TURBULENT FLUID MOTION AND SKIN FRICTION. 3 
to (velocity)?; but the working out of the idea involved various assumptions which 
are no longer regarded as legitimate. 
In these notes, the scope is much more limited. The method is applied, in the first 
instance, only to plane surfaces ; and, without further theoretical elaboration, some experi- 
mental results are examined from this point of view. 
To obtain a value of the coefficient x for smooth surfaces, take first some of the 
earlier data: Bazin’s results for water flowing in open smooth canals of great breadth 
compared with the depth. These have been expressed in various empirical formule; we 
shall quote one numerical case.* If R is the skin friction per unit area, V the mean 
velocity, v,, the velocity at the open surface and v the (apparent) velocity at the bottom 
of the canal, we are given 
Om =V (1 +1812); v= V(L— 3-626) 
where ¢ = 2 R/p V*. With a mean velocity V = 142-9 em./sec., and ¢ = 0-0044, this gives 
R = 0-0022 p V2 = 0:0038 pv? 
However, we have a more accurate expression of recent work in Lees’ formula for turbulent flow 
in smooth pipes,t namely :— 
R = p V*{0-0009 + 0°0765 (/V d)°*>\ 
This formula includes the results of Stanton and Pannell quoted in the previous section ; 
we may therefore use for the reiation between the velocity » at the wall and the mean 
velocity V the equation v = 0°475 V. Further, if we assume the formula to hold when 
the diameter d of the pipe is made very large, we deduce an expression for a plane 
surface in the required form, namely :— 
R = 0:004 p v® 
We shall use this expression to estimate the frictional resistance of a smooth plane surface, v 
being the relative velocity at the surface. 
4. In order to apply this method, it is necessary to know the distribution of velocity 
over the surface. Unfortunately there are very few determinations available for this 
purpose, although no doubt others may have been made in recent years. The only 
direct observations which have been published appear to be those of Calvert, given in 
his paper on the measurement of wake currents to which reference has already been made. 
A plank, 28 ft. long and coated with black varnish, was drawn along the surface 
of water and measurements were made with (Pitot) tubes projecting beneath the underside 
of the plank. “The speeds recorded at distances of 1 ft., 7 ft., 14 ft., 21 ft., and 28 ft. 
from the leading end were respectively 16 per cent., 37 per cent., 45 per cent., 48 per 
cent., and 50 per cent. of the velocity of the plank; and these proportions appear to be 
maintained at all speeds between 200 and 400 ft. per minute, the latter being the highest 
speed that the arrangements would allow.” 
The relative velocities at these points are thus, respectively, 0°84, 0°63, 0°55, 0°52, and 
0°5 of the velocity of the plank. The width of the plank is not stated, and we must 
assume the effect of the finite width upon the distribution of velocity to be small. 
Summing up the friction along the plank, supposed of unit width, we have :— 
28 
Total skin friction = i 0°004 pv*dl 
0 
From Calvert’s observations we may draw a fair curve showing the variation of v*/V* 
along the plank, where V is the velocity of the plank; it is shown in curve A of Fig. 1. 
*Data from Von Mises, Elem. der Tech. Hydromech., teil 1, p. 97. 
TC.H. Lees, Proc. Roy, Soc., A, 91, p. 49, 1914. 
160 
