TURBULENT FLUID MOTION AND SKIN FRICTION. 7 
It is usual, following Lord Rayleigh, to correct this to satisfy the law of similarity 
and to write :— 
R = Const. x p V? (p/V 1)?! 
It is probably true that the experiments are not sufficient to decide between these two 
forms. The present point is that without altering the empirical law as regards J and V, 
the formula can be made to satisfy the dimensional equation by writing it, for instance, 
in the form :— 
R = Const. x p V7 x (U/b)° x (v/v)? 
Similar remarks may be applied to Froude’s experiments with planks in water. For instance, 
with planks coated with fine, medium, or coarse sand the resistance is proportional to the 
square of the speed. Hence in these cases the quantity R/p V? is a function of the ratio 
b/l and of the coefficient «, which may be called the ratio of roughness; but it is not 
possible to separate the two effects in the results. 
12. Consider the distribution of relative surface velocity from front to rear of a long 
plank. Neglecting the disturbance of the edges, we may divide the distribution roughly 
into three stages; firstly, one in which the 
velocity falls rather rapidly, then a long 
stretch in which it is practically constant, 
and finally a relatively short stage in which | gpy: 
the influence of the end is appreciable. For 
a very long plank in which the middle 
stage predominates, the mean resistance per 
unit area will approximate to « pv*, where 
» is the steady value of the surface 
velocity. On the other hand, for shorter a 
planks a two-term formula may be 
sufficient, which may possibly be of the type 
p V7{A +B (v/V1)"}. 
Again, if the breadth is taken into account such a formula would be incomplete. 
Here in the extreme case of a long plank of finite breadth, the analogy of steady flow 
through a pipe is suggested; and the mean resistance should approximate to a two-term 
formula of the type just given, with the length J replaced by the breadth d. This is 
the argument which has been worked out by Professor Lees in the paper * already quoted ; 
in that analysis d is taken as the diameter of an equivalent circular cylinder and deduced 
by a certain method from the dimensions of the plank. 
13. On the analogy of the law of similarity for flow in pipes, Mr. Baker f has collected 
results on planks and models into one diagram in which R/p V® is graphed on a base 
Vljv. We have seen that certain reservations are necessary in grouping the data from 
planks in this way; but the general trend of the curves obtained is very suggestive. Fig. 2 
shows the main points in a diagrammatic sketch, not drawn to scale, but based on the 
paper quoted. 
The stage AB represents simple viscous fluid motion when R is proportional to V. 
BC is an unstable condition when the flow may be partly simple and partly turbulent ; 
after C the latter régime becomes permanently established. If the resistance R is repre- 
sented by a single-term formula f V”, it is clear that the best single power is V” in the 
neighbourhood of the points B and C. It may be noted that Froude gives V* for short 
smooth planks of 2 ft. in length, and it may be presumed that the region near C was 
then under observation. As the length is increased, the best single power decreases to, say, 
vis} near D, if we take this point to represent the limit of available data. Froude’s 
* C. H. Lees, loc. cit. 
+ G. S. Baker, Trans. North-East Coast Inst., loc. cit. 
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