A Note on Form Friction and Tank Boundary Effect 
Sir Thomas Havelock 
The following remarks are concerned with a suggestion 
made by Professor Horn many years ago for estimating form 
friction by means of the sinkage of the model, and with the 
possible application of this method to motion in restricted 
water. 
The influence of the walls and bed of the tank can, in usual 
circumstances, be conveniently separated into wave effect and 
frictional effect. The underlying theory of the wave effect is 
well known; the bed contributes the so-called shallow water 
effect, while the walls may give rise to interference effects due 
to the waves reflected from them. It is true that the actual cal- 
culations are beset with difficulties, such as occur in wave 
theory generally; but at least it may be said that the funda- 
mental causes can be specified reasonably. The theoretical 
aspect of the frictional effect seems to me to be less clear. The 
point in question is the difference between the ship form and 
a plank. A thorough analysis, theoretical and experimental, 
seems impracticable in general; though useful and important 
results are available for completely submerged solids of 
revolution. Assuming the form friction to be small, the usual 
practical method is to use the idea of effective equivalent 
velocity; that is, the actual frictional resistance of the ship 
at a given speed is taken as equal to that of a plank at some 
slightly higher speed. Failing a complete analysis of the actual 
flow, we can only make some reasonable assumption for defin- 
ing this equivalent effective speed. 
Horn [1] proposed to use the measured sinkage of the model 
for this purpose. If v is the velocity, and h is the sinkage, 
he gives for the required effective mean velocity v,, the ex- 
pression 
Vm = (v? + 2gh)”, (1) 
or if v,, = v + Oy, the relative increase in velocity is 
bv/v = (1 + 2gh/v?)*# —1. (2) 
If the frictional resistance R is proportional to vy", the relative 
increase in resistance, or the form friction, is given by 5R/R 
= n 6v/v. It was shown from model data that this gave reason- 
able values for the form friction, of the order of 8 per cent. 
In a short paper a few years later [2], I examined the theore- 
tical solution for a particular form, namely the general ellip- 
soid, including the case of a spheroid. The problem was treated 
as the motion of a double model, that is, a complete ellipsoid 
moving axially in an infinite liquid: a problem which can be 
solved exactly. 
Taking the motion along a horizontal axis Ox with the trans- 
verse axis Oy horizontal and with Oz vertical, an expression 
was obtained for the resultant vertical fluid pressure on one- 
half of the surface of the ellipsoid with respect to the xy-plane. 
If we now suppose the ellipsoid to be floating half immersed 
and if the velocity is small so that we may neglect the surface 
disturbance of the water, we can define an equivalent sinkage. 
If Z is this defect of vertical pressure and S is the area of 
the water plane section, we take h = Z/goS. The results 
were compared numerically with Horn’s value and also with those 
obtained by Amtsberg [3] for totally submerged spheroids. The 
analytical expressions for the general ellipsoid were given in 
terms of ellipsoidal coordinates; I quote now the special case 
of a prolate spheroid, where the result can be put into a 
simple form. 
The value of Z is given by 
rp rae ime lene) 
(a + b)? 
and the sinkage, as defined, ish = Z/ noabg. 
—txeabv? (3) 
In this, 2a is the length of the spheroid, 2b the equatorial 
diameter, and k, the virtual inertia coefficient of the spheroid 
for axial motion. If, for example, we take a length-beam ratio 
of 8, we find h = 0.029 v?/g; and assuming n = 1.825, we 
get an increase in frictional resistance of 5.3 per cent, agreeing 
fairly well with Amtsberg’s values. 
BR 
609 
