(ll11 32

### 5 il-t#riit)

### A Study on the Filge Keels

### Lab.

V.### Scheepsbouwçj,

### iPart 1.

### Two Dimensional Model

### Experiment)

### n iche1 Hog eschool

### DU

### By Nono Tanaka, Member*

and Toshio Hishida, Member**

Abstract

There are many sides on the rolling resistance by the bilge keel, but the authors have ,divided it into the fin resistance and the surface resistanceafter G. FI. Bryan and Prof. Sasaji!na

tine.

The present paper is the first repo.rt of these works in which the systematic experimental -results about the fin resistance and the total resistance by the bilge keel of the two dimensional model ships are shown, and an approximate calculating method about the surface resistance of the two dimensional ship is stated.

1.

### Introduction

Up to this time, it has -been assumed for convenience that the rolling resistance by the bilge keel (hereafter, to be shortencd B.K.) is proportional to an area of B.K. A (=bxl, Fig. 1) and the square of the relative velocity of water against the ship wr, and expressed as follows,

resisting momcnt=C&r3A.

### (1)

-However, many experiments have been done only tofind a coefficient C for the ship in question .every time. If the expression above were right, a suitable experimental formula would have resulted by the synthetic analysis of those many experiments. But we - have never heard of such trials having succeeded. The above measure is commonplace, but it is obvious that it is

rooted in the analogical conception "a flat plate to which the stream strikes at right angle". The function of B.K. can not be explained only by this analoglual conception as was already

b .

pointed out by G. H. Bryan') in 1900, and so it is only natural that we cannot analyse the action of B. K. satisfactorily by the coeffi-cient C only.

Mr. Bryan showed by fundamental hydrodynamics that the positive and the negative pressures which arise on the front and the back of B. K. act not only on the surface of this B. K., but .also on that of the ship, and the resultant pressure on the ship surface also produces the

resist-ing moment about the rollresist-ing axis, and that the form of the ship gives a large effect on the action of B. K. After that, although the studies along such an idea on the action of B. K. have been comparatively few, Prof. Sasajirna2) treated this problem in a more Strict and mathemat-ical manner, and concluded that B. K. plays the part of only stirring the water, went so far as to profess that the resisting moment depends chiefly upon the water pressures distributed, as above mentioned, on the ship surface. But on account of the difficulty of treating it mathemat-ically, none of these studies have yet arrived to get any satisfactory results quantitatively.

Received January 10, 1957

Assistant, University of Osaka Prefecture Professor. University of Osaka Prefecture

loo uTi l 4î : 101

And also Prof. Nato5) introduced in his argument a new coefficient rcprescnted the resisting moment by the pressures on the ship surface.

In recent years, however, with a tendency to get stability in building a ship on a more theoretical basis, many experimental data concerning the B. K. hitherto published have been re-examined by many designers. And we are required to present suitable data based on more systematic investigations.

The authors who intend to analyse the action of B. K. in a more quantitative way have engrged themselves for these several years ini somewhat wide and systematic expenir.entaI studies of B. K. along Bryan-Sasajirna line. As a part of them, an approximate calculating method to get the resistance based on the pressure distributed on the ship surface of the two

dimensional ship is found and it is repanled in the present papar.

.

### 1olljng Experiment

Here, we call the total increasement of the resistance to the ship with B. K. "the total resistance by the bilge keel", the resistance due to the pressure difference between the front and the back of BK. "fin resistance', and the une based on the pressure distributed on the ship surface 'surface resistance" separately, and the working dune during a single swing by these are respectively nanied A, AF and A5..

It needs to measure the pressure distribution on the ship surface to get experimentally A.5-directly, but as it is difficult technically, we get here A and A. experimentally and take Out As from A-AF. And especially in the surface resistance among the above three, according to.

the change of the ship form of three dimensional ship in the direction of the length its pressure amount, its distribution and the lever of its moment about the rolling axis G vary, and the subject is made to become more complicated by these facts. Tiere, in the first place some

fundamental characters uf 13. K. of the two dimensional model ship are studied.

Four model ships whose length are L=-8Scm,

- ° -- 0' and breadth B50cm are used, and thc radii of

- their bilge circles arc changed systematically as

J O L M _{shown in Fig. 2. And for the sake of taking away}

Fig. 2 _{the effect from both ends of the model ships as}

much as possible,the length of B. K. is limited to 72. 7cm.

1 The Total Resistance by (he Bilge Keel

The work A is gotten from the free damping experiments of the rolling of the above four model ships in smooth water (the heaving of the center of gravity G being permitted). The

"A" is the difference between those works donc by with B. K. and without B. K.

The experiments are held by changing the draft d (10, 17.5, 25cm t, the width of B.K. b 2,

4, 6cm) respectively, and also the position of the center of gravity KG(20, 22.5, 25cm) and the-rolling period T are changed in three ways within a possible range corresponding to the draft and KG. (experiments number about 2.000)

2) Fin Resistance

First, when the cylindricalmodel ship J rolled about the center ofsection 0 as its axis, all the pressures an the ship surface point to 0, and the momentby its resultant pressure vanishing,

only the fin resistance is obtained. Thus, the condition of experiments not only' changed according to the draft, the width of B. K. and the rolling period, but also in this case j is.

Ft .4

A Study on the Bilge Keels 101

xamine the effect of the widthlength ratio of B.K. (=b/l, aspect ratio).

### But the A, thus

obtained is for the model ship J univ and naturally from the values of A, it must be different

f

in case of other model ships K, L and M. and it is so even in case of the model ship J of which the rolling axis G does not coriespond to the center O.

However, in these ships, there is not such a special stale in which the resultant pressure on the ship surface passes through the rolling axis G, and in the usual free rolling experiment it is impossible to take the fin resistance only out of the

total resistance. Here, therefore as is shown in Fig. 3, one of the BK. (material duralumin, 75SCTe)is

j

supported on pivots fixed upon the ship to measure

the moment by the water pressure about the pivot axis

_{,,,,, ,,}

_{/}

_{/ ,'}liodel in the forced rolling. When the position of this pivot

is changed in two ways in the direction of the width Fig. 3

of B. K.,we can get the resultant water pressure acted ori the fin. Thus the influence of the ship form ori the fin resistance is brought to light.

3.

### Experimental Results and Their Analysis

### (1) A,

Of course, the fin resistance in the rolling varies by the ship form, the draft, the center of gravity of the ship, the width of B. K., the period and the amplitude, so their phenomena are very complicated. An example of the relation beteen the fin resistance and the amplitude in the forced rolling is shown in Figs. 4-7, in which the motion starts from 0= -Ox (in figures, Or.ax1S°) and ends ind=Onax. The fin resistance

8.5 p;

5.25n.

p which amounts to its maximum at the beginning

T 1.6OC. 1 z 6g-c

of the motion, decreases with the increasing of the amplitude and becomes even negative beyond the amplitude. (It was already pointed out by Prof. Sasajima.) In these figures. the solid lines arc for the front B. K. of a pair of B. K. (con--20 -iO' e o 20 -20 .10' e io .o

Therefore, A, isi) almost independent of the aspect ratio. ii) also nearly irrelevant to the draft d. But it can only be said within the range d>B/4 added to in the formula (2). and the shallower beyond this limit the draft becomes the mare rapidly the influence of the draft appears, because B. K. comes close to the water surface. Namely, a wave-making phenomenon by B. K. becomes remarkable there, and its influence on the resistance is very complicated, so it is difikult to express the resistance by siniple mathematical formula. In this case. B. K.

emerges from the water surface with comparatively small amplitude, and such shallow draft is Oot so much significant practically that it is put out of consideration. Even in the case of d>8!1

the wave-making phenomenon may be considered and observed more or less, and the fin bears a part of the work donc by this wave-making resistance. And the shallower the draft, the more remarkable this fact is to be, but on the other hand, when the pressure on the fin is relaxed. by coming close to the water surface, it is assumed both of these cancel each other in d>B/4. iii) The radius of bilge circle, the center of gravity and r etc. affect the relative water velocity against the ship in th vicinity of B. K. and the form of wake in rear of the bilge plate, and it is supposed that these factors also affect A,, but in the formula (2) these effects are

indi-cated only as the form of r-'. 1f we think of it together with T-'8, we can easily understand that the fin resistance is only proportional lo the 1.8th power of the relative water velocity. That is. A, has no relation with relative ship fornt by the rolling axis, and the same can be said with the J model, so we can assume that as far as only A, is concerned, the analogy from "a flat plate" is right.

On the contrary, these factors above mentioned, aspect ratio and the draft are expected te affect A,5, and in the experimental results of A (in the present paper, their details are omitted) their effects are discernible. Of course, the formula (2) has its meaning within the limit of the present experiment, and varies in case of other models of different scales, hence unap-plicable to real ships. Experimental 'results

### of A, for

full-sized model will be reported later.2) A.

The positive and negative pressures o ich arise on the front and the back of the bilge plate affect the whole surface of the ship, and it is supposed that the positive pressure acts on the front side (the positive pressure-side of the bilge plate) and the negative one on the back (Fig. 8). The work done by these pressures during a single swing is A,5. But as the measurement of these pressure distributions ort the ship surface is difficult, ars approximate calculating method of A.5, assuming a suitable pressure distribution, is adopted here.

The fin resistance changes conspicuously according to Revnoulds number, namely to the scale of a model ship, but it is supposed. that this pressure distribution on the ship surface is but little affected by Reynoulds number aitho the wake behind the fin may Fig. 8 change by Reymmoulds number.

### So we can infer that

if this pressure distribution is applicable to some model ships, it can be applied to other similar ships of different sizes.Now, that the experimental results A are also proportional to and r'8T-1'8 approximatelly

as A, in the formula (2 t is found, so also A5 is proportional to these factors. Therefore, from the analogy of the pressure distribution on the center line of the flat plate, the analysisses are carried out by dint of modified ones, and the following method for A5 estimation is resulted, [Assumption]

To give the following pressure distribution on both sides of the front B. K. (in the direction

### ¡

'¼

i O i 20

R

Fig. 9

form). From Fig. 9, we can infer the

following-a) the radius of the bilge. circle decreases, namely, the ship section becomes angular, Asa increases more rapidly, b) A,'ço increases in proportion to the draft, e) and is little influenced by the position of the center of gravity (KG)

### (3) A

The representative examples of the comparison between the sum of A5 by the above assumption

E

### r

X R2sCm### hG2Ocm .'

'cÇ 23A Study on the Bilge Keels 103

of the motion) of a pair of BK.

i Front of B. K.

From the formula (2) for Ap, ;ie can introduce

Pm = A,/2 erbi,

_{(3)}

where, e=rolling amplitude in radian,

is average pressure (including the front and the back surfaces of the fin) between a pair of B. K., concerning the fin area and during a single swing. The distribution on the front is as.

follo ws

the pressure is p at the joint of B. K. and then varies linearly from this to the water surface where it vanishes. (Fig. 8)

ii) Back of B. K.

The high negative pressure is expected to

### act at the joint of

B. K., and assumedly to.concentrate there, and their resultant is equal to the one on the front.

The comparison with the calculated value by the above assumption and the experimental result will be stated in the later article, but, one of the example of the calculated results for the above mentioned the four model ships is shown in Fig. 9, in which the value of Ag/2pmet (=Ag0) is shown as an ordinate and as abscissa the

radius of the bilge circle R. From the following relation

A3=p,Jx (length of the section along the girth)

x (lever of moment) x20, to

A,50 means the work done per unit pressure on the fin surface, unit rolling amplitude and unit length of B. K., and is function of only the ship form (i. e. it does not

-include the influence on the fin resistance by the ship

E

### r

X

104 R-l0cr KE- 2Om T320iec

### fin:!

### ioi

### o

ç E o., X### o

5 E L, Fig. 12 o 20 , .10### -, o

Fig 11 20 Is t,_{10}

### a

ì$ 20 O 5 IO_{Is}

### --

_{e.}Fig. 13

_{Fig.}14

andA, by the formula (2) and the_{experimental values A are shown in Figs. 1o..14, in which}
the solid lines are for the eNperimental results of A, the dotted lines for the calculated A and
the chain lines for A? obtained from the _{formula (2).} _{From curves in these figures, we can}
say that the assumption on the pressure dislribution _{in d>B/4 as above mentioned is almost}
right. Again, from these figures, it is clear _{that in the case of the J model (R=25cm) As is}
very small and negative and becomes A=AF in the _{figure, and increases more rapidly as R}

ccon1es small, and in the case of b=2cm A,5 becomes larger than A, when R=lOcm. But when b=4 and 6cm, A, is larger than A,5 irrelevant of the ship form. For reference, see Prof. Sasajirnas theory, il,Ag when R=l5cm in tue case of b-=2cm, and when b=4 axìd 6cm, A1

is larger than A5.

4. Closing Words

13v dividing the rolling resistances by the bilge keel into the resistance due to the pressure which acts on the front and the back surfaces of the bilge keel and the one based on the pressure distributed on the Ship surface, a simple calculating method of their resistances to the two dimensional model ship is found.

But the reports of a rough estimation concerning the rolling resistance by the bilge keel of fullsized ship, more detailed relation between the ship form and the rolling resistance and applications to the three dimensional ship will be reported sometime later.

It should be mentioned that the present report is a part of the researches which are supported by the Scientific Research Fund and its Subsidy of the Ministry of Education, and by the Research Fund of the 17th Research Committee of the Shipbuilding Research Association of Japan.

Lastly the authors express their cordial thanks to Mr. Il. Kitamura, Assistant of Faculty of Engineering, graduates and students of University of Osaka Prefecture who cooperated in the ¡ong process of these experimental studies.

References

1) G.H.Bryan The action of bilge keels, T.I.N.A. 1900,

2,iL H.Sasajima On the action of the bilge keel at the rolling of ships, Journal of the Society of Naval Architects of Japan, Vol.86, (In Japanese)

.3) H. Kato : On the characters of the rolling resistance of the ships and the similarity law, Journal of the Society of Naval Architects of Japan, Vol. 66, (In Japanese).

(Ilfl fit 33 . 5 JI

### By Nono 'i'anaka,

Meniber°and H iroshi K ita mu ra, Mernber

Abstract

in the present paper an empirical formula _{i} _{for hic fln resistance is obtained after analysing the}
results of the rolling experiments on the action ai a full-sized bilge plate. And a method of
approx-mate calculation of the rolling resistance to actual ships by the bilge keels is found by adding the
above-mentioned fin resistance to the surface resistance calculated by the method given in the first
report.

1.

### Introduetoii

There are only a few experinlenla] sludies about rolling for actual ships on the action of the bilge keels (hereafter, to be shortened B. K. and they are found only in the papers by Whitet), Spear

and Gawn3> and others. In recent sears, considerably wide _{esperiments of B. K. of actual ships} _{were}
carried out by the 17th Research Committee uf the Shipbuilding _{Research Association of Japan. The}
.authors engaged themselves in experimental studies _{on the action of B. K. with reference lo the}
two dimensional model ships and a full-sized cylinder model. The former study was already published
in the first report4) and in the present paper the latter experiment is to be reported. A method of
approximate estimation of the rolling resistance by BK. to actual ships is devised from this
experi-mental result and its applications to actual ships aie shown.

2. Experiment

lt was found by the twu dimensional model experiments that the resistance is almost independent of the radius of

bilge circle R, the draft d and the aspect

ra-tio of B. K., so in the full-sized model experiment a cylindrical surface is used as a model as shown in Fig. 1. The forces acting on the forced arno C. R. are measured with or without B. K. in the forced rolling about the center of the cylindrical surface, and the tin resistance, hence IF, is

-obtained from the difference between those _{(} _{25}

### /

forces.

/ The rolling experiments are carried out by

changing the width of B. K. b (0.30,0.212,0. 105 Fig. i C R.

work done A due to the lin

### ////// /////////////////f////'7/

8.

m), the rolling period T (7,10, 13 sec.i and the amplitude Oo (17.3, 10.8, 7.85 deg.) one another,
while the length of B. K, i (=2.89 in _{is constant (the width of tank being 2.95 m).}

Received December 30. 1957

*

Department of Naval Architecture, Faculty of Engineering, University of Osaka Prefecture, Japan

See Appendix

### 7'!

_{A Study on the Bilge Keels}

> Fig. 2 Fig. 3

### 'S

' i 70_{bi}

_{!4}

_{l031}

i'he following empirical formula is obtained as the result of the experiments of the fin rcsistance A.=0. 384 b1002rT l.26, (ni, kg, sec, deg ; Units)

where, r shows the a' erage distance from the rolling axis to the center of B. K. Therefore, it is found that Ay

i is proportional to an area of the bilge plaie, and

ii) is also proportional to the 1. 6 1h power of the absolute velocity of the bilge plate.

here, the wave making resistance due to the bilge plate is found to be no more than several per-cent of the fin resistance. By using the measured wave-height, the wave making resistance is calculated, assuming the wave to be long enoLigh.)

3.

### Application to Actual Ships

lt is assumed that A which is the work done during a single swing against the resistance by

B. K. to an actual ship is got from the sum of the Ap calculated by the formula (1) and the As. estimated by the same method as in the case of the model ship. There may be

many unsolved problens to apply these A and A. which are both the resistances. for the two dimensipnal ship, to an actual three dimensional ship, but here, fur the present, ' hat is called tle strip method is adopted. When the straight line

M that joins the rolling axis G and the center of B. K. makes an angle c with the

surface of the bilge plate as shown in Fig. 2, Ap is assumed to be given by A- 0.384 Wg25T_I8r26cosa.

Now, A. is calculated on the assumption given in the first report. The value of Ao is given as a function of the sectional area coeiTicient C, the hail breadth B/2 and the draft d in Figs. 3-.'5, (The ships used in the above calculation hae no rise of floor.,

'where,

As=( Aso/rb XAF'

.30 .20 o o c- - o.i d 2 3 Fig. 5

experiments for actual ships the results will

Rev e n

Ca1eulat

t.o _{(deer}Co"Jt,r)

.8 ¿39 a .2 I.0 8 .6 .4 .2

### o

2 4 Fig. 7And according as 0G increases A50/(1/2,2 de-creases linearly when (B/2)/d is constant.

(2) Examples

The decrement of roll per swing due to B. K. only is calculated by the above-mentioned method for actual ships, and when it is compared with be such as shown in Figs. 6-9. The principal iiemsof actual ships are shown in the Table. Fig. 6 is for the "Revenge" and Fig. 7 for the "Oregon." Perfect body plans of these ships can not be found, and the midship section of

the "Revenge" and the sections at the midship and the end of B. K.
for the "Oregon" are given in the literatures 1) and 2), so the other
sections are estimated appropriately from the ship of the same type. In
the same way the value of lfG of both ships is estimated, but it_{exerts}
little influence upon the A,. Lthe effective radius r is given). Fig. 8
and 9 show the results of the ships on which the rolling experiments
are carried out by the 17th Research Committee of the Shipbuilding
Research Association of Japan. Fig. 8 is for the 23 M typed patrol _{boats}
and Fig. 9 for the 800 G. T. typed passenger ship. Experiments are made
with two kinds of B. K. for the latter and never conducted without
B. K. Therefore, the residuals for the two kinds of B. K. obtained by
subtracting the calculated JG due to B. K. only from the experimental
4G due to total resistance are compared with other.

The calculated values agree fairly with the experimental results for the"Revenge" and the 'Oregon"

Toble

### " OYe3on

### t

light cond._{7.93}

_{1.002}

_{13,370}

_{16.80}Revenge 115.82

_{54.35}deep cond. 8.52 1.177 14,620 15.50 Oregon 106.07 7.07 0.914 9,790 15.66 38.5 I 1.331 1.218 56.90 3.50 Patrol Boat 22.00

_{0.306.80}F 1.271 1.398 52.43 3.60 New BK.

_{2.41}

_{1.130}

_{597.1}

_{7.35}

_{0.65s25.8}Passenger 57.00 Old 13K. 2.18 1.068 596.0 7.86 0.35'23.4

Name of Ship L m) dm(nl) GM..m W(ton) T(sec) h

A Study on the Bilge Keels 71

4 6 _{4} _{6}

60Cde5) _{O (de3)}

F

as shown in Figs. 6 & 7 in these examples. In Fig.9,

curve experimental .fO due to total resistance, new B. K. calculated JO due to B. K. only, new B. K. experimental IO due to total resistance, old B. K. ®; calculated 4G due to B. K. only, old B. K.

®; ®, T=7. 86 sec. (modified) -Ø, T=-7. 86 sec.

When the curves® and ® me compared with each other in this figure, which is for passenger ship, it is found that the calculated values are appropriate. Though the influence of the bar keel of the 23 M typed patrol boat upon B. K. must be considered, and accordingly the calculated curves may corne down a little lower in Fig. 8, it can be said that the above-mentioned method of approximate calculation for these kind of ships is almost satisfactory.

4. Closing Remarks

A method of approximate calculation of the rolling resistance by the bilge keels t. actual ships is found after analysing the experimental results on the action of a full-sized bilge plate, and applying this method to actual ships they succeeded in getting near to the anticipated result.

But the relation between the ship form and the rolling resistance on the action of B. K., and the niore detailed study- when this method is applied to actual ships will be reported latet-.

The authors wish to express their heartfelt thanks to Prof. T. Hishida for his helpful suggestion. References

White, Sir W. H. "Notes on Further Experience with First-Class Battleships" TINA 1895, Spear, Lawrence ; "Bilge Keels and Rolling Experiments U. S. S. Oregon" TSNAME 1898, Gawn, R. W. L. "Rolling Experiments with Ships and Models in Still Water" TINA 1940, N. Tanaka & T. Hishida A Study on the Bilge Keels (Part 1. Two Dimensional Model Exper-iments) Journal of the Soc. of Nay. Arch. of Japan, Vol. 101, 1957.

Appendix

In the previous paper it is reported that A1 has almost no relation with the radius of bilge circle R, but more experiments concerning this problem most be added here. A1 is obtained by the same method as reported in the Part i (See Part 1, Fig. 3) under the condition that r is constnt, the rolling period T=2 sec. and b=3 cm for four model ships as shown in Fig. 10. The fin resistance becomes

72 iil033

2o

AO (die)

larger in sorne degree where R is small as shown n Fig. 11.

A Study on the Bilge Keels _{73}

, . .s Fig.1O

_{Figli}

RC cm)
4 ,=!O°
.4 o _{zoQ}15 2

### '4:-o

lo### I,

2.o 2Ç### r

Received January 6, 1959

* _{Department of Naval Architecture, College of Engineering, University of Osaka Prefecture, Japan.}

Jiu 34 .tí

### 5 Jíitií

27### A Study on the Bilge Keels

### Part 3. The Effect of the Ship Form and the Bilge Keel Size

### on the Action of the Bilge Keel.

### By Nono Tanaka, Member*

### -

AbstractThe rolling resistance caused by the bilge keels to the actual ship is calculated for 27 ships and some examples are shown in Figs. 1---4. With these calculated results, the effect of the ship form, the size of the bilge keel and the position of the center of gravity on the action of the bilge keel is investigated.

The value of n which is giben by n=2Na'(GM/KG)°42 is shown in Figs. 9--12.

1. Introduction

A method of approximate calculation of the rolling resistance caused by the bilge keels (B. K.) was found in the first and second paper, so in the present paper, the rolling resistance by B. K. to actual ships, such as tankers (7), cargo boats (IO), passenger ships (4), and others (6), are calculated and the effect of the ship form, the position of the center of gravity of the ship and the size of

B. K. on the action of B. K. is found from these results.

2.

### Calculation and Its Result

The meaning of the symbols is as follows

Nthe extinction coefficient for the total rolling resistance by B. K. only Ns=the part of extinction coefficient by the fin resistance

Ns= the part of extinction coefficient by the surface resistance N'=N'+N5' =the extinction coefficient per unit length of B. K. N'=N' at midship

l=the length of B. K. b=the width of B.K. L=the length of ship, B=the breadth of ship.

To simplify, all the extinction coefficients shown here are at the rolling amplitude O2O. (Extinction
coefficient to all other amplitude can be readily calculated by the empirical forîuJa Noc86.) _{The}

values of N (Nr, N) and N' Ni..", Ns') are calculated for 27 actual ships whose data are shown in the Table

by the method reported in the second paper in 42 loading conditions. (The majority of them is calculated in full load condition, excepting the extreme light draught.) The typical examples of the calculated results

are showj in Figs. l--4. In Figs. (a), the distribution of N'/NZ',Ni.'/N' is shown from station No. 2 to No. 8 along the length f ship with b/B made parameter, and in Figs. (b) and (e) the value of N and N5' is plotted on the value of IlL and b/B.

As the results of these calculations, the followings are found.

(1 ) N'/N'. value which is equal to i at the parallel middle body decreases the more radicallwhen
the position is taken the nearer the stem or the stern. If Co grows less, N'/N' curve becomes sharp, so
near the stem or the stern the efficiency of B. K. becomes ext!'eniely worst. From these facts, _{}
there-fore, the efficient length of B. K. will be determined as a function of Cb. _{(see (2))}

If the ship is fine, the ratio of NF to N is large, so Ni.'/N' value varies considerably between F. P. and A. P. aa shown in Fig. 4 (a), while NF/N is small for full ship, therefore, it can be supposed NF'/N'

.02 .0I 2/L .02 .02 .01 (C) .0I

### :11 N

.4 o i 2 3 b/8 (0/.) T1 Fig. i i1O5.,i.### ()

Table .02 .01 o bi### i - N

3### _:--"-:..

NF 2 .4 .6 N## -tí

### E:.i.z_

N1 I 2 3 b/5 (e/.)### cI

Fig. 2 (b) (C) SHIP LLm) Bm) Dm) dm) . W4t) Cb Cp KGrm) GM in) T 245 32.8 18.5 13. 1 87,360 .805 .814 201 28.2 14.6 10.88 50585 .800 .809 7.80 3.72 192.52 26.52 13.87 10.424 42,976 .784 .792 7.20 3.49 T4 181. 35 25.4 13. 5 10.153 37,231 .776 .784 7.41 2.23 T 167 21.5 12.2 9.422 27,003 778 788 6.59 2.06 T.,, T7 38.5 53 11.5 9.2 5.8 4.5 5.3 4.04 3,102 1,490 .722 .739 741 3. 51 2.86 1. 41 .825 ci 152 20. 6 12.7 8.865 21,388 .751 763 7.711 485 C3 150.3 20. 5 12. 9 9.372 65 .66 C3 134 18 10.5 8.3 15,190 .734 .741 6.78 83 C4 137 18.5 7.837 15,149 .740 .751 7.038 .50 132.4 19.2 11.7 8.138 '14,922 742 .750 7.125 41 ce 114 16. 4 9.3 7.355 .740 C? 108 16.2 9.6 6.56 8,791 .744 .753 6.015 59 CB 76 12.2 6.0 5.641 3,983 .742 .766 4.63 56 CB 76 12.2 6.0 5. 183 3,608 .731 .757 4.32 1.03 Cr4 66. 8 12.0 6.2 5.448 3,257 .725 741 4. 113 927 Pl 40 7.5 3.4 2.44 398.9 - .521 .571 2.74 1.06 P2 46. 5 8. 1 3. 6 2. 032 363 457 548 3.43 .695 P3 29 5.7 2.6 1.706 152.1 .544 2.391 .488 P4 20.1 5.0 2.0 1.23 47.7 485 00 67 10.8 5.7 4.75 2, 491 .705 .735 3.64 1.04 03 60 9.5 4.75 4.111 1.683.7 .714 .735 3.64 58 0,, 04 50 27.5 8.2 5.45 4,5 2.65 4.129 1.74 1,164.9 140. 7 .682 .575 .722 3.16 2.07 .67 51 0 24 4.9 2.8 1.351 60.2 .444 .663 1. 934 1.126 0 18.15 4.23 1.69 58. 5 .51 603.02 LO .8 .4 (ct) (b) .02

### 123

Fig. 6 C2 Fig. 3is constant in comparison with N'/N' as shown in Fig. i (a).

The relation between Ni/N and C is shown in Fig. 5. When the ship is full, such as an ordinary cargo boat or tanker. N occupies the majority of N, but on the other hand if Ci, is small, N' occupies the majority of N and we are roughly right to consider the fin resistance only as the rolling resistance by B. K.

### (2)

i In case of an extremely full ship as Cb=0. 8. N value increases with ilL to ncaily l/L=0.6 and in the neighbourhood of it N value will he saturated. When Cb is 0. 6 or 0.7, the value of N will cease to increase in the vicinity of ¡/L =0.4 and the increase of! beyond this is not much effective, (Figs. i (b)..4(b)) The range of effective length of B. K. is shown in Fig. 6. In this figure. the solid line i for N'/N'=0. 7 and th dotted line for N'/N'=O. 5.

Cb
(C)
02
.0)
o
.0) _{(b>}
r

### -it

4 "C .6 * .N (c i 3 4 5 .02 .01 NJ 4 N O I 2 3 /9 (%)A Study on the Bilge Keels 2l

lo

.7 .8

30 i i, ;iil ii 1O5f

(3 ) b N and N value are increased linearly with b/B as shown in Figs. 1 (c)..4(c). The mure

C1, is large the smaller is the slope of this straight line. Therefore, we are safe to conclude that when it is the ship whose Ci is huge, we can not hope any considerable effect by increase (if the width of B. K. lt is flot too much to say that B. K. is no more than the instrument for churning the water. In case of a lull ship, therefore, it. is nut-c effective when we make b smaller and ¡ larger,

while in case of a fine ship if ;ip1iIid with the wider B. K. it is more effective.

( 4 ) KG (GM) According tu this calculation, the value of N is in case of a tanker 60% to that cf cargo boat. (We did not take into consideration theeffect of the cargo oill.) We can attribute this to the position of the center of gravity, so the effect of KG onN in a tanker (T2) and a cargo boat (C5) is examined supposing that the position of the center of gravity is shifted. From the results as shown in Figs. 7 & 8, it is proved that N value is proportional to the 0.42 th power of KG/GM. Accordingly, here, in order to estimate an approximate value of N, the value of n given by the following formula is taken tu compare.

n.= 2 N(GM/KG)°-2

(5) The calculated values of n for 27 actual ships in their 42 loading conditions are shown in Figs. 9-11. In the same way, the value of n, calculated by using the 70% effective length of B. K. shown in Fig. 6, is illustrated in Fig. 12.

### uinuuuuu

_{!iIflIII}

### IUliul

### I

YL=06 DIO 008 ooó 004 002 030 .020 OIS 005 030 020 N OIS ojo ,005 OIS c'o .1### 11111111: 'e

### -oeil

### L°

0### !IIflI

_{u}

### k-.4

LI-.03 OC 0 . .20 .00 07 lo 20 So io### i

Cs Fig. 7 Fig. 8 11 = .02 .03### !1UIUIIflI

### iuuuuu

DIO11111### IIII

= C.### 1!IN 11111111

### __...

### b

.005### auii

### auus

oo3 o .4 5 . .7 .g .9 .4 .5 .6 .7 .8 .9 .4 .5 6 .7 .5 .9 Cb Lb Cb Fig. 9### Nose

020 .0IS -DIO .007 .015 .010 7 .000 .003020 DIO ..00 4

### 7

.5 .6 .7 .8 .9 Cb T J. . .s .& .7 .8 9 Cb### n

O2 Fig. lo .4 .5 .6 .7 Cb Fig. 11### k

O2 .4 .7 .8 9 .4 .5 .6_{.7 .8}Cb

### k=o.6

8 .9 -.02### .O3

.4 .5 .6 .7 .ß .9 Cb = .03### L

_{}

### -u

### ussuusssuu

### us sus

### ussusu

### Iii

### ;:a

_{¡iuii.iiIÍiI}

### I55:5

### 11111

### 55U55S

### 0.1111111

_{55_Su 555}

.4 .5 6 . .ß .9 .4 5 6 7 s .' .4 .s .6 .7 .8 9

A Study on the Bilge Keels 31

Cb

b Lb

Fig. 12

DIO

$

32 lO54'

3. Closing Words

The above mentioned points are summarized as the followings

(1 ) N, the extinction coefficient for the total rolling resistance by B. K. only, increases in accorda-nce with b, the width of B. K.

As for i, the length of B. K., the value of N grows naturally larger as the value of ¡/L becomes larger, but we must not forget the existence of a kind of the saturation state in which we can not hope any effectiveness however increased it may be, and so there must be the most effective length of B. K.

(Fig, 6)

(2 ) The rate of surface resistance to the total rolling resistance will show a radical increase acco-rding as the ship form grows from a round one to a square one. (Fig. 5) Accoacco-rdingly in case of a ship with a large Ch, it is better to make b smaller, and I larger as well, and in case of a ship with

a small C it is more cffecti;e to make I smaller, and h larger as much as we can.

In case of such ships as tanker with a center of gravity comparatively low, the eflcct of B. K. decreases to a certain degree, so a larger B. K. is desirable for these ships, whether they are full or not.

The approximate value of N, the extinction coefficient for the total rolling resistance by B. K.

only, is obtained from the formula n=2 (GM/KG)° and Figs. 9-12.

The author wishes tu express his gratitude to the Kawasaki Heavy Industry Co. and others for the use of their actual ship data. The author also wishes to express his thanks to Professor 1-lishida for

his helpful suggestion and constant encouragement.

e

(lTi1 36 tft 5 J]

### A Study on the Bilge Keels

### Part 4. On the EddyMaking Resistance to the Rolling of a Ship 1Juli)

By Nono TANAKA, Mernbcr*

Summary

The eddy-making resistance due to the bilge keels was investigated in the previous papers. The-etiect of the ship form on the eddy-making resistance to the rolling of a ship hull is discussed in

the present paper from the results of the systematic two dimensional model experiments. It is found that the eddy-making resistance to the rolling of a ship hull is large in the case of a recta--ngular Section as much as the resistance due to the bilge keels and decreases rapidly as a radius of bilge circle increases, and that the resistance to a parallel middle body of a usual ship becomes negligibly small but its resistance to a fore nr after section of a ship becomes a considerable am--ount.

Next, the eddy-making resistance to the rolling of a ship hull is found to follow approximately the law of comparison by Froude from the results of the similar rhombic model experiment, and a method of approximate estimation of the resistance to the rolling of an actual ship is obtained.

i

### Introduction

The eddy-making resistance to the rolling of a ship is divided into two kinds the resistance to a ship hull and the one due to the bilge koch,. The latter was studied in the previous papers and

a-method of approximate calculation was obtained, and the former is irn.estigated here.

The eddy-making is generally influenced by a separation of flow, therefore, it is not only related to Reynolds' number but also to the form of a body. The coefficient of eddy-making resistance to some forms, such as a circulac cylinder or a flat plate, in a uniform flow was obtained experimentally as a function of Reynolds' number. The value of its coefficient is approximately constant at high Reynolds' number. If we limit the subject within the rolling of a ship, an isolated vortex such as

is generated behind the bilge keels makes the eddy-making resistance increase conspicuously, and the-generation of sud-i the vortex depends mainly on the ship forni and may be almost independent of Rey--folds' number. Then the eddy-making resistance to the rolling of a ship hull may be supposed to-be almost independent of Reynolds' numto-ber.

It is both difficult to measure the eddy-making resistance experimentally and to treat it theoretically since the mechanism of generation and the motion of the isolated vortex are yet unknown. In the-present paper, first, the rolling resistance to a ship hull is obtained from the systematic two dimen-sional model experiment. Subtracting the frictional resistance from the obtained total resistance, the sum of the eddy-making and the wave-making resistance is obtained, and the relation between the ship form and the eddy-making resistance is found supposing that the effect of the ship form on

the wave-making resistance is smaller than that on the eddy-making resistance.

If we suppose the eddy-making resistance to the rolling of a ship hull is independent of Reynolds' number, the sum of the eddy-making and the wave-making resistance to the rolling of a ship hull may follow the law of comparison by Froude. This is confirmed by the similar rhombic model expe.-riment and a mothod of approximate estimation of the rolling resistance to an actual ship is obtained.

Received December 20, 1960

* _{Department of Naval Architecture, College of Engineering, University of Osaka Prefecture, Japaru}

5. e-AP-i 2 23 34 45 -6 bt Fig. i 206 li f

### r 109 t

2 SymbolsThe meaning of the symbols is as follows

.Riio Roj=the edriy-rnaking, wave-making and frictional resistance to the rolling of a ship A

Rpt,e*to = Rn,c+Ra,c', C = the coefficicnt of R,, e+w),

p=density of water, S=the vetted surface area,

r=the maximum distance from the rolling axis to the bilge circle or KG for the lv. section of a ship,

T=the rolling period, T=the pitching period, O,nthe mean rolling amplitude during a swing.

R=the radius of bilge circle, B=the breadth of ship,

d=the draft of ship. a=the angle of inclination of ship side. i3=the angle shown in Fig.2 80, 40=the decrement of roll per swing for a model (or ship) without, with bilge keels, 'ôOfn,, ÔOfx= the decrement of roll per swing due to the friction calculated by prof. Kato's fora.

ula') for a model without bilge keels, for ari actual ship with bilge keels,

ôOnKa=the decrement of roll due to bilge keels only per swing for an actual ship estimate

by the previous paper2'.

3

### Three Dimensional Model Experiment

Before investigating the effect of the ship form on Roc, here, the distribution of the rolling resistance

ailong the length of the tra,v!er ship model is obtained. Cutting the model in ten parts al its square stations, free damping experiments on the rolling of each above-mentioned part have been made at the same condition as the trawler model in smooth water. A circular cylinder whose length is 50 cm and 'the radius 15 cm is fitted to the each part for the purpose of the suitable stability. The work done

per swing of free roll, A, by the trawler model having a i

io _{Troler} rolling axis at G is equal to the sum of the works, 4A.

### L'54o

donc per swing by each part about the same axis. Subtract= 28 80cm

- 5.04cm the work done by a circular cylinder during a swing from

Cd

### i

t_{i. 2.10CC}- 7.64cm

_{the work done by a model fitted to the cylinder during a}

. ia.coc'n

os

T - i.4oS. _{swing and the remainder is the work done by a model JA.}

The obtained experimental (iiI j4A)ô0=480 is shown in Fig. 1, where 80 is the decrement per swing for a trawler model and 480 is the component of the decrement of roll

per swing along the length of a model, then 480 makes

78 8=9 sss 80. In this figure, as the trawler model has an initial trim. the rolling resistance is comparatively small to the fore pa

rt, but to the after part it is extremely large.

In case of even keel, we can infer that the isolated vortex will be generated at the time of rolling behind a fore or after section of a ship, where Re will be the same order as the total resistance to 'the other parts, and that the separation will not develop into a vortex at a parallel middle body of a ship, where Rne will be small because the total rolling resistance is found to be comparatively small

spite of a large amount of the displacement.

### 4 Two Dimensiona' Model Experiment

To investigate the effect of a ship form on Re in detail, the systematic two dimensional model ex-eriments of the rolling have been carried Out and also the observations of the stream around the

__4,_- '--"---?'

-A Study on the Bilge Keels _{207}

Fig. 2

Fig. 3

(a)

bilge circle have been made. The models

who-se length arc 50cm are constructed with

woo-den pieces combined in various ways os shown

in Fig. 2. and two circular cylin1ers whose

length are 88cm and radius 25cm are fixed to both sides of a model ship for the purpose

of the suitable stability. Subtract the work done by two cylinders during a swing from

the total work dane by a model fitted to the cylinders and the remainder is the work done by the model 5E. The work done ôEse,,t by

both R5 and R,, during a swing is obtained

by subtracting from 3E the work due to the

friction, _{SE.,j, calculated by prof. Kato's }

fo-rniula):

1f _{is expressed by}

R,,ie+wj=--pCSv,

_{(1)}

he work _{ÔEs(eWt} is given by the formula,

öEs1e =

-pCr2Sr2O3/ T2 _{( 2 )}

in g. cm, sec, radian.

C valut is obtained after analysing the results of two dimensional model experiments and the characters of R,, are brought to light.

i ) _{Series A '--The experiments un the models with 's nil-side have been made. C value is shown}

n Fig. 3, as an example, for the models in which the breadth B50 cm, the draft d=l7. 5cm, KG=

21) cm and the radius of bilge circle R varies from O _{to 5cm in nine ways. C value is extremely }
la-rge at R=0 as shawn in this figure nod decreases rapidly as R increases. hut C does not diminish so

(b)
e)
5 ton _{CO'}_{t'}
A T
B .4Cm R o
d-ii 5('n-'75.Bcm
8- v,ocm
d2ocrn"-275cvn
5. 1020,30,40,50,
6O,ucm
d - i -i scm

### I FE

-iVA' ___jp j B s» C_{B'10.30. 50 cm}i o, 2C 30 d. R-' 0, I.2,3,Cm

### I_Rç

E-C i-e T, B i. 30,50 C5 )3- to, za, 30 dr. - o, i, 2,3 cm -' - D c(-'icr d 75, Z0.0 22.5 25o. 27.5Cm = I -E S. E .ptMS't*I'QS&ti - C,r_{L}\ Et C-. i ( 4 5

-f, .5 4 .3 .2

### i

i &/ 2### /

### Fi.5

3 a 208 3E)### V1 tfil

l09much when R increases beyond R=5 cm. This radical variation of C value by Ris mainly caused by R,., therefore it is found that R5 is very large when R is small and decreases rapidly in accordance with the ¡net-case of R, ond Rs,- will be small when R- 5cm.

An interesting sidelight on the motions of water around the above models ha been obtained by dro-pping blue ink into the svater so as to make the motion visible in the observation of tite rolling of

models through the observation window in the tank wall. One of theexamples of the stream around the model (B=50 cm, ¡(G =20 cm. d=17. 5cm) is shown in Fig. 4 ; (a) is for R=5 cm, (b) for 3

cm and (e) 2cm rcpecti clv. When R=5 cm, ink moves along the ship surfaceand normal flows to

the ship surface can not be observed except the free water surface, so Re may be negligibly small there. The vortex can not be observed when R=3 cm and something short of a vortex is observed when R=2 cm. An isolated vortex such as is generated behind thebilge keels is observed when R=O.

In order to examine the scale effect of the generation of the vortex, the observations have been carried out for two models whose breadth are 30cm and 80cm (KG=20 cm, d=17. 5cm in common). Bc,th the normal flows to the ship surface for R0. i B and a vortex for R0. 062511 can not be ob-served, but an isolated vortex is observed for R0.05 B. The effect of the rolling period and ampli. tude on the eddy-making is scarcely noticeable, but the eddy-making becomes conspicuous regardless of R as KG or the draft increases.

From the results of the experiments the following empirical formulais obtained for Orn is modera tely large.

### CfIl\KGorB )

### f

B KG\ -a"a### (3)

The value of fi is shown in Figs. 5 & 6, and a in Fig. 7. The calculated value of C for the above

O iO 20 30 40 Fig. 6 +1

### IShi

exa pie wh tel:. see 0. 1 esti### tr

lar r T### 's

Q s 3=060 ar5 7.66 l0 634 5 5 IO 5 20 9tde Fig. 7

exampleby the formula (3) is shown in Fig. 3. Rs.

Ocu-pies the majority of the total resistance when R is small,

where the characters of Rae maybe expressed

approxima-tely by the lorniula (3 ). Rae is comparatively small as

seen in Fig. 3 orthe observationsof theStream when

### R

0. 1 B, where the calculated C value by(3)may beunder-estimated.

lt was supposed several

### years ag) that Re to an

ex-tremely full ship such as a super tanker is aconsiderably

large amount. The calculated value of _{C,,} by the

fo-rmula ( 3) is equal to 0.04 at the midship sectionof 65. 000

T super tanker in a full load condition, and therefore R,. is considered to he negligibly small to a parallel middle

body of a full ship.

### (2)

Series B-- . . The experiments ou the models withthe sloping ship side have been made. Theresults obtained

are the following,

oi

-L 5/

Fig. 8

Fig. 9

### cí1(

### )fi)e

### (4)

where fs(a) is shown in Fig. 8, a is the angle of inclination of ship side and B is the breadth of

model at the water plane.

lt is found from Fig. 8 that C decreases as a increases within a certain value of the inclination of ship side, aç, and C conversely increases as a increases beyond ac, and that ac decreases as R/ increases.

Dr. Watanabe and other3) investigated experimentally on the effect of the flare using a model having a large radius of bilge circle and found that Rat increased as the increase of the flare. is negligibly small to such the model having a large radius of bilge circle, then R5 increases as the flare increases is found from Dr. Watanabe's experiment. To the contrary, R,)c decreases as a incre-ases is found from the result that C decreincre-ases as a increincre-ases, and Re Cannot be neglected when both R and a are small.

The a is equal to about 10' at a usual fore section. where R5 can not be neglected, and the total resistance to a fore section i.s about 70°c of the one when a=0. The value of KG is equal to 20cm in these experiments, but R5 becomcs large rapidly as KG or the draft increases as shown below in Series D.

### (3)

Series C The relation between Rae and a form of bilge is obtained in detail. The results210

.6

2 _{4}

.3-2.

ibtained are shown in Fig. 9, where 4R is a virtual increment of R by the angle shown in Fig. 2. The virtual increment JR is defined as follow; R, decreases as 3 increases, then the decrement of

Re due to is defined to be equal to the decrement of _{due to the virtual inerement JR of R.}
For an example, JR is about 1 cm when B=30 cm, ¡3=20' and R=1 cm. so C can be calculated by
the formula (3) substituting R=1 cm+1 cm=2cro. _{JR/B/2 increases rapidly as ¡3 increases and R}
is small to the form such as ¡320, then a chine line as seen in a small ship does not make so much
increase the eddy-making resistance to the rolling.

### (4)

Series D. E ._{. .The experiments on the models representing the fore or after section of ships}

are added. At first, the experiments on the models, with _{being equal to 10 and the form of the}
bottom being- parabolic as shown in Fig.2, have been made changing the draft. The form of the bottom
is determined to have the first continuous derivative on the whole surface because the discontinuity
induces the eddy-making. _{In order to explain the experimental results by the formula} _{( 4 ) an }
equi-.valent R to the above parabolic form is considered. _{The equivalent R. which is shown in Fig. 10,}

p,

decreases as KG/B increases, and Rite is conspicuous and has no relation with the form of the ship bottom when KGJB2. The value of KG/B at a usual fore section is beyond 2, then R1,0 is

conside-rably large there.

.7

1:2 :4 lb 18 2:0

Fig. 10

t. il1094i

C

d(ii) 6(om) T() Gi.i"ì M(')

.4

.3

.2

Nextly, the experiment has been made on the special formed model _{representing the square station}
1/2. The models and the obtained C value are shown in Fig. 11. Rite to this section is extremely large
as shown in this figure. where an isolated vortex such as is generated behind the bilge _{keels is }
obse-rved. R to the after section is also found to be comparatively large _{from the above trawler model}
EXperiment.

5

### Similarity Law of the Eddy-Making Resistance to the

### Rolling of a Ship Hull

A method of approximate estimation of the rolling resistance due to the bilge keels was shown in
the previous paper21. Considering _{and Rsç, that Rit,. follows the law of comparison by Froude}

Table i

t

Model L(cm) 10cm) d(crn) Wkg) KG(cmn) GM(cm T(sec T,(sec)

A0 90 20 7 852 666 0.97 152 0.72
A2 180 40 14 _{6816} _{13.32} _{194} _{2-15} _{1-00}
A 270 60 21 230.04 19-98 291 264 1.25
35 C .4 9.7 3o97
50 20 .9 2.5 44.57
4Ø 20 3.4 3.25 4H
60 30 2.2 ii-6 S.I
53 30 4.2 3.3T 40.3
45 30 7.5 .47 4Z28
.0 15 2.0
Fig. 11
.8 .5

o 8 9 EP.

Fig. 12

A Study on the Bilge Keels

### -J

9### I

B 76.5 2.5### J

4c### T

Oem ,,0cm 211was known and new formula for Rhf 5

obtained by Prof. Kato, and it is nece-ssary to establish the formula for Ra. in order to extend the rcults f model exp

criments to actual shii,.

The free rolling experiments on the geometrically similar rhombic three

mo-deis have been made in similar conditions. The smallest model is shown in Fig. 12, and the dimensions of the others are sho-vn in Table 1, in which the conditions of the experiments arc also illustrated. in these conditions R,,,. is negligibly small from Prof. Ilishida's formula4'. Obtained value of (ôO-ôOj) is shown in Fig. 13. The calculated value by means of the strip method of (ôO-ÔO1) by the use of the method of estimation of Rp.1e±w> shown in the previous article is also shown by the dotted line in Fig. 13. R/,n is negligibly small and the majority of the experimental (öO-ô01) is due to R,0. in these conditions,

therefore, is found from this figure to follow approximately to Froudes law.

(dO-501') is small compared with the total rolling resistance to a ship with bilge kee-ls, where we may neglect the scale effect

uf R5e.

In the present piper, the following law of comparison for similar ships on the

resistance to the rolling of a ship is considered;

w I5 20

e.,

Fig. 13

4ûa ÖO, _{ÔOfm ±öOfa+ôOEKa}

where the suffixes m and a are refered lo a model and an actual ship respectively. In the formula (5 ), we must obtain &O,, from a model experiment but the other terms are calculated by the method as illustrated in the article 2.

In order to assure the above similarity law, the rolling experiments on two kinds of models and actual ships as shown in Table 2 have been made in similar conditions. The scale of model is 1/25 for K Ship and 1/35 for N Ship. The obtained experimental results and the calculated results by the formula (5) of 4O are shown in Figs..14 & 15. The calculated values agree fairly, although a little under-estimated, with the experimental results as long as these ships are concerned.

6 Conclusion

The characters of the eddy-making resistance to the rolling of a ship hull have been brought to light um the systematic two dimensional model experiments. They are summarized as the following

The eddy-making resistance to the rolling of a ship hull is proportional to exp (aRid), where dL is shown in Fig. 7. R is the radius of bilge circle and d the draft, then its resistance is scarcely

noticeable to a parallel middle body of a usual ship.

In the case of the value of KG/B is larger than 2, where B is the breadth of ship at water

plane, the eddy-making resistance to the rolling of a fore or after part of a ship becomes a considerably aarge amount.

### (3)

The angle of inclination of ship side makes the eddy-making resistance to the rolling decre-ase, and a chine line as seen in a small ship does not make so much increase its resisance.The eddy-making resistance to the rolling of a ship hull is found to follow the law of comparison b y Froude from the analysis of the results of the similar rhombic model experiments, and a method of approximate estimation (the formula (5). of the rolling resistance to an actual ship with bilge keels is obtained.

The author wishes to express his gratitude to the Japan Defence Agency, the Kawasaki Heavy In-dustry Co. and the Fujinagata Shipbuilding Co. for the permisson of the use of actual ships. The author's thanks are also due to the following persons who have in many ways assisted him in the preparation of this paper ; Prof. Hishida for his helpful suggestion and constant encouragement. Mes-sers. Nomoto and Matsuki for the present of data, MesMes-sers. Kitamura and Hirano for their generous assistances to the laborious experiments.

References

H. Kato ; On the Frictional Resistance to the Rolling of Ships, JSNA vol. 102, 1958.

N. Tanaka and Il. Kitamura A Study on the Bilge Keels (Part 2) JSNA vol. 103, 1958.

Y. Watanabe and M. Inooe On the Method of Calculation of N Value, the Resistance to the Rolling of Ships, Journal of Seibu Zosenkai, vol. 14, 1957.

T. Hishida A Study on the Wave-Making Resistance for the Rolling of Ships (Report 3), JSNA

vol. 86. 1954.

212 1O93j

l'able 2

Ship L(rn Bm d,,(ni W.tom EGOn) GM(m)

### T(see)T,(,ec)

size g keelK 59-0 71 2-278 17140 2-729 0-766 6-80 3-4 I 70'18-70