6 TURBULENT FLUID MOTION AND SKIN FRICTION. 
shows the values of v?/V? for the model. Also the total skin friction, per unit breadth 
=|RdScosé = |x prdl 
taken along the straight axis of the model. Estimating the area under the curve C, and 
the curved length of Model C, we can calculate the mean resistance per unit area. It 
appears that the model has a mean resistance per unit area about 11 per cent. greater than that 
of a plank of the same length. 
9. For a three-dimensional case we take similar preliminary data from a paper by 
Mr. D. W. Taylor * on solid stream forms. We carry out the same process as in the 
previous section, and it is unnecessary to reproduce the corresponding curves. The only 
difference arises from the fact that the solid is one of revolution with pointed ends ; con- 
sequently the element of area approaches zero at the two ends. If y is the ordinate of 
the ship form at any point on the axis, we have to graph the values of yv? on the 
straight axis of symmetry as a base, instead of simply v* as in the two-dimensional 
problem. As far as the numerical approximation has been carried, it appears that the 
mean resistance per unit area for this model is about equal to, or slightly less than, that 
of a plank of the same length. 
10. The resistance of a small appendage on the surface of a ship must depend chiefly upon 
the relative surface velocity in its neighbourhood. It is appropriate to refer here to some 
experiments by Mr. Baker + to determine the added resistance due to local roughness 
of a model. If the rough area were small enough relatively so as not to affect appreci- 
ably the flow over the rest of the model, and if the slope of the surface and the direction 
of flow were known, it might be possible to deduce information about the velocity distri- 
bution ; however, one cannot analyse in this way the results to which reference has been 
made. 
In regard to skin friction for curved surfaces especially, one may venture to quote 
and endorse a remark made by Professor Lees {: “It is of prime importance that further 
measurements should be made on bodies which lend themselves to simple theoretical 
treatment in order to build up a satisfactory theory.” 
Law OF SIMILARITY FOR PEANKS. 
11. The law of similarity in its usual form :— 
R = p V?f (V Ijy) 
applies to bodies which are geometrically similar in form, and are similar as regards 
scale of roughness. In experiments with planks we may perhaps neglect the thickness 
and suppose the motion to be in two dimensions only ; but the planks will not be similar 
unless the ratio of breadth to length is constant. In other words, the general formula 
from physical dimensions is :— 
R = pV°f (b/l, Vly) 
where the undetermined function depends upon two quantities, the ratios 6/1 and V ljv. 
In most experiments the ratio 6/1 has not been kept constant, but the planks have 
been of constant breadth and varying length. Consider, for example, Zahm’s results,§ 
which he expressed in the empirical formula :— 
R = kl-°07 yres 
* D. W. Taylor, Trans. I.N.A., Vol. XXXVI., p. 234, 1895. 
t G. S. Baker, Trans. North-East Coast Inst., Vol. XXXII., p. 50, 1915. 
{ C. H. Lees, Trans. I.N.A., Vol. LVIII., p. 64, 1916. 
§ Zahm, loc. cit. 
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