Comparing (3) with Horn’s definition of the mean velocity 
we see that in this case 
Ym = v (1 + k,)[a (a + 2b)]*/ (a +b). (4) 
For most cases of interest, a/b is fairly large, say 8 or more, 
and we have approximately 
Vm = v (1 + ky). (5) 
Hence we have the simple, and interesting approximation 
dv/v =k, ; BR/R=nk,. (6) 
For example, the virtual inertia coefficient for a spheroid 
of length/beam ratio of 8 is 0.029, and 5R/R = 0.053. It 
might be going too far to apply this to ship forms, where the 
inertia coefficient is itself subject to uncertainty; however, 
assuming an effective virtual coefficient of 5 per cent would 
give a form friction of about 9 per cent. 
Of course for a spheroid the velocity distribution is known 
exactly and we might take some other suitable definition of 
the mean velocity. For instance, it might be obtained from 
the mean of the square of the tangential velocity per unit area 
of surface. It can easily be shown that this leads to the same 
approximation (5) when a/b is large. The point of Horn’s 
definition is that the sinkage can be determined experimentally. 
Coming now to the corresponding problem in restricted water, 
the tank boundary effect or the so-called blockage effect has 
become important in view of the need for greater accuracy 
and certainty in interpreting experimental model results. 
Reference may be made, for instance, to two recent papers: 
the B.S.R.A. experiments on the Lucy Ashton [4] (Conn, 
Lackenby and Walker), and the scale effect in Victory ships 
and models [5] (van Lammeren, van Manen and Lap). In the 
discussion on the former paper, Professor Horn referred to 
his method of using measured sinkages to estimate form fric- 
tion and suggested that it might be used to determine the 
necessary correction due to the boundaries of the tank. How- 
ever it seems that, at least for the Lucy Ashton, the differences 
in sinkage were too small to be determined experimentally with 
sufficent accuracy. It might be of interest to extend my previous 
calculations to the similar problem in restricted water. Consider 
a spheroid half-immersed and moving along a tank of breadth 
B and depth H. With the same limitations as for unrestricted 
water, we consider the motion of the complete spheroid in an 
enclosed rectangular channel filled with water, B being the 
distance between the side walls and 2 H that between the upper 
and lower walls. We require to calculate the quantity Z of (3), 
that is the resultant vertical force on the lower half of the 
spheroid. It is possible to obtain analytical expressions in a 
series of terms involving spheroidal harmonics; but they become 
very complicated and it is difficult to assess the degree of 
approximation numerically. The particular case of a sphere 
can be worked out in more detail, but the spheroid is com- 
plicated by the additional parameter of the length-beam ratio. 
Taking only the first step in the approximation I give now 
the result obtained for the quantity Z; it is 
z= Aneaby!] (drape 
(a + b)* 
ab? be 
1 + (Ll + k)) ——a } — 1 7 
pep) er @) 
If we write 
* q = (m? B? + 4n? H?2) / (a2=—=1b2) ; 
B = 4 fq + (q? + 4)"*], 
the coefficient a is given by 
PPP feel esamilba guP ey celle 2. 
a LD (eee -s-F eR (8) 
where the double summation is taken over all positive and 
negative integral values of m and n, excluding the pair 
m = 0, n = 0. This summation arises from the doubly infinite 
series of images involved in the solution. This result may be 
subject to correction if the analysis is carried to a further stage, 
and the range of applicability is uncertain on that account. As 
before, we may simplify the result if b/a is small; we have 
approximately, 
b2 
Vm =v (1 +ky){1 + — (+k) a (9) 
a 
and, instead of (6) for unrestricted water, we have 
5 b? 
ak tf = GPa. 
2 
Vi am 
(10) 
Numerical computation has been made for a few cases for 
the spheroid with a = 8b. We have taken B = 2H as a usual 
tank ratio and it also simplifies the computation. For B/2b 
equal to 12, 8, 4)/2 the approximate values of the coefficient 
a are 0.065, 0.160, 0.392 respectively. If we define the 
blockage coefficient as the ratio of the maximum cross section 
of the half-spheriod to the sectional area of the tank, this 
coefficient is 0.005, 0.012, 0.024. From (10) the percentage 
form resistances at these values are 5.46, 5.84 and 6.93 
respectively, the value for unrestricted water being 5.29. 
The differences are negligible for small values of the 
blockage coefficient. It is not worth while attempting any direct 
comparison with model results meantime. The calculations 
were made for a spheroid under the limitations specified; 
moreover they refer only to the effect on form friction and 
take no account of surface disturbance or wave effects. 
References 
{1] F. Horn, ,Hydromechanische Probleme des _ Schiffs- 
antriebs“, p. 94 (1932). Also ,,Intl. Conf. Ship Tank Supts.“, 
Berlin, p. 20 (1937). 
[2] T. H. Havelock Zeit f. Ang. Math. u. Mech., 19, p. 202 
(1939). 
[3] H. Amtsberg, Jahrbuch der S.T.G. 38, p. 177 (1937). 
[4] J. F.C. Conn. H. Lackenby, W. P. Walker, Trans. I.N. A. 95, 
p. 350 (1953). 
[5] W. P. A. van Lammeren. J. D. van Manen, 
A. J. W. Lap, Trans. I.N.A., 97, p. 167 (1955). 
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