2 TURBULENT FLUID MOTION AND SKIN FRICTION. 
magnitudes involved may be illustrated by some numerical cases. Taking indirect calcula- 
tions first, we may quote an instance from Lamb’s Hydrodynamics (6th edn., p. 666). 
Assume that the resistance per unit area of the wall of the pipe is given by xpv?, 
where p is the density and v, the mean velocity of the liquid. Also suppose the velocity 
to be approximately v, over the cross-section, except in a thin layer of thickness 1 
in which there is laminar flow. Im order to obtain the same resistance per unit area, 
we must have pv,/l = «pv, or 1=v/kv,, where pw is the viscosity and v the kine- 
matical viscosity. For water moving with a mean velocity of 300 cm./sec., this gives 
I = 0:024 cm. 
For the cognate problem of the motion of a solid through a liquid, take an example 
from Froude’s data for planks. The resistance of a 2-ft. plank at 600 ft./min. is given 
as 0°41 Ib. per sq. ft.; the thickness of the equivalent layer for laminar motion giving 
the same resistance is found from pv/l = 0°41, or 1=0-°007 in., approximately. 
But these are indirect estimates, and we turn now to experimental determinations 
of the velocity. Here the velocity is obtained by means of a Pitot tube, and it is obvious 
that the nearest point to the wall at which an experimental value can be found depends 
upon the dimensions of the Pitot tube. For the motion of a plank through water we 
have Calvert’s measurements of frictional wake.* In this case the Pitot tube was one-eighth of 
an inch in diameter. It was found that the relative velocity at the surface of the plank 
decreased from full speed at the front end to about half that speed at the aft end of 
a 28-ft. plank moving at about 400 ft./min. For turbulent flow through pipes, passing 
over the earlier work of Bazin and others, we may take an example from measurements 
by Stanton.t The Pitot tube was of rectangular section, the external dimension in the 
direction of the radius of the pipe being 0°33 mm. With a smooth pipe of 2°465 cm. 
radius, the velocity at the axis being 1,525 cm./sec., the velocity at 0°025 cm. from the 
wall is given as 592 cm./sec. Further, the mean velocity is about 0°81 of the velocity 
at the axis.{ Hence we may deduce that the (apparent) velocity at the wall is 0°475 of 
the mean velocity. A similar result is obtained from other cases given in the papers 
quoted, the value of Vd/v being in the neighbourhood of 50,000. 
We shall assume that we can refer to a relative surface velocity which is sufficiently 
definite for certain purposes, the limitations being indicated by the numerical examples 
which have been given. 
PLANE SURFACES. 
3. We wish to see if the frictional force per unit area on any plane element of surface 
can be expressed by «p v*, where v is the relative velocity of the fluid and wall at the 
point, p is the density of the fluid, and « is a non-dimensional coefficient of roughness. 
One of the earliest attempts to analyse turbulent fluid motion, by Boussinesq, involved 
a surface friction of this kind, together with a constant effective coefficient of eddy 
viscosity, or of mechanical viscosity as it was called by Osborne Reynolds. Experimental 
results on flow through pipes can be fitted more or less by a scheme of this kind, but 
it is generally recognized now as only an approximate statement. In the first place the 
mean friction on the walls is not simply proportional to (velocity), but depends also on 
the diameter ; so that the friction on an element of the wall may include a term involving 
its curvature. Further, the effective eddy viscosity is not found to be constant over the 
cross-section, though it varies little except near the walls. A similar theory has been 
applied recently by G. I. Taylor to the turbulent motion of the atmosphere and the skin 
friction of the wind on the earth’s surface. 
Rankine, in his method of augmented surface, assumed a skin friction proportional 
* C. A. Calvert, Trans. I.N.A., Vol. XXXIV., p. 61, 1893. 
} T. E. Stanton, Proc. Roy. Soc., A, 85, p. 366, 1911. 
+ Stanton and Pannell, Phil. Trans.. A, 214, p. 205, 1914. 
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