A study of full ship forms in view of the wave making resistance theory

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( 39 5 -CIM)

"A study of full ship forms in view of the


making resistance


By Masatoshi Bessho, Member


Recently, the theory of the wave-making resistance of ships has been applied successfully to

reduce their resistance.

This is one of such study in this direction.

The author picks up the full ship form such as the mammoth oil tanker with low Froude number,

gives some methods to attack this problem and results obtained, and estimates the possibility to reduce their resistance theoretically.



In recent years, the ship form grows fuller and fuller with the appearance of the huge oil or ore camer.

Such full ship form has not experienced satisfactorily in the past experiments,

so that it has

thrown many questions to the researchers.

The increase of the residual resistance by its fullness is one of such problem and many researches bave been carried out in many ship research facilities.

The author, too, has studied this problem in the theoretical point of view of the wave-making


Now, the theoretical prediction of the wave resistance is very wrong quantitatively at the present stage of the theory, but it is also well known that the theory predicts its character precisely.

The object of this paper lies in this point of the research to grasp its qualitative characters and to obtain the direction of the future development.

For this reason, we suffice to use the usual approximate theory, which gives the wave-making resistance R as follows.

At first, introduce the resistance coefficient as

C=2R/pV2B2, (1.1)

where p is the water density, V the speed of ship and B the breadth.

Nextly, take a new unit system in which the half length and the speed of the ship are the unit.

Then, we have5) f12

C,= (2g4/r)f

F(g sec2O, O)Jsec5OdO, ¡.0 ¡.1 with F(k, O) = (1/b)J J 7(x, z)exp(kzikxcosO)dxdz,



which we shall call the amplitude function and where g means the gravity constant, b the half

breadth at midship, t the draft and ?(x, z) the half breadth of the ship lines.

The mathematical representation of the ship surface is one of the difficult problem, but since the

draftwise distribution of the displacement volume does not very much contribute to the wave resistance, we may represent as follows. )

ij(x,z)/b=H(x,z) =H(x)+(1+2z/i)h(x), (1.4)



* 21 (1.2) (1.3)


with H(x)=A(x)/2btC (1.5)



where Aix) is the sectional area, C the area coefficient of the midship section and H.(x) is the half breadth of the water line.

Using this notation and interchanging the order of the integration, we have from (1.2)

o i


H(x,z)G(x,z)dxdz, (1.7)

-t -i

with G(x, z) = 2


f' H(x' z')P5(g xx', g z±z') dz'dz'


where P, function is defined as

P2(x, t)


exp(t sec O) sin(x sec O)cos2r OdO,

P271(x, t) = (_1)n'1j"exp(_t sec2O)cos(xsec O)cos2'OdO,


which has the next asymptotic value when z and x2/4t»1.

P2(z, t) cos


t)J(1) Vr/(2x) exp(t)5(xir/4)


Method of the influence function2',5

Consider the increment 4H of the half breadth at the surface point (z, z) and take the variation of (1.7), we have the variation of the wave resistance, neglecting the higher order term,

4C=G(x, z) [24H(x, z)dxdzj. (2.1)

Namely, the increment of the wave resistance is proportional to that of the displacement and the

function G, which ve shall call the influence function because of this property.

If this function takes the positive values, the wave resistance increases with the increment of the

volume and vice versa.

Consequently, if we know the influence function of the ship considered, we may find easily its

wave-making character.

Fùr example, consider that the draft of the ship is infinitely deep and the half breadth is draftwise


By this simplification, the problem reduces to the one of the shape of the water line curve but the principal character will be preserved.

In this case, the wave resistance is given as


Cw=(2g2»Z)f If(gsecö)I2secOdO,

with ((p) =f' H(x)exp(ipx)dx, (2.3)

where Hçx) represents the half breadth of the water line.

Define the influence function in this case as follows



G*(x) = (2


H(x')P1(g xx', O) dx', (2. 5 where P1 function is proportional to the Bessel function of the second species, that is,

P_1(x, 0)P_1(x) = (r/2) Y0(x).

The function G* has also a property as in (2. 1) but for the area variation.

Assume here that the function H(x) is successively differentiable in z, and integrate (2. 5) partially. we have



A study of full ship forms in view of the wave making resistance theory 23

G(x) = (2/) g[H(x')PO(gxx')+H'(x')P1(gxx')/g+H"(x')P2(gxz')/g._



where P1 function are tabulated in the reference 1. and especially

P2(0)=0 and P2.1(0)=(-1)''(2.4.

. ..2n)/[3.5....(2n±1)]


When g is very large, that is, the ship speed is very low, the successive term in the right hand side in (2.6) is smaller than the antecedent, and since H(x) vanishes usually at := ±1,

we have approximately at F. P.

G*(1)_(2kr) [H'(1)P1(0)H'(-1)P1(2g)J.

Using (1. 10) and (2.7). it is also


(2/7r)H'(1) ±H'(-1)sin(2gr/4)/Vrg(2/r)H'(1)<0.

(2.8) Accordingly, the influence function o! the ship shape considered here takes usually negative value

at F. P., so that the water line curve near F. P. may be better to be taken more full form, that is,

we may reduce the wave resistance by adding small area at F. P.

This suggests that the round cylindrical stem may be prefered for the wave resistance.

In the same way, we may deduce usually the same conclusion about the bulbous bow form from (1.8) and (2.1).

In the reference 2, we can see a numerical example, and this tendency was verified

in the

experiments by some stage. 4)

This approach was proposed by S. W. W. Shor too, with more explicit numerical procedure.


Analysis of the experimental data

Generally speaking, even though the computed wave resistance is much larger than that of the

experiments in low speed, but the characteristic correspondence between them is remarkable.

Thence, we might explain the experimental facts clearly on the theoretical foundation, and that if the reliability of the theory would be fairly good, then we might find the method to reduce the resistance by using the same theory.

For this purpose, we take up the systematic series experiments of the Shipbuilding Research Association of Japan,3) which cover the full ship forms with the block coefficient Cb=O.80 and 0.82

with the bulbous bow and the ordinary bow.

The models are

Series of Cb=0.8O, B/L-0. 1362, 2d/L=0.111,

ÍOR (Ordinary bow)

CB8o J BO (Bulbous bow type but no bulb)

BO FU (Bulbous bow type with full shoulder) BO FI (Bulbous bow type with fine shoulder)

Series of Cb=O.82


CB 82- ) OB (The letters mean the same of the above)


and CB8I (CbO.8l, B/L-0. 1476, 2d/L0.113) which is not involved to the above series but taken up for the reference.

At first, prepare the prismatic curves, which are shown in Fig. i and the load water line curves, read the functions H(x) and h(z) in (1.4) and represent them as follows,

H(x) = E Ax'7, h(x) = E ax1, for

= 1<x(a0<0


6 A.P. 7 2 6 1 2 O CB 80-80 ' CB 80-80-PI

C) CB 80-80-PU A CB 80-OR «-C8I 8


Fig.1 (a)

( CB 80-80


0 CB 80-BC-Fl p Ii CB 80-fC-FU ¿' C3 EU-)R


FIUPI i (b) L'w(x) Fig.1 (b) 9 3


4 C CB 82-BO CB 62-80-Fl n CB 82-80-fl t C8 82-OR ---CB 81 1.0 J / OOQ

O CB 82-BO O CB 82-80-PI (J CB 82-BO-FU A CB 82-OR ---CB W

Fi0UR I (d) Hw(x) A.P.

Fig.1 (c)

Fig.1 (d)

7 8 2 A.P. F.P 6 4 o 0.5 4


A study of full ship forms in view of the wave making resistance theory H(x)=1, h(x)=O,

Hz)= EBx'1, h(x)= Ebx',

?40 p1_U Fig. 2

for ax<as \

for a3xa4=1


Then, we may compute the wave resistance by (1.2) and (1.3), but it is very laborious and difficult, and moreover, instead of the difficulty the result is merely of an approximation.

Thence, we will suffice with the asymptotic computation.

It is easy to obtain the next approximation,



E (-1)



, > ,,

where A(u) = [P3(O, O) 2P3(O, u)+P3(O, 2u)]/(1e)2, (3.3)

= VC2+D2,

= tan(D/C),


C, = [H' _H"/g2+H(B>/g4+B(gt) (h'_h"/g+h(6)/g4)]r.

D= (1/g) [H"_H(l)/g2+H(6)/g4fB(gt) (h"_h(4)/g2+h(B)/g4)]rg for n=2 and 3, - (3.5)

D= [DO.],,g[H+B(gt)h]xa,

for n=1 and 4 )

B(u) = [1(2/u)



The results are shown in Fig.2 and 3.

As usually known, the computed values are very much larger and sinusoidal compared with the

experimental ones.

However, the general tendency is very similar.

For example, in Fig. 2 the computed resistance of OR-series is higher than the one of the corre-sponding BO-series, and corresponds with the experiments, and then we may expect farther that the

Model CB81 has a good shape, because its computed resistance is lower.

If this may be true, the conclusion of the preceding paragraph may be true also at some extent, for the last model has a small cylindrical bow.

Nextiv, in Fig. 3 the ship form with fine shouldes takes the larger resistance in low speed and

smaller in high speed in computation, but the test results of CB82-BO series are not in this order.


IC -

- VZ -0.05 o 6 CA LC. U L .4 TED WA VE RESISTANCE O-Is

Fig.3 (a)

R. = vi'jt, 0.20 a22 25


0.20 -0.20 0.02 RESI DUAL




o 0.01 -o Io oI C IOBo C.BloC R CU2BO o o CD ¿o n ciO so ri. C ro o ri) CBI2OR CB ti i', o." tL= O 20 0.22


'020 o o_Io 40 RESI D.JAL FS ¡STAJC ¿ CB28O -. I CS82 BO FI A BOF I o. 0.02







0.It 020 0.22

However, ve shall be convenient to take the

theoretical conclusion for the general property

because the test results too ma not be reliable

owing to their small ditlerence.


Transverse wave-free



The formula (3.2) says that the amplitude of

the transverse wave or the value of the amplitude

function (1.3) at O=O rules over the wave


If this is truc,

we may reduce the wave

resistance easily by selecting this amplitude


For the limit of this reduction ve may have the vanishing transverse wave, that is when

0) =f J" Hx. z)exp(gzigx)dxdz=0.

Fig. 3 ( b ) Moreover, if we assume the draftwise

uni-form distribution merely for the simplicity, it is sufficient to be

f(g) =f H(x)exp(igz)dx =0.

(4. 1)

Now, introduce the auxiliarly function M(x) instead of H(z) by the next equation

H(x)=M(x).4-(1Jg2)M"(x), (4.2)

put this into the above and integrate partially, then we have

f(g)= (1/g2) [igM(x) ±M'(x)]exp(igx)]i1 (4.3)

Accordingly, if we select for M(x) the function with the boundary values

M(±1)=M'(±l)=O, (4.4)

the transverse wave really vanishes. For example, we put

M(x) =1-4x'3±3x'4, for 1xb, =1, for bxa,

=1lOx'3+15x'4-6z'6 for ax 1

(4. 5)


x'=(xb)/(lb) for 1>x>b and x'=(a--x)/(la) for 1<x-(a.

If we take a =0.6 and h =0. 5, we have the area coefficient as,


f'H(x)dx=o.55--o.2b+o.25a=o.80. (4.6)

\Ve show these distributions in Fig. 4. .0

The curves show full ends and their fullness

increases by the speed.

In the other hand, we should design the ship shape as the way in which the load water line

curve lits to the low speed and the prismatic curve to the high speed.

Thence, we may expect the efficient shape by

taking the appropriate cylindrical bow with the

combination of the bulbous part in bottom.

F;= 022% o 204 0

i+ 5íi?z.

- , 7 FP AP / 2 3 Fig.4



A study of full ship forms in view of the wave making resistance theory 27



q a.2.i6'-JQjp.2Z3(. 02 0,4 Fig. 5


Minimum wave-resistance form'


Generally speaking, the minimum problem of the wave-making resistance has no solution, and that we could reduce it to zero with suitable combination of singularities.

Hence, we do not ask here the question of the munimurn problem but merely a method to reduce the wave resistance by the solutions of the minimum problem which ve have.

Now assume here once more that the ship has an infinitely deep draft of uniform section.

Then, the wave resistance is given by (2. 2) and (2. 3) and the minimum wave resistance form is

obtained when its influence function (2. 5) becomes constant between z! <1.

But this optimum form is very fine and not practical in low speed, so that we must put another restriction to obtain more full ship form.

Solving the problem for the given area coefficient, we can obtain the doublet distributions as shown in Reference 7 and 8.

These are all the doublet distribution and do not represent the actual ship form, but since

it is

very difficult to obtain the ship form accurately, we show here the two dimensional stream lines in Fig. 5, namely, the stream line at great depth and not near the water surface.

These all curves correspond to those

of the area coefficient 0. 8, which is defined

for the total sum of the doublet, so that the real area coefficient may differ from these values as we see.

These curves show also clearly the

round cylindrical bow and full shoulder to

be optimum.

As we see in References 5, 8and 11,


the optimum doublet distribution is the same as this for the very shallow draft

ship too, so that we may say the same conclusion in such case and so generally.


Cylindrical bow form

The all preceding analysises show the preference of the cylindrical bow form. Now, here is an experiment about this connection.

Table 1 0 20 Al?ACtffFr FV rW&ET .'45TN. 080 -


.Io I&I.Er fS7R. STfN1 L/W

The parent form is Model CB81 and models are derived by cutting off or extending the bow part slightly, as we see in Table.

The resistance tests were carried out at the Yokohama national University tank, and are shown

MODEL No. 1 No.2 No.3 No.4

Diameter of stem (m) 9.00 6.00 3.00 0.40 Length on i. w. i. (m) 204. 16 205.67 207.04 208.17 Breadth (m) 30.05 30.05 30.05 30.05 Draft at ;. Full load 11.52 11.52 11.52 11.52 Light load 7. 16 7. 14 7. 13 7.12 Displacement Cm3) Full loda 56.981 57. 055 57. 089 57.094 Light load 34. 033 34.075 34.100 34. 072 Block coefficient 0.820 0.815 0.810 0.805


> r°'

.-io--.- -je--.- H.31 [o-- N.4 FULL LOAD 's, 020 Fig. 6


/Q3O ..J3tf I.. 030 in Fig. 6.

It is surprising to see that the form with the round

LvE5 0F ES;'AL. Y stern has not very larger resistance especially in low


soì't / speed.



fThis tendency

is not predicted by the direct


culation, but it suggests the reliability of the preceding


¡n the other hand, ve may suppose also that the

frictional resistance will be affected and may be reduced by these deformation.

If so, the prediction of the resistance of the actual ship may become difficult because of the lack of the actual ship data like these forms.

7. Conclusion

The author considers that the full ship form like the modern huge oil or ore carrier may be not yet studied out at the stand point of the resistance and has tried to study this problem in view of the wave-making resistance theory, and obtained the conclusion that the round cylindrical bow and the bulbous bow form may be preferred theoretically

and partly from an experiment.

But the experimental results suggest a question for the amount of the frictional resisiance among

the total resistance.

Thence, he awaits for further experimental studies of such ship forms with special reference to

their frictional resistance.

Lastly, the author must notice here that §1 with Refrence 1, §3 and §6 are a part of independent

researches of Uraga Heavy Industry, Hitachi Shipbuilding and Japan Steel Tube Cos. respectively with his collaboration, and moreover, the formula (3. 2) and (3. 3) owe to Dr. Iwata, Meguro Model Besin,

Japan Defense Agency.

The author thanks to them all for their kind permission to use their results freely.

Ref enece T. Jinnaka, Journal of Zosen Kiokai, Vol. 84, 1952. M. Bessho and T. Mizuno, Fune no Kagaku, July 1960.

The Shipbuilding Research Association of Japan, Report No. 31 November 1960. T. ¡nui and others, Journal of Zosen Kiokai, Vol. 108. December 1960.

M. Bessho, The Memoirs of the Defense Academy, Vol. 2, No. 2, April 1962. M. Bessho. The Memoirs of the Defense Academy, Vol. 2, No. 4, Jauuary 1963. M. Bessho. The Memoirs of the Defense Academy, Vol. 3, No. 1. August 1963.

H. Maruo and M. Bessho, Journal of Zosen Kiokai, Vol. 114. 1963.

9 S. W. W. Shor, "Trial calculation of a hull form of decreased wave resistance bs- the method of steepst descent" Read at the Seminar on theoretical wave-resistance,

August 1963 at the University of Michigan

10 B. Vim, "On ship with zero wave-resistance" Read at the above Seminar





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