Some applications of the slender body theory in ship hydrodynamics

SOME APPLICATIONS OF THE SLENDER BODY THEORY IN SHIP HYDRODYNAMICS

SOME APPLICATIONS OF THE SLENDER BODY THEORY

IN SHIP HYDRODYNAMICS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAG NI FICUS DR. R. KRONlG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE,

VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 4 APRIL 1962 DES NAMIDDAGS TE 2 UUR

DOOR

GERRIT VOSSERS SCHEEPSBOUWKUNDIG INGENIEUR

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR: PROF. DR. R. TIMMAN

STELLINGEN.

De golfweerstand van een slank schip wordt bepaald door de vorm van de kromme van spantoppervlakken en het getal van Proude.

2

Peters en Stoker hebben de mogel~kheden van de theorie van de slanke lichamen voor het gedrag van een schip niet voldoende onderzocht.

A.S. Peters en J.J. Stoker - Comm. on pure and applied math,

1Q,

1957, P 399-490.

3

Het berekenen van de hydrodynamische krachten op een slank schip in golven door gebruik te maken van het concept van de relatieve beweging tussen schip en wateroppervlak, is een correcte methode voor een schip dat in recht voorinkomende golven vaart met een golflengte van gel~ke orde van grootte als de scheepslengte.

4

Het in rekening brengen van de snelheidsafhankel~kheid van de dempingskoppeling tussen de stamp- en dompbeweging is niet in overeenstemming met een gebruik van een striptheorie voor het berekenen van de hydrodynamische krachten.

B.V. Korvin-Kroukovsky en W.R. Jacobs - Tr SNAME,

22,

1957, P 590-632.

5

Door toepassing van het theorema van Green en een geschikte Greense functie, kan men een logische afleiding geven van de theorie van slanke lichamen in een onbegrensd medium.

6

Het bew~s dat in golven recht op de kop geen vermogensvermindering ten opzichte van het stil water vermogen kan optreden, is nog niet geleverd.

Discussie van Weinblum op Swaan en Vossers, TRINA, 103, 1961, p 297328.

-7

De optimale plaats van het drukkingspunt uit het oogpunt van weerstand is afhankel~k van de model schaal.

8

Het ontbreken van representatieve golfspectra voor verschillende oceaangebieden, is een ernstige belemmering voor het beoordelen van het gedrag in zeegang van een scheepsontwerp.

9

De aanwezigheid van moderne rekenmachines kan er toe leiden dat grote bereke-ningen worden uitgevoerd op grond van t~felachtige hypotheses.

10

De negatieve waardering van de invloed van de techniek op mens en maatschapp~ door sommige cultuur filosofen berust in vele gevallen op een onvoldoende kennis van de techniek.

11

De bezorgdheid van enkele leden van de Eerste Kamer omtrent de afwezigheid van I.e.~ . . . etenschappel~ke faculteiten aan een technische hogeschool, dient zich evenzeer uit te strekken tot een bezorgdheid omtrent de afwezigheid van een technische faculteit aan een universiteit.

-Verslag van de besprekingen in de Eerste Kamer ter gelegenheid van de stichting van een derde Technische HogeSChool,

Aan mIJn Oudere.

CONTENTS. Summary.

Chapter 1 • Introduction and general outline.

Chapter 2. _{Formulation of the problem. Linearization.}

Chapter 3. _{Veloc1ty potential of a elender ehip in eteady mot ion} for high :Froude numbers.

Chapter 4. Wave reeietance of a slender ehip in eteady motion. Cbapter 5. Velocity potentialof an oeci11ating elender ehip at

zero spee"d.

Chapter 6. Velocity potent1al of an oecil1ating elender ship at forward speed.

Chapter

7.

Equat10ns of motion of a slender ship.

Chapter 8. Exc1tation forces on a slender ehip in waves. Referencee. Overzicht. p p 17 p 33 p 57 p 60 p 70 p 76 p 87 p 94 p 96

•

•

SUMMARY.

The tbeory of a elender-body ie applied to tbe bebaviour of a elender ebip on a free eurface. A elender-body ie defined ae baving a ebape witb botb beam and draft emall compared to tbe lengtb and tbe problea ie attacked by formal developmente witb reepect to tbie emall beaa-leagtb parameter.

In tbe firet place tbe case of tbe steady mot ion at conetant forward epeed is etudied and a formula ie derived for tbe wave resietance of a elender _bip in cala water.

In tbe _.eond place tbe uneteady bebaviour ie dealt witb and it i_ ebown tbat tor a eerta1n range of frequeneiee of mot ion and forward epeede tbe bydrodynamic oeeillatory forcee on tbe body can be calculated by meane of a strip tbeory. integrating tbe two-diaene10Dal torcee per unit lengtb on transveree sectione of tbe ebip. Por tbe bebav10ur at forward epeed in bead eeae w1tb a lengtb ot tbe eame order of magnitude ae tbe ebip lengtb. tb. ezcitation forees consiet of tbe buoyaney effect of tbe dieplaced volume by tbe undisturbed wave and a strip-wise correct10n factor depending on tbe eectional added mase and damping valuee.

CHAPTER 1. INTRODUCTION AND GEHBRAL OUTLINE. 1.1. Preliminarx remarka.

A complete treatment of the behayiour of a freely floating r1gid body on tbe free eurface of au infinite ocean ie in general beyond tbe

capab111tiee of modern mathematica and in aD early etage eevere restrictioD. on tbe type of equatione and inital and boundary conditions bave to bs impoeed.

One of tbe first restrictione normally being mad. i8 tbat tbe treatment remain8 within tbe realm of tbe potential flow tbeory of au incompressible medium. Tbi8 exe1udes consideration of viscous effects. A eecond restrietion is tbat tbe amplitude of tbe eurface wavee wi11 be small, wbicb requiree eertain conditione for tbe èbape of tbe body, tbe forward epeed and tbs aJlp11tude of mot.1on of tbe body. Under tbeee aeeumptione a suitab1e 1inear1zat10n of tbe problem can be acbieved.

Varioue restrictione on tbe eMp. of tbe body bave been used. Poesibilitiee witbin tbe elaee of sbip-ebaped hulle are illuetrated in Pig. 1.1.:

(a) tbin ebipe (b) nat sbips

, baving a beam emall compared to tbe lengtb of tbe sbip

, baving a draft sma11 eompa~ed to ~be lengtb of tbe sbip

(c) yacbt-type ebips, being a combination of a tbin aud a flat sbip (d) elender sbips ,baving as we11 a beam ae a draft ema11 compared

z to tbe lengtb ot tbe sbip. x y ----6-:..-tr~ z (C) Pig. 1.1.

i

i

z (b) z x y---677777~---(d)

Tbe .tudl of tbe thin ehip originated witb the famous paper by Miohell (1898), wbo inveetigated tbe wave reeistance of a ahip aoving in atill water at a conatant forward epeed. A aequenee of notable papers

followed thie original paper. Por a review see Wehausen (1957).

The study of tbe unsteadr behaviour of a thin ship in wavea, etarted witb a paper ~y Baakind (1946) and waa followed by a seriea of papera by himaelf (1947), (1954) and by Banaoka (1957).

Bo.ever, Petera aad Stoker (1957) sbowed that for a thin .hip the foroea due to hydroatatic preeeure and body inertia are of lower order with reepeot to a beam-len,th parameter than the bydrodynamic forc.a representing the ao-oalled da_ping and added maea terme. This difference in order of magnitude haa-not been taken care off in Haskind's and Banaoka'a a priori linearisation.

Bewa&D (1960) tried to cireumvsnt tbe difticulties by introducing the assumption that for resonanee condition tbe amplitudes of motion of the sbip are of lower order witb reapect to tbe beaa-lengtb parameter tban the amplitude of tbe inooming wavea. Altbougb be obtains in tbis manner a consiatent theor,J, the assumption of tbe amplitude of motion of the ship being of lower order than the amplitude of tbe incoming waves is queetionable in view of available experimental evidence of actual ahip-sbaped bodiee.

On this basis it must be concluded that tbe thin ship approximation is not a satisfying model for tbe investigation of the unsteady bebaviour of a sbip, altbougb for tbe atudy of tbe wave resistance in atill water considerable sucoess bas been aohieved.

A next step to formulate a traetabie attack on tbe unsteady behaviour of a sbip-sbaped body would be an approximation in tbe direction of a flat ship. Tbis approximation corres~onds to a givan presaure d1atribution on tbe .ater aurfaoe. Bogner (1924) studied in connaction witb tbe wave resiatanoe ot a sbip the effect of constant pressure distribution _oving at a siven forward apeed. Peters and Stoker (1957) and KaoCamy (1956),

(1958) formulated the proble_ of an oscillating flat ship on tbe surface of a fluid. Por suoh a flat sbip tbsre is aconnection with tbe diffraction of water .aves against a flat plate - tbe finite dock problem. Unfortunately, tbe . . thematical analysis of flat plate problems is ratber complicated, since tbe solution of a singular integral equation is involved. No numerical reaults for sbip problems are available. Altbougb tbe difficulties witb tbe th in ahip conceming the relative order of magnitude of tbe bydrostatic and bydrodynamic forces are avoided, it is not aure wbetber tbis type of

approximation oorreaponda auff1ciently witb tbe bebaviour of an actual ebip to warrant a detailed numeri cal treatment.

The yacht-type model bas been proposed by Petere and Stoker (1957) as a more sbip-ebaped approx1mation than the tb1n or tbe flat sbip. The difficult1ee in arriving at suitable numer1cal results will be even larger for tb1a type of approximation tban for a flat sbip and it is not to be expected that tbia approacb w1ll lead to usable results 1n the near future.

The final suggestion of a sbip-sbape approximation, aa given in Pig. 1.1., is a slender ship, baving a. well a beftm aa a draft ... 11 compared to tbe lengtb. One may wonder wby tbis type of approximation haa received auob a amall attention in tbe hydrodynamics ot tbe bebaviour ot a ahip, sinoe 1t appears to lend itself perfectly to tbe presentation ot tbs ,eometry of a ship. Pract1cally all existing sbips bave lengtb-beam ratios bet •• en 5-12 and length-draft ratios between 15-25.

In the aeronautical literature tbe concept of a slender-body bas been very powertul, since it originated witb the paper of Munk in 1924 on -The aerodynamic torces on airsbip bullsn.

-Ä multitude of papers bas appeared in tbe meen time. ae well for eub.onie flow as traneeonie and supersonic flow. Por a review of tbe slender-bod, tbeory in aerodynamice see Ward (1955). Milee (1959) end Lightbill (1960).

Tbe prinoipal idea of a elender-body tbeory is tbat for a body witb transverse dimensions emall oompared to tbe lengtb dimension. tbe flow in tbe neigbbourbood of tbe body can be approximated b1 neglecting the firet term in the potential equation

~xx + ~yy + ~ zz = 0 (1.1)

wbere tbe x-direction correeponde witb tbe direotion of elongation of tbe body. Tbe flow near the body is adequately deseribed in that case by

~yy+ ~zz = 0 (1.2)

With a suitable boundary condition on tbe bod1 iteelf, and eventuall, a free-surface condition. tbe equation (1.2) posee a two-dimeneional problea in tb. traneveree plan •• Tbe solution will TarT fr om section to eection. i.e. witb x end can be written as ~ (XiY,Z). Por tbe forcee and aomente on tbe total body tbe two-dimensional reeulte are integrated in tbe x-directioa. tbe eo-called "strip-tbeory". In tbis way a considerable simplification of tbe problem bae b~en achieTed. since for the two-dimensional plane tbe well-known metbods of conformal traneformation of tbe oomplex function tbeory are availablei also a direct numerioal attaok appeare to be more promising for tbe two-dimensional problem tban for tbe complete three-dimensional oase.

Tbe development of a slender-body tbeory in aerodynamice ie near11 completed. In tbe field of sbip bydrodynamics. bowever. only a few atteapt. for developing a Blender-body tbeory are known. Tbe presenoe of a free surface condition complioates tbe Bolution.

Intuitively some of tbe oonceptB of a slender-body tbeory. namely that tbe flow in eaob cross-Beotion normal to tbe longitudinal axie. 1e indepen6mt of tbe flow at otber seotions. ba. been applied already 1n tbe calculatioD of tbe oscillatory forceB on aD uaBteady Bb1p: eee Grim (195'). (1959). (1960). Haskind (1954). Korvin-Kroukovsky and Jaoobe (1957), Kaplan (1957) and Pay (1958). Some justifioation of tbe use of tbis etrip-tbeory ba. been given by Grim (1959). (1960). but no treatment bas been offered witb tb. same r1gour as Peters and Stoker (1957) carried out exemplary for tbe tb in ebip case.

Sucb rigour ie neceseary ~oo for tbe slender-body tbeory. eepecially for the foroee in the longitudinal direotion. It .1e known fr om tbe slender-body tbeory in aeronautic~ tbat one sbould be cautious in coneidering ths eolutions ~ (Xi y, z) of tbe two-dimensional problem ae tbe complete eolution of tbe orig1nal tbree-dimensional problem. Por tbs velocity potentialof a Blender-body in an unbounded medium it ie obvioue tbat a term g (x). depending only on tbe x-coordinate. can be add~d to f (Xi Y. 11) witbout imparting the two-dimensional formulation.

Tbe only satisfying way to derive the tunctlon g (x) is to etart from tbe original tbree-dimensional formulation end to derive tbe required firet order term by an aeymptotic exp8Dsion from tbe solution of tbis three-dimensional problem.

Sucb a program bas been carried out for supersonic flow for bodies of revolution in an unbounded medium by Von Karman (1935). (See also Ward (1955)). In tbis study we wil1 carry out a similar program for a slender-body on a free-eurface for a certain eboice of tbe order of magnitude of tbe forward speed parameter and frequency parameter.

1.2. Outline of tbe analrsis.

We consider in tbis paper several coordinate systems; tbe principal ones are indleated in 11gure 1.2.

:~,

- - V

z

Xl

Fig. 1.2

Tbe system X, Y, Z is a system fixed in spaee witb tbe X, Y-plane ln tbe undlsturbed free surface and tbe Z-axis vertlcally upwards. Tbe x, y, z-system ls a moving eystem witb tbe x, y-plane coincidlng witb tbe X, Y-plane and movlng witb tbe mean speed V of tbe sb~p in tbe positive X-dlrection. Tbe system of coordinatee x', y', z ls fixed in tbe body. Tbe origin of tbis osclllating system wil1 be represented by a vector witb tbe components xo, yo, zo, denoting respectively surging, ewaying and beavlng of tbe sblp. Tbe angular displacements around tbe x, y, z-axes are indicated witb tbe modified Eulerian angles ~ , ~ ,X , d,not~ng iespectively rolling, pitcbing and yawing of tbe sbip. Tbe system x , y , z ie assumed to coincide witb x, y, z-system wben tbe sbip and tbe water are at rest in tbeir equilibrium position. It is assused t9at tbe center of gravity of tbe ship coincides wttb tbe origin of tbe x', y , z'-eystea; a lees restrioted approacb may be formulated witb tbe re sult of lntroduoing some addltlonal terme wbicb are irrevelant for tbe purpose of our .tud7.

We lntroduce tbe equation of tbe sbip's bull witb respect to tbe primed systea of coordinates in tbe fora

F ( x', V', z') = y' + ~ (x',z') cr 0

wbere f1 (x', z') is a two-valued (corresponding to tbe port and starboard eide of tbe sbip) funotion of tbe coordinates witb continuous der1vatlve, except, possibly at tbe bow.

Purtber.ore we introduce dimensionless coordinatee, ae well in the primed ae tbe unprimed .y.tea:

(1 .4)

-'lba equaUon. of 't·be-sbip' s bull will be represented in tbl.

dlmensionle ••• ystem of coordinBtee by

(1 .5)

As a cbarac'terlstic lengtb tbe balf-lengtb ot tbe .bip ~ bae been

2

cbosan. 'lbe parameter 0 ie a meaeure of 'tbe traneTeree -extent of tba

boundariee of tbe ebip ànd correeponde to tbe beam-lengtb ratio of tbe ebip:

a.

B/L.

lor a elender ebip tbe vertioal and tbe transveree extent of

tbe boundariee of tbe ebip are of tbe eame order ot magnitude and botb of

order

a

compared to tbe longitudinal extent of tbe boundariee.

'lbe motion of.tbe water isassumed to be given by a velocity potential

~. (:I, y, Z, t), wblcb is a solution ot Laplaoe' 8 equlltion

~xx+ ~yy+~ZZ" 0 (1 .6)

and appropriate boundary conditions at ~be. free surface, on tbe bull of

tbe ebip and at infinity.

'lbe boundary eonditione on tbe bull ~f tbe ebip depend on tbe motion

of tbe ebip, wbicb in ite turn ean be found by determining tbe forees aeting on ·tbe ebip tbrougb tbe preeeure of tbe water and by tbe solution

of tbe ditterential equations tor tbe motion of 'a body witb eix degreee

of freedo ••

We want to linearize tbe problem and tbink ot tbe velo city potential

t

and tbe quantities wbich determine tbe mot ion of tbe ebip ae tunetione of

tbe elendemees parameter a •

In our application the veloelty potential ~ (:I, Y, Z; t, 0 )=

i

(x, y, z; t; 0 ) ie aeeumed to posseee tbs d~velopment

-_{t(X,y,z;t;Ol.O. (x,y,z;t)+O ...} 2 (t) 3.1.(2)( _x,y,z,t ' . l ₊

'lbe free eurface condition in tbie moving syetem of ooordinatee will

bave tbe following form for tbe firet term ~(1) I

(1) Z (1) (1) (1)

.tt + V ~xx -2V

'tx

+ g tz

=

0 (1.8)

Witb tbe introduotion of a dimeneionless velooity potentlal

(1.9)

and a diaenelonless time

wh10h reducee for harmonie motion to

~= lilt (1.11)

where

(1.12)

represente the reduced frequeney. we ean write the free surface condition ae:

where ~o repreeents the forward speed parameter

We can identify in tbe free aurface condition eome well-known dimensionless parameters:

(1.13)

(1.14)

(1.15)

(1.16)

!he parameter ~L is a dilDensionlese frequeney parameter important for tbe generation of surface wavee. Sometimes it is more convenient to use

(1.17)

!he parameter ~ has been introduced by Baskind (1946) and is of importaDce for the desoription of the wave pattern generated by a moving oscl1lating bod,.. Por a -value of

r

~ Y4 tbe generated waves are oonflned to a seotor behind the moving bod,.; tor a value 1 <:

1/4

waves are found betore tbe body too.

Witb these parameters tbe free surfaoe condition can also be wrlttea aa

t·o

(l.H!)

-order of magn1tude of

»

If we eboose tbe

and ~ 0 .. 0 (a), tbe

o

(

d ) and b1gber:

free surface cond1t1on beeomes. delet1ng terms of

(1.19)

W1tb Laplaee's equat10n 1n d1mens1onless form, also by del.t1on of terms of 0 (0 2 ) ,

(1.20)

and a su1table boundary cond1tion on tbe body tbe equations

(1.19)-(1.20)

pose a two-dimensional problem wbere tbe x-w1se ebange of tbe flow pattern is not present.

If the order' of magnitude of ~o=

0 (1),

tbe equation

(1.19)

still bolds; bowever, terms of 0 ( 01/2 ) are deleted, wbieb may cause too mueb

error. A cboiee o! t B . 0 (0 ) degenerates tbe free surface cond1tion to

(1.21)

wbieb applies for tbe low-frequeney approximation. In tbe same way a cboiee of ~B = 0 (a4 ) degenerates tbe free surface condition to tbe bigb-frequency approximation

,=0

(1.22)

Tberefore it is of som. interest to restriet the cboiee of ~B to values being of 0 (1), sinee in tbis range tbe most important contributioDS arising from bydrodynamie d~ping due to tbe emission of surface waves are to be expected.

Tbe ebo1ce ~ B" 0 (1) eorresponds for tbe case of forward speed

/30 • 0 (a ) to

and we pose

S1nee

~2

-lf- .

w1 tb 1 tbe lengtb of tbe generated waves. tbe ebo1ee

g B = 0 (1) oorresponds to

TtB=0<11 l

or tbe 1engtb of tbe generated waves 1s of tbe same order of magn1tude as tbe beam of tbe sb1p.

In terms of tbe well-known d1mens1onles8 Proude number

(1.23)

tbe ebo1ee ~ o:a 0 ( a ) eorresponds for a beam-lengtb rat10 a between

0.08-0.2 to a value

Fr ~ 0,2-0,3

wb1eb covers praet1eally tbe whole speed range of ex1st1ng sbips.

Por tbe case of steady mot10n tbe free surface eond1t10n degenerates wUb 130· 0 ( 0 ) or even ~o = 0 (1) to

(1.24)

We d1seuss 1n some deta11 tbe boundary cond1t1on on tbe eb1p's bull. We assume tbe 11near d1splaeeaents of tbe OG of tbe sb1p and tbe angu1ar rotat10ns of tbe sb1p to be smallof order 0 2 and we put for harmon1e mot10n:

2L

g

-i" _{Yo,,02J:. iio e -i"} 2 L - _i-3

}

)(0= (j 2" oe

2 zo"o "2 ~oe

(1.25)

lj).o,_{2iji o e-}i" Ijl =02~oe-i" Xz02Xo e - i"

Us1ng tbe appropr1,te transformat1oD formulae for tbe ebange of tbe pr1med system

:I',

1', r; 1nto tbe unpr1med system of eoordinates x, y, z, tbe k1nemat1eal eond1tion on the bull of tbe ab1p tbat tbe part1ele der1vat1ve sbould van1sb, ean be formulated as followe, deleting terms of

o (

a )

and bigher:

(1.26)

-where n is tbe normal on tbe bull.

Tbis kinematical condition coneiste of a time-independent part, given

by f1g ,tbe longitudinal slope of tbe hu1l, and a tima-dependent part,

compr1a1ng tbe swaying (11 0 ) , yawing (X 0 ) , beaving (t 0 ) and pi tcbing (Ijlo )

motion. Tbe rolling end surging mo~ion are not present; tbey wil1 appear

tor a elender-body in a bigber order approximation.

Tbe equation (1.26) comp1etee tbe set of equationa (1.19)-(1.20) (or (1.20) and (1.24) for the time-independent case) required for the two-dimenaiona1 eo1utions in tbe transverse planea perpendicular to tbe direction of e1ongation of tbe body. Tbey do not aerve ua for finding tbe funotion g (x), wbicb may be added on the two-dimenaional eolution,

altbougb tbey are of some be1p to indicate wbetber two-dimensiona1 eolutions are to be expected. Por a complete solution of tbe problem we bave to return to tbe original tbree-d1mens1onal Laplace equation and tbe free eurface condition {1.8).

The normal procedure for formulating an integral equation for tbe unknown velocity potential, ia to apply Green's tbeorem

!(G*A

t-~

AG·) dxd ydz

=-/~G*n

-

G"tn) d 5 (1.27 )

to tbe velocity potential ~ and tbe adjoint of ä Green' s function G,

appropriate to our prob1em. The boundary eurface S around tbe volume R conaiate of tbe wetted eurface of tbe eb1p S, the free surface outside tbe sbip and suitable surfaces at large diatance from the abip, wbieb are allowed te go to infinity. Becauee the eingu1arity of G in xl, "1, Z 1 ie of

type l/r, with r 2 = (X-Xl)2 + (y_Yl)2 + (Z-Zl)2, we f1nd, if tbe Green'e

function eatiefiee tbe free eurface condition, witb tbe c1assical argument the fo11ewing integral equation for the unknown ve1eeity petential:

(1.28)

Tbe tota1 velocity potential coneiete in principle of the fo1lowing terms:

(1.29)

where

~o tbe velocity potentia1 of the sbip in eteady mot ion in

etill water

~ 1 the velocity potential of tbe p1ane progreeeing

incoming waves

~2 tbe velocity potentia1 of tbe dieturbance created by tbe

Por the time-independent motion we can write tbe equation (1.28) b7 introduetion of a dimene10nleee Green'e fUnction

(1 .30)

and use of tbe dimenaionlees notation (1.4), (1.5), (1.9) andtbe bounda~

cond1tion (1.26):

'oC el' 'I1l·tl)" -

2Tt

I {.

pcr

'l1-. flt

rt) - r·f

_le

1

d~dt+

OCol

wbere the integration sbould extent on botb sidee of tbe bull and appropriate values of tbe integrand sbould be cboeen on each side.

(1.31)

Por tbe ti . . -dependent case we have in a eimilar way. by deletion ot

tbe time tactor e-i" in

(1.32)

wbere

.zn

ie given by

(1.34)

Tbe value ot

.zn

euggests to split

.2

in the following components:

(1.35)

(1.36)

and tor eacb component we can write an integral equation

(1.37)

-witb

px..-g

(1.38)

(1.39) -a

Tbe ve10city potentiala ~2 repreeent tbe velocity potentiale due to tbe oscillatory motion of tbe sbip in tbe a -direction. Tbe velocity potential ~~ represente tbe influence of tbe ebip form on tbe incoming waves or tbe diffraction potential.

When suitable Green's functions are known. tbe integral equations (1.31). (1.37) and (1.39) can be solved.

For eteady motion tbe following Green's function bas been derived (see for instanee Haveloek (1923). Timman and Vossers (1955). Peters and stoker (1957):

Go

=

+

V(X-Xl)2 + (Y-Yl )2+ (Z-ZI)2 V(X-Xl)2+ (Y-Yl )2+ ( ZtZ l)2

(1.40)

j

Tt/2{ P[(Z+Zl )+Î(X-'Xl) cos-1)]

+~ Re d-1) dp e cosf P(Y-Yl) sin-1)] y2 2

o L 1-p_ cos-1)

g

Tbe patb L encircles tbe pole p = ~ sec2_-1)

y with a small circle

via tbe poeitive half-plane.

For unsteady mot1on at zero speed we have (see Peters and Stoker (1957»:

2

wbere the path L enc1rclee tbe pole p= ~ v1a tbe negative half plane. Pinally we bave a ratber eomplicated Green's function G2 for the

caee of uneteady motion with forward epeed, for which we refer to (6.6)-(6.7) of Chapter 6. (Peters and stoker (1957».

It w111 be tbe purpose of our analysie 1n the subsequent cbaptere to derive an asymptot1c expansion of tbe integral equations (1.31), (1.37) and

(1.39) witb tbe appropriate Green's funct10ne GO' G1 and G2 for small valuee of tbe parameter 0 • Reta1n1ng only the firet order terms 1n tbe developml!!ltll we can eimp11fy theee integral equatione eoneiderably and we sball be able to identify a function g

(x),

if tbere is any.

When tbe 1ntegral equRtione are solved in tb1e way. we can calculate tbe prse8ure on tbe bull by aeane of Bernoulli's law

wbich can be wr1tten by means of the dimens10nless notation

(1.43)

Por tbe case of steady motion we can calculate tbe wave res1etance of ths ehip by meane of thie pressure w1tb the formula

(1.44)

Herewitb we are able to give a new formula for the wave resistance of a elender ebip. (See Chapter

4).

Por the eaee of uneteady motion we have to uee the pressure on the huIl to formulate the dynamieal equatione of motion of the sbip, einee the cbange of momentum of tbe rigid ship e~uale the force aeting on the ship and the change of angular momentum equale the moment acting on the ship. Theee principles allow us to write four equatione of motion in the oscillatory eomponente yo. X , Zo and ~

( 1.45) I d2X +mXXd2X +NXX dX+mxvd2yo + NXY dyo. GXe-iwt

dt 2 dt2 dt dt 2 dt

-(1.46 )

or tbe1r d1mene1onless aequ1valente.

K repreeente tbe maa. of tbe sh1p, I tbe long1tud1nal maes moment of

1nert1a, wb1le tbe eoeff1e1ents ma~ and Na

p

follow from tbe 1ntegrat1on

of tbs dynam1e pressure over tbe huIl aurfaee. Por 1nstance we have

ol 0

mZl+ i

~Z

=

o2pi-3

~1!dglidtl

f

_{t1 <g"t,)}

~i

_1 _d

wbere m zz 1e calléd the add ed !DaSS for heav1ng and N z z the damp1ng

eoeff1c1ent for heav1ng.

(1 .47)

The coeff1e1ents B a~ follow from the hydrostat1c contr1bution of the

preaeure and represent the restor1ng forcee and momente, wh1ch are only

preeent 1n beave Rnd p1teb. Pinally the coefficients

Ga

denote the

_ 0

exei tation forces and moment. and follow from the contr1butione ~ 1 and ~ 2

in the veloc1ty potent1al.

The equat10ne (1.45) &nd (1.46) represent two systems of coupled

equatione of motion; the first system is a coupled system for swal and

yaw &nd the second aystem a coupled system for heave &nd pitch. Prom these

equations the ampl1 tude &nd phase of 1₀, X ' t 0 and cjJ ,or iio·"xo, ~o and ~o in (1.36), can be calculated, wbich completely determines the eolution of tbe uneteady problem.

Pinally we like to discuss in this intt'oductory chapter somewhat core

in detail tbe éxcitation forces. The firat part of thie force is derived from tbe velocity potentialof the ineoming waves, wh1eb can be written in the x, y, z-eyetem of coordinates

~ .~2 . .

(1l ig W gZ+lg<xeosa+yslOa)_lwt

., = - h e 11

wbere ti '"' o2fi(11 repreaents the wave amplitude, 11 ia the wave

( 1 .48)

trequeney, a the direction of travel of tbe wavea with reapect of tbe

x-axie and w tbe frequency of encounter

2

In the dimensionless system of coordinates, we have

(1 .50)

with the dimensionless wave amplitude

snd the wave frequency

(1 .52 )

The expression (1.49) may be written in the following dimensionlese form

(1.53)

Since we specified in our applications the frequency of encounter gL to be of order 0-1 , i t is clear from (1.53) that in general g?, should be of order 0 -1 too or in other words

(1 .54)

In that case we find the exciting forces in beam seas ( a = 900) to be of the same order of ma~nitude as the other hydrodynamic forces. However, in head seas ( a

=

1800 ) tbe exciting forces are of order 0 compared to

tbe other hydrodynamic forces and in the equations of mot ion no excitation will occur. This is clearly understandable since for a wave lengtb small compared to the ship length, the contributions in the excitation forces along the length of the ship cancel each other.

If we suppose 0,' < 0 < 0.2 , we may assume in head seas wi th a ship at forward speed that g?,=O (1) ,although gL=O(o·l) • "hieb ie illustrated in Fig. 1.3. Under this assumption the velocity potentialof the incoming waves is written as

... jo - f?,(o~-jf l_i" 0 ( )

"'1·.:.z. v e 1" 0 I1V

(1.55)

-and we can expect to have a contr1butlon ln tbe excltatlon forcee of tbe

eame o~der of magnltude ae tbe otber bydrodynamlc forces, 1.c. added maes

and damp1ng.

Tbe same argume'nt applles for excl tatlon comlng from tbe dlffractlon

potentlal ~CZ

,

for wblch we refer to Chapter 8.

Reetrlctlon of tbe value of tbe parameter 0 on the lower a1de, l1mlts

tbe accuracy whlcb can be obtalned 1n tbe calculatlon of tbe excltatlon

forcee 1n bead eeae. However, w1tb tbe llmltatlon of 0 ~ 0.1 etlll a

reasonable approxlmatlon can be acb1eved.

15 10 5 0.5 10 1.5

_{- ) . A}

10 5 4 3 2 Fig. 1.3

In tbe 8ub8equent cbapters tbe outline given in tb1e introduction w111 be dieoue8ed in more detail.

In Cbapter 2 we deal wi1jh tbe foraulatlon of tbe problem and a suitable eboio. of par . . . t.re in order to be able to indicate wben a series of two-dimeneional probl.me can be expected.

In Chapter 3 tbe velocity potential is derived for a slender ebip in et.ad7 aotion with a forward speed parameter

poz

0 (1). HereiD a

derivation of tbe funetion g (x) i8 given, whieb allowe ue to formulate an integral for tbe wave reeietance of a elender .bip. (Chapter

4).

. Ae a 8eeond exaaple we apply our aDalysis to tbe velocity potentialof

~ oecillatiD, elender-body at zero epeed with a frequency of motion

~8 • 0 (1). In tbat oaee we find a etrip tbeory to be applieable and tbe reeulte publiebed for the damping and added mass of two-dimen8ional

eeotione oan be ueed for tbe prediction of tbe bydrodynamio forc •• on a elender ebip. (Chapt.r

5).

Even at forward epeed wUh a epeed paraaeter

fJo.

0, ( 0 ) it ·appeare

in Obapter 6 tbat tbe oeeillator7 bydrodynaaie forees are given by the s&ae equation. a. for zero .peed, altbougb tbe nature of tbe generated 8urfaee wavee ie coaplete17 different for thi. oa.e ( r - 0 (1» from tbe situation at zero .peed ( 1 -

0).

In tbie Caee no first order eontribution in tbe funotion g (x) ie found.

froa ehip

!bie

.bip

In Chapter 7 tbe equations of motion are discussed in more detail. P10al11 in Cbapter 8 the exeitation foreee are ealeulat~d. It appeare tb1s Chapter tbat in bead .eae we can caleulate tbe influence of the on the Wave (tbe diffraction potential) by means of a strip tbeory. effect i. aequivalent to the concept of tbe relative motion between and water, wbieb bae been proposed intuitively by some autbore. Our analysi. bae not been exbauetive, sinee a large number of other cboio.e of the order of magnitude of tbe frequency and forward speed paraa.tere ie poee1ble. Por instanee the eboiee ~o· 0 (1) and gL-O (1)

&&y be of ~ome intereet for high speed ehips in long wavee.

-CHAPTER' 2. PORMULATIOl'i OP THEPROBLEII. LINEARIZATIOl'i.

Coordinate syeteme.

Por the formulation of the problem we need several eoordinate eleteme whieh we ehooee as follows:

a) .coordinate system O-X, Y, Z (pig. -1.2), fixed in space, w1th tbe X, Y -plane in the undisturbed free surface, tbe X-axie in the direetion of travel of tbe sbip and the Z-axis vertieal upwarde (all eoo~dinate slsteme are supposed to be rigbt~handed).

b) Coordinate system O-x, y,-z (lig. 1.2), witb the x, y-plane eoine1ding witb the X, Y-plane and moving in the positive x-direetion witb tbe constant Qpeed V of the shi,p.

The transformat10n of tbe O-X, Y, Z-system into tbe O-x, y, Z-sletem i8

'therefore given by

x ~= x-Vt }

Y

=

Y z = Z

(2.1 )

c) Coordina'te system O-x" , yn, z", witb th,e axes parallel to tbe x, y, z-axee, but with the origin fixed to the center of gravity of the ship. The center of gravity is supposed to make the following motions:

surging Xo swaying Yo heaving Zo

The transformation of the O-x, y, z-system into tbe O-x", y", Z"-sY8tem is given by

x" .. x-xo } y"

=

y-Yo zIt z-zo

(2.2)

It is assumed that the center of gravity of tbe sbip eoineides in tbe rest position with tbe x, y-plane .in order to be able ~o omit eome extra eoupling terms in tbe equations of motions. Tbe analysis might be earried without this assumption, ~ut the extra terms are irrevelant for tbe purpose of tbis study.

d) Coordinate system O-x', y', z' rig1aly attaebed to tbe sbip. Tbe orientation of this system relative to the O-x", yn, z"-system i8 def1ned by tbe angular motions pf the sbip, for wbieb tbe modified Eulerian angles X , Ijl and Ij) are used:

yawing : X

pitehing : Ijl

rolling : Ij)

These modified Eulerian angl'es can be visuatized by traneforming tbe system O-x", yn, zIt into tbe system O-x', y , z' by means of tbree successive rotations.

P'ir8tll • • rot.tion tbrough an angle X (poe1tive in accordance witb tbe rigbt-banded coordinate 8ystem) around the Z·-axi8 bringe tbe coordinate

87stell O-:z-. y", z· into. tbe 87stem O-:z- , 11

" , S M I • A rotation cjI

around tbe 7"' -a:zis bring8 tbe coordinate sy8te. O-:z'"

,1"

,z·'

into

tbe elstem O-:z 1r " . , z. • Lastll, a rotation tbroqh an an,le op

around the:z. -axis brings tbe coordlnats eysts. into its final position,

the O-:z',

1',

Z'-s7stem. Tbs total transforaation of ths O-x-, yM, ZM_

s7sts. into tbe O-x',

1',

z'-system i. thersfors. written in matrix

nota:UoDI

[ ;:]';[LJ [;::]

z' z"

(2.3 )

witb tbe matrix

[L]:

cos op cos X cos cjI sin X _ sin 1\1

sinopsincjl cos X- sinopsinl\lsinx+ sin op cos cp

(2.4)

l Ll ,. -cos op sinx + cos op cos X

cos opsin I\l cos X+ cosopsinl\lsinx- cos opeos cjI +sinopsinx -sin op cosX

In tbe subssquent analysi., it.will be assUJIed tbat tbe motione of tbs

8hip will be small, s&y

Xo=t ~'

Yo =ty~)

111 2o·tZo

(2.5)

wbere t represents a small parameter, to be identified later on.

Inserting tbese developments in tbe above-mentioned transformation formulae, we find tbe following tran8formation of tbe O-x, y, z-system into tbe O-x', 7', Z'-s7Ste.:

(11 111 Cl) 2

1

x'=x~t(-Xo +X y-I\I z).Ott I y'= Y .. t( _ y~lI+,n~_ XC1

k )

+ O(t2 )

111 (1) Cl1

z'.zn(-Zo .. 1jI x-op y)-tO(t2 ,

(2.6)

Pundamental equations. Boundary conditions.

We assume tbe fluid to be non-viscous and irrotational. Tbe mot ion can

bs deacribed in tbat case by a velocity potential ~(X, Y, Z, t), giving

for tbe apeed in tbe X, Y and Z-direction respectively:

u"'fX

It ie kno"n that t

equatioD: aati8fies in tbe balf-space

- 18

~XX+ tyy+ ~Zz

=

0 (2.7)

From tbe equations of motion for an 1rrotational flow, it is readily verified tbat tbe following form of Bernoulli's law is valid, wbicb bas to be ueed ae a dynamical boundary condition:

(2.8)

In tbis equation prepresents tbe preesure in tbe point X, Y,

z;

p ie tbe deneit1 of tbe fl~id and g tbe acceleration due to tbe gravity. Tbe constant C (t) may depend on t but not on tbe epace 'variables X, Y,

z;

we wil1 take C (t) =

o.

A eecond boundary condition ie formulated by ueing tbe general law in continuum mecbanice tbat once a particle ie eituated on ~be eurface of tbe fluid (eitber an interface or a rigid boundary), it remaine on tbis eurfacs. Tbie kinematical condition ie written, if'1 (X,

r,

z,

t)

=

0 represente tbe formula of tbe surface, ae:

(2.9)

Botb boundary conditions (2.8) and (2.9),bave to be applied as well OD the free eurface of tbe fluid ae on tbe boundary of tbe ebip. These

boundaries are, bowever, not fixed, but movable. The fr.ee eurface ie determined by the incoming wavee and tbe motion of tbe ebip, wbile in turn the mot ion of the ebip followe by integrating tbe preeeure on tbe bull by meane of Bernoulli'e equetion (2.8) and eolving the reeulting eix equatione of mot ion for the ehip.

It is neceesary for arriving at an unique solution, to etipulate soma additional conditione for the bebaviour of tbe velocity potential at

infinity. In the eubeequent analyeie it will be aseumed that tbe water deptb will be infinite, wbich impoeee tbe condition

tz

= 0 for Z_-_. On tbe free eurface for Y ___ and X _ _ more trouble some conditione are required for guaranteeing an unique eo1ution. In tbe caee of harmonic

oscillatione it ie cuetomary t9 lmpoee a "radiation condition", whicb allowe on1y waves to be radiated from tbe body to infinity and euppreesee any eolution giving travel of energy from infinity to tbe body. In tbe time-independent caee, one requiree tbat the motion far abead of tbe ebip vaniebe& No probleme ariee witb the formulation of these conditione, wben tbe problem ie coneidered ae an initia1 value prob1em aod the motion prior to t

=

0 ie euppoeed to be zero. Tbe rigbt eo1ution for tbe harmonic oecillation or tbe time-independent motton followe by tbe aeymptotic bebaviour if tbe time t _ ...

In our analysie DO direct use of either of tbese two metbode will be made, since tbe Green'e functione belonging to tbs varioue problems are eupposed to be known.

-A derivation of these elementary Green's functione bas been given by Peters and Stoker (1957) and in tbeir analysis tbe appropria~e conditions at infinity have been uaed. Tberefore there is no need in our analys1a to consider tbis matter once more.

Linearization of tbe free surface oondition.

In .order to linearize tbe boundary conditions, it is assumed that the velocity potential ~ can be expanded as a power series in terms of a emaIl perturbation parameter 6 :

(2.10)

Berein we follow the suggestions of Stoker (1957) and Wehaueen and Laitone (1960). We are not identifyin~ at tbis stage tbe parameter 0

We assume tbat the free surf ace can be expanded in a similar power series:

11l 2(2)

FeX.Y.Z.t )cZ-Zw(X.Y.t)=Z-6zw- OZw _ ... =0 (2.11)

Ineertion of tbe equations (2.10) and (2.11) in tbe Laplace equation (2.7) and the boundary conditions (2.8) and (2.9), reeults in tbe following equatioDs, collecting tbe coefficients of tbe first powere of 6 :

(1) (1) (1) tXX+tyy+~ZZ=O Z< 0 (2.12) (1) (1) ft -gzw

=

0

1

(1) ... (1) 0 - Zwt + '!tZ = Z=Zw(X.Y. Z.t) (2.13)

It i8 good to remember that .for t(l) ,whicb bas to be evaluated in

(2.13) on tbe free 8urface (2.11), the following expaneion is assumed to be valid' 111 (1) • (X. Y. Z • t I

=

t

< K.Y. 0 • t I + Z ~ Z ( X • Y. o. t ) + ... ,-(2.14)

ww

=t(X,Y,o.t 1+ 6z w

tz

<X.Y,o.t) + ...

Thie indicatee that tbe condition (2.13) has to be aati.ried on the fixed bOUJ'ldaZ'1 Z • 0 ineteacl of the moving 'boundary Z • zw.

-Combining the equatione (2.13) we arrive at the well-known free surface condition:

(tI (1)

4>tt+gtZ=O Z=o (2.15)

It is convenient to transform the above-mentioned Laplace equation aod'

boundary condition into equatione for the moving coordinate eyetem O-x, y, z, ueing the traoeformation formula (2.1).

We introduce

(1) (1) (11

• eX.Y.z.tl=t (x+Vt.y.z.tl=+ (x.y.z.t)

Ueing the relationehipe for cllange of coordinatee, we rewrite the Laplace equation and the free eurface condition ae followe:

z

(0

z=O

Boundary condition on the ehip'e hull.

(2.16)

(2.11)

(2.18)

We Buppoee the equation of the ehip'e hull in the coordinate eyetem rigidly attached to the ahip, O-x', y', z', to be preaented 'by

F(x'.y'. z' I .. y'+ ti _{ex ', Z'l}

=

0 (2.19) We aeaume f 1 (x', z') to be a two-valued (correaponding to right and left aidea of the ehip) ~unction of the coordinatea x' and z' with

continuoue ~erivativee.

With (2.6) we traneform thie equation into the one for the coordinate syetem O-x, y, z, which in virtue of the amallnees of the motion parameter

1: becomee:

FCx+tp,y+tQ. z.qrl=FCx.y,z)+tpFx+tqFy+tr Fz +0(1:2),.

In th1s mov1ng system of coord1nates the k1nemat1cal boundary cond1t1on on the sh1p'e huIl is g1ven as follows, by transform1ng (2.9) and us1ng (2.10):

Th1n- nnd slender-body approx1mat1ons.

We introduce several typee of 11near1zat1on, us1ng a metbod of analys1s of K1lee (1959).

We introduce tbe follow1ng d1mene1onlese coord1nates:

(2.21)

(2.22a)

(2.22b)

(2.22c)

(2.22d)

by Purthermore we aeeume tbe equat10n of the eh1p'e bull to be preeented

(2.23)

-~1nal11 we introduce tbe dimens10nless veloc1ty potent1al

cj>(J) VL

Ix.y.Z,tl:

2'

cI>(g,"I\.t,~)

Por an osclllat1ng sbip at zero speed it is 1nappropriate to nond1mensionalize tbe velocity potential witb tbe forward speed, and tberefore we introduce

Tbe following conneetion between botb potentials exists

(2.24)

As a ebaracteristie lengtb tbe balf-lengtb of tbe ship L/2 bas been eboeen. Tbe parameter 0 is a measure of the horizontal extent of the

boundar1ee of tbe sbip and corresponde, therefore, to the beam-length ratio of tbe sbip a .. BIL.

Tbe parameter ~ ~enotes tbe vertieal extent of the boundary sueb tbat 2/~ l .. 0 (1) in tbe neigbbourbood of the sbip. Two ehoiees of tbe

1

parameter ~ are of 1nterest:

a)

b)

~«o , wb1ch 1ndieates a tbin sbip (Miehell-ship)

~"Olol, which gives a slender ship, baving breadtb and deptb dimensions of tbe same order of magnitude. (See Fig. 1.1a

and d).

Tbe parameter K is a parameter defining the time rate of change and

correeponds to tbe redueed frequency

K:wL

2V

if tbe motion is simvle barmonie.

(2.25)

In tbe foregoing seetion we introdueed a small parameter t defining tbe order of magnitude of tbe motions of tbe sbip and a small parameter ö

for linear1zation of the velocity potential. Tbe dimensionlesB parameters

E t K t tand 0 are to be determined by tbe boundary conditions and in relation to tbe, by de!1nitioD, small parameter a , denoting tbe transverse e:ztent of tbe sbip.

Subetitut10n ot tbe d1mens1onless notation

(2.23)

and

(2.24)

in tbe X.aplac. equaUon

(2.17),

tree surtace condi Uon

(2.1

·

8)

and k:ineaatical boundar.r oond1tion on tbe ship hull

(2.21),

re.ulte in the foliowing ·.quation.z t40 (2.26)

(2.27)

(2.28)

on 1) +

f

f I (~. t )

=

0 (2.29)

24

-(2.30)

Tbe parameter

Po

ie connected witb tbe Proude number

Fr=~

einee

(2.32)

We require botb sides of tbe equation

(2.28)

for tbe time-independent ease to be of tbe same order of magnitude, for wbieb we ebooee tbe follo.ing relation for

Subst1tution in tbe Laplace equation

indicatee tbat two poseib11itiee of the cboice of 6 and 0 are of interest: 4) b) 25 -(2 • .,5) (2.3')

!be f1~et poeeibili ty givu 0 . 0 and 11- 0 (1). wbiob. in view ot

tb. two eboieee of tbe parameter 11 • dieeuseed on p 23. indieatee that tb1. po.e1bility ie app11eable to tbin sbip ••

!be eeeond poseibi11ty givu 0 • 011 and 11-1 • wbieb 1ndieatee tbat tb1e poeaibility 1s applieable to elender ebipe.

Betor. we .ub.titute tbe •• two poeeibi11t1ee in tbe boundary eond1t1on on tbe bull. w • • bould aeeua. on order of magnitude of tbe mot1on paraaet.r

1 • A rea.onabl. aeauapt10n about 1 ie that it ie of tbe same order of aapitud. aa 0 and we eboose ~ - 0 • ~bie ie aequivalent to tbs aaeumption tbat tb. ebip motione are ot tbe same order of magn1tude as the wave beigbt. Ile.entarf eODa1deratione on tbe motione of a ebip on tbe free surfaee aake tbie aeeuapt10n plaueible. Por a tbin eb1p otber oboiees have been proposed. wbieb we re diecueeed in tbe introduetory ehapter.

SUbet1tution of tbe above-mentioned eetimatee of tbe order of magn1tude of tbe parametere 11 • 0 and 1 in tbe equations (2.26) and

(2.28). givee the followins equatione. omitting terms of 0 ( 0 ) and bigber:

a) ~b1n sb1p b) Slender ebip

.1\-

-)lB~+ fl~-X

\

.~~

t .'11'11 t

~2

.tt

=0 11-0 ll)

·1J+~·tflt·-)l11(8~+*C~fl~)+fl~

} .'1'1I+.tt -0 11=0(0) 26 -(2.37) (2.38)

Por botb typea o~ abip tbe ~ree aur~aee condition

(2.27)

remain. applioable.

We abould make an assumption eoneeming tbe order of magnitude o~ 'Il and

Po •

Por tbe tbin-abip approximation tbe onl, intereating posaibi1it7 18 'Il - 0 (1) and P o · 0 (1). Other eatilllates lead to a degenerate ~ree surface condition ~. 0 or ~t· 0, wbieh eorresponda to tbe bigb-, reapeotivelf low ~requeney approx1mation. One exeeption muat be made:

'Il .. 0 ( 0 J; Po· 0 (1), whieb, bowever. corresponda to the atedy lIIotion o~ tbe sbip.

W1 tb 'Il = 0 (1); 130 zo 0 (1) we have, posing 11

=

1:

a) fhin ahip. (Kichell-ahip).

~~~~ ~1J1J'" ~tt =0 t~o

'Il "'.0 UI (I) (I)

~1J.

L;;

+'IlX

~f-'Il'P.ot+fl~-X

_(2.39)

2

'Il ~o~.o.o+ ~0~~~-2'1l~04l.o~+ 4It" 0

Por tbe alender sbip another estimate o~ tbe order o~ lIIagnitude o~ 'Il and ~o is of intereat: 'Il" 0 (0- 1 ); ~o '" 0 (0 ). Putting

~ 0 " 0 ~I and 11 = 0 we bave

b) Blender ahip.

or us1ng tbe d1111enaionleall veloe1 ty potential ~ jO (aee p

23),

we have

wblcb sbows tbat tble approzlmat10n 1s aleo app11cable for V·O.

We 1dent1fy tbe follow1ng well-known d1menelonless parametere:

(2.42)

:ror tbe tbin sb1p ~ L and I are of 0 (1); for tbe slender eb1p we

lnYeatigate tbe cases gL= 0 «i-I ); ~B" 0 (1); ~ - 0 and 3- 0 (1).

fbere le a non-eeeential difference 1n tbe formulat10n of the boundary cond1t10n ae g1ven 1n (2.39), compared w1tb tbe formulat10n of Peters and Stoker

(1957).

In our formulation the center of gravity of the sb1p 1s

oscillat1ng in tbe sW&y1ng d1rect10n w1tb respect of the coord1nate system

O-z, y, z; 1n Peters and Stoker's formulation the coord1nate system 1s follow1ng tbe eway1ng motion; theretore in tbe1r analys1s the term w1tb

Yo~ ie not present.

It ls clear trom tbe boundary cond1t10n on tbe sb1p's bull tbat only

tbe "811ng mot10n, tbe yawing motion and tbe rol11ng mot10n are contr1buting to tbe tiret order approz1mat10n. Tb1s 1s clear fr om an

ele.entar1 analysls, by visuallzing the tbin sblp as a vertical plate w1tb

lengtb and draft of tbe same order of magnitude and the th1ckness small

compared to tbe other d1mens10ns. The other motions are not present ln tbe

first order approziaation and tberetore no contribution in the veloc1ty

potent1al, i.e. added mass or damp1ng terms of the pitching, heaving and

surg1ng motions are to be ezpected. These remarks of Peters and Stoker are very important and attect the foundations of Haskind's (1946) and

Hanaoka's

(1957)

calculations.

It is known that the equations (2.39) tor the steady behaviour of a

tbin sb1p witb zero yaw angle were al ready used by Kicbell

(1898):

-$ ~~ ~ $1111 + $ ~~ = 0

1

$l1=fl~ 11 =0 (2.-43 )

~o$~~+$~=O ~=O

The equations of the slender sh1p (2.41) show that a part of tbe determination of the unknown velocity potential 1s reduced to the solution of a series of two-dimensional problems in the transverse planes

~ = constant, since ~ -whe change of the velo city potential is not shown in these equations.

The boundary cond1t10n on the hull discloses tbat bydrodynam1c contributions to the velocity potential may be expected not only due to swaying and yawing oscillations of the sbip (which is similar to the

boundary cond1tion for the thin ship) but also due to he~ving and pitching. This boundary condition should not be applied on the plane y

=

0. as is tbe case for the thin-sbip approximation, but on tbe sbip bull in the rest position. since the motione are of 0 ( 0 2 ) and the transverse extent of the hullof 0 ( a ).

The slender-body theory for time-independent motion of the ship

$1111 ~ $~~ = 0

}

$l1+$~fl~=fl~ l1+fl(~.t)=O (2.44)

$~=O ~= 0

has been used by Cummins (1956) to investigate the wave resistance of a slender ship. Although he was the first to realise the importance of tbe aeronautical studies of slender-bodies for the wave residtance of slender ships, his analyses are inconclusive and are not· used in our investigation.

It appears that the orders of magnitude of the various parameters used in our analysis are reasonable in view of practical application to technical problems.

-\Te bave aesUlDee! 'IC.. 0 (0- 1)

h=oCo-

l ) I" 0(1) or 0 ; ~o"O ~1J • OCo) or 0 ( ; gS-OIl)

r

(2.45)

Tbe parameter 0 indicatee tbe beam-lengtb ratio and varies tor most sbipe between 0.07-0.20 •

. Tben ~o. 0 ( 0 ) corruponde wUh

~r 0.20 - 0.30

in wbicb range most ahips are operating. Wave beigbts ane! motion amplitudee ebould be ot order 02 • ~ull scale wavee ot practical importanee bave in general dimeneioDe aequivalent witb

..5...,...

0.01 - 0.02

À

wbere

ti

repreeents tbe wave amplitude and À tbe wave length and tbese

Taluee are in agreement witb our aeeumptione. P1IDar-body approximation.

Altbougb tbe planar-body concept is not used in our in~estigation. it ie of eome intereet to pureue the same systemat1cal analysis ot tbe

boundary conditions ae tor tbe th in and Blender-body.

A planar or flat eb1p (ct lig. 2.1) can be repreBentsd by tbe following equation

(2.46)

Tbe parameter 0 denotee tbe vertical extent ot the boundary

z

1;:'

IC

i--

.4

r--

-4 (4) (b) lig. 2.1 30

-of tbe ebip and correeponde, tberefore, to tbe draft-length .ratio of tbe

ebip o . 2~ • Tbe parameter E , wbieb baa -been introdueed in (2 .22b) ,

denotee tbe borizontal extent of tbe boundary of the ebip, auob that

-_Y-.O Ol

E

Lh

in tbe nei.gbbourbood of tbe ebip. Tbree ehoieee of E are of intereet:

a') E-O(O); E« 1 ,wbieb ie equivalent to tbe alender-body approx1l18tioD given above

b) E

»

0 ; E.O(l) , wbieb indieatee aplanar ebip (Pig. 2.1a)

c) , wbieb we eall aalender planar snip (Pig. 2.1b).

Tbe detaile of tbe analyeie are omitted, but on eimilar linea ~a has

been done in tbe foregoing eeetton, it ean be deduèed tbat tbe aeeumptioDs b) and e) lead to tbe following equatione

b) Planar ebip

c) Slender planar ebip

t1Jll + .~~ • 0 ~ , 0

.~= 'M.IZO~ -'M.I.o~e +fl~

Lh

on A of ~ .. O

(2.47)

out~ide Aof ~·O

on -a<y<a of ~=O

(2.48) on y>a and y<-a of ~·O

Tbe equations (2.47) were already derived by Peters and stoker (1957) and used in tbeir formulation of tbe motion of aplanar ship. It can be

seen that ino tbe firstO order terms of the perturbation series of the

.,eloci ty potential contributi1ons of the beaving. pi tching and rolling

motione are found. A similar formulation of the problem has been put forward

by YacCamy (1956. 1958); however. essential differences with the approacb

of Peters and Stoker are present. whicb require further investigation.

A combination of tbe planar and thin ship approximation° has been

proposed by Petp.rs and Stoker (1957) as a suitable scheme for receiving

hydrodynamical damping and inertial contributions in all motions. except

surging. They call thie a yacht-type approximation. (See Pig. 1.1).

The equations (2.48) were used. in the caBe of time-independent ~otion

of a slender planing craft by Tul1n (1957). The form of equations (2.48)

Buggeet that an extension of hiB analysis may be possible for an oBcillating Blender planing sbip. where again a two-dimensional approach may be

BuitabIe.

-CHAPTER 3. VELOCITY'POTENTIAL OF A SLENDER SHIP IN STEADY MOTION FOR HIGH FROUDE NUMBERS.

As a first-application of the concepts intrGduced in tbe second chapter, we will calculate the velocity potentialof a slender ship in steady motion.

We start with the application of-Green'e theorem for two functions, one of the functione being the adjoint of the Green'e function Go belonging to the free eurface boundary condition for eteady motion, the other function~o

being the unknown velocity poteniial:

(3. ' )

R denotee a three-dimensional region and Sits boundary surface. Let us

firet discuss the Green's function appropriate to thie problem. This function, Go' is a solution of tbe equatione:

2

'L

G_{oxx ...} G_oz = 0 9

z-o

lim Go - 0 2 - - 0 0 (3.2 ) lim Go - 0 x-co

Herein Go(x, y, z; x" y" z,) represents tbe required Green's function, which depends as well on the location of the o·bserver (x, y, z) as on the

location of the singularity (x" Y1' z,); ti is the Dirac delta function. The free surface boundarycondition follows from (2.'8) by equating to zero terms depending on the time derivatives. The third condition of (3.2) eneuree the boundedness of the solution at infinite depth.

A solution of the equations (3.2) has been given by different methods. (See Havelock (1923), Timman and Vossers (1955), Peters and Stoker (1957)). We give the result only:

Tbe patb of integration L ie given in Pig.

3.'

p -plan. L

Pig.3.' Since

Go! x. y. z ; X,.y,.2,) "+ Go! x. y. 2,: X.Y.Z)

we ehould be careful 1n app~y1ng 1n Green'e tbeorem tbe adjo1nt of

Tbe flrst term of

(3.3)

repreeentethe Gr~en'e !unct1on correepond1n6 to

a polnt souree 1n x" 1"

z,

for an unbounded med1um. Tbe aecond term o! (3.3)

follow8 trom an 1mage 80urce w1th oPPo81te s1gn 1n tbe upper hal! space,

re!lected wlth respect of the tree eurface. The lntegral term in

(3.3)

sommate8 the contribut1ona g1vlng r1se to the sur!ace wave pattern •

• t

-Appl1cat10n o~ Green's theorem.

We apply Green' s theorem· (3.') totbe volume R bounded by the ·surfaces (eee Pig. 3.2):

.here

So

S,

part or the eurfnce of the eh1p below the water plane z = 0 part of the free eurface Detween tbe water11ne·of tbe sb1p Lo and the arb1trary c1rcle L,

the surface of a vert1cal c1rcular cy11nder between the c1rcle L, 1n the free surface and a bor1zontal plane at the bot tom

clos1ng plane under water of the vert1cal circular cy11nder.

z

Fig. 3.2

Prom Green's tbeorem and tbe 01ass1cal argument 1nvolv1ng tbe s1ngular1ty 1n x" 1" z" we bave

+O(X,y,zl-

-21t

11('ioG~n -G~ionldS

sa+

S,+S2 + S3

On tbe free surf!ce S, we find, since the free surface condition of (3.2) applles as well to

.0

as G;

.""

- Go +oz ldxdy· n.7)

2f"' .... '"

21"'··'"

L

<toGox-Go+oxldY-L <toGox-Go+oxldy 4ng 4ng Lo L,

The integral along L, vanishes when the radius of L, goes to inflnlty. Also tbe integratlons over",S2 and S3 vanlsh for these surfaces golng to lnflnlty, since G~n and ton tend to zero of eufficlent order for lnflnlty. Therefore the only contributions left trom (3.6) are the lntegration over tbe 8urface of the sbip So and along the intereection of the ehip with the free surface Lo.

On the eurface of the ship the following relations are valid:

•

G_{n •}

•

•

•

Gx F l( + G Y F Y + G z F z

V

_Fl( 2 _{+ Fy} 2 _{+ Fz} 2

with F<x.y.zl= y+f,<x.yl

belng the equation of the ship's hull.

36

-(3.8)

(3.9)

(3.'0)

Therefore the integral over the surface 8₀ in (3.6) can be written ae:

_ 41ft

/1

Ho(G: F'I.

+G~

Fy + G; Fz) - G"( +ox F'I. + ;oyFy +

~ozFz)}

dl! dz (3.12) So

where x-wise integration should extend from - ; UIl

+t

'

L being the length of the ehip and the z-wiee integration over the longitudinal plane of the ehip from z

=

0 till z

=

T with T being tbe draft of the sbip.

Pollowing tbe argument of Chapter 2 we introduce the following dimensionlese notatien: x = J.. ~ 2 p

=..z..

q L 2 Ilo

=

l:!....

gL y

=

a.la.1J ; Z

=

a l

t

2 2

Since f 1( ~.t ) ie a two-valued function, we aeeume for a eymmetric ship

fl ( ~ •

ç )

= - f ( ~ •

ç )

= +f(~. Ç)

}

Furthermore we assume tbe ehip to have peinted ends, whicb gives

(3.14 )

(3.15)

It ie not neceesary to aeeume this condition in the slender-body tbeor.y; eepecially square-ended eterns may be tolerated and are of some practical interest.

Tbe parameter a denotes the beam-lengtb ratio ot tbe abip: IS .~

L

(3.16)

W1tb tb1s DotatioD we ma1 write tor the equation (3.6), using tor tbe

boundar,y eODdition OD tbe sblp's bull tbe expreBs10n (1.26) aDd delet1ng

terms of order a and h1gber,

(3.17)

wUb

(3.18)

aDd

(3.19)

Tbe Green's tunetion (3.3) transtorms into

l . ',2,3 (3.20) (3.21)

·

,

~2·-~====================

V

2 2 2 2 2 (~-~,> +CJ (11-11,> +a (t+t,> lt/2 ' ,

·

~

J

j

q[CJ(t+t,>+i(~,-~>cos~]

[ ( , "']

r

_{o 3 ..} ~ Re d~ dq e 2 cos qlS 11-11" SlO v O L ' -~oq cos ~

,In (3'.17) tl)e 1ntegral (3.7) along Lo ia rlot present, einee it 1a ot

We introduce the follow1ng notat1on: with and with ± -± K2l

=

± rOl f~(~.t) Finally we call which reduces (3.17) to ~o • 1: 1:. L;l +

1:.

L

L kl k l k l

1

; k = 1,2 l . 1,2,3 (3.22)

It will be the purpose of the subsequent analysis to develop systemat1cal11

the integrals of (3.24) in an asymptotic series of a in order to find the

first order terms wb1cb bave to be retained in the f1nal formulation. +

A,symptotic expansion of the contributions L kl ; k - 1.2 ; l · 1,2

Tbe term L 21 + is presented by:

(3.25 )

Ueing -d

{l09(~,-~

+

V(~_~,)2+a2p32)}=

d~

+ 0(a2)

V(~

-

~,)2+

a2p32 and for ~,-~<O +

we ma)' wri te for L 2'

o

~1 L;, = -

4~

jdt

/-f~(~,t)d{l09(~,-~+ V(~_~,)2+(12p/)}+

-d _,

,

+lf~

(~,t)

d

~log

(~-~,+

V(~ _~,)2+0'2p/

)} + 0«(12)

=

~ I 0

~,

•

- ...!...

Jd

t {- f

~ (~

, t ) log (

~,

-

~

+

V(

~

-

~,

)

2 + a 2 p J 2 )

I

+

4yt -d _, 1

+f~(~.t) log(~-~,+ V(~_~,)2+a2P32)1.

+ ~,

+

1:~

'II'U'

'og

'f,-I

+

V,

f- f," . .

'p,"

--,

-1~~ f~~(~,t)log(~-~,+V(~_~/+0'2P32

)}+0(0'2). ~,

o

• - 4'yt

J

o t

(f~

(-',

t) log 2

(f,.1)

+

f~

(', t) log 2

(1-~,)

-

2f~(~t;

t) tog 0 P ₃

}_j

-1 (3.27)

f

O/f

t

fa

1

- 4'yt dt

d~ t~(g.t)tOg2(g,-e)+21t dt/d~ tg~(~.t)log2(~-~I)+O(02)

- d -I -d ~I

-+

With the eame argument we can reduce _{L22 and we have}

(:3.28)

with (3.29)

The funct ion f and f ~ ehould be evaluat ed in ~1 and not in ~ I and I

therefore. the expreeeion (3.28). being i·nd ependent of ~ I repre.ent. a part

of the two-dimeneional solution of the problem.

The contributione L 21 + L 2-2 can be traneformed in a eimilar w~ 1nto

with and with

o

L₂₁ +L 2-2 '"

dTtJd

t

f~(~1;t)(I09<TP1-lo90'P2)+O(0'2)

-d +

The contribut1on L 11 has the following form:

L

~

=

...!..

f:~j:

t ___ cr_

2

_G.:,.3 _ _

4Tt

{

2 2 2 3/2

-1 -d ( ~ - ~ 1) + cr p

1

which traneforme by meane of integration by parte into

L~

=

,~

f:t

:f+

0""

-d

(3.30)

(3.32)