204 
MESSRS. T. E. STANTON AND J. R. PANNELL ON SIMILARITY OF 
before the Society in 1911 # the results of experiments on the radial variation of the 
velocity of air flowing through smooth brass pipes of 4‘9 and 7'4 cm. diameter were 
given, showing that the velocity distribution curves were only identical when the 
values of the ratio vd/v were identical. It was also found that the distribution of 
velocity was only independent of the values of vd/v when the surfaces of the pipes 
were so roughened that the resistance varied as the square of the velocity. 
On examining the results of Darcy and Threlfall no very satisfactory evidence 
of a variation of the ratio with variations of vd/v was forthcoming. This may be 
due to the fact that Threlfall’s observations are all included in a range of vd/v of 
from 66,000 to 300,000, and as the variation in the ratio may be assumed to be 
small, the difference between the results would not be detected very easily, and also 
to the possibility that Darcy’s observations were made in a region over which the 
curve showing the variation of the ratio with vd/v was nearly horizontal. 
As an accurate determination of the ratio was essential for many of the proposed 
friction experiments it was decided to make an extensive series of observations with 
a view to tracing the variation of the ratio from its value 0’5 below the critical value 
of the ratio vd/v to as high a value as possible. As the use of an accurately 
calibrated gas holder was available it was possible to make the determination for the 
case of air flow as well as for water flow. The pipes used were 07125, 1'255 and 
2'855 cm. in diameter, and the method of estimating the velocity at the axis was that 
previously described. It may be mentioned that in this part of the work comparatively 
small errors in observation would have obscured the effect to be investigated, as its 
whole amount in the region of eddying flow is of the order of 5 per cent., so that the 
greatest care and accuracy was necessary. 
The calculated values of the ratio of mean velocity to the velocity at the axis are 
plotted in fig. 1 as ordinates with the corresponding values of log v mAX . d/v as 
abscissae. The values of the velocity at the axis have been adopted merely for 
convenience in the use of the curve to determine rates of discharge from a pipe by 
means of a single observation at the axis, and experience has shown that in this work 
a better graphical representation of the results is achieved by using a logarithmic 
scale for plotting the values of vd/v than by using a simple one. 
The observations were continued for smaller and smaller values of vdfv until 
stream-line motion was set up, below which it was not considered necessary to go. 
The values of the calculated ratios are tabulated in Table I. It will be seen from 
the mean curve drawn through the plotted points in fig. 1 that the value of the ratio 
ranges from 0'5 at the critical value of the speed, i.e., at vdfv = 2500, to 0‘81 at 
the value vd/v — 70,000. For the comparison of these results with those obtained by * 
other experimenters the mean curve through the National Physical Laboratory results 
is reproduced in fig. 2, in which the results of Darcy and Threlfall are plotted. 
It will be noticed that the curve passes through two of Threlfall’s points whereas 
* ‘ Roy. Soc, Proc,,’ A, vol. 85, p. 366. 
