1918.] The N.Z. Journal of Science and Technology. 155 
a theory of fluid friction applicable to turbulent flow and sufficient for the 
purpose of a working hypothesis has been in existence since the year 1883, 
when Reynolds* * * § published the result of his theoretical and practical 
investigation into the question of fluid friction, whilst further experimental 
proof of the applicability of the law to practical conditions is furnished 
in an ample manner by the work of Stantonf and Stanton and PannellJ, 
and by the work of the Aeronautics Committee of Great Britain ; but 
hitherto there has been no recognition of the law by hydraulic engineers, 
whilst the text-books invariably content themselves with supplying a list 
of empirical formulae for the reader to choose from after a more or less 
brief account of Reynolds's works. 
The object of this paper is to call attention of engineers in this Dominion 
to the existence of an accredited and comprehensive law of fluid friction, 
and to urge upon them the desirability of adopting it in practice. An 
attempt is at the same time made to review the experimental data available 
in regard to the flow of water in pipes, and to make deductions therefrom. 
Briefly, the law in its summarized and generalized form amounts to 
this—that for geometrically similar surfaces the following relationship 
exists, viz.: R /pv 2 is a function of vl/v, where R is the resistance per unit 
of area, p the density of the fluid, v the relative velocity, l a dimension 
of the surface, and v the kinematic viscosity or the ratio of the physical 
viscosity to the density of the fluid. This law applies to pipes or channels, 
moving surfaces, and vanes of any form, and to any fluid, liquid or gaseous, 
and affords a means of co-ordinating every possible condition which enters 
into the problem, and furnishes a physical parameter, vl/v, in terms of 
which to express the values of R /pv 2 . If now the law be restricted to 
pipe-flow, and the expression ~R/pv 2 converted into a form which is. more 
usual in engineering practice, we obtain id/4v 2 as a function of vd/v, where 
i is the hydraulic gradient, d the diameter, and v the mean velocity. 
It was first suggested by Lord Rayleigh§ that vd/v should be plotted 
as abscissae, and id/iv 2 as ordinates ; which method has been adopted 
by Stanton in plotting the results of his experiments, except that it is 
found necessary, because of the extremely wide range of observations, to 
employ the logarithm of vd/v instead of the number itself. 
The following is a summary of the more important results of investiga¬ 
tion to date, and of some conclusions arrived at from a study of the 
subject:— 
1. The critical velocity, or the velocity at which a fluid changes from a. 
linear state of flow to a sinuous state, is such that the value of vd/v is a 
constant.* 
2. The distribution of velocity throughout the section of a pipe is such 
that the average velocity is a function of vd/v, where v is the velocity of 
the centre filament.f If the flow be linear the ratio of the average velocity 
to the maximum at the centre is a constant—viz., 0*5. If the flow be 
* Reynolds, Phil. Trans. Roy. Soc., vol. 176, p. 935, 1883. 
f T. E. Stanton, Proc. Roy. Soc., A, vol. 85, p. 366. 
J T. E. Stanton and I. R. Pannell, Similarity of Motion in Relation to the 
Surface Friction of Fluids, Phil. Trans. Roy. Soc., Ser. A, vol. 214, pp. 212-15, 1914. 
§ Lord Rayleigh, Phil. Mag., vol. 34, pp. 59-70, 1892. 
