On the increase in the resistance of a ship in regular head sea

1

On, the Inoreae ii th Roajetance

in Re1

Hoad Sea

Bibliotfeek van de

Onderaf doling derScheepsbouwkunde T.chnisch.

HogschooI, Deift

p

by

Eitoahi..p,jj

DrY Eng.

Z'akei Pakajiaaj

On the Increase

in the Rociatanoe of a. Ship in Rou1ar flead Sca

* *

flhtoshi FUJII and Takeshi TAYJ.HASflI

Abstraot

in the previouc

study, the authors

presented a method for ooniputor calcu-.

lotion of the longitudinalmotion of asbip in wavésand compared the recultc

with experimental 4ata.

In the present

report, aa

the

second ntep, an attempt

ha.n been made to ootabliuh an' approximate method to oáloula.te tho resintanco increaoo of a chip in wavon.

-The incthod of calculation was doveloped'Ori the basis of the exact solution. according to a linearized theory which wan devoloped byProf. ]1aruo.

Numerical

calculation woo performed by a computer and the rosulto woro compared with .experimontal data. The model

tents wexe

carried out on Wigley's mathematical V

'hip form, a cargo'ohip and a

tanker.

The results of the theoretical

calculatiOn, show as a

whole a tondoncy.

Olinilar to

the

expbrimeita1, data, though areoment betwccn theory and

exporimont

varies wth Froudo number and wave 'length/ chip length ratio. In the

practical

rano, agrOomeflt i

eatjafactbryaMthtoá1ethOd will. proide

agoapprox.i-ation'fo±px'otioalj'urposec. V

le Introduction :

V

With thâ progress of the

ship hyd.rodyna.mics in

recent yoaro,.tho wave

making resistance of . chip in Ctill water han been reduced. extremely and the

projulcive performance hac been improved.. Then, 'the

seakeeping qualities of' a

chip havo beoomo to be taken up gradualy as a succeeding problem. The

Vrcerohes, on

the seakeeping

qualities, ship motions, rositance increace and

others in waves

have'been

developed

omarkàbly,utt'ov]uatethe'Derfornanco

cf a chip in wives-is

very difficult and complicated., :1he 'author presented

a

method i'or.computer

calcült1onof the longitudiralrnotionof a ship inwavoc

and now thimethod in put into praotioal'ucc atthe initial etage'of a chip

denign. An theccoond. stop, an attempt ha; been mad.e 'Oota.blish an

approxi-mate method

to

calculate 'tho

reoietanôe

increase of a. chip in aroc.

Speed

drop al' a ship in roigh sea depend mainly V0fl the 'resistance increaco,

though iie

omó

mae?formed.' intenttonallr

avoiding he severe chipmotion.

a..

* _{xporimonta1 Tank, Nagasaki Teohnioal Institu1e,} '. V :

V

knowledge on the performance of a ship in'voyago is very nocescary for

coti-iu.ting the sea margin and designing the hull. form that has better caekooping

LUOlity.

Therefore, many sort of 'theoretical invootigationc for'the resictance norca,;e have boon oond.uotcd. by various researchers, the wave reflecti.on theory

1reitner, ilavolock ), the drifting force theory ( ilavelock ),. the nonuniform' rave resistance theory ( llanaoka ) and'others were reported. ' ow&vr,' no theory

ould. give a reasonable dosbriptioñ', 'because each: -theory only explains the one

ido.of phonomena 'om a different point of view.: "

Recently, Prof. Maruo presented. the coneio'tont theory that explains the all. om)onents of. the reei8tance increase of a ship in waves, applying tho energy 'elation in this problem, asoumed the fluid. around. a ship is the perfect fluid. that is, when the ship penetrates into waves, she is osOillate. by waves an&

L1Sturbo the free nurfaoo,' so tie afresh waves are gone'ated 'by the disturbing ffeat of the ship and. moreover ocean waves are also disturbed by the existence f the ship. Accordingly, the energy 'given to the fluid appears a the energy fflux on the cóitr1 piano oonsiclerod at a great distance from the ship.

'hei'foro, the amount of work done 'bJ the ship to'

the

curroun&in'fluia is equal

0

that of work whIch znakoc the hip go ahead agaist"the resitance, so the

esistance incroaccan be calculated as'aninorcaoe oftheravezeIotc.nce.

misreport describe the calculation method whioqas doe1àpod ont1e

asis of the exact polution according to Maruo's theory arid th'o reults of corn-.

arison with th numoioal oaloultion performed, by a computer arid the eiperi-ontal data.

Theoretical Calculation of ItesiBtance of a Ship in. Regular Waves

The donsiotont description of the reistcnce increase of a ship in waves

s'pr000ed

by Prof. Maruo as the increaso' of tha' wave xsictance and an exact

olutionaccording

'to the lirearized .theorywao givn.

In this report, an

ppr'oximate

calculatior fonTula

that assumes a chip' aV a.cle'Mor body arid the

echnique of. nuinerical ca ltion i&'o shown. The shi motions are treted a

he knowii mat tors because the approximate oaloulation of tthern wa already

atabliebod

bytho. trip nàthöd Therefô',,:theprodicti6n' of the ropid'tancc

ncreas'of ash'ip In regular'boad sea comeà.to be. possibloontho:whOle by

the

heoretical calculat3on. I

.:lAppraximateFormulaof Roaotance Increase in Regular Waves

Amount, of 'the. rosistance inoreaso of a ship in re ular'.hod sea or

follow-ng sea, R*w , io given

by Prof. raruo as fouiowi.

4irf

cff)

IHtC)12d (i)

2--U velocity of ship

w

oncountir circular frequency

wave length.: . ... -.-.,

the function deternined by

the

_{singularity distribution that} represents the ohip's hull _:

± sign in eq. (i) _{1.upper sign indicates the head sea and lower the} following -,

Uavelook asoumcd two vertical Ip1nes, _{on of which .Infront of a ship} moving on a äalm sea. With uniform, velooit _{and the other far behind it, and.} derived. tho wave making &'oeiB'tance from theenrgy _{rolatioiin the fluid}

..containod between thetis

planes

_{and calculated the.valuc:of.onc. ccnzirlering the}

rizlaritydicitribtrtion instead of the.ship'ahull, _{Equation (1) that}

rcrrceont the xesistanco incre'ace juct coreciponds _{to Bavelook's formula that}

roprecents the wave making rcaitance.ri a calm sea. Inotead Of the vertical planeA, a cylindrical surface with vo'rtical axie._{and. large radius 'currour4in,g}

tho hipio taken a.c a control

_prarie

to derive. he zcsistance increase. _The

energy efflux

which

_{'is carried' away b' the fluid going across the surface is}

ovaluatèd., then the increase of

wave resistance is calculated

_{considering the} singularity distribution instead. of the s1iip's hull. _{As the.. motion of the shij,}

and that of the fluid change

periodically,

_he

_{energy relatin .ohc.ngo frori}

-time totiine. _{Aacoxdingly,±n such acaso, the energy rclation mut be taken} the time average of it.

Sinoe the fluid _{otion around."tho ship ic considered. as the linearized} uuperimporsin with the inoident wave motion and the

wav motion that geiiated

'by the disturbing effect of the chip, tho-velocity potential is exproceed as

follbwr..

tiherp

4,,, t velocity potential of incident wave c1istur1anoe velocity potential

ic represented by the distributed _{ingu1arities and. Eatiafies the linearized} fxeo sufa.ce cond.ition,.'and4 must-be determined, so as to satisfy the surface condition on the

1thip'8

hull. Accordingly, what sort of approximation is adopted to present the diaturbing effect by the

key point in tis-pro'lem.' Aacumin

_{the1o2endoDbody& the}

source diz3tributlon

o() at each section i represewtod by the Concentrated, one at the vertical position _{(x), and. mOreover this}

_{position is a constant levol}

Diong the ship length, (x) becomes

=

-c'r0

3-where

S2

whore

_h

k,-,eey

,zc

-

hqS(nIeX

$

arnplñtudo

of -. , q '-. '-' . r - -

-.

fi--

t incident

wave

and the- motior 6f the

ohip by

-

+3iCOS&Jet

0 =

Os

,9(M Wt

COS tJ where

Bciii

_cosEü

Ej

armplitude and phase

O4p

amp1itud.

and phase Using. the lineariZod approximation for the

We

make

ihip

oatisfyiig

the boundary ond.itjoi.

cx)}e

whore -

r ..- .

1) _-

tRfY

1

Oc(X)

₌

_CX)äys(x)

u!-{8(x)yCxfl

In tho eq (3), BCE) ic the bradthof coo'tion at load wterlino and ic the rerticai relative dip1acoment to the nuburfa000fwave at

gf each section.

i

rocd

rr follows.

vs Su tJet

3y CoS Wet

whore

-

_S-ICCZG>9

e'Lscx).

yc =

_{+ (x-z)e}

e"iicC)

-* distance from

the

mjdshj-

to centOr of gravity

to wave of heaving at center to vavo of itchin

and

oirnp]ifying

the velooity potential

oici1iatingin thregular waveG, -the function HO")iB obtained

L

*

1cnth of ship

_:

at free surface

e

cSLt

= QA

3

each tern to the dimenijionlsi.Eom,'

- -L

-.

r,Zq.

f

2ki

-- =

B01-breaàthof ship

of

gravity

whoro vertical prinrnatio coefficient draft of

nhip

The equation of

uburface of

wav.o at . is'expresuod by

and dAfPte tha

_

r

A

) )

A

r)

JL

tcOS(i)

COSC" ;)J

dx

'Jc I

K5

sjCje)X 1

<)

St. [c0±i

A

fsiC')

.

cOsc)Jdx

Lic)

r sc')

L5'

COSCX)

ubotituting o. (3) to (2) and caloulatin the fnctton

_*

I-I(r) , concidering

the"cond.ition &x) 0

t both end. of th hull,

HcI

beoome

A A

I

A

)Q.3E r4+We)

()

-47t

whore

(i;i. x:)

,,

02

A

+ Z

_{

_{(I, 1, + IcT)}

_{C L -}

_{(ieJ - L ,Te ) 31}

_{(Ej - Lv )J}

I 2e

C {1.(Lcc

L,1) -. I (LCI±L,C)J Sine,

.r{ic(L,,;Lee) 13(L,,tL.cs)}co5J

± 2 &' C

31 ( Lc. L) -

C Lc. Lsc) } .si.v

*{J4(L1j.) "J.(L56±Lc1)JtoS

The coefficient of thc Desiotarice increaco

ic ciofined ac it ic

u'o;or-tional to thc oquare of wave amplitude by the linoaxizedtheorf.

yuw

RAW

I

(5)

Uoin

eq.,(1) and (4), KAW to reDe6entO

a

follows,

=

+ 0,. 8A

+ 023

C.à,JM

+ OJ BA

(6)

whore

-cJ,rt(e-e9)

0, CoSCEa.6)

Di O, .siv E

-

D,

Stfl E..

+

_a

-

_IM(,

I,t Iid

'% A

-

e2f MC.)(&

+

where

ohiir motions..

1ntegal 'óz'der, far example

-P33

e2

e

_M

(ka.

i<') d,.

eff'1C(2)cosA(

S(2 '

-

f .stt (i

2) 1

-C()

J

=

CoS((2)

jd

'I

2.2 )Iothod

of

Numerical

domputation"

'i'he 000ffioicnt'of, conponenteof the resistance _{increae, 0j, is obtained}

b cring out the' intoratiori

with reopect to

,ut

_{this computation requiroc}

muoh trouble if it is just as the present

form, so we transform

to the

following

forms

fox' the convenience of the numerical oomputation.

_Changing

_the

Ii

and if the values ,of the functions S() and' c.(2, are previously _{btained. to the}

dimeneionl'soe parameters,

1oigitudina1

_{coordinate '.I.}

, wave lergth/shi

length

ratio4,.. .,' Frouda number

and. .vertioal position of singularity distribution

r_,,

the double thtegrale with

respect

_{towiabeabletopeorm'eacilyby}

making use of these values and. it is convenient 'to use theso values as

the

program constants of oomputer.

From eq.

:.'f

p

(8)

U M

.

-

)

-

MCr.JIrd.

Oa

-

2e5 M(rK f LK.)d

-

±

2fM(JKcfcKc)du

-

zi"J.

Al C3 X

+Jc

<)L

'i

(ii+(

the ourfjxoc of the coefficient D%j roprosont

that the cuffix 1

_{howo th}

quantity concornod to, the heairi, the

suffix 2. t

hepidhinan'dstd'fix 3

'to the wave

arid. eq.

(6)

_'shove

that the ooffioen:t

bf'th6ret1o'ance

_nàreaoc

7

b'

'

-(I-2j/,_4)

inoo the denominator of4 becomes _{an imaginary numberfor the interval}

corresponding

to _{in the integral in eq.(8),}

this Interval must except from the domain of integration.

_Then

C.

L

when _{*< .ca}

-

J+ 5

--

_a. * when

O(Q<

.,,

As the iritogra].e with respect to _{have the infinite intervals}

and

the

losed intrval in which the

interanddIvoroe;

ineuoh an Improper interal

he oonvergenoe must

be

_{checked and the numejc}

_computation

munt o carried ut carefully.

_{The intora1 in}

_{eq. (8) oonveroo except}

1

- , an hreafter

stake ca#*

One example of the _{computed values of S() arid Cci) for the parameters}

/L

p

and , .is shown in Pig. l _{SCi) and. Ct) have}

he osoillatory form for he.parameters

_/L

_{, Fr and}

, and the oomplioated variation especially for nd. it seems impossible for _{S) and .C.) to fit the simple}

curves. It is

eoeseary to divide the ahip's. hull to _many

parte,

bgcause theyrapidly oscillate

ox the parameter _{as F. becomes smaller values.}

The oomputiné time in required o ]onger as th _{number of divisions inoreaBe.}

Investigating tho condition of

nvergenàe of _{KAW for tho number 'of}

Ivio'iono and considoring the accuracy' of

mputation, we divided the ship's hull equally _{40 èectiono to compute the}

-motion S(2) and

C()

Input;data are the partiCulars of hull

_{form Lr}

B. ,' T. , C, , x and

'ead.th of seotione B(- _{at load water line, the wave conditions}

ip speed Fv and the

corresponding

_{ship'Inotio304,}

and.

_Ee

It i

'oessary to compute the ship motions prev-iou8ly (

because IBM 7040 ic lack of .paoity to compute the ship motions _{and. the resistance increase}

to&ether ).

ough iiie;impossible to fit the S)

and. CC2) by the simple _{curves, :the} eponce unction _{are not necessary to be computed for the arbitrary values}

each parameter;

>1L'

and. F

, so, in order to siinljfy the computation, o

inputs _{with the}

function 5(i)

_{and. .c}

_{which wore computed for}

pred.etermnjned'

lues of parameters. _{KAW has not hump and hollow with}

respect to Fe.. as the

effioieitt of the wave making resistance in still _{water and changes imply. for}

4.

.

_{Accordingly, itis enough'to,eotimate the.kAw"by thjamethod.}

Comparison of Caloulatod..ana Measured. Resulto ' '

a ship in

rogzlar head. sea,' uning a

digital computer.

Though in the cac of

calculation of the longitudina)

.ohip motion by the strip

otho& the fairly

good a.grnementbetweefl

the'caloulatodaZ4 the experimental.

valuon Mr boon

shown, in the C5Q

of' calculation of the rciotanCe

irrorcane

the good.

agree-ment seems to be ions expected, becaune the oa1culatio

ir based on

nuch a

rough treatment s.c acummjng.aChiP as aslender bod.y replaced b

a line

oingularity.

Accordingly, it ic

coecary to.exarnine the appricability

of

this

coraputingmethpd,

Comparing with te experimental 'data.

In

theoaloula-tiorL

ottho,'resitanOO increano,'

phaSe: difference, of, the zthip motions' to waves

arc as 'important an

the arnplitud.c'Of mot1onc

no it requires the

complete

data of them.

Mi'tcubihi 'experimental .tank oonduotod,the te3tc in waver

urually

in the ie1f-propelled.

condition.

However,',it ohou]4 be better to corpare

the

oomputaiOn with the

experimezit not ln.the thi.ict

.nCrCs.sO but in

theenitaflCO

ioreaoo. ?ór this

urpose,

the resistance tecitn'In'reula1' waves

wore condotod.

3l Model 'Experiments

:

' :

'

An. the typioal'hull. forms,

Wigley,'s mathematical 'ship form

(the parabolic

vator line

nd frame line), a cargo

hip and a.. tnaksr were

oloctod.and the

resistance teeti.wCPS

oonducted..

'The prinoij.al.ParticularG of

the

tcted

models are shown in Table

1.

.'. -

'-"The wavécoMitions were.

0.75,

1.0 , 1.25 ,

1.5 for eaöh model

aid eupplementalY

05 for the tanker model. ,n the linarized

theory

'the

ipmotiôns'arGpr0P0rtb0l to te'wavehei.ght

and: the reintanoe Increaco

is"proportiOnal o

the cquaro of that.

To. oertifythin

rlations,

two kind

'of. wave

height

were chocon. 'The

resintance inceao wac

meanurod

by .'tho

gravity dynamométer, and the' ship iotIonn and wave height

were mcaurod by

the .cuctomary: technique

in

our tank

, '

.1

The,roei9tanC

Inoreane jn'wt4VcAw is

obâthcd'by nu.b&traoting the

renistanoo in

still

ater Ro from' the total' .renizrtanOe in aves Rt.

An

exaipIe of moai;urcd total rooistano6 is shown in Pig.'2.'..

Asit io

uppo;ed

theoretically

tMt some

tmunual phenomena rhall, occur

at the critical point,

tenth wore

oonducted.

extenä.Ing to

the range, 'whloh'ig' F

0.1 inthe tected.

iave length and. out of

unual1 praotiOal øpad.

3.2

omparisOfl between the

computed

and the

14easured valuoc

Experimental

resulto ,of'

the chip

motions

d.theroiirtafl0e inoroae in

re'ular' head; wavcr were

.taken d.'irnOnuiOfllS55 form

and

compared with the ooutád

values.

CopaatiVe picturob 'r shown in P.3

5 for eaoh, model.

Conoex'nin&with tho'ship motions, itcanbe' idthat'he agrço:nc:t between

the computO

and

tio meacurod Valye are goncrally.

good for praotioal

9-purpofloo, but unusual phenomenon which,occu.rs at the lower cpeod range of F

0.15 cloe not appear in the 'iitripwise paloulationof the

ship

motions, it is threo dim'nt;ional phenomonon.

Concerning with the recistanco, the'mooured va1uor seem no ccattcred in

total'.resistanco. 'as is

shown

'in Fig 2, but if the incromeiare taken out

and'non-dimcnsiOflalized

b;tho cuaro

of"thoavo height, some amount of scattering come to

stand

out. n spite of that, the computed. valuca can be

oaicLt000thcidoViththo moarured valued as a who1e. Tho reistancc inoreaso coeffioientsKAwof

each

model for different Fnas a function

ofL,

are

chown in"Fig.' 6. For

Wiglcy'3

mathematical mode1, the good agreement is most expected because. the

form of this

moc3,el approximately eaticfies the tbeorotical assumption replacing a ship ao'a slendór'body.

Ahis shown

n Fig. 6(a),

the

agroemont in fairlygood from

the

lower to the higher speed range, 4xcopt' the

:nbourhood of

1.0.

Thej'oason:of this discrepancy is cooid.crcd. that the difference of pitch synobronism' 1n the computed valued and the

experi-mentalvalueo affactfi on thô reoictanóe ncreac ooeffioiorxt. KAW. When the

resistance increase wat computeduing the'meaL3uredship motions instead of 'the 'compute& ship motiths, a good 'tendency to approach the experimental results has been shown. flowever here ti11 exiats com d.isrepancy, and it

oónsidored .that'thieapproximate method itself )as come questions. For the-cargo model and. the tanker model it can be said that 'the resu.to of the oalou-lations on

the

hole

agree with experimental data, 'but in the small A/L. range

of .the tanker model, rnoa,sured K ohows pretty:large value in spite of, the

small amp1tude of the chip motion. In fuller sipu, the wave reflection has a

considerable effect .on' the resistance1inoroaae in waves, .but in this 'approi mate', tretment the tranAverne boundar' condition on the ohip ouxface is not' .,'.

adjusted nu.ffioiently1 and' this seems to lead. ouch a difference. Generally speaking,' one may un&erntand that _AW oO.13'be, estimated 'by th theoretical

oalculations'inthe usual range of

Fn'and..in.thoran.Of 4t

in

which the

ratio of''the reaistance' increase is larger, even' if there exists the tendency "4hat the"cornated.valUe is oyorestimation at the lowor.irarge of'F,<O.j5 and

.undorostimatiofl'at he"upper range of ' '

For the Series 60 models,'0.,J.Sibu1 conducted tho minute exporithental researoh ontho resiotanoe'inorease in

rogular.wavo1,7i7'ohOW

the

oompari-on between our oomputatioompari-on and. his experimental data. As there is no

des-orlption.on the phase angle of 'ship motions,'InSibul'oPapor, another

expori-mental

data of" chip motions whiohbad been published byGerritma and others 'wero usedin' thin oompi ationo,'The'resulto.of.the oomputatiOio.shoW

là'

threo models 0.6, 0,7., 0.8.. However, thcre is the namc tendency, as

:was ehoi.rn previously in the 'cargo model or tho"tankermocicl, that the computed value is undorotimation in the higher range' of Froude number.

'3.3

The 1henomena in tho Neighbourhood of the Point Q1/4

The point

4

gives the critical' opoed. at which the disturbance

progresset

ahead of ship, and at this point the integral formally diverges 'in the theoretical calcul.tion of the recistanco increase. And, it has 'bcen

o'oneid.orcd. that ome unusual phenomenon will occur in the fluid motion, beoaue the damping of motion becomes 'infinity at 'this point. We can not dis cribs

whether it is due to the mporfeotion of this approximate mothod,' if the: boundary.value prob3.em is not solved exactly and 'the nature of tho function

Hr('n)

i's not oloar. Deoa,use there may exists the case 'that the integral b000inea finite according to the form of the1 function

In' rogular head waves, ship speeds aro given in following table wh'on

0.75 1.0 _{].25 1.5} 2.0

F

.: _{0.072 0.083 ' 0.092 0.101}

0.117

The unusual phenomena appear in' experiments as followo. Namely, the measured

values of ship motion seem to be much ncatterod at the first glance in the

lower range of

F

< 0.15, 'but carefully obsorved there exists some tend.enây'. that the amplitud.e of

pitching

böcomoø larger and.. that 'of heaving becomes

smaller in the. neighbourhood 'Of the oritical,potht SZj/4,"and the out of

that point the amplitude of pitching. becomes smaller. This tendency is remark-able in the fuller ships. At the' ma1lor Froua,e number range, 'the effect of pitching is predominant on the resistance increase.' Indeed, in model oxporimont, surging motion becomes ccvoro.and" unstable in the neighb9urhood of this point -andit was yery difficult. to measure the resistance Gf the model'. Thin

un-usual phenomena experimentally occur at'O.Ol higher point'in Froud.e number

than the above table. Thin phenomenon has somo -relations with-a ship in voyage., -'and oomeone oupposos thiephenomonon as one of 't}e reaooz for a ship to start

'pitching

in an apparently quiet ocean. Thio' phenomen should bernore

.investi-gated theoretically and experimentally',.' oono6rning to; the', problems of the

,resistanoe inore,se and. the slamming. ., . '.

'4. ' Relations of Hull Forms, Ship Motions and. Res.stanoe Increase

Relation between the hull'form paramtors and. the reriistanoè inoroase

is very cprnplicat'od..' The, resistance increase ooeffioient- cortoiets of the

terms which oorrenpond.to eoh.oOmponent of ship motions, as shown ineq. (6),

'figure t3h0wn that in lower npeed,rangO

the

offeot of pitchim i

prec1oinant,

but in higher epeed range the effect of hoaving'bOoomee. predominant. However,

from a. rioint of view of the dinturbingoffect, heaving 1nd pitching motion3

hove the came contribution to the roniotanoc .iriorease in

WnV and which

bocomec prcd.ominant differs accoraing to the hull form, 'tho longitudinal radiuc of

gyration, wave length and chip speed. .'

From the results of èomputationc.ath experimont oi many ship models

including three mod.el tetod hero, the fol.lowing remarks can be roughly said

on

the

relation botwoen the hull forrnn and the rociotance inorearo.

(i) In mathematical ship model symmotrioal fore and. aft, the effect of pitoh-"

ing is predominant, and the direct e.ffoct of wave' reflection is li'ttle.

(2) In models 'of' aotual typeD of 'ship cu3ymmetrical foro and. aft,, the effect

of pitching is. predominant at lower speed range whOn wave length iecthorte,

and on the contrary tho,offoot of heaving ii pro&ominant at higher speed.

range 'when wave length is longer. Generally, tho pioture of

the change of KAW

resembles to the change of the ampli±udc of

heaving.

(3).

In full chips, the wave refleot.on

has much,

influence on the resistance'

increase :whon wave longth is short and' the computed, yallies aro less than the experimental values.

-'(4) As the resistance increase

àoeffioient includes the terns that are pr

portional to the ouare of the amplitude of motiont3,. the resiatance

inrcas

,i much affected by

tho,

slight difforenoesOf ship motion. Genorally speaking,

smaller hip motion loads t:o leer' resistanos inorcase.

The longitudinal radiOus .of'gyz'atiOn:

affeots on the ship

motions., Sra1ler

gyration leads to smaller ship motions at the

synobronus

pont and results

n

tho lose rocisto.nco'inC1'Oae.

Iri.soriee-' modelo, it is found 'that iprnotion":beOOXflos smaller an Cb

,b000inoc

larger, but the value of

K4weronOarlY'O.Uaiat the came Froude

nuinbQr 'so far as the experimental'data'Ofl the Sorieo6O, hence the smallor C's. the'

ighor the resistance increase per unit displacement.

The

caioe of the resistance increai;o in waoo icdue,:tO tho disturbance

of the fluid surface including the inctd.enco waves

'by'ehiprnotioflc, and. it

nubrte.ntiallY

differ from the'roistaflCe in still water. So, it irrational

'to

giVe

the sea margin as the

ratio t'o

tho resistance in still,water.

Thic

fact should be rdmembor

for the orsti;nat.on of

the sea margin of the recent

high' speed. cargo chips. ,, ' ''' .,

It i very difficult to give', the rclátion between tho' hull forrAs and."the renietance inoroase,'afl huge time anti expense' will, be needed if one seaks

-

12-'by the theoretical oaloulationc3, it will be very wefu11. in the doin of hull

:forr, as much data will be obtained yctematically. For thia

sake, more

improvement for

the ca1ulation method 'is

required, so that the coincidenoe of the theory and tho, oxperirnentc beoomeo better than now.

-5,

Coyioludin

Remarks.

The, cxaot solujionof the rer;istanooinorearo in waver; is giveh in the forms of the inorcos of the wave rnakin renie'tanco

under the anumption of

linearized condition and the solution is obtained a the singularity diotrtb.i--tion is given.: Jurtae the problem of the correr;pond.ence between the hull form

and the

singularity d.ictribution

in

the theory of wave making resistance of steady motion,

thâre

ciinto -the problem hOw to approximate the offectof

din-'

turbance due to chip motion by the singularity dthtribution.

In thii approxiiflatO oaloulation, a ship is acumed. ac a elendor body :whioh is roprecontod. a line singularity located at an adequate depth. Ac for

the boundary oondition on ship surface, only the draftwiso velocity of each section is adjusted. at the representative point of the rection, ignoring'

the ship

surface.

Theroforethis method.is fairly

rough

approcimation.

This 'approximate oaloulation corrospondc'toMicholl's theory of wave making resistance. instillwater, and fully explains the-physical phenomena of the.

resistance inorea'-e in wavoc. However, oornparirig the oompatcd values

with

"tbe experimental values, fairly good agreement is shown in the cao

of the

.inathematicalship model, 1,ut'in the ca'e of the raotical models the calcu-, lation does not have always eufjioiont accuracy.

-There ieio' oloar'physioal moaiiing on 'the location of the line singularity in this calculation, and merely it is due to an approximate treatmot. The improvement of agreement between theory and. experimot cóems to be poosible 'if the location

of singularity

io'ohanged, nearer-to

the

free surface

when

the

wave length becomes ehortor or the speod. of chip beoomes higher, considering

the correepondenoo with experiments. Ofooureo it should. be desirable

direc-tion of the improvement that the boundary

condition on chip

surface is eatisfi--edmore exactly, and those improvements are left ac the future problem.

Even-if the calculation .rnethod does not give tie aticfaotory results in all range of Fri and. _rp paoticaly.neadod, it givoe'.a key o the -thoorotical ectithation -'of the reciistanoO.iflOrtafle in

wvec'andalQo gives

the

direction of practical ::uso.

:

--:,::

_':

:,

-The, authors gi'ealy appreciate 'any. suggestions d: tachine givon to

Ref oronoe

H. Naruo ; "Rec.i:;tanco in }lavo", 0th anniwtrrary rorien vol.8, the Sooioty of Naval Architeote of Japan, 1963. p.67

H. Maruä ; "On the Increaco of

the Rociotance of a Ship

in Rough Sea"

(2nd

report, The origin of the add.ittdnal reoirtanco),Journal

of the Sooiety of Naval Architooto of Japan, No.108 (1960)

p.5

H. ?ujii & Y. 0awara ; "Calculation on Heaving and Pitching Motion3 of a

Ship by the Strip Method" , iournal of the Society of Naval

Achitoots' of Japan, No.118 (1965) P36

0.J.Sibul; "Ship Recistanco in Uniform

Wavec", Tnitituto of

EnincorinC

Recearoh, Univoraity of California, (1964).

13

-(.5) j. Gerritsma. ;

"bip

i'OtjOfl

in LonGitudinal Wavoz", 1.5.?. Vol.7 No.66

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