Sir T.E.Stanton (1865-1931) and J.R.Pannell.

Stanton, who was an established physicist, and Pannell worked at the National Physical Laboratory in Great Britain and published their work in 1914.[1] They said that, for pipes, "no systematic series of experiments appears to have been made for the purpose of establishing a general relation which would be applicable to all fluids and conditions of flow".

 

They said that the object of their paper was "to furnish evidence confirming the existence under certain conditions, of the similarity in motions of fluids of widely differing viscosities and densities and, by extending the range of the velocity of flow, to investigate the limits of accuracy of the generally accepted formulae used in calculations of surface friction."

 

They also noted that Rayleigh had observed that Reynolds' relationship was only another form of a more general relationship attributed to Rayleigh :-

       where  was the resistance to flow per unit area of wetted surface, which applied to any object immersed in a flowing fluid and pipes come into this category as a pipe is really an object wrapped round the fluid. But Rayleigh had also noted that true geometrical similarity would extend to the imperfections in the surfaces as well as to the circular shape.

 

Stanton and Pannell recognised that they could not satisfy the surface finish requirement and decided, fortunately as it turned out, to use "commercially smooth-drawn" brass tubing. They report that they used pipes of bores ranging from about 3 to about 100 millimetres but in fact they used pipes having nominal bores of 1/8², 1/4², 1/2², 1² bore and one of 5² bore although it is not clear whether this was a brass pipe. From an engineering point of view, applications where the two small pipes might be used do not normally require an accurate foreknowledge of the friction loss, the others are sizes which are commonly used in engineering.

 

The first four pipes were tested with air and water and a sixth pipe of 4² diameter was tested with oil although the results for it do not play an important part in the outcome. The range of water velocities was astonishing by ordinary engineering standards reaching 60 m/s in the small pipes as a result of connecting them to a commercial high-pressure water main. For some reason the maximum air velocity was only 55 m/s and that is very low for ordinary engineering application. The measurements were, by any standards, made very accurately. The data for the small pipe with water flowing in it is given but not plotted on the graph.

 

They adopted Rayleigh's expression and plotted [2] against  for laminar and turbulent flow and the plot is shown in Graph 7-1. Originally various symbols were used for the different pipes but I have changed the symbols to dots and added the range of  for each pipe. It is then clear that only the  mm and the  mm pipes were tested over a wide range of  and these with water. The other ranges all overlap with the two pipes. It is evident that the plots of  against  for the several pipes all lie on a single curve except in the region of transition from laminar flow to turbulent flow. The coincidence in the laminar region was to be expected because Poiseuille's expression can be rearranged to give . However, Stanton and Pannell had shown that the pipes that were tested had, over the range tested in the turbulent region, the same relationship between  and . This confirmed Reynolds' results and Rayleigh's suggestion that his plot would show the dynamical similarity between different pipes at the same Reynolds number. The outcome did not show the limits of applicability of the various expressions. This is of no consequence as the range of  for the practical range of the ratio of maximum velocity of flow to minimum velocity of flow in a given pipe is not often more than 10 and usually very much less because, at the low velocity, the pipe is too large for the application and so unnecessarily costly, and at the high velocity end energy is needlessly wasted in friction. So a given pipe only spans an increment of  of less than  and probably nearer to .

 

Stanton and Pannell undertook some retrospective plotting of data from other sources in the new way proposed by Rayleigh and added these to their plot. The most important of these was a comprehensive set of experiments by Saph and Shroder[3] on 15 pipes of "probably identical surface" ranging in diameter from  mm to  mm. These results, when plotted in the new way, totally confirmed Stanton and Pannell's results. Oddly some measurements made by Reynolds fell well below Stanton and Pannell's curve and they attribute this to a higher roughness even though they selected the figures because the lead pipes used by Reynolds were thought to have a similar roughness to those of Stanton and Pannell. We now know that Reynolds' results fall into a region that is physically impossible. It is idle to speculate on the reason for this, although the addition of  to Reynolds' figures would resolve the difficulty, but it does tell us that Stanton and Pannell were not aware of the fact that their curve was the lower boundary for points on their plot. The work of Nikuradse was needed to show that this was the case.