TURBULENT FLUID MOTION AND SKIN FRICTION. 
By Professor T. H. Havetocx, F.R.S. 
[Read at the Spring Meetings of the Sixty-first Session of the Institution of Naval Architects, 
March 26, 1920. | 
INTRODUCTION. 
1. Iv is generally admitted that our knowledge of the laws of skin friction for a 
solid moving through a fluid is not very satisfactory. This may be ascribed to two main 
reasons: in the first place the inherent difficulties of the theory of turbulent fluid motion are 
great even in the simplest cases, and in the second place most of the experimental data 
which are available have been gathered, not with the primary object of building up a 
consistent theory, but with more immediately practical aims in view. 
Although no general investigation is attempted in the following notes, it is hoped that 
they may be of interest as a critical discussion of certain aspects of the problem. The 
work may be summarized briefly as fcllows :— 
(1) An examination of experimental results with a view to defining or estimating the 
(apparent) velocity of slip of a fluid in turbulent motion past a solid. 
(2) The expression of the frictional force per unit area at any point of a plane surface 
in the form «pv, where v is the relative velocity at the point; determination 
of the value of « from experimental results. i 
(3) The calculation of the total frictional resistance in the case of a plank for which 
the distribution of velocity is known; remarks on the distribution of velocity 
for a long plank. 
(4) Two numerical calculations to illustrate the assumptions involved in applying a 
similar method to curved surfaces. 
(5) Connection with the law of similarity ; the effect of the ratio of breadth ta length 
in the case of planks; remarks on the extension to long planks and high 
velocities ; general problem of ship resistance. 
RELATIVE SURFACE VELOCITY. 
2. When a liquid flows in steady turbulent motion through a pipe it is usual to 
express the resistance of the wall in terms of the mean velocity over the cross-section, 
because it can be defined precisely and measured accurately. Further, in any theoretical 
study of the motion, it seems necessary to assume that the fluid velocity at the wall 
is zero, there being no slipping of the layer actually in contact with the wall. However, 
in many cases it is found that the velocity varies little over a large part of the cross- 
section and is an appreciable fraction of the mean velocity at points very near the wall; 
this occurs when the turbulent régime is well established, either because of high velocities 
or of large diameter of the pipe. It may be then. for some purposes. a matter of prac- 
tical convenience to treat the motion as if there were a velocity of slip at the wall. The 
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