On the turning of ships

1 NOV. 1972

ARCHIEF

nische Hoge;chOO

By

Shosuke ¡NOUE

With the Compliments of the Author.

Lab.

y. Scheepsbouwkunde

Teclinische. Hogeschool

Deift

e

Reprinted from the Memoirs of the Faculty of Enginee,ing

Kyushu University, Vol. XVI, No 2

-f

On the Turning of Ships

By Shösuke INOUE Professor of Shipbuilding (Recèived 28 March 1956) Contents Page 51

Chapter I Wing Theory of the Turning of a Rectangtilar Pjate with Small

Aspect Ratio ...52

Introduction ...52

The integral equation that decides the strength of vortex sheet 53 * 3.

Solution of the integral equation ...54

Normal force and moment ...58

Comparison of calculation with experiments ...59

Conclusions...61

Chapter II _{Normal Force and Moment of a Ship Sailing Obliquely ...61}

§ 1 Introduction 61 § 2 Normal force of the rectangular plate . 61 § 3 Moment of the rectangular plate .. .. 63

§ 4 Normal force upon a ship

...

.. 65

§ 5 Moment upon a ship 69 * 6 Experiments in oblique motion 70 § 7

Conclusions ...74

Chapter III _{NormalForceandMomentofaTurningShip (Even-KeelCondition) 75} § 1

Introduction ...75

§ 2 Inducement of the non-linear part of angular velocity ito the theory

of a flat plate ...-

75

§ 3 Normal force upon a rectangular plate 77 § 4 Moment upon a rectangular plate 77 § 5 Normal force upon a ship 78 § 6

Moment upon a ship ...79

§ 7 Calculation of C, Co by means of simple-formed ships ...79

§ 8 Experiment of turning and its comparison with theoretical value 84 § 9

Conclusions ...88

Chapter IV Normäl Force and Moment of a TurningShip (Trimméd Côndition) 89 § 1 Introduction 89 § 2 When a flat plate moves obliquely 89 § 3 When a ship sails obliquely . ₉₁ § 4 When a ship turns ₉₄ § 5 Conclusions

...

99

Chapter V Course Stability and Steady Tiirning of a Ship 99 § 1 Introduction 99 § 2 Course stability of straight course .. 100

§3 Steady turning

...

_{.. 103}

§ 4 Course stability of steady turning _{.. 105}

§ 5 Conclusions _{.. 107}

Conclusions _{.. 170}

50 Shösuke IÑOUE (Vol. XVI,

Nomenclature

L length of a ship

B breadth of a ship

d draft of a ship

dm mean draft of a ship

r trim

C, water plane coefficient

C; midship section coefficient Cb block coefficient

S. center vertical plane area

k aspect ratio of a pair of center vertical planes when

they are placed symmetricallS against water line

and represented as 2S/L2 a drift angle 2 angular velocity U linear velocity p water density R turning radius w L9/U_L/R N normal force M moment

Cpz normal force coefficient =N/(1SU2)

Cjv() CN of only thê case of oblique motion

Cyo Civwhenk0

CN,' C due to the wing theory of Prandtl

moment coefficient =M/(-- LSU2)

Cm(a) Cm only when sailing obliquely

Cm' Cm about

Cm Cm about center of gravity G

Cm when k=0

Cjiti Cm'o when k=0

CieL Cm about leading edge

x,y.axis axis x liesin the longitudinal direction and axis y

in the transverse direction through the center of

gravity G

M virtual mass of direction x

M virtual mass of direction y

i

virtual moment of inertia around the vertical axis

1957) _{On Ihe Turning of S/zips}

51

INTRODUCTION

Movements of a ship in water surface

in-cluding such problems as course stability,

turning and steering may be solved by

evalu-ating Euler's equation on U, a and 2. as

follows:

Ma,(ÙcosaUsjna â)

±MÜsina.

M71(Ùsin a+Ucosa â)

Ma, Ucosa. 2= Y

I ¡ = M,

where, it is provided that X, Y and M

re-present, as shown in Fig. 1, the x, y

com-ponent of external forces acting upon a ship and their moment about the center of gravity G respectively, and these involve the thrust

of propellers, the fOrce due to rudders and

the hydrodynamjc forces the ship-hull suffers

from the water surrounding it.

Fig. 1.

The problems that should be counted for

in solving this ecuation are the virtual mass

Ma,. M the virtual moment of inertia I and the force and the moment X, Y; M.

It is

also possible to study X, Y; M under two circumstances: to consider them separately with the ship-hull only and the rudder only

or to watch them at mutual interference.

Therefore, the problems can be divided into the following four parts:

the problem pertaining to the virtual

mass and the moment of inertia of

a ship

the problem of the rudder only

the problem of the force and the mo-ment acting upon a ship-hull only

the problem of the interference between

the rudder, the propeller and the ship. hull.

As for (i), a rough calculation of it has been made possible by the latest studies of

Dr. WATANABE,C') Messrs. DAVIDSON and SCH IFF(2), and others, though it was in the same incomplete state with the equation of

motion of a ship. About (ii), various calcula-tion formulas have been introduced by many

since the time of Jössel's experiment, but

Dr. AKAZKI made it almost possible by :his

experimental equation. With respect to (iii), though no theoretical evaluation has so far

been made on it, measurement is being carried on some ships. Taking up the experimental study by a model for consideration to begin

with, there are two kinds of method in it.

One is to depend on a curved model, which was applied to the study of airships by Mr.

JoNas and others. With the case of a ship, Mr. WEINBLUM has evaluated normal force and moment on some ships. The other method

is to carry out evaluation by turning the

model directly.

On the case of an airship, the force and the moment are evaluated by turning it in

the air.

The same experiment had been

carried out on the case of the wing by Mr.

WIESELSBEROER. The experiment witha ship

is carried out in a turning tank, and, though

Messrs. DAVIDSON and SCHIFF have written

papers on the 'result of their experiment, no

necessary value has been obtained both in America and France, as it.is not long since

it was started.

The reference made above is mostly 'On the studies in laboratóry, but it is also con-ceivable to obtain normal force and moment

by analyzing the turning test made on an

actual ship. In this case, however, since

both rudder and propeller work together, it

duplicates with (iv). Though it means to

seek the force and the moment of a ship-hull

by presuming the force acting upon the

rud-der on the basis of the drift angle and the

turning radius which are observed at the

time of' steady turning, there arise various changes according to the hypothesis of the

force working upon the rudder. Therefore,

it is possible to put the results of

52 Shosuke' LNÓUE (Vol. XVI

to the presumption of1 turning radius, but

there are still problems to be solved before using them. for evaluating the force and. the

moment acting upon a ship-hull. Mr.

Hoy-'GAARD 'and messrs. TANIGucHI and MAÑABE have adopted this method. for their analysis.

With respect to (iv), there is a study made

at resitance-tank byMessrs. BAKER and

BOT-TOMLEY(S) beside that of Mr. GAwN,

but it

is necessary to make a though-going study on the case of (iii) firîst of 'all, because it

holds the key to. solve (iv).

The reason for the' difficulty hitherto felt in making any theoretical study on the force and moment acting upoñ the shiphull is that the ship-hull's aspect rätio was too small to

be. applied to the' wing theory' that developed

on large aspect ratio in. its original size.

Therefore, .the wing theory of small aspect ratio should' be. made. clear at first, and then

enlarging it to the casé of turning as wéll as to the case with 'breadth 'the évaluation

'with the case of a ship is to' be got.

Accord-ingly, the author will begin his reportwith

the theory of BoLLAY-KARMAN in Chapter

I,(26) proceeding. next' tothe cases of oblique

'motion and turning, the calculation of the

force 'and the moment acting upon a

'rectan-gular plate on the basis of -wing theory 'as

well as comparing 'the result of calculation

with the experiment. In Chapter II, he

will give expressiOns to the normal force and moment of .a

rectangular plate that

moves obliquely by using 'the calculation

value of Chapter I in a 'simple' way, clarify their physical meaning and apply the result

to a ship to compare. its calculation value with the experiment in a model-tank. In Chapter III,

the author will take up the

cases of turning, comparing them with the

experiment of turning by introducing the

non-linear part of angular velocity which he

could not refer to in Chapter I and by en-larging it to the cases óf a ship. In Chap-ter IV, theoretical and experimental studies

are made on the cases of the

turning of

a trimmed ship. Chapte. V30 gives calcula-tions made on the course stability of a ship and steady 'turning on the basis of'the values obtained.

By the way, the following theory deals

with, the wing theory of turning, tak'ing up an ordinary cargo-ship for its objective but

excluding such a high-speed ship that changes

her posture according to velocity.

Chapter I

The Wing Theory of the Turning of a Rectangular Plate with,

'Small Aspect Ratiot26

§ 1. Introduction

In dealing with the force and the moment

acting upon the ship hull while a ship is

turning standing on a theoretical point of view, it is likely that we apply the theory

of aerodynamics to a low speed ship, because

its free surface effect is small.

However,

the aspect ratio* of a ship is so small as 0.05

0.2 that 'the w'ing theory of large aspect

ratio is not applicable to it as it is and it

becomes necessary to introdüce that of small

aspect ratio. In that case, our thought

natu-rally turns to a ship of an extremely special type, namely a rectangular plate with small

'aspect ratio. As the case of a ship is con-sidered to be solved by studying the above and enlarging it, the rectangular plate will

be taken up first for study.

The wing theory of 'a rectangular plate

with small aspect ratio has been studied

by Messrs. KARMAN and BOLLAYt1° with the

result that their theory coincided with ex-periments. However, as they provided the distribution of vortex sheet along the plate

as twodimensional or having infinite aspect ratio, it goes well with normal force, but its - assumed distribution leaves a question. That

is. it is presumable that the calculation Of

the moment or the' center of pressure cannot be managed well, but nothing about the

mo-ment is referred to in their papers. On the

other hand, with the progress of high-speed

airplanes, the theory of small aspect ratio itt a straight course has been taken up by

* In dealing with the force acting upOn the ship.

hull in the light of wing theory, consideration is

given to 'the image of the ship.hull against the water

line regarding it to be moving through the infinite

fluid. Therefore, the aspect ratio is 2S/L2 of nearly 2 dm/L.

1957) On the Turning ofShips 53 Mr. LAWRENCE(IT) and others, but it is

con-sidered that it is not applicable to a ship.

As to the force and the moment of the

rectangular plate when it is making steady

turning, Messrs. GLAUERT(I2>U3) and

WIESELS-BERGER

are engaged in the study of the

case of large aspect ratio. However, since

they have not included the case of small

aspect ratio in their study, the author intends to deal with the theory of steady turning of

a rectangular plate with small aspect ratio

in this chapter and, as one of its special

cases

evaluate the moment of a straight

course which has not yet sought by Mr.

BOLLAY.

§2. IntegraI, equation deciding the vélo-city of vortex sheet

Supposing that a rectangular plate of t in length, b in breadth and k in aspect ratio is making steady turning in a large turning radius, o will be taken in the center of the plate, axis x to the longitudinal direction and axis z vertically against the plate. In that case, the plate will be moving forward

at the rate of Ur=R2 while turning at the

augular velocity Q. Though the plate can

be replaced by the distribution of a certain

vortex sheet, but its distribution, as Mr.

BOL-LAY considered on the basis of Mr. WINTER'S

experiment, does not change along the breadth

of the plate with such a small aspect ratio as it is now, but changes only in the

direc-tion of length, on which assumpdirec-tion the

dis-tribution of the vorticity of vortex sheet

r(E)

will be decided so as to satisfy the

boundary condition. Therefore, when both

bounded and free vortexes passing a point E as indicated in Fig. 3 are examined, they are

found in U-shape of the vorticity r(E), and

as the plate is turning, free vortex is almost

in a circular arc. And Messrs.

WIESELSEER-GER _{and GLAUERT'2) (13) have proved in their}

experiments and calculation that the velocity induced by the arc-shaped vortex is

obtaina-ble by taking the length of 180' from the.

starting point of the vortex and that it is

the same with the semi-infinite linear vortex

if its radius is large enough. Consequently, if integral equations are evaluated to decide

Fig. 2.

Fig 3.

(E) on such terms that normal velocity that

is caused both by bound vortex and the

semi-infinite linear free vortex from the tip of the plate and that of' the velocity of fluid against the plate are equal, the result will

be. as follows:

r()dE

+.p- 1' cost9r(E) _dE

27rJ1k2+sin2(x E)2

cos26(xE)r(E)dE

2irJ [k+sin2t9 (x E)2] /k2± (X

-=Usina+x.

(1.1)

where, it is provided that e represents the angle formed by free vortex and the plate,

and in the above equations, the leading edge

is transformed to 1 and the trailing edge

±1.

The ist term of (1.1) is by the bound

vortex and the 2nd and the 3rd are those

produced by the free vortex, coming from

54 Shsuke houE (Vol. XVI

k l'i'

4g )_

)-Since the above result is the same with

Mr. BOLLAY'S solution, 9 can be decided after his idea so as to make the average direction

along the tip of free vortex equivalent to the direction of the mean speed at the tip.

Accordingly, in (1. 1) 9 becomes independent

on the integral concerned to E. Now, with a view to solving (1.1), put

which will be substituted into (1. Ï) as follows:

r

1 casO

X

L(cosO cosç) /k2+ (cos ço - cos O +k2-f- sin2 O (cas O - cos)2

+

cos2O(cosOcos)

ldço

(k2 + sin2 O (cas O - cas ç)2} /k + sin2 O (cas O - cas )2J

s1na+- cosO

and the calculation of the left-side term may be reduced to the following three integrals

in the end r cas n ;o

J(cps_casO)/k2+

(còs_cosO)2 'r cosn k2+Sifl2E9(casp_casO)2

'r (cosç9 - cosO) cosnço

dço 2gJ,L

2n'i

,r(E)dEdx

)/k+ (xE)2

pL ti I I cosOr(E)

d d

4irJ1J1k2+sin29(x)2

X E

+--

II II

CoSOr(E)(xE)

4gJJ [k2+sin29(2_E)2J/k2+(x_E)2

= Usintx [k2+sin2O(cosçt_casO)21v/k2±(cosc_ casO)2

As it

is difficult ta calculate (1. 3), (1.4)

and (1. 5) accurately, following equations

may be introduced,

P!v±Qv

i

P2v2+Q2v4 k2+sin2Ovz k2

x=cosü,

Er cosco

(Both O and ço are between O'7r).

As it is also made clear in Mr. WINTER's

experiment that there is a hump of pressure on the leading edge, r(E) will be assumed as

r(E)_r(c)_1

_tan-±

_A,,sinnç

U U 2

ni

(1.5)

i

P!vH-Qv2

[k.±sin20v2]Vk2±v2 k3

provided. y = cosco cosO with V placed

be-tween ± 2.

And P1, Q1 and others will be decided to make both right and left sides equal at y r 1 and y = 2.

first.

As it

is difficult to solve them: as

§3. Solution of integral equation _{variables, average will be obtained from} In solving (1. 1), 9 should be considered them, i. e., integrating (1 .1) on x and getting

their average,

(1.3)

1957) _{On Me Turning of Ships 55} Namely,

P

32(k2+4 sin9)/4+k2

1 -2

+31

2

31

i1+k

2 k

(k2+sin2)/1+k2

2 k3 1

1 +1

1 V1+k2 2k

2(k24SÎfl29)/4+k2

4 1

51

sjfl2@) k2+sin28+ ₄

_i

. 1

(k2+sin)V'1+k

+2k3

Accordingly, (1. 3), (1. 4) and (1. 5) will be calculated as follows:

1

1

1

11

212(k2+4 singe)

3

k2+sin20+k2

(a)

_{L= J}

cosnçodço

(cosç' -cosO) i/k2+(cosç -cosO)

therefore,

Lor=irQjcosOPj(2O--ir)

L1 =

f(1ci.cosO)+--.Qt

L

_ksinê

lrSinflO(i _Ç!sinO) c1 =--P1k

n 2V and putting cos(2n-1)O zrn=l

(2n-1)-?( i

i

_)cosnc

2V 7Z1 2n-1 2n+1 the resultwill be

L0=[_P1_Q,2V]cosOP[c03O+c0O±...]

cos2O±1 cos4O+

L2 =3_r(2_-_4_C1)cosO+&c1_kL (0.4 cos3O +0.0952cos50+

32V 2V

L4 f[(2_L) cosO +(2-4-) cos3O +-0.12i2 cos5O

4c'

_{(2-cos3O+2i'cos5O+...]}

L=--[(2_--1 COSO+

272V) 112V) 1

J

COSflÇ9 do+QiJcosnco(cosc_cosO)dc9 k _cosçocosO

Pi(Jcosncodco

_{f cosflQd4)}

"o therefore, cos4e.] M1 =r[(P2-31Q3)

cosOcos3O]

M2

=?[-9+Q2+-Q2

cos2O]

M3= -

càsO M4=f6Q2 214'5=2W6= .. =O

-N'z _{[k2sin2(cosq' _cosO)21 V'k+ (cosçocosO)}(cosq'cosü) cosnq'dq'

NnJ(cosc

cosO) cosnçp-dq' ± Q3 (cosq'cosO)3 cosnço dç

k30

H

P:z[Je

j

(COSO - cosO)2 cosnç2 dço

-

J (cosç' - cosO)2 cosn q'

dio]-What is to be consideied is the third terim as it was with the case of (a), whose integral value is

when n=O

(?r2O)(iCO52O)

f.sin2o

when n=1 (ir-2O) cosü

(f +c02O)

when n=2

-

(i

28) + (-

cosO + cos3ô) sinO

sinnOi 2

2(n2±2)

other cases sinO i_{L sinO 'n(n2-1)}. i

+

_{n(n2-1)(n2----4)} cos2ô

(21)(24)(COS(fl+1)O+COS(fll)O}

therefore, as it was with (a),

_cosOirQ(- cosO ±+ cos3O)±(

cosO+O.5689cos3O+O.O116COS5O ±

N2 = .!irQ, cosO +-'(2.8333 cosO O.4063cos-30+O.0271 cos5O

-4 -ir

56 Shösuke Isou (Vol. XVI

(b)

M=

CSflÇ9

k2+ (cps2 cosO)2 sin2ø

1957) On the Turning ofShips 57

N. =--Q3---(0.2844-0.3841 cos2O+i.1354 cos4ô+_{8 ir} )

N4 P3(Ò.O406 0.0677 cos3ü + 0.0307cos50

-N5 ?P3(0.0042 0.0351 cos2ô + 0.0382 cos4O

-N6 = __?_P3(O.0019 cosû-0.0062 cos3O+0.0166 cos5ô ...)

Substituting the above L, M, and N into (1.2) and comparing the coefficient.of cosnO,

simultaneous equations with respect to To, A, A2 and so on can be obtained, i.e.,

a60r0+a61A1+a62A2±aA.+a64A.1+aA1+ ... = siria

a1r0+aA1±aA2±a53A3+a51A4+a5A6±

₌

a40To+ a41 A1 + a42 A2 + a4 A3 + a44 A4 + a45 A5+ ... O

a30 T(+ a31 A1 ± a A5+ a33 A3+ a34 A1+ a53 A5+ = O

a2 To± a21 A1+ a22 A2± aA3± a24A1+ a25A5+ ... = 0. a, To+ a1 A ± al2 A2+ a13 A3+ al4 A4 ± a3A5± ... = O

And though (1. 6) is infinite simultaneous equations, all that is necessary for the actual

calculation of numerical value is to solve

simultaneous equations on the variables such as Tu, A, A2, A., A1, A5 and the coefficient

a and others of (1. 6) will become as

fol-lows:

a=r

---}ci

_+tQ

+cosE)(firP2 +_irQ3)

,

(ir

9

P64\

(5 P2 ± cos- (9

±

- ir Q -

-- -)

a41 = cos E) ir

Q

-a50 = --P1'± irQ1 + cosE) ir(3Q2Ps) a3[ = -0.3999

ir k

ir9

±COS2E)(

±--irQs_ff)

4e

i'. P2

a4=--

1+cos@ir2Q2---8P

irQ2 = ± cos &

9ir

2

ao = ± cos& irQ2+

cos90.0813

15k 8 ir

a1o=---

_cos2(9O.0116 25ir a61

coset( _F'2++12)

a51=--c1 ---P1+--Q1

±cos2E)( 5.6944 + 3irQ3) + cos2 (tQ P10.4876) a21 = cosE)ft

± cos2(9(--irQ3P:1128) aIl = -0.0952

_-

-4-Ç +cos2E)90.0077

8c

ir,-1

a62=--

¡--'t

± cos2 Q:l

0.5688)

+_{COS2Ø (} 3.4133 Q:

58 Shsuke IZoUE (Vol. XVI, (P2 S

a= cosf9ir---7Q2

a42

_{= f-}

- + C0S2f9(! 1.6143

-

--irQs) a32 = - - Q LOS E) a22 = - 0.1524 - cos2 ) _0.1084 a12 O

ir(

P2 a = cosE) --- -i--8 ci

(3

a53 = jj + cQS f9tj-ir 3 a43 = cos (9 ir a33 = -- - -0.4285 ± C0S26) 02709 k k ir = O a1 = -- 0.1270 cos2 (9 0.0443 k ir a84 = - 0.0711 ± cos2 (9 ( 0.1464

_{- j}

Q)

a5= ---Q cosE)

a44 r ! 0.3048 cosf9 0.2272 k ir a.3 = O a2 = -- - ---- 0.5079 + cQS2f9: 0.1059 k k ir a1.1 = O a65 = ir cas (9 a55 = ! 0.0762 + cos2 E) 0.0387 k ir a45 r Ø a35 = 0.2116 -'- cosE) 0.0739 k ir

a=0

a15 =

-

- 0.2424 + cos2) 0.0474 k k ir

3

11.4573)

§ 4. NormaI force and moment

The distribution of the vorticity of vortex sheet having been clarified, the normal force and the moment about O will be

N =

J

(U cosa + U'z) T(x) dx .-(1. 7)

M=

PJ

(Ucosa+W )XT(x)dx ...(1.8)

where, it is provided that Wz is the X

com-ponent of the velocity induced on the surface

of the plate by vortex, and since Wz is

krl'

cosf9r(E) _dC

w = tan 9[

J k + (x - C)2 sin2 9

f'

cosf9(xÇ)r(C)

J[k2+(x_E)z sin26J/k2+(x_)2

the following equation will be given by (1. 1)

i- . 12

W

tanf9LUs1na+j-x

k

By substituting X = cosO, =cosço and

re-peating the same calculation as §3 getting

thereby CN = NJ(f b t u2) Cm

M/(b t2U2),

we can obtain

7 A1

CN =

ircosajro+-+ ir tan 9 {sin a (To ircosajro+-+ ji)

12/

A2\ ir To)To2J (L9) ir

i

A0

C=COsCXTo

ir (sincr( A3 tQ( A, A3 ---(TO2---AITu 8 ir2 (1.10)

1957) On £he Turning of Ships 59 provided that

( A'\ç To A2

i (

To A4\ A2

(4j){ro

(A2 A4\ ( i ( A2\

i (

A4\ A2

1YT0+) +f

To+ _{) +}

-§ 5. Comparison between calculation and

experiment

With a view to comparing Mr. FLACHS-BART'S' wind tunnel experiment, the case

(A4 and Ar are extremely small compared with To. A1. A2, A)

Accordingly, the distribution of vortex

sheet may be calculated by

TotanOJ2+

A sinnü, which will be shown in Fig. 5.

On the whole the distribution is fiat,

except-ing

the part near the leading edge, and

forms a low hill in the latter part of the

center. This distribution signifies that A1,

A2 and A: are more conspicuous making considerable difference from that of k which is also found in Mr. WINTER'S"4

ex-periment.

As the next step, the comparison of the

calculation of C, and Gp = 1/2 + Cm/CN Table I

of k = 0.2 has been calculated on the basis

of (1.6), and the values of To, A1, A2, A3,

A4, A5 have been sought with the result

like Table 1, which is shown in Fig. 4.

which represents the center of pressure, in the case of straight course on the basis of (1. 9) and (1. 10) with experiment will be

shown in Fig. 6. However, it is considered

to suffice the purpose for us who takes up c < 100 for consideration in the turning of

a ship. The calculation of R r 5 will be

also shown in Fig. 6. Gv increases by

turn-ing while the center of pressure is located

nearer to the leading edge. The comparison

with the turning experiment will be

con-ducted in the later chapter.

By the way, in

(1. 9) and (1. 10), drift

-1 To A1 A2 A3 30 0.0232-0.1964 0.0211+0.5428 0.0153-0.1220 0.0084+0.1411 5' 0.0376-0.1962 " 0.0303+0.5384 " 0.0269-0.1192 " 0.0149+0.1425 10' 0.0768-0.1996 0.0355+0.5364 " 0.0597-0.1042 " 0.0285+0.1526 15' 0.1152-0.2027 0.0134+0.5376. " 0.0963-0.0816 " 0.0427+0.1641 20' 0.1531-0.2050 " 0.0259+0.5356 " 0.1368-0.0697 ' 0.0555+0.1689

60 Shösuke INOUE

angle which is necesary for us will meet our requirement if it is below 10'. And as

this theory is based ón the circumstance in which R is large, the second term may be

Fig. 5.

neglected in comparison with

For exathple. if an example

a =10' is represented by k = 0. like Table II.

Table II

the ist term.

of calculàt ion Z it will look (Vol. XVI,

U0

t9

U5

i

CN

ist term 0.1858 0.2283 2nd term 0.0028 0.0036 Cm ist term 0.0313 0.0135 2nd term 0.0007 0.0002 0.4 0.3 02 o.' o

'Accordingly, both CN and Cm' thai be

approxi-'mately as follows:

Cj'ir

cosa(rOE±!)

/

A2

Cm-cosaru-'

In other word, it means that the velocity

pararelling to the surface of the plate may

be looked upon as U cosa by' making wj= O

in (1. 7). and (1. 8).

§ 6. Coñclusion

In Chapter 1, the normal force and the

center of pressure of a rectangular plate

with small aspect ratiO was sought beginning with Mr. BOLLAY'S assumption and, as a

re-sult, it was made clear that it proved to be fitting, äs far as our reqúirement goes, iii the case of straight movement, CN increased

in the case of turning and when a was

small, the center of pressure moved

back-ward. beyond the center. By the way, when

a is small and R iS large, C. and Cm will

be expressed as follows:

i

A\

_cosa

7r/ AA

C- -o---2-) cosa

Chapter II

Norma I 'Force and Momént

of a Ship Sailing Obli'quely(27)

§ 1. Introduction

In the. preceding chapter, the author tried

to explain how to calculate normai force

and moment thät act upon a' rectangular plate of small aspect ratio when it moves obliquely

or turns in a large turning radius. In order to apply the above to a ship by enlarging it,

it will be divided into two parts of oblique sailing and turning. The first will be

dis-cussed in thi chapter and the second in the

next.

Now, while it needs Only tO calculate the

normal force and moment upon a rectangular

plate, that moves Obliquely putting Q = O in

the calculation of preceding châptér, the. author will devote this' chapter' to the clari-fication of that simple method of calculation

as well as. it physical 'significance and to the solution of the normal force and moment

acting upon a even keel conditioned ship

by enlarging it to the case of a ship with

breadth, thereby comparing it with the

ex-periment in ä water tank.

§ 2. Normal force of a rectangular plate

If the diStributiOn of vôrtex sheet r on the ceñtral line of a plate is expressed as

follows:

*=Totan+Amsin

...(1)

the normal force coefficient may be drawn from (1. 11) as follows, as fär as it is

neces-sary for the drift angle of the turning ship;

CN=7r

cosa(ro+)

Therefore, discussion will be forwarded on

the basis of this formula. As the first step,

ru will be divided into two parts; TUL that

is drawn from Prandtl's wing theory and

To' of the rest as follows:

To = Toz + To' (2. 3)

To-

_{1'2/k cota}

sina

And as Tu,' A1 can be calculated from (1. 6)

to any k, it will be possible to draw out Ti' from (2.3). Now if 2r', -A +4/ar tana sin cr

is calculated by k = 0.2 for iristnace, it will resült as in 'Fig. 7, which makes it possible

to conSider both are approximately equivalent

when a is below 10'. Since this can be

Fig.

(2. 2)

1957) On he Turniñg'of Ships 61

(1.11)

62

applicable to other k, following expression Therefore, substituting (2. 3) and (2. 4) to

may be approìimately possible:

A1 2 To'±

tan a sin a. '(2.4) c, c. 0.4 0.3 0.2 c_ c.. 0.4 0.3 0.2 0.1 O C.. C_ Rectangular plate h=1/3 ® Experiment Theory lo. Fig. 9. co C.' Shösuke INOUE Rectangular plalc h =0.2 ® Exerhnent-...--Theory r, s

(2.2), C1 will become as f011ows with Tu'

disappearing:

C.(a)

sina cosa+2 sin2a

i+kcosa

_(2.5)

However, though the ist term of (2. 5) is

the value drawn from PrandtFs wing theory which has been studièd from various angles

up to date, the ist term of (2. 5) may be

expressed as follows in a broad sense if the inclination of lift coefficient is expressed by

2a1, when k = co,

sina cosa

...(2.6)

rl"

0.5 0.4 0.3 0.2 0.I 2a0 2a0

1+ cosa

_irk (Vol. XVI, c_i 0.6 04 03 02 0.1 . Rectangular plate 0 0.5 10Eopmimaot - Tbecry .Sm fo of Fig -C0 -ISt

r,

__

y

0.4

::

01 Rectangular k0.I34 10 Experiment

They

Scé feotnoleu I plate t

'iiI

---_. -o-.

Show the points moved the experimental 10' 20'

points ® in order to be zero at a = O. Fig. 12.

10 20 Fig. 11. 10' 20 Fig. 10. 0.1 0 10 Fig. 8. 20' C, C.,. Cn C, a

10

Fig. 14.

On the other hand, the 2nd term of (2. 5) which is the normal force coefficient when k = O is the same with Newton's resistance

law on a plate and equivalent to the force

which is. concentrating attention exclusively

on the front of the plate, calculated from

the change of mômentum. Accordingly, C

may be expressed as the sum of the value obtained from Prandtl's wing theory and that of k = O. What are shown in Figs.

8-14 are the result of the comparison reduced

on the above and the experiment of both

Messrs. WINTER Sand FLAcESBART15> and on

the whole they are indicative of its

appro-priateness. However, on the case of k = i,

it becomes larger than experimental value, but, as this theory makes k < 1 as its

objec-tive, it is conceivable that there will be sorne

disagreement. When k = 1, the disagreement

occurs even with Prandtl's wing theory which

makes the case of a large k its objective. Any way, it is fàr from being accesible re-jecting most of the efforts to get at it.

By the way, with a ship whose aspect ratio

is 0.05O.2, the ist and the 2nd term of

(2. 5) approach each other so closely in their order that it is impossible to express the

nor-mal force acting upon the plate, even if either

of them were neglected.

§ 3. Moment of the rectangular plate Though the distribution of r in the chord

direction may be expressed by (2. i), all that has to be done in order to find out what kind of normal force is related to each term is to substitute (1. 7) with each term of (2. 1), the

normal force being sought by (i. 7).* In the

beginning, Totan (ço/2) may be divided by

(2. 3) into To tan (ço/2) and ro' tan (ç'/2), and

from the former comes the force N which is the same with Prandtl's wing theory at

the pOint tJ4t from the leading edge. From the latter cornes the force N' at the point of t/4 from the leading edge just as it was with

the former case. A sinç may be divided by

(2.4) into 2ro' sin çc and (4/ir) tanc4 sina sin cc,

* Since c is below 10', it will be made as W= 0 in (1.7).

t Center of pressure will be obtained by

calculat-ing the moment puttcalculat-ing w=0 in (1.8) and

divid-ing it with normal force.

.0.2

Rcthn8ubr pbte

5Eperi't

- Theo,y

- See footnotes of Fig. IO

--..iiiiioii

01iauu--0.8 0.7 0.6 0.4 0.3 02 0.1 Restongulor pinte ®Eoperooeoto O Prondtlo wgg theery Theory

- See footnotes of Fig. 10

uuuiu

u.riu

r

UdJU

1957) On the Turning of Ships 63

o lo. lo.

-64 i Shôsuk

and from . the former comes the force N'

WhiCh is' the sathe in size .with the case of .-ro.tan(o/2), and at the center of the plate in he reverse directioi. From the' latter..comes the force N0 which contains the coefficient

2 sin°a .of normal force of' k = O at the center of the plate. The normal force does not corne

..fröm A2 sin 2o, . but there occurs a couple

by (1.8). The fact tht the terms A3.sin3ço,

A4 sin4 etc.. have influence neither qn.

nor-mal force nor - on mOment . can, be detected

by (1. 11) .and. (1. 12). .It miy ..be illustrated

as :shown in Fig.. 15 where the normaLforce

becomes N ± N0 and independent from N'

as it has been. already stated..in §2. . As for

the. moment. about the center of the plate, it is added with the i4iothent due to N' and

0.5

-Li

TO_A

1ct

cL N0

Coupe due to'A, sin 2'p

t/4

e INOtTE . (Vol. XVI,

Fig. 16.

the couple from A2 sin2ço, and when k is small, it grows unnegIigiblè against N 1/4. Now the coefficient of' moment will be as

follows from (1. 12):

/

A2

Cv-7 COSroj

and, since To and A2 are almost in proportion

with asina like the calculat-ion example

of Fig. 4, both risma and A2/sina can be

regarded as the function of' k. Fig. 16 shows

r(/srna and-A2/sina. ro/sincx increases with

k -and A/sina has a maximum point at

kO.1. However, (ro Az/2)/sina increases with k on account of To> A2/2. Now put

- (To-'--_{4e)/smn a}

f(k),

then f(k) will become the function that

in-creases with k, and if C, is expressed

mak-ing the moment -in the direction where a

increases as plus, it will look as follows:

Cm(a) =f(k) siria cosa ...(2 7) The result of the comparison made between the value due to this calcülation and the ex-periment by Messrs. WINTER and FLACHSBART

0.7 0.5 0.9 10

k 0"

1957) On the Turning of Ships 65

is shown in the series of flgures* ranging from Figs. 8-14, which is favorable on the whole. The case of a rectangular plate can be calculated by (2. 7), but a study will be made here on the moment upon the plate

without taking circulation into consideration

as a preliminary step for enlarging it

to

a ship. In that case, if the additional mass of z direction is expressed by M, the mo-ment about the center of the plate will be

represented as follows:

4M2U2 sin2a.

Now assumiñg J as the ratio of additional

mass when it is considered according to three and two dimensional flow,

M =fib2J

Therefore, the coefficient of moment based on the fore-going idea will be

Cm(iX) =

irkf sina cosa

...(2. 8)

Accordingly, putting the ratio of the

theo-retical value (2. 7) with (2. 8) as e 2f(k)

ir kf

where it has less value than one as -shown

in Fig. 16. However, J will be calculated as

an elliptic plate and will be expressed as

follows by the ratio of three and two

dimen-sional calculations.

J=

i

101 3'820₊k2 siñ0 dO

§4. Normal force of a ship

In most cases with a ship, as we can

neglect the effect of free surface when the Froude number is small, we have only to take the effect of the breadth of rectangu-lar plate with its aspect ratio k 2d/L into

consideration.

Now as it has been already stated in. § 2, * As moment coefficient about leading edge Cm,.

has been introduced into the experiment, C,,,1 was

calculated for theoretical value by using Cm and Cay.

since C., represents the total of what has been obtained from Prandtl's wing theory and the case of k = O, it may be approved

to consider independently.

At first, as the

breadth effect of the ist term of (2.5), or of (2. 6) in a broader sense, is included in a0, it will be expressed as fôflows making q

a given constant:

B (to)

a0 = ir

(i

_{- q}

Therefore, all that has to be done is to

-substitute it to

(2. 6), but as the aspect

ratio which is necessary for a ship is small

and

-irk

the effect of breadth will not bethough C, gets small to some- degreeso large -as the

wing of an ordinary airplane with large -as-pect - ratio, which can be almost neglected.

The next is the one which is equivqlent

to the 2nd term of (2. 5) and

it can be

calculated as follows by applying Newton's resistance law just as it is. In Fig. 17, ships

side equation of thé front will be expressed by y y2(x) and that of ship's rear by y =

y1(x) against the flow

Fig. 17.

This case can be considered in two ways; when - a is larger than Ou

(the half of the

cut-water-angle)

and when it

is smaller,

and applying the rule of momentum change to the entire ship's side facing the flow with

the former case and to the interval of QQ'

(Q and Q' show points of contact of the

tangential line drawn on the ship's side in

the angle of a against the central line) of

the thick line with the latter case,.

66 Shsuke .INotfl (Vol. XVIi

whena>00

_{Cyo(a) =}

Lr

_{a sin2a}

bL rL/2

N p I

U2 Sjfl2 (a 02) dx ...(2. 9) lo i see2 a sin 2 a J_I.!2 _g (1'+4(BIL) when a<00 Xof Q

N=rJ

U2sin2(a-0)dx

- ¿/2 z of Q' J - P I U2 sin2 (a o5 dx (2. lO) 112

providedthat tan0i=!, tan02=.

dx dx

Therefore, (2. 9) and (2. 10) are

when p 12 U(sin2a ±Y2t2

± cosa

sin2a-

Y2' 1±Y2'2 when a<00 ab.sciazofQ '2 Nr= p j. U2(sin2a1

±Y2'2+ cos2a1±Y2'2

L12

sin2a

.Y2' _)dx 1 ±'Y2'2 z ',t Q' U2 (sina1 +Y L12± cosa1±Yi'2 - sin2a1 _Lyj'2) dx.

_..(

12)

As a result, if y = yi(x), y = y.2(x) are

clari-fied, calculation is possible as follows:

Br

Ix2

(1)

Br

Ix 2

Y2

L'-from (2. 11) and (2. 12), when a>00 L

12B.

C(a) = -

_{tan -Z-sin2a}

- -

tan]cosa

...(2. 13)

when a<00

...(211)

±(tanaa)

cos2cx] ...(2.14)

therefore, when both a and B/L are small,

from (2. 14) the following expression is possi-ble:

where, the term of a came out, unlike the

case of the plate. This makes its appearance

when the point of symmetry Q" is made to

Q' in Fig. 17 and Newton's resistance law

is applied to the interval between Q'Q" and

Q"Q", and a2, cf etc. appear when Newton's

'resistance law is applied to QQ". This is

not only limited to this case but applicable to the ships of other forms in general.

C;-0.4 0.3 0.2 0,I 4

a

'pp

AIA

MUWIA

rivairi

A#iUlfl

_'r

10 Fig. 18. B/L=1/3 B/L=0.2 BIL=0.15 BfL0.1 ß/L=0 a A

C,, (a) -A

1957) On lhe.7'urning of Ships 67

Again, since (2.13) and (2. 14) coincide

each other in their value when a

0 as

well as their tangential, they are connected

to (2. 13) smoothly from (2. 14) with the in-crease of a. Fig. 18 is the result of calcula-tion of (2. 13) and (2. 14) on various BIL.

(ii)

Br

(x\4

Y2[1-7)

When calculation is made by (2. 12) putting,

ar=/tana

when a<U0

CNO(a =2ab cos2cx{2 tana

/

1+v'3a±a2

+ 2

+ tan

(2a + /) ± tan_L (2a

-b . I

1±2b2±b'

1b2±b4

0.3 0.2 0.1 0

C ()

/

I0 Fig. 19. B/L=0.2 B/L= 0.15 B/L=0.1

logl+2a+ a4

1a2 + a4

±2v'

tan'

_{(-I +}

2/ tan1

-2 v'

tan (2 a +

+2vtan'(2aV)}

(2 15)

In this case, since 0 tan' (4B/L), it is not

necessary to consider the case of a> 0

within the range of a we need.

Fig. 19

shows the result of calculation given to

(2. 15) on its various B/L.

(:1:) The form of a ship with parallel part

in

its center and its sharp edges pointed

rectilineally

Fig 20.

Putting I

to the length of the parallel

part as in Fig. 20, we get L-..4L1±l±4L2

In this case, we can obtäin the following by

the application of the càse of a plate, even

if (2.12) is not availed:

when a<00

C0(a) = [2 sin2 (a ± 0o

2 sin2(a - 0;)] ± 2

sin2af

s 2 sin2O0 sin2a + 2 sin2a

(2.16)

If it is with a ship formed symmetrically in its fore and aft, since

4L14L21

_LL

_C

_1-2C

₁

"

L

(2. 16) will be

CN(a) 4(1 - C) sin 200 sin cos a

+ 2 (2Cm 1:) sin2a

(2. iT)

On the other hand, if it is a> 0 and

68.

C0(a) r 4(1 - C,) (sin2a COS2Oo

+ cos2a sin2 Ou5 ± 2 (2 C,, fl sjn2a. (:2. 18:)

Figs. 21 and 22 give the coefficient of sina

4(1C)sin

r0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 o o-JO Shsuke INOUS Fig. 21. (Vol. XVI, X cosa in (2.17:) by changing its B/L and

Cu,.

The 2nd term of (2. 17) has not so

much effèct as a plate when a is below 10'.

C being evaluated as above, C can be

ex-pressed as the total of C and Ci,, due to

\cw

1.0

1957) On ike Turning of Skips

4(1C.,)sjn 20, 1.0

0.5

the wing theory of Prandtl. However, with

a ship of common form, since the . form of water-plane changes according to draft, and even with a ship of cylindrical shape whose

water-plane does not change with draft, since

the effect of its bottom is indistinct, is

multiplied by a given constant .

Accord-ingly, the result will be

Cfri)

C., .- ir C.vo(a) .. (2. 19)

where, ir is the value determined

experi-mentally.

§ 5. Moment of a ship

As stated in § 3, the moment is expected to

be evaluated when the distribution of force is known, but since the efféct of breadth for

N' determined from ru' and moment from

A2 sin2ço namely that of ru'AI2 is

un-known, the moment upon a ship will be

evaluated starting with the following equa-tion which is equivalent to Munk's,:

sin2c (2. 20)

M will be evaluated, as it was with Mr.

LEwis' vibration problem, by calculating

2 dimensional additional mass of each

sec-tion, by integrating it in the direction of

the ship's length and by adding the effect

of three dimensions to it, i.e.,

Fig. 22. 0.5 0.6 0.7 0.8 B/L M1, =PJd2csJ1idx

where, it is provided p2rd2Ca/2 is two

dimen-sional additional mass of each sectiòn and C8 is the function of B/(2d) which can be

approximately calculated on lines.

J is the

effect of three dimensions. which is,

if it is

a ship of common shape, the ratio of the

additional mass in the three and two dimen-sional cases that were evaluated -on a ellip-soid of revolution7 or an ellipsodial body»

For instance, in the case of a ellipsoid of revolution, when L/(2d) 10, J

98 and

when L/(2d) 6, J1, 0.92. _Accordingly,

put-ting MXIMI, = e, the moment coefficient due to (.2. 20:) will be expressed by

-J1,Ck(1---e5 sin2a

provided that Jd2Csdx

CS-. Thät this

value is large compared with experimental

value is recQgnized from the case of the plate

in § 3, and that is also maintained by Mr.

WEINBLIJM comparing it with his experiment,

actual C will be evaluated by multiplying

it by that was provided in § 3, as expressed

by the following equation:

Cm(a) -= (1 - e) sin2a. .(2 21)

70 Shsuke IÑ0UE (Vol. XIII,

And in the above approximate calculation, since no consideration has been given to M6

about the effect of free surface nor M has

any definite value, e will be determined

ex-perimentally.

By the way, with the case of a ship of

cylindrical shape. C8 is being calculated

analytically by Mr. LEWIS117 because of its

rectangular section. As a simple example, evaluation will be made on the moment of a cylindrical ship with a parallel part in its cnter and both of its sharp edges pointed

rectilineally.

If the length of the parallel

1.5 1.0 C 0 _1.005 0.5 0.666 0.75 01 0.133 0.15 C, f' 3d/B

part is changed, C,,, changes, and as indicated

in Fig. 23, the result of numerical calcula-tion of C and Ci,, on the constant (2d)/B is

linear. Therefore,

c=..5()+2f(!j):Co.5

providing that fo.. f,

are the value of C

when C,,, =. 0.5, 1, which is shown in Fig. 24. When it is expressed by numerical equations,

it will be as follows:

f.(?j)

1.329-2 X 0.064[

-

i]

= 1.457 -0.l28!

() = 0.183 2 x 0.025[_ i]

* The models whch were used not oñly for the eicperiment this time but for the experiment of turning are recorded at the same time.

No. of \

modéls \ Length(m) Bieadth,(m) .

Form of water plane

C

Form of direction

of draft Remarks

1 1

-

-

-- Rectangular plate

1115 . i 0.15

With sharp edges and shoulders Cylindrical type. C,,=2/3 Table III 0.5 06 07 08 09 18 Fig. 23. C = f,,+ 2f,(C-O.5) f .# 0.233-0.05. f, f f,, L451-O.128-0.5 1.5 =: 0.233 - 0.05-Accordingly, from (2. 21) Cmn(a) = k [1.457 - 0.128 () ± {o.466 - 0.1 (c,,, - 0.5)]

xJ60(1e) sin2a

(2.22)

where, evaluation of e will facilitate that

of Cmn&X) at once.

§ 6. Experiment of oblique sailing

With a view to give the above-mentioned

theory an overall examination as well as

to determine the value of and e, an ex-periment was done. The experiment was

carried out by the resistance tank of' the

Kyushu University, Faculty of

Engineering,-Naval Architecture Department

and the

models8 used were as follows:

1.0 1.5

Fig. 24.

fi

The result of the

experi-ment is shown in Figs.

26-32. On the whole, both

Ci., and Gma can be.

re-garded as linear against x

when it is below x 10',

and the effect of velocity

below x 10' is

consider-ed to be none as far as

Froude's number

neces-sary for a merchant ship as observèd in Fig. 26 is

concerned.

Gma can. be evaluated

by (2. 21) orif it is a

cylin-drical shiptype of Fig. 20. by. (2. 22>.

When it is the model

lI and d 5cm,

Cm()

0.277J?,(1 - e) sin

cos

0.227 X 0.72 x 0.98(1 - e) sin

cos

=0.160(1e) sinZ cosa

=0.120 sinx cosa (experimental value) therefore, C 0.25, Model VI (L=lm)

._..._.u.eu.uuuuI.IUuIIIitiIIiv

&trnilIi.,I,1UlÍ

iuiiiiwiriu i

IIIIIi.uII ui1ti1II

iuiiiiiii,uiiuirirniii

ILIUI1uII II1I1MIUi.l

MII1I1tuIuiTDiitui

uuuuii iwwrniu

I.,.

0.5 W. L.

I

BASE LINE Fig 25.

when it is the model 1115 and d = 9 cm,

= 0.385 eJ5 (1 - e) sin cos X

= 0.273(1 - e) sin x cos

=0.205sincos

(.experimental value) therefore, e = 0.249,

No. of

models Length(m) Breadth(m)

Form of water place

Cw Form of directionof draft Remark

-Ilio 1 0.10 Do. Do.

IlI 1 0.15

With sharp and shouldersedges

C,, = 3/4

Do.

IV 1. 015 Quadratic parabolicform '

C,,,=2/3 Do.

IV[(, 1 0.10 Do. Do.

0.15

Quadratic parabolic C. = 2/3

Quadratic parabolic form

Eq. of ship surface

,2_2[4](d_Z)

a

B shows the value at

d=0.075 m VI

VII

1 0.148 . Cargo ship type. The lines are showed in Fig. 25

0.75 Similar model of 1V15

1957) On the Turning of Ships 71

Table Ill. (cont'd)

10 W. L

8.0cm WI.

6.0 c W. L.

4.0cm W.L.

when it is the model llo and d = 5cm, Cm() = 0.2183sJ0(1H e) sinix cosa

/

Model U, d0.05m(k=0.1, ® Froudes number 0.135 u u _01.6 0.20 c_C C, C' C.0 C.. C,

=0.1506(1-e) sinX cosa

= 0.1126 sin x cosX

(experimental value)

C, C..

therefore, e = 0.252

when it is the mode III' and d = 5 cm,

0.4 0.3 0.04 0.03 Cm() = 0.231J0(1 -- e) sincx cosX =0.163(1-e) sjn:X coSX = 0.1225 sin cos (experimental value) 0.2 0.02 0.1 0.01

72 Shsuke INôU (Vol. XVI,

o 10'

Fig. 26.

Model Il. d0.05m (k0.1)

therefore, e = 0.249e

when it is the model V and d 5 cm,

Cma(cr) = --kJ0 0.88 1 - e) sin cos X = 0.098(1-e) sinr cosx

0.074 sin cps

(experimental value) C, c therefore, e = 0.245, 0.4 0.04

when it is the model V and d = 7.5 cm,

0.3 0.03

Cm() =

EJ010.90 (l'-e) sin X cosX

0.148 (1 - e) sin cosX o.:. o.o

0.111 sin cosi

(experimental value) 0.0 0.01

therefore, e = 0.25,

when it is the modelVI and d=5cm, Cm5()

_--

k eJ111;366(1 - e) sin X cos X

0.1465(1 - e)' sin x cos zx C., C.,

= 0.110 sin

cs

0.5 0.05

(experimental value)

therefore, e = Ò.249 0.4 0.Ñ

and it needs only to put e0.25. Thefefore,

(2. 21) will become as follows: 0.3 0.03

:0.589JyCkE sin2x (2. 23)

0.2 0.02

On the other hand, Co in the case of a

cyl-indrical ship, may bé ealuated generally by

the value obtained by dividing (2. 12) with 0.1 0.01

pJ2LLP, and in the càse of a ship with

knucle shoulders, by (2.'lT) and Figs. 21 and

22 and the value of ' will be as shown in

Table 4. o 10 20 Fig. 27. Model Ill. d=0.05m (h0.1) o 10' 20' Fig. 28.

1957) _{On the. Turning of Ships 73} c 0.6 0.06 0.5 0.05 0.4 0.04 0.3 0,03 0.2 0.02 0.1 0.01 C, 0,4 0.04 0.3 0.03 0.2 0.02 0.1 0.01 c-1, Mode! Il,, d=ß.09m (h=0.18) Model V d=0,075fl (k'0.l5) e Table 1V

(evaluation made by the value at aod0.1 rad5.r)

C. 0.3 0.03 0.2 0.02 0.! 0.0! C-. Model V '0.05cm (k O.1) lo. Fig. 32. a 20 C'., « Experiment CNP NCNP CNO CN-CNR. CNO Il d=0.05 m 0.067 0.030 0.037 0.058 0.64 1115d=0.09m 0.089 0.052 0.037 0.058 ' 0.64 1110 d=0.05 m 0.054 0.030 0.024 0.037 0.64 11115d=O.05 rn 0.065 0.030 0.035 0.055 0.64 V d=0.075 m 0.044 0.044 0

-

O V d=0.05 rn 0.030 0.030 0

-

O VI d=0.05 m 0.043 0.030 0.013 0.052 0.25 10' 20' Fig. 31.

C,, C'., Model VI d'O.O5m (k-'O.I( 0,4 0.04 o 10' Fig. 29. 20 10 Fig. 30. 20 C, C., 0.3 0.03 0.2 0.02 0.1 0.01

ir 0.6 0.4 0.2 Value of ir Model V o Cylindrical ship Weinbium's experimU

Model VI (Cargo sh p)

* Precise evaluation will be obtainable from (2. 12),

but here the value of a ship with knucle shoulders'

with same CIL, will bè used.,

0.7 0.8 0.9

Fig. 33..

(2) The moment coefficient with respect to

the center of

the plate

is C(rx) =

f(k) sincX cosa.

Then, enlarging the result of the rectan-. gular plate to the case of a ship form,

fol-lowing equat,iiji btained for CN,

CN(x)=,/,/ 7t sp :X cos X ± ir C(cx)

where, ev ation ir is made possible as

the coefficient o C 1f rom Fig 33, and CRO from (2. 14), (2d5) or (2. iT) by applying

Newton's resistance law to a model-ship. As for the moment, as it was impossible to

en-large directly the equation of a rectangular plate, evaluation was started from Munk's

equation and c of the case of a rectangular

plate was introduced as mediation with the result of

C(x) =1.l78JCk sincx cosi

where, it is provided ,J, is calculated from

ellipsoidal revolution an'd C from the drawing

based on Lewis' graph. When it is the case

of a ship with its water-plane

in knucle

shoulders shape and its direction

of draft

cylindrical, C will become as follows from' (2. 22):

74 Shosuke IÑÖUE (VoL XVI,

With the case of a cylindrical

ship,

it will be

practically

possi-ble to put ir90.64,

irrespective of

draft,

water-plane coefficient

and B/L. With

the thodels V and

VI, since it is

complicated to

calculate them

separately

be-cause of the

dif-ference

of C

at each water-plane, ir will be obtained by eva1

uating on the

_0.5 0.6

water-plane of

t.heir

draft,

which, with the Model V, becomes ir = O

independently from draft, and with the model

VI,* ir = 0.25. Therefore ir is considered to

have a considerable effect to the shape of

bilge of a ship, but here' ir will be expressed

as the function of C as shown in Fig. 33.

That is to say,

ir rapidly grows smaller

with the decrease of C, dwindling almost

to zero at Cx < 0.925 I'ig. 33 gives ir that was evaluated in Mr. WEINBLUM'S experiment,

where ir 0.

§ 7. Conclusion

Before dealing with the problem of oblique

sailing of a ship, oblique motion of a rec-tangular plate was takén up with the

fol-lowing result.

(1) CR becomes the total of the value

coming out of Prandtl's wing theory and the coefficient' of the normal force in the

case of k = 0, i.e.,

27r

CR(ct) = sincX coscx+2 sin2

2

1957) On the Turning of Ships 75

C 1.457 0.128(2 d/B)

± (0.466 0.1 (2d/B)}(C 0.5).

Chapter HI

Normal Force and Moment

of a Turning Ship20

(Even keel condition)

§ 1. Introduction

In Chapter I, theoretical evaluation was carried out on CR and Cm of a rectangular

plate that turns with a large turning radius,

and in ChapLer II, physical meaning of ob-lique motion was clarified as well as simple

expression was given to the normal force and the moment acting upon a rectangular plate and then, method of calculation was explained on the normal force and the mo-ment by enlarging the preceding result to a ship form. In this chapter, though it is

meant to deal with the problem of normal

force and moment acting upon a turning

ship, it will be preceded by what was left

ôut in the evaluation made in Chapter I,

the problem of a rectingular plate that turns

with a small radius. The evaluation of it

will be based on the deduction of the

physi-cal meaning referred to in Chapter II, then

enlarging it to the case of the turning of

a ship form and lastly compare it with

theo-retical calculation by means of the turning

experiment.

By the above, the author considers that the calculation of the hydrodynamic fOrce

and moment which is necessary for the

evalua-tion of turning and maneuverity has been

made possible.

§2. Inducement of the non-linear part of

angular velocity into the theory of

a flat plate

As it was stated in Chapter I, in the wing theory of a rectangular plate, it was proved

that the non-linear part of drift angle x can

be introduced as Mr. BOLLAY maintained, and

that its value was 2sin

at the normal

force coefficient as shown in (:2. 5), but here

a consideration will be given on the case of

a plate being turned, i.e., how 2 effects it.

The wing theory of Chapter I only deals

with the case when turning radius is large, but when the coefficient of tS2JU of To,

A1, and A2 is examined by Table I, it

can be regarded as constant irrespective

of x when cr is below 10°, which may be

also understood from the fact the

differ-ence between

t 2/U = tIR = O and 1/5 in

Fig. 4 is independent of x. Accordingly,

the coefficients of 12/U of T, A1, and A2

are considered to be independent of x when

it is below 10',

and the

function _{of k.}

Therefore, the coefficients of t 2/U of CR and Cm that can be evaluated from (1. ii) and (1. 12) are also the function of k, and

putting them respectively as gi(k), h(k), the result will be as follows:

C = g (k) (3.1)

Cm'- Jz(k) ...(3.2)

Fig. 34 shows the result of the evaluation

givén to g(k) and h(k) depending on the

calculation of (.1. 6) and putting

g(k)

2

i+

where, g(k) is approximately 0.4 at the aspect

ratio necessary to a ship, while, though k(k)

is considerably different from that which was applied to the case of a small k by

en-larging the theoretical value irJ8/[i - (i/(i +

(2/k))} in the case of GLAUERT'S'3) large k,

h(k) has a tendency that gradually coincides with the increase of k.

CR and Cm that can be calculated from the

wing theory of Chapter i ïñ accordance with

the above method are limited to the linear

term of 1.11/U, and this wing theory makes

it possible to evaluate only the linear part of 2. The reason forth this is considered to

be due to the neglection of the effect of the

curvature of free vortex, i.e., the part

in-cluded in the 2nd approximation, considering the free vortex as an arc of 180' after Messrs.

W1ESELSBERGER5 and GLAtJERT,'2°3) a large

76 0.4 0.3 0.2 0.': h' (k)

induced velocity against the plate änd

re-garding the free vortex

as semi-infinite

straight line;However, tvith the case of the

turning of a ship, it is necessary to take

the value corresponding to the 2nd approxi-mation into consideratibn, because the

situa-tion is different from that of an airplane

where the turning radiüs is not always large,

sometimes amounting twise as long as the

length of a ship at rudder angle 350, For all

that, as it is very dicult to evaluate the

integral equation in determining vortex sheet,

turning centre,

Fig. 35.

Centre of gravity G is situated at the point s

f ron the leading edge.

Shsuke 1Nov

g(k)

h(k) k

i

2 k

i

(Völ. XVI,

calculating the effect of the curväture of

free vortex, it is necessary to depend on

somê other method.

Now the non-linear part of a rectangular

plate is 2 sin2cx and can be evaluated from

Newton's resistance law. With the case of turning1 since it includes forward velocity

as well as angular velocity, it may be prob-ably considered to be possible to iñtroduce the non-linear part of zx as well as that of Q with the introduction of Newton's resistance law.

In Fig. 35, the plate with the length t is considered to be turning at the velocity U

'in the direction of x against the plate and

at Q around the center of gravity G. Taking

the origin of x on G, the mass of the fluid

dashing against dx for the unit time will be

p(Usin±X2) dx

and the velocity perpendicular to the plate

at x is

UsinEX+x2

therefore the force suffered by the part dx

o

01

02 0.3

04

0.5

will be as follows according to Newton's

resistance law:

p(Usintx+xQ)2dx

and perpendicular against the plate.

Accordingly, the normal force acting upon

the plate, if the fact that the signs change at the foot of the vertical line drawn from

the turning center to the plate, namely pivot-ing point. is taken into consideration, will be as follows: t12-1-3

N=p)

(Usinrx±xQ)2dx - R RIfl Raina - p J (U sinX -- x2)2 dx ...(3 3) t/2-f-8

where, it

is provided that th

2nd term

exists only when the pivoting point is located

within the plate, i.e., when it is R sinX <

1/2s, and it is O when R sinr>1/2s.

Accordingly, from (3. 3) C0=

2sin2+sinx

2 1 3s2\,'l.Q\a

+--+ -j)-y)

4 1 ( . (1

s\tQ3

(3. 4)

.U.

provided that the 4th term will be put as O when it is R sin > t/2 - s.

Again with the case of oblique sailing,

the moment about stemming from

New-ton's resistance law is O, but it will become obvious when Q is introduced, because the distribution of normal velocity U sin X + XQ

becomes unsymmetrical against .

In the

same way (3. 3), the moment about G will be

t/2-l-S M=_pJ x(Usinzr+xQ)2dx R sin a R siu

-

p J X (U sinX ± x Q)2dx... (3. 5) t 121-8 2s 2

4(1

s-'\ . 12

Cmüo

-

sin

-

_{-- y-}

+3

sinZ

_IW?2

_1\.2

_I2AU

sin:X

1 (1 Ss\2(t2\211 sin4rx _{(3 6}

22

) "U)J

3 (tQ\2

u)

where it is provided the 5th term will be put as O when it is R sin x > 1/2 - s. That is to

say, by 3. 4) and (3. 6), the non-linear part

of x and Q has been introduced, but when it is 2 = O, it is no doubt the same with the

case of oblique sailing.

Normal force acting upon a

rectangu-lar plate

C.(x) in the case of only oblique motion

may be expressEd by (2. 5). but with the case

of turning, since it only needs to add the

linear term of (3. 1) to (2. 5) and use (3. 4)

instead of 2 sin?cx of (2. 5), the result will be

c=

2ir

_{cossin+g(k)!}+c}

1±--cosX

_k U _(37)*

1f G is taken at the center of the plate, we

get s = O, therefore C8 will become as

fol-lows f rom (3. 4):

C =2

+ ()

41 (.

1tTh

_'(3.8)

U

where it is provided that the 3rd term will be expressed by O when it is R sin > 1/2.

Moment acting upon a rectangular

plate

When only oblique motion is dealt with,

the moment about the center of the plate is

xpressed by (2. 7). With the case of turn-ing, when only linear term Q is taken into

consideration, it needs to add (3. 2) to (2. 7), i. e., it will be as follows:

Cm f(k) sinX cos X h (k) ... (3 9)

however, when the. non-linear parts. of and

Tó the term t il/U of (3.7) is included cose

and it is different in form from (3. I), but since either k or ai is small in this case, it is virtually

the same with (3. 1).

Q are included in it, it needs to transform

the moment about Gof (3.6) into the moment

coefficient Cmi about the center of the plate and it to (3.9). Therefore,

Cm =J(k) sincosh(k)

± Cm'So (3. 10)

where it is provided that, in such a case of

ordinary turning where the pivoting point is

within the plate, it needs only to put s = O

at (3. 6) and

2 1 (tQ\2 i sin4

Cmu sin

Tj'Jy) +3

_{(Q 2}

(3.11) Similarly with the case where pivoting point is outside of the plate, it will be expressed as

Cmi -

snx

(3.12)

and when the pivoting point is at the leading

edge of the plate,

i. e., sin x 1/2(tQ)/U,

(3. 11) is connected smoothly to (3. 12).

§5. Normal force acting upon a ship It has been already stated that the term proportioned to sinx in the ist term, when (3. 7) is enlarged to the case of a ship, is almost independent of the effect of ship's

breadth, but whether or not g(k) of the

term related to co is subject to the effect of

ship's breadth is yet unknown. And while

C is the value that can be determined from

Newton's resistance law and those concerned

only to X among them have been already

solved in Chapter II, as to the term in which co is included, there is no other way but to

determine it experimentally together with

g(k).

Next, evaluation of C,yu will be made.

Assuming O as the turning center, R1 and

R2 the distance between O and the elementary length ds of the ship-hull, O the angle formed

by ds and x axes and ço the angle made by

R1 and ds, the outward normal velocity

against the hull will be R1Qsin(r/2ç) =

* While it is (t 12)/U with the case of the plate. it becomes (L2)/U =w with the case of a ship.

Fig. 36.

R12 coso, and the mass of water dashing

against ds, within a unit time, pR1 Q cosç'. Accordingly, the force perpendicular to the

surface of the hull ds are represented from

Newton's resistance law as pR12 22 cos2ç9ds,

the force perpendicular to the center line

may be expressed as follows:

pR cos2!2 dx

and putting x and y as the abscissa of the surface of hull, it will be

R1 cosço s cosO (fi siflX±X - R cos x tan O - y tan O) therefore,

N=_pJco&'O[Usincz±XQ

U coscx tanO yS? tanO]2dx

± p JcosO[UsincX±xQ

Ucosx tanOyQ tanO]2dx

as a result,

CNO= - J2 cos2ü (sin x±-w

cosLX tanOw

tanO)

±

2 cos2O(sinX±W

yB

\9dX

_cosX tanOjco tanO)

-i-(3. 13)

1957) On the Turning of Ships 79 where it is provided that Yt and Yz represent

the integral of the part that has outward

normal velocity on the cufvature of the ship's

side.

Moment acting upon a ship

3. 10) is to be enlarged in this case, but

Cmo will be evaluated before that. The

mo-ment concerned to will be as follows by

taking the moment about due to Newton's

resistance law of § 5 into consideration

P J cos20(xy tanû)(Usincx+x.Q

- U coscx tan0 yQ tan0)2dx

+.o J cos20(xy tan0)(Usinir+xQ

- U cosa tanO yQ tan 0)2 dx

therefore, the result will be

C0= 2f cos20(sina±w

co: tanO ---w

tanO)2

tanô dx

L L

+2f cos!0(sinX+-w

cos tanö -

Ct) tanO)

xy tan0

L L

where, just as it was in the case of § 5, the term concerned to oblique motion will be substituted by the value of oblique motion

(2.23) in Chapter II considering in connection

with the ist term of (3. io:) f(k)

sinzcosr.

The value related to the linear term of

w

needs only to be determined experimentally along with h(k) of (3. 2:) of the plate, there being no alternative as in the case of oblique motion.

Calculation of C20, Cmo by means of

simple formed ships

As the evaluation of an actual ship is

com-plicated, evaluation will be made by a simple

and geometrical form of a ship.

(3. 14)

Fig. 37.

The force 4 N that acts upon a part on the

ship's side Al will be Es follows if there is

outward normal velocity: (Fig. 37)

4N= -

_L

-

U cos x tan O - y Q tan O J2 dx ... (3. 15)

and as y=y2-f-tanô(x2x)

the result will be

4N= p Jcos2ocP+XQ)2dx

providing that

P = sin - cos

tan O - (Y2 +x2 tanô) XtanO (Q/U)

Q = Q/U sec28

where (x1, y1)(x2, Y2) are the coordinates at

the both ends of 41, (X,y) coordinates on 41 and 4x component of 41.

Accordingly, 4 C0 = AN

2 cos2O4[P2+PQ(xt+x2)

±Çx12+xtx2+x22)1 ...(3i6)

Similarly the moment 4M about G will be,

4M= pU2

tcos2ô(P+xQ)2(x

y tanô)dx

80 ShOsuke 4Cm0o 4M

-

A cos2O['

X1 X2 kL2u2 L 2 L

+

+ - (L Q)2(Xi 1+x2)(x12+ X22)1

_tanOY2±O4Ci,

4x Yz

tanö-[xi+x2

_tanO

_9(p+X +X2)(4X)3

_{(3 iT)}

Secondly evaluation will be given to such

a ship of simple shape as shown in Fig. 38

using (3. 16) and (3. 17).

Fig. 38.

.length of the parallel part of a ship,

where forward of G is expressed by £/2s and abaft of G by t,f2

+ s or other signs (i, l

O ...angle of inclination of the ship's

side on the front part

O ...angle of inclination of the ship's

side in the rear part

4L1 ...length of the inclined part on the front part of a ship

4L ...length of the inclined part on the rear part of a ship

L1 ...length cf a ship forward of G

L2 .. .length of a ship: behind G

When this ship is engaged in turning, the parts that have the outward normal velocity

to be evaluated are represented in Fig. 38

in thick lines as h, ì, .gh, and hi, where. b, d, f and i can be evaluated by drawing perpendiculars from the turning center on the ship's side.. Fig. 39 gives an example

on a ship of C=2/3, where ab and other

were determined for thj. case of X = O. And

Fig. 39.

if cx is increased by R constant, b,

i and

j increase respectively as far as e, e and g. Ji does not change in an ordinary turning

while 1h decreases into O. If R decreases,

increases, but ef decreases. (i) Evaluation along eg, gh

It is the case of

and ji with outward

normal velocity.

With the case of eg, itbecomes _{as follows}

in the both eqûations of (3. 16) (3. 17:):

P=sincx-f-coszx tanOl±± tan2O1w

Q =- sec2Oi coiL

x-=li

x2=L

Y2=O which will be put as follows

4 = 4uCjio, 4Cyn(i()= 4?LCmí,II With the case ofj, it will become

P=sincxcoscx tanO1--- tan?8tco Q SCC2Oi w/L

i

x= L1

y=O