TURBULENT FLUID MOTION AND SKIN FRICTION. 5 
CURVED SURFACES. 
7. If a body is moving through a liquid we may suppose the force on an element 
dS of the surface to be resolved into a normal pressure and a frictional force RdS; 
the latter will be in a direction opposite to the relative velocity and, if we suppose it 
to make an angle @ with the direction of motion of the body, we may define the skin 
friction as [Rd Scos 6, taken over the wetted surface. 
For plane surfaces we have shown that there is some justification for taking R equal 
to x pv, where v is the relative velocity ; in the general case one would probably have 
an additional term involving the curvature of each point. Consider first the case of “two- 
dimensional ”’ flow, when the longitudinal cross-section of the body is of ship-shape form. 
Here each element is curved in the line of motion, and if the curvature is small and we 
assume R = «x pv”, the effect of the curvature is to be found in the distribution of velocity. 
The effect of this kind of curvature has been discussed by Mr. G. 8. Baker by estimating 
the distribution of velocity in stream-line motion. It should be noted that it is not the 
same as the effect of the shape of midship section, for there the curvature is at right 
angles to the line of flow. Naturally in three-dimensional flow both effects are superposed, 
and cannot be disentangled. No experimental determinations of surface velocity appear to 
have been published, at least for ship forms in water. The extension from plane to 
curved surfaces is thus to a large extent speculative: however, as the extension has been 
made already in other methods, two numerical examples are given here to illustrate the 
various assumptions. 
8. For two-dimensional motion, suppose that the model js 28 ft. long, as for Calvert's 
plank, with a longitudinal section shown, as to the upper half only, in model C of Fig. 1. 
This is a form for which Baker and Kent * have calculated the pressure distribution in 
stream-line motion; from the curves given in that paper we can draw a curve of the 
distribution of v?/V2 in stream-line motion, v being the relative surface velocity and V 
the velocity of the model. Now, as an arbitrary assumption, suppose that in turbulent 
flow v? diminishes for the model according to the same law as for the 28-ft. plank ; 
that is, we take a reduction factor at each point from the curve A of Fig. 1. We obtain 
thus the curve © of Fig. 1 as an estimated distribution of relative velocity, or rather it 
* G. S. Baker and J. L. Kent, Trans. I.N.A., Vol. L., Pt. I1., p. 37, 1913. 
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