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(1)

D

fl

a12.

'13.

u15.

'16.

¶ ,. JAPANESE DEVELOPMENTS. ON THE THORY OF

WAVE_MAKINJ

AND WAVE RESISTNC'E

By Takao'Inui', Member S.N.A.J.

Skipsmodelltankens meddelelse nr. 34, April 1954

Summary. In this repprt the author attempts to give a general view of the developments in

the theory of wvè-making and wave resistance whIch have been achieved in Japan during the

last ten years

(1944 - 1953).

Reference is made to more than seventy papers which are grouped hère into two parts according to the motion treated being steady or not.

In Part I, for steady motion, the highly developed theory of a planing surface or hydroplane

(3)

and the elaborate solution of higher approdmatión. tò a submerged prolate spheroid

(5)

are noticeable. The lift and drag of a submerged wing or hydrofOil are e,camined (4) in the same manner as 3.

With regard to displacement ships, the second approximation

(6) is

proved to give us a reàBoflable comparison with experiments. This enables us to form a sound conception of the effect: of viscosity on the formation of stern waves. As to the direct application of Michell's theory a method of mathematical repre5entàtion of hull forms which is appropriate for applying thé theory, is proposed (S7). Numerical calculations are made

with Tylor's standard model with special reference to "resisting pressure" distribution

similar to Guilloton's [Al] 'conventional pressure" (7). Havelock's theory is applied to

a wide variety of problems (,9), including several papers dealing with the effects of shallow water, restricted water, and tank- or canal-side walls (8). A few rmarks aré also given to the effect of finite amplitude (lO).

Part II, under the heading of non-uniform motion, is commenced with a methodical development of the theory for wave-making resistance in harmonic oscillation of ships

(ll,l2). Noticeable results are derived with the rolling of ships (ll). Pitching, heaving, yawing as wèll as rolling of ships in ahead motion are also discussed in a more generalized manner (êl2). Concerning, the problems of accelerated motion or any other

arbitrary motiOn, sorne fundamental contributions are made and the virtual mass effects are discussed (l4.,l5). Prçiminary calculations are also given for the initial wave

resist-ance in shoal water (alo).

Contents

Part I. Uniform Motion §1. IntroductiOn

'2 Basic theory

3; Theory of planing on the surface of water

4. Submerged wing

n5 Exact èolutlon for a submerged prolate spheroid

'6. The second approximation to displacement ships

«7. Direct applications of Michell's theory.

Mathematical ships' lines

Resisting-pressure distributio4 along ships' hulls

. Effects of the restriction of water in depth, and width,

.9. Miscellañeous applications of Hàvelock's theory

u9A. Trimming moment of a submerged prelate spheroid and a9B. Lateral resistance of a displacement ship in motion

iñcidence

'9C. Mutual interference between a prolate spheroid and a

for a propeller

'10. Effect of fimite amplitude

Part II.. Non-Uniform Motion Wave-making resistance to rolling of ships

Non-uniform theory of wave resistance Wave-making resistance among waves

Fundamental formula for wave motion caused by moving bodies

Virtual mass effect

-1-sitial wave resistance in shallow water

PART I. UNIFORM MOTION

l. Introduction

It was in 1929 that Sezaia'f 383 presented a paper dealing with the shallow water effect on the wave resistance of a.submerged ciréular cylinder which might be, regarded as the! earliest contribution by Japanese to the theory of wave-making resistance. During the first fifteen years until 1943 our. output of work was relatively small. Shigekawa, apply-ing Machell's theory, discussed the hull forms with least Wave resistance [5] and- the

effect of the position of LCB f41, [5] , as well as that of midship section forms f33. He

V'

and 'their combined effeöt a displacement ship with a small angle of

point sink substituted

ZAssist. Prof., Department of Naval Architecture, Faculty-of Engineering, Tokyo University The Society of Naval Architects of Japan (The Z6sen Ky8kai)

(2)

..2..

also made a comparison between calculated azd measured wave profiles with a deep-draught cylindrical model whose entrance and run are O 30 L in length and parabolic in form, the breadth-length ratio being O 10 [6] Have].ock's early method of travelling impulse was

applied by Tokugawa andKito [29] to an axially symmetric submergéd ovoid. Discussions

were also made ori the approximate extentlon of the theory to a submerged body of fish-like

form.

-- - After this preparatory step. of development., the present stage has been opened

particularly by Kinoshita [39] and his successors In the following, descriptions are confined to this stage of the last ten years. Onthisoccasion,jt ought to be remarked that throughout the former half of that period Japan was perfectlyisolated from any information on progress throughout thworld But these unfavourable circumstances have been fairly improved now, which enables us to understand our own actual position in

relation to world progress Especially to the latter, outstanding contributions are added by Havelock, Wigley, Guilloton, Lunde, Sretensky, Weinblum, Barril].on, Shearer, Kotchin, and others

Throughout these noten, if not otherwise

stated, Cartesian coordinates

moving with the body have been assumed. The origin 9 has bèen taken in the undisturbed water surface, withOc vertically upwards and 'x parallel and opposite to the direction of the motion

The notations adopted are the same

as those used by Havelock and Lunde [A2]

eiceptirig that the direction of motion is reversed and that the velocity potential is denoted by,

Cot by

, which denotes here the acceleration potential. Further, we:,'..

denote the velocity of advance by V instead of

c , which has

been adopted so far

peculiarly in

wave-making

theory.

Japanese papers with the number C J

are listed in the bibliography which is cotiled

by the present author. All other papers, which are referred

to, are grouped, for

con-venience sake, into two parts

with the marks I:w

J and

(A

3 .

The reference number

(w 3

is

the same with that of

the bibliography, which

was compiled by Wigley and published at the

end of Guilloton's paper [A. 4]

,- where [4 ]denotee the additional references.

§2.

Basic Theory

On the basic theory for steady motion of

waterwith

free surface, the method of

analysis has been considerably improved by Hanaoka [1],

[23.

He (13 introduces Prandtl's

[A

5]

acceleration potential into wave-making problems and shows that for some cases this

affords

a. means of analysis more convenient thán the usual method, where the velocity

field alone has

been takeninto coisideratjon.

Folloqjing his propositions the acceleration potential caribe more easily analysed under given c9nditions of the problems than the velocity potential, which can be derived from the former through a known transformatiôn. In this procedure a possible merit is also anticipaté.d for a further comprehensive understanding of the dynamical properties of

surface waves. r

Denoting by the velocity potential, and by the acceleration potential, the

equation of motion in the linearized theory cañbe written in the form

XList of

symbols is given on page

23

(2.1)

with

(2.2)

while the

(2.3)

______)(=ò,

z=ò

with the velocity field, and that the fOrmer is 'transformed into the latter in. the usual From (2.1), (2.2), and (2.3) he:.ehows tiat the acceleration field Just conforms

way

r=JØdf

(2.4)

Further, this method of analysis by acceleration potential is suggested as being pre-eminently suited for the case of the wave motion due to a submerged wing or a gliding plate on the surface of the water. For example, the acceleration potential which Is representative for a two-dimensional gliding plate with infinite span is written

as

(2.5)

(x,z)-Considering the

disturbance

of the free surrace, we may write

(2.6)

z)=&fp(é/Kre.xP(Kz) ca.s/«x-)dkd

From this the well known velocity potential is easily obtained by (2 4), as follows

(2.7)

1

d/,1p'xPT/Ç1Z.#i

cx-)jJd

$::;

,

z-o

(3)

-

-ihere tke real

part alone should be

taken

In

the

1gîthadT8idi of (2.7).. Furthér,

the

above velocity potential may be transformed into )the form

(2

8)

I)df'

).wz*

Observing thit thé firSt. terrn:of thé right hand side of.(2.) is. sifflulár to.that of.a,.thin

wing in uniform flight, he shows a gliding plate is to be reprsent1d by the singular points

similar to a tnin wing in

anInflniIiffdT.

he

TÍopointsoutthat in this case the

wave motion

is represented by some

harmonic distribution of sources and

vortices

To find a suitable distribution of singular points proper to be substituted for a ship-like body in ùniform motion is of practical importance. Valuable suggestions are àlso giveñ by Hanaoka [2] in like

manner

'mentionéd above.

Assuming

that the free surfacé i an equi-poteatial surface in the accéleration field without 8urface

wave,

the acceleration field de to a ship alone may be represented

by the linear combinations of following three kinds of potentials through Green's theorem:.

V/k(-191dJ2

4

(x,q,z)=-j/(

j-- -.-4d

03 (x,q,z) = jJ'Ø-.)[4

t)]d5

where J?. andB denote the

free surface of. water and the verticaimedian plane of a ship

respectively, with

4

and

4

denoting

Ø

on the right and left side of S

.

In an

ordinary

case of symmetrical ship

form; Ø

consists of

and

.

For an extremely

shallow-draught ship a simple expression by

is sufficient; on the contrary, for an

extremely deep-draught ship

alone

lé also sufficient.

Either-pcxi in (2.9) or(Ø-,) in (2.11) is given as the solutionof a certain

integral equation, but when a ship form is

given by the equation y'.y(x.z)

, we have a

simple relation in (2.10) as

-(2.12)

Adding the corresponding wave

terms

to (2.9), (2.10), (2.11) and transforming thém into the velocity field, he obtains thé velocity potentials, as follows:

(2.13) =

J2//epKcz

#iJad/d9.

v- (V

#)-

-Fiñally, the a follows: (2.16) (2 17)

-

?)dJ/I sec9dKd9

g'

I'

g

*

¡

pCI,

'')dJZ1

exp[

9

(z

(2.14)

/rj.tM-

- í-,'d$#

dI/ exp/iç

a[-zt ¿)

#

dB

(2.15)

-f

4-u[

4.

;9 da

-

.se8f(z

#»u}J

9ea

d

corresponding wave resistance is given,, for instance, by pressure integration,

*

e/OdXdû'

(4)

(2.18) ..

where g-(y,.#

) , with

.

and b denote the breadthwlse ordinates of the

. corresponding right and left hand side of the unsymmetrical ship's surface respectively. When the direction of motion is parallel with the vertical médiân plané we hvej-o for

a sy-inmetricäl Ship, an4 in such a case =-C,.-e) must be substituted for VOÇ,Z) j

(2.17). A proper coibinatlon of (2.16), (2.17;) and (2.18) is shown to be selected for the

calculation of wave resistance ofa ship-like body according to given circumstances.

- .

-§3. Theory of P]ningonthe Surface of Water

Two theories wére formerly presente4 fOr: the piahing on the surface.of water, neither of which satisfied simultaneously the boundary condition on the planing surface together with that on the free surface or the effect of. gravity.

In the first theory, originally propouided by Lamb [A 6) and based on. a method of a travelling disturbance importance was placed on the gravitational effeät.., and the

condition right on the planing surface was satisfied merely with a ro h a proximation, despite tJe definite contribu,tions b7 Havelock [W li) , W Ï2 , Hogner1W ]',

(w l3,

Weimblum LW 27) , and Miki [161 , [17), (181 ...

On the contrary, in the second theory, a- total:or-a prtial neglect was made òf the gravitational action of the. free surface, while- arigorous treatment Was given to the condition of the planing body.

The theory of discontinuoué motion is applicable to the latter case and some calculations and experiments were made by Wagner [A 7], [A 81, Green [A

93,

Weinig (A 10), Franke [A il] , and Sambraus [A 12).

Under these circumstances an outstanding contribution was made by Maruo through

his methodical investigations . (19)', [20) , [211, [22),

[231,

[243, [25) , where he succeeded

in finding an exact solution which satisfies both the two boundary conditions above mentioned simultaneously. -

-- We can write the pressure condition on the -surface of the water referred to as the

moving -axis, which is fixed to the planing body of constant velocity 1/ , in-the following

form, by introducing the fictitious frictional force pröportional to the velocity

according to- Rayleigh. .. ..

(3.1)

v-j

The pressure p may be assumed, for simplicity, to be zero on the free surfacez-o , while it takes a. definite value Çx,) on the planing surface .

In the linearized theory of the wave motion, the slope distr1bution is given by the equátion - .

(3.2)

-and- shoúld take the. deflnitè value 9 (x', V) which- is given beforehand on- the surface --In thé twò-dimensional fluid field, the preSsure distribution on'thè planing surface is determined by the following integral equation deduced from (2.7),

(3.3) . (j)d

cx-I)J,4J

.

-where the real part alone should be taken in the right .hand side of (3.3).

Maruo found an analytióal solution of (3.3) after a transformation similar to the wing theory [21] , [22). The solution is of the form,

. . a 4 a,,

with the relation

and the coefficients in the right hand side of (3.4) are some complicated functions-of

1(1

defined by a kind of recurreùce formula. It must suffice here to show general features alone in' Fig; 1 for the case of a flat plate or a piane of constant slope expressed by the equation,

(3.6) . c.o,ué

where

fi

is the angle of incidence

The lift coefficient is given by the formula,

Ç

L/('.v2L.a.ka,)

and, for a flat plate, it j. proportiöñal. to the angle of Incidence. In Fig. 2 are shown the numerical results derived from the values given in Fig. 1. He also show8 that In a part, of the smaller values, of

('

O,j)

the follöwing approximate line is' to be

substituted with sufficient, accuracy, .

(3.8) ..

.

'/,=*

Marmo showed that the total pressure resistance consists of two elements, the splash, resistance and the wave making resistance [19] , the coefficients of each

(5)

-ç=

/'-Cvj.. ffa

ç= ç/(?v)=

rt[(a

(/,t)t%E (-fa,

(')Y'fa (i0t)6

-

(IY)17

where (Çt) j the Bessel function of the first kind.

. .

In the caseof a flat plate, /L is equal to.unity as shown in Fig. 3,where the relative Importance of R and

Ç

is distinct for the different value of Froude

number p ..

-When the span of the planing surface is finite, the subsequent fluid motion is naturally a three-dimensional one, which presents by far greater difficulties and

complications in its theoretical treatment,. compared with the two-dimensional motion. above

mentioned. ;

: The integrai equation corresponding to (3.3) is written as

( 3.10) (x,q)

'

;rff&a '')

d1d7J

d9ÍK.5L;z:L»

Because of the serious difficulty of finding an. axant solution oi (3.10), Maru devised ah approximate method of solution which is analogous to Prandtl?s lifting linê theory. This

method naturally applies only to the asç óf a large aspect ratio and is based on à

simplification of the integral equatIon (3.10) by means of the two-dimensional solütlons previously obtained.

After some appropriate transformation of the Integral with respect to k , the right hand side of (3.10) can be divided to two parts.

(3.11)

where 9

is derived from the local disturbance in the vicinity of the planing surface,

while 9 Is from the free: wave motion in the rear.

Further, it is shown that on any longitudinal section of the plane 9 (x,q) can be

replaced practically by the corresponding slope (X) whih would be realized in the

assumed two-dimensiohal fluid field of an infinite spanned planing surface whose cordwise pressure distribution is just the same as that on each section parallel to the direction of motion.

On the other hand, the remaining part 9a (x,q) which is derived, from the free wave motion is affected considerably by the finite span, but has little variation along the longitudinal section or the cord óf the planing surface. Therefore O2x,qJ may be replaced

by (q') which is determined only by the spanwise distribution of the lift.

These relations are similar to those in the lifting upe theory of a wing, as qt'x)

Is analogous to the characteristic of a profile and

02&)ls

similar to the induced angle. The only difference is that, in the case of an aerofoil, the induced velocity Is caused by the trailing vortices and does not exist in. the two-dimensional fluid field, whereas, in the case of a planing surface, it is due to the wave motion and does not vanish even in the two-dimensional motion.

By virtue of the above considerations and adopting the relation (3.8) he projected an integral equation of only one variable, which deteinines the spanwise distribution of the lift of a planing surface-whose longitudinal sections are all straight.

(3.12) Oty) &) -[p4,,

#a+771t-0JM)# 7/PK 4-7) d7

where t

(3.13) (3.14)

x

'W

/3X (4A'

ai

K2

A

0(q)

is the angle of incidence of each section and

K

, K2

are the modified Bessel

functions.

-For a planing surface of arbitrary section, for Instance a cambered section, a similar consideration may be applied. Assuming, at.first, that thé following relation is found for each section by the two-dimensional calculations,

(3.15)

-

y»)/1,v ij

4 c'y)

- -

-then the integral equation correspônding to (3.12) becomès

-(3.16)

# f77

,J1(y)4(q) [?

o,477./GJ

#

'7)X4i-'7)d7

The-method of solution of the equations- (3.12), (3.16) is similar to Glauert!s calculation of the lift distribution of a wing, and the numerical examples are reproduced

here from f23) foza broad rectangular plane and a triangular plane such as shown in

Figs. 4 (a), 4 (b). Fig. 5 shows the spanwise distribution of lift in, each case. -Computir the total lift we find approximate rèlations as follows:

(3.17)

AO/CL X,,33A

for rectangular plaie, (3.18)

2IS,4=x2#4322

for trièngular plane,

(6)

-6-where 2 Is the a3pect ratio and X, ,

X

are the functions of

k

, as given in

Fig. 6.

Maruo also disc.zssed the shape of the planing surface of a given wetted area whi gives the minimum resistance under a constant total lift [253 . In these circumstances, t frictional resistance may be safely assumed to be constant. By the aid of the varlationa method, he expressed the condition of minImu'i pressure resistance in an integral equation

§4. Submerged Wing

The effect of the free surface of water upon a submerged wing or a hydrofoil has practical importance to the immersion-characteristics of a screw propeller blade, and in this respect a considerable improvement was made on the theory by Maruo (263 and Nishiyam

[27] , [2e).

In these papers due consideration is given to the effect of gravitational motion,

of water which was hiterto neglected by Weinig [A 10), Havelock (A 13) , Tomochika [A l4 and 1mai (A 14), (A 15).

Nishiyama's papers(27] , (2e) are devoted exclusively to the two-dimensional problems, while in [26) instructive discussions and numerical calculations are given also to the'three-dimensional case of a horizontal or a vertical plate hydrofoil of finite spai

The general conclusions are that the causes of the efficiency drop of a submergec hydrofoil acting closely under the free surface may be attriìuted chiefly to lift reductic and partly to drag or resistance augmentation.

The lift reduction is the combined result of a negative horizontal velocity eÇ

and a down-wash , where u1 and denote velocity components of a, local

disturbance right on the hydroplane, of which the latter has a greater importance than the former.

On the other hand, the drag augmentation is due to the free wave in the rear, as formerly shown by Dickmann [w 84].

For the two-dimensional case of a plate hydrofoil, the first approximation is

given [26] , [27) by substituting a single vortex for the hydrofoil.

Subsequently, the second approximation is given [26) with the known distribution of circulation

(4.1)

7(x»..ì(_

)

where r denotes the total circulation.

And lastly, the third approximation is obtained by aid of a complex velocity potential and conformal transformation [283.

Fig. 7 is from [28] and gives a comparison'between the first and the third approximate solution in the case of f-2( and (= 5° , where Ç is the depth of immersion.

From this we find that the effect of the'free surface is inverse according to

I or

) f . In the lower range of (cl) ', the free surface acts somewhat like a rigid

boundary and lift is increased slightly. But at a higher speed this resembles a surface of discontinuity or vortex sheet asymptotically and its effect is always negative.

Consequently, lift is reduced by about forty per cent at ' . The effects on

drag-lift ratio and on drag augmentation are also shown in Fig. 7.

For a three-dimensional case of a plate hydrofoil whose circulation has a semi-elliptical distribution such as,

(4.2)

r()= Ç(f-3), (y)

an approximate solution is deduced by Maruo [26] from the lift-line theory, as follows: The mean down-wash is given by

(4.3)

where b is the-span, and is the mean circulation

f iÇ'

The drag is also given,

(4.4)

R=

-;

where ' is the same as induced drag for infinite'fluid field without a free surface.

'The dimensionless coefficient

k

in (4.3), (4.4) are as follows: A Horizontal Plate Hydrofoil,

(4 5)

k -

_/;[21

2,

/eq[-Kcu'. j77kAa V'7)

v'

i

A Vertical Plate Hydrofoil,

(4.6) _/

/ZJ%)4' # /[-'/?

f)J47Á (u2#í)J

¿u,1)4

da

where

7

,

f

, ¿re the Bessel functions, and

f

denotes the immersion of the midpoint

of a plate.

Fig. 8 is from [26] and gives the general featucps ofk, and A . From this we can understand that the three-dimensional effect of a free surface on a submerged hydrofoil is quite similar to the two-dimensional one, and that this effect diminishes rapidly when the immersion gets larger than the breadth of the hydroplane.

(7)

where

-7-,5. 'Exact Solution' for a Subme'ged Prolate Spheroid

In the wave resistance theory', where the water isassunied to be frctionless,the fluid motion, is specified by a velocity potential

#

satisfying. given boundary condition

(1) the normal-fluid velocity on the solid is equal to the normal velocity of the solid at each point, (2) the pressure is constant at the free surface of the water. The solutioh

is generally pictured as an infinite series as follows:

(5.1) .

where satisfies the conditiOn -(i), - is a-correcting potential so that g,

satisfies the condition (2), is again a correcting' potential t maintain the condition

(1), and so n. 2 .

-Unfortunately, in most problems this continued process is too-difficult to be carried óut further than the first two terms #. . In fact, the only. case up to the

present is that of Havelock [A 163-, [W

82),

by2coplex velocity potential, who gave the

complete successive solution for a submerged circular cyl-i-nder, excepting that he [W 12)

[W 83) also obtained for a submerged sphere.

Under these circumstances, the complete-solution of a submer8ed spheroid,

including a submerged 5phere ini the simplest case, has been ably obtained quite recently.

by Be'ssho

[32] .

- -

-- The-scheme of his analysis is the sane as that of Havelock [A l6J and a little

different from the alternately successive approximation mentioned above. Beginning with the more generalized case of a submerged ellipsoid, he puts into the form

(5.2) . - -

-eío

"

- -

-d

)

(,% )exp[Kf(#/ú)Jd5d/(d9

with. , -and by ) denoting the ellipsoid of immersion

f

.

-If we write

(5)

--

a442'7

,

'y,

¡)eçt'[Kfz-1).iiJd$d/(d8

thesum of

(ej*(e is a ways self-consistent with the condition (2) for any arbitrary

source distributiòn

d(i,7,')

.

-This-source distribution is naturally to be found in regar4to the remaining

condition (1), and it may be easily written down explicitly in a well known form if we content ourselves with the -first approximation.- Wher,eas1 in our complete solution, this

is not- the 'cáse but it nee4 a ]engthy process for computing the exact expression of

ò(,,7, .) .

At first,'è(,T,J'

is expanded into ellipsoidal harmonics wit-h

indeterminate coefficients A7' , which are settled afterwards by the condition (1). Thoúgh he obtained a general solution applicable to any submerged spheroid, here the final result for a submerged sphere alone is given as

-(5.5)

c,

=

ç. [{r-Lt'rf;2J-*fL-

1;3)K(f,3)/K(fa.

- -

# --'fjL (r)/ç'2f)#

(/)A(13)JJ

(5.6)

CL= f

[(í-,vL (y;2)-

/i (f, f;4)

-f [j/!(t

f.)

1./Ç2(1,) K(O,3)/L't'f,

t;3)jJ

where

Ç.

, C are the first apnroxirnatlons, given as [w 12)

(5.7) ..

-

ç

(f,)

(5.8) -

CL =

¿!L (i,f;3)

with A=

Iaf

,

Z

, and K(,'-y')

L (m,n;)

are some definite functions of connected with Whittaker function #á()[A 17]. Above formulae (5.5), (5.6) are verified to-give practically exact values of , and CL as far as the depth of immersion Ç is not less t-han

1.5 a

(orA-' 11.3 ). -

-Fig. 9 is an example for the three-dimensional distribution of sources and sinks

on the sphere, by combination of the vertical and horizonta]' sections, for / 0.707 and,

1.291 ( A=7 ), ähowing a different effect of the disturbance due to wave-making with

varying Froude number . -.

Concerning the resistance and the vertical force,

and CL

are shown in

-Figs. -10 and il for the case of A'j

(115a)

- -and

(/aa )

. The general

features are very-analogous to those of a submerged circular cylinder [A 16], but the magnitude- -of the disturbance on the solid, due to the existence of the free surface-, is noticeably diminished in the three-dimensional fluid motion. This fact is of great practical importance, with regard to the second approximation of the displacement ship

mentioned in- the following paragraph.

-- Further, further light is expected to be thrown on this dark point of

(8)

46. The Seàond Approximation to Displacement Ships

. Before going into details of.this subject, we should make lt clear. that the term$

of the "first" or "second" approximation here used have a physical meaning somewhat different from those in the preceding paragraph Concerning a displacement 8hip of

ordinary form, it is not easy.éven to find correctly the firstterm

ln(5.l),which

. specifies the assumed motion of the ship in an infinite liquid. The word "exact" or

n approximates? here used should be strictly confined to this primary stage If we use some hydrodynamical expressions, this may be reduced to the problem of finding the "exact" distribution of the sources and sinks substituted in place of the ship "without" any consideration to the existence of the free surface of the water For the first

approximation, in the present meaning of the word there have been introduced three or mOre kinds of methods by Miòhell W 6) , Mavelock tw 32) , [W 63 , Hogne 1W 31) , W 341

Guilloton [W 623 , [W 67] , (W 72] , and Lunde 1W 733, EA 2].

First of all, the approximate source distribution given as

(6.1)

with

q..y(x,zL>O

denoting the equation of the hull surface, is well known to be safely adaptable when l4ichell's condition for a slender ship is sufficiently satisfied To

larify to what limit this first approximation may be safely appIlcéble is of particular importance to a cori ect understandiri.g of thé, effect of 1scos1ty on the formation of stern waves. The present author, [l3 , [15] has examined this, with special regard to the fact

that the measured resistance gets larger than the calculated resistance in the vicinity of the last hump speed ( F

- 46

with almost ali, models ever treated by Wialey [W g) ,.

[w 19]

,

(w

243 , (W 63] , (w 663 , (w 693 , and Weinbium (w 26] , '(W 37],

[w 421.

It

may be natural for us.to suppose that at hump épeeds the calculation for ideal fluid should, if it were correct, give more or less a higher resistance than an experiment which is actually affected by the viscosity of the water.

In this connection, with due comparison between theory and experiment, another three'points of importanòe should hé noted. The first is the method of experiment or, in

other words, the effect of the trim of the módélin the run. The second is' the correct

evaluation of frictional resistanàe of the môdel, including the socalled form effect

together with the correction for a change of the wetted surface, either of which gets

more and more important with the incresing breádth of the hull. The third Is the disturbance around the ship caused by the wave motion, which has already been discussed

in the preceding paragraph and still remains, as the greatest difficulty in the theory.

(Symmetrical Models with Infiñite Draught) Let us confine ourselves, in the meantime, to a cylindrical model of infinite' draught E13] . This rather fictitious form of model is peculiarly appropriate for our present purposè in discussing the accuracy of the first approximation (6.1). With such a cylindrical model, the water-line function yx)and.th

source-sink distribution function 0' (x) are connected with each other by the simple

equation ' . . .

(6.2)

with

(6.3)

G(x,y,'I)

=ia,,-"('), '

(o.9

ii)

In (6.2) and (6.3) the coordinates

q and

are all cOnsidered as to be dimnsionless

for simplicity, the half length of the model being taken as unity.

Equatior (6.2) may be regarded'as: (1) an integral equation with respect tod(r), or (2) a direct integral expressiOn for q(x) , according tÓ which of them is given

beforehand. Confining our interest further to symmetrical models for simplicity, and substituting the following pOlynomial expressions 'for

á'd)

into (6.2)

FOr fore-body, (),=

-fJ'O

Fr aft-body, a' (t)=

(a1*a4#aja4/',,,

oE t

we can easily carry out the following añalysis:

(1) To obtain thé "exact" distribution q'(x) , correspon4ng to the "exact" given watér-line fòrp yCx) , '

- (2) To obtain the "actual" water-line forñi y (X) , corresponding to the

"approximate" distribution (X), given by (6.1).

These resultS, combined with the final comparison between the "exact!' resistance R, and the approximate resistance , enable us to get a definite understanding of the limiting

aCcuracy. of the first approximation (6.1). The numerical: results are as fòllows:

(a) The effect of hul-1 breadth: ' ' ' ' ' '

Two parent water-,l-ine formé are taken, ,

(6.5) '. -' ,

()

:

(6 6) q ¿(r-xi)

with the breadth-length ratio /L(' varied as ¡ 0.05', 0.tO, 0.20 br' /.8'20, 10, 5. Fig 12 is for a comparison of the söurce-sink distributions and water-line forms with (6.5), (6.6) and Fig. 13, 14 are a comba'ison of the wave-making resistance with (6.5). Further Fig. 15 shOws the general features of the index n in the 'relation

(9)

'6 7)

-.

RóR

n

.

for the parent form (6.-5). With the first approximation (6.1) the well-known relation

a ¿

Is always consistent without any. restrictions ònthéparent forms, Froudenumbe,

or 4.

, but this is not the case for an "exact" solution

(b) The effect of water-line forms: .

Havelock's calculations on the effect of later-line fórms [w 201 are revised. In Fig.. 16, (a) is reproduced from [W 201, while (b). is the present result. .

From these results it may safely be concluded thàt witl infinite-draught models Michell's theory or' the first approximatiôn gives a- somewhat -larger resistance than the "exact" theory or he second approximation in the lower part of Froude number until the

last hollow speed (f0.3]

-

0.34) is attained, 'but that In the higher speed the above relation is reversed, and Michell's resistance falls remarkably lower than the "exact" resistance in the vicinity of the last hump ( F 0.50 - 0.54) as might be expected.

(Symmetrical Models with FIniteDraught) In the three-dimensional case of a model with finite draught, the mathematical difficulty Is so much increased that we are obliged,,for

the present, tocontent ourselves with the inverse method of analysis (2), Instead of (1),

mentioned above Four "exact" hull forms with finite draught have been obtained [151 with - the given three-dimensional source-sink distribution, such as.

o(j,o,)=

v-ìÇ (1i ()

V

where

iÇ(1)=-'9/,

(-ftf)

'.

(-e-'o)

.

with (2 and t shown in the Table L

Fig. 17 shows the general features of theexact hull forms

Table 1 '(x,z) in comparison with the corresponding water-line forms y.(x) for infinite draught ), and Y,(.(,z) which -is regarded Model -a-1 as being represented by the given distribution (6 8) in the first

approximation,- and denoted by L-101, 102, 201, 202 respectively. S-101 0.4

o.

In other words, the water-lines (,z) for finite-draught .

s-102 0 4. 0 2 cylindrical models L-series conforms with (6 6) where B/L.6). 0 10, S-201 0.8 . 0.1 0.20 combined with

(t%)

= 0.05, 0.10.

S-202 : 0.8 0.2 . .

(Effect-of Viscosity) - The main object in carx'ying out the rather laborious calculations

with (6.8)was simpÏyto make a direct comparison-between thèÌy and eïperiment.. In this connection,. special-care must be taken to the method ofthe experiment. All the experiments]. work previously published by Wigley, Wéinblum, and othèrs except by Shearer LW 75J , were made with the usual -"free" guide In which the model was run with free bodily sinkage and

free trim. . -

-. . Shearer[W.7,5] uses a-special type of guide so as to maintain an even keel

condition of the model in rins, but-he does -not make any initial adjustment of draught in each run corresponding to the bodily sinkage. Consequently, his model is subject to a

certain amount of lift at higher speeds, resultingin an effective reduction more or less with the displacement Recently, a special type of "horizontal guide" has been

designedand producedat the ToyoUniversity Tank, with which we can easily make such a

resistance-test that-our modèl allows for free bodily ainkage, but not for the least trim.

Our experimental workon the abovementiozèd four models ic stillin progress. Fig. 18

is from [15] and showsthe result of theearlier preliminary testwith a 2.33 m

model-S-102, where 'a swing-frame was used combined with the initial adjustment of draught for each run to the previously calibrated'mean, bodily sinkage.

To'return to our present subject:howto correlate calculated and measured

resistances with each other, or inotherwords, what kind of corrections should be made to

the theory if we introduce the actual viscosity of water : Three kinds of effects of

viscosity must be considered in this regard, namely: - - - -

-Change f wave-making characteristics of the stern, or the deformation of the stern wavea-,. sometimes also of the aft-shoulder waves with lesser importance,

- which is analogously represented by the incomplete recovery of the pressure in

- the assumed cSSe of infinite liquid, combined with the confuSed turbulent flow

patterns in' the--rear of the model.

-Effective increase of breadth due to thé additive thickñess of boundary layer

along

the sidè.s of

the

hull.

- -

-- (.3) Decay,

of pre-produced waves in their after stage of propagation. -

-Among 'the three, (1) is -the most important as

Haveloök

has pointed out in [w 463,

y 7]) ,

[A .3].

He [W 71] also discusses (2). and shows that a consideration of the displacement thickness of the boundary layer Is advisable, but is by far insufficient for the complete rea'poning of the existing discrepancies between theory and experiment. -

-'As to-(3), a-mathematical approach isavailable in like manner as Lamb fA 18),

if we confine ourselves to molecular viscosity, neglecting eddy viscosity. Wigley [W 56)

has introduced'a decay factor which denotes,the ratio ac/. considered at-the

position of the' stern, where 4 and-

are

the bow wave height for ideal or fr viscous

fluid

re5pectively. -

-- - -

(10)

lo

-(6.9)

eflLr.;]

where denotes Reynolds number, and P Froudenumber.. The present author [15] and Sakao (12] have obtained also the expression of

which is applicable to Havelock's

elementary waves W 5 in the three_dimensional free wave patterns, as follows:

(6.10)

ø(=ex,ß[

/2Se

(9i'9-8ftJ

,here the upper sign is for transverse wave ( O O''-35I6'), and the lower for diverging

wave ( O-.6°i'-' QO° ), with G denoting the direction of propagation of any elementary wave. In the case of 6= a , (6.10) gives the least value for

, which is just the

same as (6.9).

As is eastly shown, ø in (6.9) or (6.10) is safely taken as unity even for any

small model with ¿

¿n'

and V 4bm,

or with

'

1,°b

and

a/s

within an

accuracy of one per cent ( 0.99-1.00) , so that Wigley's assumption

Q , where is the reduction factor for aft-body inclination dY/ax roughly corresponding to the

correction (1), may be rather an overestimatiOfl for . .

-Concerning the remaining correction of the most pract4cal importance (i), Wigley's empirical formula LW 56]

(6.11)

flexp(-.001/p')

is well known. The above expression (6.11) gives a widely varying value of

3

with P

from ,'

0.210 for F

0.20 to , O.94 forP

-0.50, which is rather an unexpected result, when we consider the fact that is in nature a simple function

of R

practically less affected by

r .

This discrepancy may be chiefly attributed to the errors óontained in the first approximation or Michell's theory, because the general features of

the differences in calculated resistance between the first and the second approximations above mentioned, with infinite-draught models as well as those of the differences in measured resistance between Model S-102 and L-102 shown in Fig. 19 are somewhat similar to those of the correction derived from in (6.11).

With regard to the correction (1), it should be further remarked that the actual viscous flow patterns, so confused and turbulent, in the vicinity and also in the rear of the stern are to be fully reproduced or, in another expression, to be equivalently

translated into, terms of nonviscous hydrodynamics by approximate modifications. Remembering that neither Havelock's nor Wigley's modifications succeeds in explaining the known phase lags of

hump-

or hollow-speeds between theory and experiment,

the present author [15] has proposed a practical method of

correction (i), as follows: (1) The free-wave systems orresnonding to ideal fluid are to be obtained at first.

(2) Among

all free-wave systems, the stern wave alone is to be modified in two points

The wave height is to be reduced by a reduction factor f9

(somewhat

different from Wigley's

ß ).

The starting point or origin is to be shifted abaft by or more generally by 61. , where is a shift-correction factor, thus, for example, in our models S-series, the free wave systems, consisting simply of the bow wave and stern wave, may be represented by,

(6.12)

(6.13)

f)ca.

B

ty &i'

8jJd0

(6.14)

f[/c5eBf(x- i-gS)casG

J»2jJd8

4f

co.s 9)

cos[A see2B/(x f-è6)eos

Z

The corrected wave-making

resistance, corresponding to

the above modified

free-wave systeme,

is easily calculated by

Havelock's energy method

[W 43), tW 5), as follows:

(6.16)

-fafl/L

(ft#)2Z&)(1J(1)]

is introduced in place of dimensionless ]ength 2, and

1(6)

f(kízased8

(6.17)

.7(6)

/?Çz4,I)z2O.Gde

with

- ,ig('/Çd)

K[(f()Ai2ft6M1

,V(K,6)

where

(6.15)

where the actual ship length L

are given as

(11)

with

(6.25) y(x)

'4*4('x)

-

il -where

K

For sufficiently large /('

or lower Freude Number F

the second term of the right hand side of (6.18)

may be negligible and we have approximately

(6.19).

c

(q)

(1' 6)

(-,),

¿4q

ja(,,)p

where (6.20) .

ç-=

Taking advantage of the fact that the

shift of the humps or hollows in calculated resistance curves is dependent exclusively

üpon 6

without any relation

to /3 , we can

determine the most suitable f . beforehand,

in iriew of the actual phase differences. Once

o-determined, we óän easily draw the corrected

resistance Cur-e Ç, for any given value of ¡3 which should be finally

determined by taking into account the generá]. features of the calculated

resistance in comparison With the measured-resistance As mentioned, the reduction factor as well as the shift-correction

factor 6 should naturally be taken as constant for a widely

varying range of Froude

numbers with any fixed

model. . .

-In Fig. 18, for Model S-102, the corrected

resistance coefficieht5c,

(4O.o5;

/

-0.7.and 0.8) cothbined with the uncorreted Ç,

(

i ) for ideal fluid are

also shown. For the present

case 6

0.06 with ,. O. seems to be most appropriate. These values of 6 and jB

are of the order to be expected. From this we may safely conclude that in our

second approximatIon of the theory, a fairly

good agreement with the experiment is possibly attained

without introducing any unreasonable assumptions into the correction for the viscosity of the water.

Complete experimental work with the above mentioned

"horizontal

guide",

together

with wave profile

measurements especially in the rear of the' models and

form-resistance measUrements' by submerged double

models are now In progress.

(Further Improvements and.Comparjsons with Observéd Wave Pro'jles)

Okabe and Jinnake [l4J

have added very

valuable Contributions to

our present 'questions. First, they found that

in solving our integral

equation (6.2), the

successive approximation is widely adaptable

with sufficient accuracy, as follows:

(6.21)

.

'Xff. d(-4'1)

d&-)

Applying this to the model with' which

Shigekawa (6J has masured the wave profije,

they obtained the result

shown In Figs. 20, 21,

where eveh their second approwimatlon proves of fairly, good accuracy, and the third

approximation has an error no more than one or two per cent.

The expression for the water line is,..

for entrance

(-l)ç

-0.3) and

'6 22) .

run (O.3x.l),

y(x)

0,7,

' for parallel body

(O.3x

0.3)

Thereafter, a calculation of the boundary layer

was Carried out, with

special reference to

the rapid Increase of the displacement

thickness just before the

stern 'is approached. An

approxinaté formula similar

to Millikan?s [A 19J for the surface of

revolution is applied, as follows;

(6.23)

aa983Re4uM/LI

where

2

(s)

is the velocity of inviscjd flow at

any point on the surface of the model with

arc-length

.s

from

the bow.

Finally, introducing the dimensionless expression for

the displacement thickness

taken

a

norma], to ds

(6.24)

d.(i)'

j-

6(&

we have the new contour of

the model ship effective

to wave-making In the form,

without the

least consideration to the senaratjon of flow,

(6.26)

'a/

Numerical calculations are made for three different

values of V

or

Re with L

5 n, Corresponding to Shigekawa's model.

Fig. 22 is for V

l.80'%

( 7.888 .lO6).

Taking advantage of these results

they obtain the wave profiles on t.e vertical

median plane ' O in thc cas

of an ideal and a visco water.

Comparisons are made at

P

0.207, 0.257 and 0.371 among thefo].lowjng: Observed wave profiles

- Originally measured by Shigekawa

[6J.

Calculated wave profiles...

First approximation without boundary layer. Calculated wave profiles

- First approximation with boündary layer. Calculated wave profiles

- Second approximation without boundary layer. Calculated wave profiles

- Second approximation with boundary layer.

Fig. 23 is for F 0.257 ( V= 1.80 'z), showing the results: With the second approximation the slight

difference in the phases of the waves between (A) and U3) or.(C)

is ithproved fairly well. But, somewhat

pre-exaggerated wave amplitudes in (B) and (C) are further augmented noticeably with (D) and (E).

(12)

12

-(3) With regard to the viscous correction, the reduced stern waves in the observations can be approached to a certain extent by taking into account the rapidly

increasing displacement thickness of the boundary layer near the stern. With respect to the above result (2), we can not presume definitely, for the prezent, from what kind of sources this discrepancy is brought about. Lastly, it may be useful here to note that their further work gives the separation point in the neighbour.:

hood of .5= 0.94 L.. (or x- 0.93L from the bow), suggesting further necessary modifications, more or less, on the simple result given by (6.26).

7. Direct Applications of Michell's Theory

7A. Mathematical Ships' Lines

The idea of representing ships' lines by a certain system of mathematical formula has been tried repeatedly by Taylor [A 20) , Benson [A 21) and others. Weinbluin [W 261,

[w 4.4) has made a noticeable contribution in this respect, with special reference to the direct application of Michell's theory for any actual hull forms of ordinary type.

Watanabe [7) has troposed another ingenious method, also fitted for wave-making calculations. Improvements are made on two point5: o.) applicable to any special type of bows or sterns, including Maier forms, bulbous bows, or cruiser sterns, (2) the varying length of the parallel water-lines at each draught is taken into consideration, resulting in fairly good reproduction of bilge circles.

In his method, a given hull form is divided into three parts: entrance, run, and parallel middle body. The length of the parallel body is to be taken at its minimum, usually at the lowest water-line. The entrance and run are treated separately. For the present, we shift the origin O on the midship side end of entrance (or run), taking

X = 1 at the fore (or aft) perpendicular, and Oz vertically downwards. Thus each water-line at any draught is represented by

(7.1)

q(x,z(zr(Irz)#77xn'z)#

Wx)w(z)*fl(x,7'(z)

where 3(x),

F(x)

, 7?x) , W(x) are certain algebraic polynomials in the form

a,

x#a4x'.'08x',

with coefficients

a

, a,, ,

...

given as Table 2.

Mx)

is an additional ftinction of

)( for the correction of CG , without any disturbance on 6(z),

Çczj,

1(z) ,

given as

#3X

#0X'O8X 1-aX

with a = O (upper water-lines) or

a,

O (lower water-lines) according to the draught of

a given water-line. Further, ez. ,LYz) , t(z) , wc), m (z), are the given conditions

specifying the features of each water line

yúc,z)

, such as

¿(z) =y(o,z), Ç(z) j(t,z), (7.2) t(z)fr (1,Z) -'q (x,z)a'x, o /575

fl7'(z)-3Oan,(Z)

6OJ(z)-/(z)_--_W'Z)*1(z)

where

777(&-/;y(x,z)dx

given as the moment of the water-plane. The values of functions

3(i)

,RxL-'o ,

...

, ,ilcxj being as given in Table 3 for square stations X 0, 0.1,

0.2, ... , 1, we can easily obtain a perfectly fair water-line at any given draught

Zz

arid then finally determine the whole of the hull form satisfying the given five

conditions (7.2). Therefore, all the hull forms are simply specified by five draught-functions (z), (z) , 1(z)

W(Z) ,

,n(z) . Usually, these draught-functions are rather

complicated and it -is more suitable to show them by curves, though their máthematical expressions are not too difficult to obtain.

Table 2

given as the maximum half breadth

given as the half breadth at the perpendicular given as the entrance angle of the water-line

given as the water-plane area

Remark: Usually, ,4x) is recommendable for the upper water-lines, and fi(x) for the lower water-lines.

a0 aa 08 Ql0 3cx) 1.000 0 -30.000 56.000 .27.000 0 f(x) O O - 0.625 1.750 - 1.125 0 r(x) O 0 39.375 -78.750 39.375 0 W(x) O O - 9.375 22.750 -12.375 0 O 0 2.000 - 8.200 10.800 4.400 /Oc) 0 -0.400 2.000 - 2.800 1.200 0

(13)

(7.3a)

The above mentioned method has been applied to- several kinds of cargo ships by Watanabe [7] with sufficient accuracy. Fig. -24 is an example for a cargo ship

L B T C = 136 in x 17.70 in 8.00 in Q.7l2

whose draught-function

aré shownin

Fig. 25. Taylor's standard model [A 223 has

been aisOanalysed by Jinnaka [a] as shown

in Fig. 26. The draught-functions, in this case, are given by

Fore-body:

f (z)-O

w(z)= O6.37_O.O6óZ_O3fgZ#_ O.a52z3°

m (z.'

o

O.2a6-o.oz_o. f27z-aos7z'°

Aft-body:

c3(z)- f-o.a7sz- a725z

/71)-Ô (7.3b)

z' (z) = O.fa7- O/7z

-

w(z)-in(z)

'2a6o-o.,Qoz

-O.Oe7z...oO2.3Jo

Simple modified expressions by the Fourier series in three tfrms as

(7.4)

YÇx,z)'Ó()z)#Wxz),&M'A)mL)

where

.'(x), W), H)

are in a form,

4thrx164 eosfX#keO/fx

with coefficients k0 ,

4

, h given in Table 4, are studied byJinnaka [8].

It is interesting to note that (7.4) sometimes proves more favourable than (7.1), especially in case

0G-

'[j )0.27

FIg. 27 is from [SJ showing a comparison between the calculated wave resistance obtained by Jinnaka and with the measured residuary resistance derived from Taylor's charts [A 22].

§ 7B. Resisting Pressure Distribution along Ships'

Hulls

It is well known in thetheory

of wave resistancefor steady motion

thatamong

the two different kinds of wave systems (1)

the local disturbance in the immediate vicinity of ships, and (2)

the free wave patterns in the rear, only the latter is related to finite wave resistance

from the former. Taking advantage of this, and simplifying hie discussions by treating the infinite-draught cylindrical models, symmetrical fore and aft, Jinnaka ES)

has shown instructively how the résulting wave resistance come!, from the total sums of the lengthwise distributed local wave resistance, which is given as

Table 4 k0 - 2.930

Wyj

11.523 j M'(x) -19.046 -34.852 I 34.852 69.705 Ï-69705 - 13 Table 3 -in Michèll's

X

.B(x)

F(x)

77x)

W6)

fr1 (xl fr/e(x) 0

1000

0 0 0 0 0 1 997 - 001 - 0001 0039 0002 - 0038 .2 .956 - .014

-.0009

.051. .0026 -.0l3Ò .3 .796

-.060

- .Ó039 .264 .0108 -.0218 4 444 - 155 - 0096 711 0234 - 0235 5 - 106 - 279 - 0161 1 384 0316 - 0153 6 - 729 - 361 - 0182 2 090 0220 004' 7 -1 171 - 288 - 0090 2 459 - 0098 0240 .8. -1.138 .048 .0140 2.090 - .0434 .0306 .9 -. .545 .612 - .0357 . .933 - .0370 .0167 1.0 0 1.000 0 0

6

0 (7.5) with

d(x)

oPÓcf A

(7.6)

F())

ck,o;z)dz and (7.7)

6p (x,Ò, z)

[,o (x, o, z).

(-x,o,z)J where in (7.7), ,.o(x,oz) , is the fluid pressure peculiarly derived from

(14)

with coefficients Q ,

Table 5.

Table 5

- l

-theory or belonging to the free wave patterns, and

extended sense of the words, is called "resisting pressure" Guilloton's "conventional pressure" EA 11

With varied forms of water-lines, he [81 discusses distribution of the "resisting pressure"

PCx)

is related to then clarifies how they are influenced alternately into the

ad finally their important rôle in the total resistance

R

The water-line forms treated are as follows:

(7.8)

y(x)

(f.'x2#a4x4)

together with water plane area coefficient

Ç. ,

are given as

Fig. 2.8 is for the total wave resistance. Fig.

29

shows the distributions of "resisting pressure"

Pc'Q

with model A for a widely varying

range of Froude numbers. Fig.

30

is for a comparison

of P(xJ together with the local wave resistance

IR(x)

showing how they are actually influenced by the

given water-line forms, in case of P

0.215

(hump-speed).

Finally, he alsó shows that the Yourkevitch form [A 23] for least resistance is justified of a safe theoretical basis from the above mentioned

poiTtt of view.

The optimum position of the point of inflexion C

empirically recommended by Yourkevitch [A 233 , as

(7.9)

X f3.o4.3P-o.5°°,

(ç(o.4g)

where

X

denotes the distance from PP to the infledon point

C

measured with L

2.

. In Fig. 29 these positions of point C corresponding to F=

0.265, 0.290, 0.316

are also given, showing that they approximately come on the same position of X, where the curves of

P(x)

almost cross the base line

Px)

O with their initial signs positive

(hump-speed) or negative (hollow-speed) at the entrance.

We should like to take thià opportunity of referring the extremely interesting experiment of Okabe and Taneda [10] . It is well known that there exists a complete analogy

between the ideal fluid pressure p and the electro-static potential ? , and also that, with infinite-draught models, the transverse waveis more dominating than the diverging wave. Takingadvantage of this, they have reproduced in an electrolytic basin

(900" 860

mm), a roughl7 corresponding two-dimensional electro-potential field. A metallic plate, as one of the two poles, is corrugated into the corresponding wave profile. After probing the equipotentials, they obtained the pressure distribution around the hull surface, and thus, after integrating, the wave-making resistance. A comparison is made with

Wigley's Model

1254 LW 45] ,

whose results are shown in Fig.

31.

Some useful simplifications are also suggested by Jinnaka [il] with respect to the numerical evaluation of Michell's integrals, where a finite range of integral.is

replaced for the infinite upper limit. Numerical comparisons with the exact integrals are given for both infinite- and finite-draught models.

§8.

Effects of the Restriction of Water in Depth, and Width, and their Combined Effect

(Calculations of Wave Resistance for Shallow Water) The problem how the wave-making

resistance ofa ship is affected with the diminishing depth of water is of practical

importance on official speed trials. This has been attacked repeatedly, as follows: Numerical-work for a submerged sphere: Have].ock [W

77]

has revised his early

expression (W 76] for a submerged sphere in shallow water but the rjumerical work remains.

Kinoshita

(39]

has done this and corrected Havelock's early result [W

76),

which is also reproduced in Baker's text [A

24].

A submerged point sink substituted for a marine propeller: Dickmann's approximate

method

[w 843

,

[w 85)

of estimating the efficiency-drop of a marine propeller simply from

the torque augmentation due to wavemaking, without any deliberate consideration to more important factors as mntoned in § 4, has been applied to the case of the finite depth

of water by Kinoshita L40i.

Numerical work for a submerged prolate spheroid, and displacement ships: General expression for shallow water resistance of any arbitrary source-sink or doublet distri-bution has been obtained by Kinoshita and the present author [41), which is consistent with the later result, the equation

(15.18),

of Lunde LA

2].

The numerical work was done

with

A prolate spheroid whose ratio

5,

where 2a ,

26

denote the major and the minor axes respectively.

Two displacement ships, given as

(8.1)

(x,z)6(ffr)oS (fx)

with t

1/16

or

VL

1/32

Model .A (shallow draught)

¿ =

1/8

or T/Z 1/16 Model B (deep draught)

Model.5 is similar to Wigley's model No.

755 [w 183.

Fig. 32 shows the total resistance

curves for Model A with length

L=

1.5

ni. Interpolation formula for the wave resistance

in shallowwater corresponding to Hogner's expressions [W

31], [W 34.)

for deep water is

also proposed there.

z), together with

F(x)

in an somewhat in a like manner to in detail, at first, how the the water-line form q(x), and local wave resistance '(x),

on the load water-line is

Model

a

C,,1

A -1.0 0

.667

B

-1.2

.2

.640

(15)

, 15

-.Effect ofsea bottom properties: Hoshino [45)discusses approximately the effect of

the sea bottom properties on the wave resistance of a submerged sphere by regarding the sea bottom, which is üsùally covered with soft seawêeds Or dirt-beds, as the subsurface boundary

of a certain perfect fluid of large density

f

,whose.ratlp .P7y is varied as f/f=l'-5.

(Side-Wáll Effects in Deep Water) ApplyingSretensky'sexpr.ession Lw 79]

for wve

resistance in a deep water canal, Bessho and the present author [343 have carried out numerical work and tank experiments with Model S-102 mentioned above ( 6). Fig. 33 is the result for b./L

= f ,

where

4

is the width of the water, showing a more exaggerated wall-effect on the part of the experiment.

(Twin-Hull Ships) As a kindred application of the theory, the wave-making characteristics of twin-hüll ships were also studied by Kinöshita-Okada [30] , and Yokowo-Tasaki [33]

, [351

Especially the latter have carried out valuable experiments with a twin-hull model of S-101, together with a single model. S-201.

Fig. 34 is from [35) , where ¿4 (the transverse distance between the centre-lines)

is varied, as /L=O.2, 0.3, 0.4, 0.5, aO .

(Restricted Water Effect) For shallow water, the length of the wavès caused by a moving body is remarkably increased.

Subsequently, thé side-wall effect gets more and more important as the depth of the water diminishes. With regard to this interesting combined effect of finite depth and limited width of a water way, here called "restricted water effect", the present author [42) has obtained the same expression as Sretensky [w 80]. Numerical workwas carried out with a submerged sphere and also with the above mentioned Model A

(shallow-draught). Fig. 35 is for the latter. In this connection,'the interesting discontinuitiès on the resistance curves for shallow water and for restricted water are pointed out, as

follows:

-For shallow w.ter, the inclination, or the first derivative

dR/d1

, iè suddenly

decrèased at the critical velocity (= V'. = 1), by

[J=-z

///o1I.a1IdJddJ

For restricted water, the wave resistance itself is suddenly decreased at the critical velocity, by

(8.3) R» =

1ef

ffJaii.

o,

JJdIa/

2

where o'rJ,o,?) denotes the distribution function of sources and sinks. Further, there exists the following relation

(8.2)

(é.6)

(8.7)

For.modelsA and3 ,

we have

i-(Rh),

(A.1

(c,,» ¿., ()?'4-)

(8.4) ..

)-and, for cases of a relátively large

4/L

approximately

(85)

The conception of HaveloCk's [W 51 elementary waves is shown to be eminently effective for a complete understanding of where these discontinuities come from. For instance, ¿ in (8.3) is easily shon as belonging to the first elementary wave

0

0,

which vanishes' in nature for) 1.

The ratio can be easily evaluated,and from (8.3), (8.4) this

affords a practical criterion as to the needed width 4 of model basins for shallow water experiments, as follows: Putting / (8 ) R», ì._ , c'R )»-G-d t we have approximately,

where C denotes the blodk coefficient, an C the wave resistance coefficient. Because the theory gives rather an exaggerated effect at the critical velocity, the

selection ofa sultéble allowance for is somewhat ambiguous. If wé take tentatively = 0.30, (8.7), glvesthe needed width of water ¡ shown in Fig. 36 for the two models

'A and .B . If 0.10, for example, proves more favourable, three times /L in

Fig. 36 should be taken.

-(Components of Wave Resistance) ' The present author [43] has shown the relative importance of the, two components of wave resistance:

(i) Wave resistance due to transverse waves.

(2) Wave resistance dud 'to diverging waves, for the three cases of (a) deep water, (b) shallow water, (c) restricted water.

(16)

16

-Fig. 37 is the result for Model A The calculation for (a) is analogous to Lunde's A 2) result (Fig. 7 of his paper).

9. Miscellaneous Applications of Havelock's Theory

49A. Trimming Moment of a Submerged Prolate Spheroid and a Displecement Ship

It is a well known fact, In the theory of wave-making, that the local disturbance around a ship has nothing to do with her wave resistance, even 1f the form of the hull is unsymmetric fore and aft. But when we treat the bodily sinkage or the trimming moment of a ship on a run, this system of waves, more complicated than the free-waves, can be no lonCer neglected in general cases, except for the first approximation of the trimming moment of a longitudinally symmetric body.

Kinoshita and Abe t30] , having regard to the above mentioned fact, carried out a numerical work on the evaluation of the trimming moment of a submerged prolate spheroid, and of a fore- and aft-symmetrical scout.

Calculations were made not only for deep water, but also for shallow water, and the results were compared with some experimental work, previously published. The cases

calculated were as follows: Prolate spheroid:

Slenderness of the spheroid:

'/

L/

/r

i

Depth of immersion: C/ or e/aa = (% -O.f25

Depth of water: h/1 and , or

h/

0.25 and

Fig. 3 is a comparison with the deep water experiment made by Tokugawa and Kito [A 25] with a model L 2a 1.8 m ,

T 26

0.15 m ,

f

0.225 m , where the

calculated curve for shallow water is also shown. These results with a submerged prolate spheroid, together with the calculation of the wave resistance by Kinoshita and the

present author [4].) may be regarded as a preliminary step to the latest and more elaborate work of Havelock [A 26) , and Wigley [A 27] . It is Interesting to note that in this case

of a submerged spheroid, the sign of the trimming moment is negative (trim by bow) for a wide range of Froude numbers.

Symmetrical scout:

For a rough comparison with the shallow water experiment, made by Taylor EA 28], a symmetrical hull form was selected, as follows:

(9.1)

y

(x,Z)

6 (i-

5X2#

05X4)(f-IJ)

with

The results are given In the case 4/L 0.15 in Fig. 39, showing a rather

exaggerated effect of the shallow water on the part of the theory, which might be expected,

considering the factthat the experiment was made with a false bottom instead of with t}ie

actual bottom of the model basin. In Fig. 39, for an example of a shallow water experiment without false bottom, the result of a destroyer (1.500 mx 0.1351 mx 0.468 mxo.5l8

= 0.164) obtained by Izubuchi and Nagasawa LA 29] Is also referred to. 9B. Lateral Resistance of a Displacement Ship In Motion

with a Small Angle of Incidence

When a ship is moving with a small angle of incidence, she is always subject to a certain amount of lateral drifting force. For a lower Froude number, this drifting force can be attributed exclusively to the bound vortices or the circulation around the ship, but for a higher Froude number, some part of it comes from the unsymmetrical wave-patterns.

Nishiyania [36] has investigated this problem by substituting an appropriate

distribution of doublets, for the ship, where the direction cosines ( / ,n ) of the axes

of the doublet are given as,

(9 2) J._., A(.

n

A9m

where ( , ,, ) are the direction cosines of the relative velocity of the uniform flow i',

and A

, are some numerical coefficients depending on the form of the ship, or

more generañy, a moving body. For instance, in the case .of a prolate spheroid, they are

given ae

(9.3)

A'=

1eZ

a

(e)

41

¿e(/-f)

(-a)

=

f

For

%

5, ''/'A is about 1.8, and this tends asymptotically to the

limiting value 2 with the increasing ratio a/é .

-The general expression for the longitudinal resistance and the lateral drifting force , both due to wave-making, are written as

(9.4)

R- ¡R5,

87tf

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