Osborne Reynolds (1842-1912).

Towards the end of the nineteenth century two incompatible groups of expressions for loss of head to friction in a pipe were in existence. In the first group, due to Darcy and others working with large pipes, the loss was found to be mainly dependent on the kinetic energy head i.e. on velocity squared. In the other, due to Poiseuille and others working with small pipes, the loss was found to depend directly on the velocity. The Poiseuille expression was known to apply to flow at low velocity but there seemed to be no simple way of deciding when to change from one expression to the other.

 

Then in 1883 Osborne Reynolds[1] demonstrated that water flows in pipes in two quite different ways. He injected dye into water entering a glass pipe through a trumpet-shaped intake from a tank of still water (see picture 7-3) and found that at, very low velocities the dye formed a "beautiful straight line through the tube" (see figure 7-4a which, with 7-4b and 7-4c has been copied from Reynolds' original paper). This line was parallel to the axis of the pipe. The implication is that water can flow in pipes in an orderly manner in paths that are parallel with the axis without mixing. This type of flow is now called laminar flow. (We have already seen this in the Hele-Shaw apparatus.)

At some higher, although still very low, velocity, the stream of dye was seen to break up intermittently and, where it was broken up, fill the glass tube (see figure 7-3b). Reynolds viewed this break up in flow in the "light from an electric spark" and observed that "the colour resolved itself into a mass of more or less distinct curls" and represented them as shown in Figure 6-4c. Had there been more than one stream of dye of different colours, it is reasonable to suppose that these curls would have been seen to intertwine and ultimately to mix.

 

At all higher velocities the dye simply spread to fill the tube and it can be inferred that the water was mixing in some way as it flowed. This mode of flow is now called turbulent flow, although it is hard to associate this rather dramatic word with the glass-like surface of a slowly flowing river, which also flows with turbulent flow. Perhaps the appearance is deceptive and that, if it were not for the effect of surface tension, it would be seen to be in a violent state of agitation. The word "mixing" is probably a more apt description of the mode of flow.

 

These observations by Reynolds on the behaviour of water, can be confirmed by the observation of the discharge of oil from a horizontal pipe. At low velocities when the flow in the pipe is laminar the oil forms a "jet" which is depicted in Picture 7-5. It looks like one half of  the tail fin of a fish. At higher velocities the appearance of the jet is shown in picture 7-6. There can be no doubt that the shapes of these jets are caused by two different modes of flow in the pipe. Examination of the "fish tail" jet shows that the main flow comes from the centre of the pipe and, as it clearly has the highest trajectory, must be the higher-speed part of the flow. The low speed part comes from the outer layers and the low speed oil from the top can be seen to flow round the higher speed core flow and to combine with the rest of the low velocity flow to produce a bead of oil which has a low trajectory. In between these two streams of oil a thin web of oil is formed. It seems to be likely that the bead is the result of two effects, the speeding up of the low velocity flow as it moves over the core flow and the surface tension pulling the low velocity flow into the bead. The transition from the laminar flow of picture 7-5, to the turbulent flow of picture 7-6, is produced by increasing the flow through the pipe and, during transition, the jet has alternate slugs of the two types of flow. The jet shown in picture 7-6 is for oil flowing at a velocity at which the transition is just complete. The surface of the jet could well be imagined to be the result of Reynolds' intertwining "curls" of oil tending to break up the jet, yet being prevented from doing so initially by surface tension. The jet does eventually break into large drops and spreads. The oil flowing in the pipe was pumped and therefore entered the pipe with considerable disturbance yet it changed quite quickly to laminar flow. We must conclude that laminar flow is a stable pattern of flow, a fact the Reynolds probably could not demonstrate with water.

Reynolds went on to show that, if the density of the fluid is , the diameter of the pipe is , the velocity of the fluid determined from  is , the viscosity of the fluid is  and all these quantities are in consistent units, the group of variables  takes values less than 2,000 for laminar flow of ordinary liquids and gases in pipes and values greater than 2,300 (with a range up to about 10 million) for turbulent flow. The group of variables is now called Reynolds number, and denoted Re. Picture 7-6 is for a flow at a value of  of about 2,600. For much higher values of , for say water flowing from a pipe, the internal flow is by no means as evident, which suggests that the mean size of the internal flows is much smaller relative to the size of the jet.

 

In Reynolds’ day, order in experimentally derived data was sought in the use of homologues. The idea was to try to find combinations of appropriate physical quantities that would give graphs that would display the data in an economical way, frequently as a family of curves. Reynolds also showed that, for geometrically similar pipes, the values  were the same at the same values of . Newton had already drawn attention to the concept of dynamical similarity between two mechanical systems, that is, to the idea that, provided certain conditions are fulfilled, two mechanical systems will behave in the same way. The most simple must be that of a force acting at the centre of mass of a body. If the ratio of the forces and of the masses is the same for two dissimilar bodies they will have the same acceleration in the line of action of the force. Reynolds' work clearly suggested that, in the much more complicated mechanical system of water flowing through a pipe, a condition of dynamic similarity between different flows might exist in pipes. If this were to be so, then the value of the non-dimensional group , would be related to the flow pattern in the pipe. Reynolds thought that the general relationship between the relevant variables could be written in the form:

                                                         .