4 TURBULENT FLUID MOTION AND SKIN FRICTION. 
The integral can now be evaluated approximately from the graph; applying Simpson’s 
rule with intervals of 1 ft., the integral of v?/V? along the plank comes to 10°2. With 
a velocity V = 400 ft./min., this gives a total resistance of 3°51 lb. 
The resistance of the plank was not measured by Calvert. However, we may obtain 
another estimate from W. Froude’s results (Brit. Assoc. Reports, 1874). Using Plate II. 
of that report, we can read off from the curves the resistance of a 28-ft. varnished 
plank at 400 ft./min.; it is 3°51 lb., as nearly as can be estimated. Naturally one 
need attach no importance to the coincidence; except that with a constant coefficient 
«x = 0°004 and taking account of the actual distribution of surface velocity, the value 
of the total friction is in agreement with direct measurements in similar cases. 
5. It must not be supposed that this method means that the total skin friction is 
proportional to the square of the velocity V of the body. From the theory of physical 
dimensions applied to similar bodies we have :— 
R = p Vf (V Iv) 
On the present statement, the only difference is that it is the relative surface velocity 
which is some undetermined function; for instance, in the graph of Fig. 1, if x is the 
distance from the leading end the graph must satisfy an equation of the form :— 
v/V? = F (2/l, V1/v) 
After integrating along the plank, we obtain then R in the general functional form given 
above. 
6. Assuming the value 0°004 for x for smooth planks we may deduce some informa- 
tion as to the fall of surface velocity, for the mean resistance per unit area divided by 
kp gives the average value of v* over the surface. 
Taking Zahm’s experiments * on varnished planks in air, using a suitable value of 
p and taking the results as they are given in the table for the resistance of planks of 
various lengths at 10 ft./sec., we obtain the following :— 
Length .. ¥ at 2 4 8 12 16 
Average v*/V?.. so OSB OSB ORIG — Osea 0°49 
From the similar tables of W. Froude for planks in water at 10 ft./sec., we find— 
Length... ae Be 2 8 20 50 
Average v?/V?_.. Pens 29 0:419 0°359 0°316 
There is a much quicker fall in water than in air, but of course the VJ/y values do not 
correspond in the two sets. Froude gives a column which is said to be the resistance 
per square foot of the last foot of plank; this is, one may suppose, obtained as the 
difference in resistance of two planks differing in length by 1 ft., and it obviously 
assumes that the addition of 1 ft. to the rear of a plank does not alter appreciably 
the distribution of velocity over the rest of the plank. Taking the figures as they stand, 
we may deduce the average value of v* over the last foot of plank for various lengths ; 
they give :— 
Length Ge at ne 2 8 20 50 
Average .. 58 eee OsI503 0-340 0-309 0-291 
the second row being the average value of v?/V? over the last foot. Taking the square 
root, we may estimate the relative velocity at the end of a 50-ft. plank moving at 10 
ft./sec. as about 0°54 of the velocity of the plank; and this estimate will be on the high 
side. It may be compared with the value 0°475 which we found for the similar ratio 
in flow through pipes when the steady state has been reached. 
* A. F. Zahm, Phil. Mag., 8, p. 58, 1904. 
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