pannell's curve which has become known as the smooth pipe curve. However, as the velocity in a given pipe increases and the thickness of the laminar layer decreases, there comes a point at which the crests of the irregularities of the surface start to protrude into the turbulent flow. At this point it is possible to think now of the fluid filling the hollows of the surface but not the existence of a layer of fluid moving with laminar flow.

 

When this stage is reached the crests disturb the turbulent flow just as the irregularities in the kerbstones of a road disturb the flow of water along a gutter[4]. This disturbance is propagated towards the axis and adds to the intensity of mixing. As the velocity increases still further the fluid in the hollows of the surface are increasingly scoured out and more of the surface exposed and the disturbance caused by the surface increases.

 

Nikuradse's curves show us that there is a further phase where  becomes independent of . Then the Darcy expression reduces to  where the constant equals  and  is dependent on the value of  and not on the properties of the flowing fluid. In this phase the flow is said to be hydraulically rough.

 

This has a most important consequence. Suppose that two pipes of different diameters have been manufactured in the same way and can be taken to have the same internal surface texture. If one pipe can be tested in the hydraulically rough region the value of  could be used to find a value of  on the Nikuradse plot by interpolation. As  will be known, a value of  can be found for the pipe surface. As the surface texture of the pipe is almost certainly not the same as that of sand this  is a measure of the surface texture of the pipe as an equivalent sand grain roughness. This has given us a way of quantifying the surface texture through its affect on turbulent flow and not by some process that depends on direct measurement of the surface profile. The equivalent sand grain roughness is given the symbol  and values of  for commercial pipes are given in data books. Of course the fact that the scale would have been different had Nikuradse used sharp sand to coat his pipes does not matter; we have a scale that works. But, going back to the two pipes, a value of  for the second pipe can be found by finding the new value of  and reading  from Nikuradse's plot.

 

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Graph 7-3
It is instructive to see, in Graph 7-2, the plot of  for hydraulically rough flow against  derived from Nikuradse's published figures. There are no figures for the range of values of  below 0.001 when most practical pipes have values of  in this range. It is hard to see how Nikuradse's work using sand coating could be extended into this range.

 

There is a piece of information missing. We know how pipes coated internally with sand behave during the transition from laminar flow to turbulent flow, what we do not know is how real ordinary pipes behave.

 

 

 

 



[1] I set my micrometer screw gauge to have a gap of 0.01 m m and I could not see light through the gap. It is very small.

[2] See “New Scientist” 21.6.1997 p40

[3] A high head hydroelectric power station was built with Pelton wheel turbines. During the first few months after commissioning its overall efficiency gradually improved. When the opportunity came the inside of the delivery pipe was examined and was found to be coated with a fine growth of algae. What was that algae actually doing?

[4] Look for yourself. It is interesting.