Hydrodynamic interaction effect in waves

Deift University of Technology

Hydrodynaniic Interaction Effects

in Waves.

J.A. Pinkster

Report No. 1025-P

ISOPE-1995, Offshore and Polar Eng.

Conference, The Hague, The Netherlands

TI.J Deift

Faculty of Mechanical Engineering and Marine Technology

The Proceedings of

the Fifth (1995)

_{International}

OFFSHORE AND POLAR

ENGINEERING CONFERENCE

VOLUME 1.11, 1995

NUMERICAL WAVES, WAVE MEASUREMENTS,

WAVE BREAKThG & STATISTICS, WAVE-BODY IÑTERACTIONS, HYDRODyNAr4Jc

FORCES,

DYNAMIC RESPONSES, HIGHER-ORDER EFFECTS VORTEX & VIBRATIOÑS, COASTAL

HYDRODYNAMICS, LABORATORY & OCEAN MEAStJREMENTh

edited by:

Ji11 S.. Chung, Colorado School olMines, Golden, Colorado, USA

Hisaaki Maeda, University of Tokyo,

Tokyo, Japan

C.H Kim, Texas A & M University, College Station, Texas, USA

presented at:

The Fifth (1995) International Offshore andPolar Engineering Conference

held in The Hague, The Netherlands,.June

_{li-16, 1995}

organized by:

International Society oíOffshore and: Polar

Engineers

sponsored by:

International Society Of Offshore and Polar Engineers

_«SOPE)

Offshore Mechanics and PolarEngineenng Council (OMPEC)

with cooperating societies and associatiöns

International Society of Offshore and Polar Engineers (ISOPE)

Copyright ©

1995

by IntemationalSÓciety ofOffshoreand PolarEÌigineers,

Golden, Colorado. USA. All Rights Reserved.

Internacional Standard Book Number: ISBN 1.880653-16-8(Set) ISBN 1-880653.1.9-2 (Vol Iii)

Library of Còngress catalog Card Number:

94.737%

ISO PE Board of Dii-ectors

J.S. Chüng, USA, C.P. Ellinas (Chairman), UK, K.M. Han, Korea, M. Isaacson, Canada, A.M. Lopez, USA, B.J. Natvig,

Norway, M.S. Triantafyllou, USA, and J. Wardenier, The Netherlands

CooperatingOrganizations:

-CanadianAssociationofPetroleurnProducers(CÄPP)

American Society of Civil Engineers (ASCE)- Engineering Mechanics Division

Korea Committee for Ocean Resources andEngineering(KCORE)

Canadian Societyof Civil Engineers (CSCE) Engineering Mechanics Division

Chinese Society of OceanEngineers (CSOE)

Chinese Society of Naval Architects.and Manne Engineers (CSNAME)

ChineseSociety òf;Theoretical and

AppliedMechanics(ÇSTAM)-RussianAcademy of Sciences

Singapore Structural Steel Society (SSSS)

Norwegian Petroleum Society (NPF)

The Institution of EngineersAustralia(IE Australia)

Kansai Society of Naval Architects,.Japan!(KSNAJ)

Technical.ResearchCentre of Finlan&(VTf)

1RO (The Netherlands)

The Society. of Materials Science, Japan (JSMS)

The OffshoreEngineering Society (OES), United Kingdom

The publisher and the editors of its publications assume no responsibility, for thestatements or opiniOns expressed;in papers or

presentations by the contributors to this conference.or proceedings.

Printedand'boùnd in USA

Abst;ract

Iii luis IHIPCE a coinputatioiial method for determining the

hìydrodyiu

interaction between several bodies in waves

based omm time application of 3-dimensional linear diffractioim

theory is treated. A review is given of the theory which is

based on the application of distributions of sources on the

bodies. Methods to obtain time solution of the equations in

the uiiknowii source strengths are discussed. in this context

im mactimed based on successive approxinmatioim of the source

sl.icíigtlms is also treated. This method is applicable to

imitmiti-body problemas when interaction effects between time bodies

mio umot i,mehiimle strong stamidimig wave effects. Results of corn-p mt.atioi is of I iy i rodymin uni e ii itejactiomi effects are comm i corn-pared

vithi previously pumi)iishme(l data for a cyliimdrical body next to

ii mect.muumguilnrpoiml.00mm. Results of coml)mmtations of mHeauI drift

foires omm two tankers moored iii tandem are given which show

time a))hicnh)ihity

of

time nmethod

of

successive approximnatiomis

to such cases.

Ime,' 'tTor(ts:

I )ilirartioii theory; muimilti-luumly; drift formes; taiidemmm Il Iii 'lI 11g.

i

Iiìtroductioii

()lfsl!oie opni;itioI!s olteimimivolve llore limaim omm floating or fixed strum-lii res uvi rk 11g iii vInse urox imi i ty Lo cadI other. Ty ial operations mmm ay

jii'mmivm' Lue transfer of eumiipiument. or supplies froimm one vessel to tue

millier, or, For ¡mmstammue imu time eased ujilshtorecommstrmictiomm tIme transfer

i mf i al-ge heavy k ids fromm i a t ramisport I marge lo a crane vessel. iii smmcim

cases lime wave frnq nemiry relative immoLions between time vessels play

im liii I)nrlmmml role iii lime leasibilily uf time operatioum and will imm imlaimy

i ei .

li till I in work ilslmty Iii oilshiorr ioidimmg/immiioidmmug oper mt ions

immvmmiviiug uerummummmeimlly immoored FloaI.immg Slum-age a-mid ri-omiimetiomi

ves-seis ;uiuii Slimiliku Ia.imkers, lime Slitilik tanker is olleim iumoored in tamokmmm iwiuiiimi lime l'i S() uvimimim wealimemva,imm's about a Single Pòiut Moorimmg s'simiuu. lui siiiIi rases thin- relative- mimol.ioiis of lime how of time .Shmmml.tle

tanker muid -liii slemum of lIme iI'SO are of immmportauiue. Tue reimtive

uuiuiliiimis uvill he dmmmmuilm;LIusI huy tIme 1(1W finiliemmiV Imom17oiilai imiotioims

HYDRODYNAMIC INTERACTION

EFFECTS IN WAVES

l.A. I'iimkslc,

S iii p II yd rommmecimami irs Laboraloiy

I )cl ft ti vers ity of Teclm mmoiogy

- 'flue Nel.herlammds

in(immce(I by how frequemmeysecommd order wave drift forces amid wiiud amid

current forces, Wichers (1988). Nowadays there is Sohle imiterest in

floating breakwaters as a imuealis to re(lmice wave artiomu alommg (oasI-hues. Preiiimuimmary studies have licou carried mjiil at; thin-Civil

Engiliecu--ing (ieh)artmimeimt of Lue Deifl, (Jmuiversity of 'i'erlmimology imivoiviimg several

iloati mug struictim res amici w Im idi are ai mmmci at inn-k i img ami asscs.su miemi t of time immflmieimce of suiclm striictmmres omm- beadm erosioim. Such stmmdies require imiforimmatiomi omm thiecommmhi!med effect of several! iloatimig striuctumres

sitim-ated close to cadi otimeromi time waveI'meid, Emigeieui (i995). 1mm a wave

elm u' i roh mumeim t ami aspeu:t of i mmm portl1lile is- the extemm t of Imyd rO(hy mmalui ir

iiiteratiomi betweemu Lije vessels imivoiveci. Defmlidilig ulm the relative

1)051 liomu of L!me strmmctui res Lii is i mm lera(:tiomi ramm resu I I in a red mmction or a_Im iiirrease of wave imm(iuced mimotions of -lime slrIuctiIreÑ, on lime second

ouder wave d rift forces or on tIte wave field -ar(mimnml time strmirtmumes.

TIlls mimay Imave ami imim portamit effect omi time efficiency of the operations i mu wii lvii tue vessels are imivolvecl.

Nielliods to evaluate thin liydrodymiaummir imulerartiomm betweeii several ih)atimmgorFmxe(I strumrllt res imm waves iia'e-hneim developed based ou

liii-ear lii ree-ui immmemusiommai poteumlial theory. See, for i nsl,ammre, Oortmmìers.sen

( t 979),Ooitmmierssnim ( i 98 i ) , Lokeuu ( i-9M I) ami Ferreira ( i !l91). ixf)eriImmemmtal veriflaliomi of results of roimmpuita.liomis imms bueim iimmm-ited to bodies wiLli simimpleshuapes., Oortimmersseiu (1979). Sudi comuupar-isomis ivave iiouvever shiowim LImaI 3-D diirractioui mumetIiocis gemierahiy cati

bui al)piied lo stich cases ammd are therefore a good tool to imuvesligate

such effcrts.

um this paper a briefdescripliommisgiveii of tIme 3-I) cliifractiomi

pro-grammi DELFI?AG developed at time l)elft Ijmiiversit.y of Tcciimmòiogy

by mmmeamms of wimich Lime Iuydrodymmammmir. interactiomm effects hetwnemm

sev-eral free-iloatimmg or fixed or imitercoimmiected bodies ramm lie caii;uuiatcci.

imi solvimmg time imimmiti-hody d ilfracliomm/radiatiomi proh)leimI dillememml

sohli-Liomm teclimmiqumes are emimployeci. A novel immetliod imivoives successive apI)roximmmalion of the imilcraclion eflect,s startimig frommm thieresimit.s of the

mndivmcluai i)odmes witimomit uimleructiomm effects Resimlls of commipuilutmoiis

of first order wave freqimeiicy c1umaimtities are ucimmupareci wiLli available

experi immeim Lai and colmi pii ted -data for Iwo si mmi pIe sii apes. Fi mmahiy solite cmmmmmplmle(i exammmples of interactiomi effects betweemi Iwo tamukersmmmoored

in tandemim are presemilnd lui this cuise time resumiLs m;omicermi the uumealm

secolm(i order waved rift forres.

2

Theory

'lie thcc iry is i veti for au arbitrary nit iuhr 'M of ari u trarily shaped

hiniui lu wit ii its owic p Lrtluular slrtpus tiid iIuetuu positioui and head

ing iii the li,irizoiital plane. tjse is uivade of ariglit-hatuded, earth-fixed

O - Xu

X. X3 systeiii of axes with origin at the mea!I water level

uicd .\ axis vertically upwards. l'ho body axe.s'G, - Xml- Xc,,2 2cn3

cii cauti of tice N lluiatiicg or fixed bodies has ils origin in the reuter of gravity of tice body, :r,,u axis towards the 1)0W, r,,,2 axis to port and

axis vu'rt.ivaIiy uujiwards. iii the iciest, positioui the cnuiterof gravity

icc cii w oc is inca leu al. X,,,,, ,X,,,,,2 X,,,,,:1 relative to the eartii bou 11(1

5\'stc:iii 'ci lixc'cl. axes. 'l'tue hceauliiig ;cicgio iic,,, ola l)o(ly isgiveui relative

ci tui () -

i

'i\is

/i ro lie RiIli

iiigle siicilic', th31 the C - T

xis nf the icciciy is i>araiicl to theo i axis.

hi ivi cii y ilion 'uicui

iii

pcit nIt ii', cri refereici ed ki the fixed

Sysleiii cl ixes.

I 'liase a,cgies of lccuiiy iiiutuotis a.reri:fereicc:ed t,o Ihr ciiicljstii ilced wayc' elnv;c.I,icuic at tice origiii of' the hixed axes.

iii rc'gci la i- h ciig-ri-est,vc I waves lie ci ini stil r bed wave rievatioti is as

Ic clic ors:

((Xi, X2, t) = C,, rcA(Xl '««+X co)-ciii

(I)

in wiiiulu:

I: = wave iuciuuiici

ís= wave cliii'c:tioii, zero for-wave travelling ici tue positive X1 (Iirc'ct,icìn

w= 'rave fic'ci cuc'ucc;y

= v;ive a ucijulitcicie

'liii'

ciiccl,ic,iis cil

lie in-tinily iii

tue, j-cico,ie relative to its body axes

;Ire gi 'eci icy:

c,,, (t) =

e

'

(2)

icc wicic:lc th,p-ucverliiueii,,hj,ates tIce c;ouccplexaicciituidr rif ticiacccot.iccn .

lic: l'ciiiccwiiug, tice overlicce wicieh also-applies to the couuipiex i)OteuitiaiS

et,, is ccc'glc'c:t.ed

'l'iceflicici ,cotjouus-;cre ciescrjheil l'y lie Iot,a.l potecct.ial 'i' as follows:

cI'( .-\_{i .I'2, .k:u, t) = /,( N i, .N2, N':,)} _cii (:1)

'I'iiecuciiiulex lcict.l'cctiici /, fcciiows li,uici lue sucicerilositioci of tice

'iii-clisliurhci'cl wave ixcteuiti;ui i/ci,, tice wi'esiiiFi-ac:t.ioui i)uit1'tilii 511(1 I-lie

i,uctc:Iiti;uis if,,,,j isscu'iit,ed willi tice j'tuiuuieirof u,uolioui of t,icc M 1)0(1105:

M 6

'I' = i.,.)

_{(/'ic + '/'i)(. +}

> i,,,ix,,,3 }

(t)

iii=1 i=i

'l'tic:-vi'i, cri t.y iiot.u'ii tisi ass,wiateui vitli t!ie ii ii(iiStui rico(l kcclg-i'reste(l rc'gici;cr wave iii water cil c'ciiist,aict, depth lu is giveci icy:

q costi !'(.\':, + lu) _{icc i)} = 2 _icishi

I,'!,-ihucici pressure fniiows froccc I kriuunclhi?s law:

,u(Xi, X2, X3, t)

=

= ;u(X, X2, X3) e'

willi:

- M u;

¡c(.'1, "2, X:c) = ¡' = ¡ci..22{(l/',c+ i) C,, + i ii: Ji,,,j :c:,,,)

ii=1 j=i

'11cc' i'lici;cIicous uil' icciticcuc ud tue hoily u, iii lice k-iccode iiiclicdiiig

tui' c'ii'i'c:t.s cch' hiyii'ucciyic;ciiuit iictoi;cu:t,iocis are giveic by:

A!

> (

__(,,J2

( Ai ,c&-,, f (L,,&.,,,) - iwb,,,,,1, + :r.,,,J = .Y,,. -(8)

willi:

= wave forre oli tite n-body iii tice k-tiiode

= iucert,ia itcatrix of thur 71-body for i icertia c:ou pli tug iii tic,'

k-mode for acceleration in tIce j-mode = O for n

n'-= clamping utiatrix-for the forro ou n-body ¡tu the k-mode chite to velocity of the ni-body in the j-iuiode

= spring matrix for the force on tice n-body in tite k-iuiode (lite to motion of the u,i-body in the j-mode

Tice and C,,k,,cj iciatrices may cóntain co,itribiitiouis arisiicgfroucc iciechcanicaidauiipiicg and spring roiupiiuigs betweeic tIce variotus i)odies.

'I'hie wave forre X,,& is given by: X,,k

=

L!71nit dS,, = _pw2/f(i/cn

+

) flA1S,,

in -hidc S,, is titean wetted siu rface of n-l)o(iy acid1 ui,, is di rertloic cositce ofsurface eleiuicict dS, for tim k-iuiode.

Tice direction cosines are -defined s.s follows:

'l'ue a(ided ucinas auch clatic pi ucg eon pi c,g meli Iu:ieu,Ls -are deli tied a.s fol

-lows:

=

'I/'

",,t i-1$,,]

=

_[pwJji/c711j

71-,k dS,,]

Ior these u'oefliciu'uu Is i t u's-ui lue show tu tic ;c t the fol lowi ucg s)' i,, theIr reiatioicsicips apply, see Oortiiuerssemc (h 979):

=

(t-,,j,,k

- b,,k,,,

=

bc,cj,,k (12)

Thue-contribcutibns of the bodies to tice velocity poteictish are based riti a- dist,ribuutiotc of ,soiuri-es over Ice body siiifac:es. Tice potemitiai at. a

point (Yi , )(z, .N3) chile to tice -body i, iii tice iccoche j is giveuc by:

l,,J(Xu,X2,X3) =

-!_JJr,,ii

i, A2, A3) c(x1, X2, X3, Au, /12,

I1) dS,,c (il)

IiI which ,cj(Ai A2, ít3) is the source strength at a i)oiuct witic

eartit-lìxe(h co-ordinates (Ai, A-2, A3)occ tite in-body duce to the tucotiomu of

tice ct-body in the j-tiiode. G(..) is tue (reet, f,iiictioti1 or itifluietice

futucctiouu of a sou ree satislyicig tice equation of couctiucitity, the hiucea-rised

iioiuuu(iary couichitiotis on tite free siurfau'e ami oui the sea-floor arid tice radiatiouc conclitiouu at iiclìicity.

TIce ,unknówti solirce stre!ugthssî,,1(..) are dcter!uiiuied based oui tice luorilcal velocity botuiudary coticiitioui OU carli of the -M bodies:

-

= -a,1j(Xu,X,,X3)+

,

LÍ"

(Ai, A2, A3) "G(X:t, X2,-X3, At, A'2, /13) dS,,,( 14) lui -the above equatioucs-, the operator signifies tice grarliecut ici

tice di rertioti of tite ucorutuah t,o the -slu rfame of body un.

2 =

n,,3 =

"-"'i

=

=

'I,,,,;

=

A'J1C, -01.

2..

(lit' suitittuit uif (lie itiotiohi i)uteiit.ials /, 1,114' rigluL-ha.tid side

ut' (Iii' aI)(iVfl uIi1L?,iti (14) is given l)y tue (hiPelioli coSities (lC[IIIP(I uy ettiaI.tutt

(i(I)

Liii' suiuut,i(iit of (lue duifrartioit i)otetiIiai 'bd (lie riglit-liatid side is giveii by:

¿)çbj ¿NIe _r

(Ia)

du du

Suivi tug the integral eq tiation (14) resulLs i n (lie u uk itowtu source stiu'utgflis.

Siibtitufioii of (liese iii equation (li) and in eqiistioti (9)

yields tite a(ldpd inas.s and dauii ping coellicieiiLs and flic wave forces

tes i u'.' ti vely. Fitually (lue titotiouis are deteriti i nod Irotii flue sol ii fiuti

iii eq utatiutu (). 'lito tutean 500011(1 order wave drift, forces oit the

iii-ulivi.luual i'i)(ll('Siut'e cuutu ittuted by (lie tut'ai'-liekl pressure integration

tiit't.luud . Sis' I iuikster (I

3

Nuinei'ical Aspects

I t ilisc'rt'tisiiug lue tutean welled sui rfur of a lanly un. into N,,1 patils

uni w diii (lue suai rus' sti'euug(lis a io huiuuogeiuoouisly distributed, integral

equtut.iuiuu (I 'I) is lra.uusfuriite(l iiul,o a su't o1siiiuuultuieouis linear equtafiotis

iii N,,, tin kuuowui complex soit roe sfreuigflus:

[(iPi

t

(7uil

(i7ift

(hiAIftI J

\ ,u

\

_{11M,u,j J}

"n22 ( ,'ii11

( I 6) I tu (Iii' ultuuvu' ('(luta.tiu)Ii itul atliitiuuuiaI iuulex lois leeuu added tu) (lie (l's niud lui' ii 's iii (uiiI('i ((I sluiiv that For a given und.' ut, j, (lue norujual

'i'Iui,itii's uuud suit rie sluu'uugl.Ius a it' uu'l;ut,'ul t,p lite i,tt.uticuilau' l)ouly tutuder

umiiusiuloi'atiu,ui. lui (lut' tuuuii-tlisu'u'r't,ispul qui;uli1uuus (lie disliuuutioui is iuuaile liuuu.uiglu (lic' ('aul,lu-iix('ul 'cc-uu'diut;tt,e itulicaliotu.

l"tiu'(luou'utui cuu': (r,,,,,,, =sucitrt'u' stuc'iigtlu vi'tt,tui' vit,iu length ìV,

rol)-ri's.'iuliuug lue suiuu tues out liii' ltu-l)(u(ly. rrtuoruuial velou'ily

'is'lor

%'il.lu louuglli i\',,, ro)tesuuuliuug (lie tiouuiittl vu'liiu:il.y ilisttii,uul,ioiu nut lite

uuP-lcu,(IY uluie lo (lue iu-luociy itt liuu' j-uuioclu'. (c'li,,11, rr N,, u A',, tuuat,rix

uul i'uauu lilu'X iucll iuutuoe csn'(liu'it'uil.s u,iul:iiiueil fuujuut (Sluuilliiuii (

i 'I) iii _ ii

siguuilìi's lite su'If-iuulluueuuçe of a Icusly a.uuui un n. siguuilies tite

uhu-t'uil' tuf siuturu'u's cit ilu(ly lu Ott tite uiiiruuua.i velocities oui body mii. lui

!)lI!'I?.'%(.'

use (lue M l'i' rotitiute l'lN(flEEN for the evaluuatioiu

iii' tIiu' ( ru'c'ut l'uutuctiuiu auucl its (lerivat,iv('s. Sur Newttttt.ti (I 9.S5)

i 'lue u I btu' 'cte soi t rce stretugtli vis' Lors o,,,,, a re solved l)y Pit (ti uig

all ii,,,,,, v;tlutes isiuu;tl to zero for un tu. for j = 1.6. 'Flue (lilfrartiout

ru'lat,od soit 't'o siretuglIus 7,,1d are solved by ta.kiuug ituto account, all 11,,d v;uluii's siuuuiilta uueotisly. 'luis set, (if eqnaliouus cati 1)0 solved icy direct

stuluufiutiu lc'c'lu iuiq ups or by iterative solvers. 'l'la' cotti putttitiouial effort iuvtulveil willi iterative solvers is quuaulra.tic itt (lue uuuituuber of iii kuuowtus wiuiiu' b ir direct, suulvers (lue eli'ort is a cubic fiuiurlloui of (lie tuiuuuti)('r of

uuuukuuucwuis. As a l'uuturliouu of (lue untidier of bo(lies, however, (lite to

titi' laut, thaI t,iuc' tttuuttla'r of cases to he solved is proportional lo tite

uuiutuubu'r iui hiuuilies, 1,1cc' colui puitafiouual elfoiL for ali iterative solver also

l)ei'cpttui' tu u'ttluir lttitu'tioiu of lite total uiuutuil)er of uuikuiowtis. Ntuuuuerical

c'xpe ui tu toit is w i Lii va ryi tug it it ut1 iou's tuf udii i ilar 1)0(1 P5 t'oit li ritt fItis tretud

As a lui l5S lilo alter tta.ti ve tu u u si tug ei tIter of tin' above tuieuutioited

soittliouu tuti'l,hituuls we louve i tivestigaled a. uitel,ltod vluicli is ai)plica.hle sueu'itiu'uu.liy tui tite utuuult,i-ltudy prcibleiti. Titis titeiliod, based out

stuc-c'essive tu lupruuxiiuutu.ti(utl of (lue total sotu tue slrcuigtIts iuichuuditig ali

inter-u inter-u lii inter-un 'ib'it,s , starts frouu i tite sul it (,i(in of Lite soti rue stretigt,ius oit

a. luiu.iy wil,liuiuit, iiulhuueuiu'e frutti l,lue oUter lioulies loused out (lie soluil,iouu ti, (bu' l'ctIlutviiig set, (cf eqtua.tioius:

(1 _{,,, ,, =} (17)

'Fuis oiierat.iouu is carried out fur ail iuuclies aiuti ituodes and Lakiuug advauutage of auuy attd ail syitttuuetries preseutt iii (lue bo(iy geouuum't.ry.

At, the saute l,itiie (lue iuuverse utiatrices Gn; are cotuupuutc'd and stored. 'l'ue thus obtainetl source streuugtlus are uused Io compuite new itoriva.l velocities oit ali other (hixed) 711-bodies uisiuug tite followitug relatiotusluip:

On

=

V7(m,I,j = (''",,.,. (18)

Based out these uiormal velocities, corrections to flue source strengths

are coni pitted whticlu, in tuirn generate tuew correctjoius to (lue normai velocities oit all oIlier bodies. Tite corueu:tiou,s to the sou roe streuugtiis

follow fiotti:

(1mi,j = G7Ç,1 . - V ii,11 (I 9) Knowiuug the c_uu, n, j values oui all îuì-1,odies for (lue j-uuiode of luody u, (hue uuoruuia.l velocities oui a body k follow fiotti:

M

= >

Giq,,,, . (20) 1=1

Alteruuate a.pphicatioit of eqitatiotu ( I 9) and equation (20) rest,lt,s iii successive correctiotus to the soturce sireuuglius which are at eachu s(np

a.(ldu'(h to tIte i tu itial vai lues. h n tIte khutal case (lue suicus'ssi ve roureu:t,iu tus

to tite original source stretugtlus wiil vanish vitiu itucreasilug utiuuiuher

of steps ut Lite approxiuuuatioiu. lit tite lituuit (lie same solui(,iotu will be

reac:hie(l as is tite case wi Lit (lue prov iouusly i uueuuiouuec h solo fiori tiuethuoi Is,

'l'ho u'ouiu i)it tatiotual elfori is of tIte saluti' niiler as for tite previo tisly

iiteiutioited uuuethuods. A tu ìspec.t of soucie i uuterc'sl, iii (lue niethiouh of

suuux'essive a,l)u)roxiutiat,ioui lies iii the re.luuussl aiuuouuuut of dat.t itt core tut

auuy tinte uf tite l)roi'e(huure auud iii tlue ios.sihili(y t,o Iiuuui( (hue uuttuuulier

nl 5(01)5 depeitdiuug oui the relative hucafiout nl (hue huodies. if tIte bodies

are relatively far aparl few steps are re(iuuire(l.

'l'ue tuueLhuoch cati be directly rehuctecl to thun 1)huysit:al behaviour of

wave syuìteuuis creaLed by a body and tite way iii whuichi these waves

are reflected back Lo tIte hódy and scattered by ahi ol hier l)ochies.

if

(lue geouuuetry of tite bodies atud their relative i)osiliuuu tire sticht (,list sigiiilicattt stauudiiug waves uutay occur Icetweeui (,hie boilies, tice uutethuod

(if suuti;es.sivui al)I)roxiuua.tioils wihh hot i'ouuveuge. if, out tite l)Lhtt'I luatud

tIte positiciuu of the huu)(Iii's is studi that, sIrucuig sLuuuuuhitug wave olfeu'(,s

ate not, likely to occur, (hue tuietluitih is expected to cu)uuvergi'. This

shec.t Itas sottie aLtrau:tiouu silice we nitty, froutu physical cuuusideratiouus

related to thun body geometry stud tite reiaiivp positiouu of (lie various

haudies, have soute imudivafioiu of (lie likehy stutu'ess of tIte mutelhuod. lit

(lie following soctioiu, exattuples are given of restult,s obl,aiuicd by (lue

uuuethiod of successi Ve a.i)i) roxi iiiatiouu coiti p''' I wi flu testi It,s o bLai ucd

utsitig a direct methuoh.

4

Va1idatoii of computed results of

hydrody-namic interaction effects

iii tIte h)reviouts sertiout ait outline of (hue couuuptutalional uuuetiuoch auud

(lun tlicoreticai asI)ec.Ls Itas heetu given.

iii

titis sertioti sottie resuull.s

ol iiioheh tests wiLli two floating bodies are cctuiupare(h wiLli results of

cotti i)ittatiomus I)a.sed oit the method of flue reviouus sectiomu.

'['hiere is a decided scarciLy of modeh tesi resiulLs when

roiusider-itig iiiuuhLi-luody luydrodynautiic iiute.ractiouu of lloatiitg structures.

Oort-muunrsseuu (1979) was otte of (hue first Lo carry out systeiuiaLic uuiochel tests

wiLli two siniple shaped l)odies comusisting of a vertical cyhiuuclrical (hock

alt(i a rertauiguihar box. ltestult.s of uuuochel t,esLs were compared wiLli :3-D (liifrartiomu cottipuitatiotus. 'I'huese iuuodel tests were carried out to cheteriiu i ito, aiiioiig others, thin ad(ie(h tiia,ss auud (lain ping iii Lerartinui

u'oehlicieiit,s sud tIte effect of ititerartiouu oit lirst order oscillatory wave forces and oit ineatu secouuci or(icr drift fccrce. Ilestills oiitaiiueh Icy

vaut Oortimuersseuu which show tite effect of luydrodyuiauuiic iuuterac(ioui

iuuost dea ny tire tite added mutuas uu,uu(l daiui pi iug iii teraction roelhicieui Ls.

f\V";

₃

(Tu,,,,j

Iuir luis rea.soIi the roiiiparison of eoiiipiiled results willi re.sijll.s uf

tumid losts is liete resirictoti lo those roehhideiil.s. Specifically, tito

sui u-ge ;idulu'd uttass and datut ping interaction coellicient.s are COflsi(lered

lui tito tuoxl. section results of coiil1)uItatiouus for realistic body shapes

a te gi velu. A t. lit is ti i tun ito rosit i is of n todd Lests for stich sit apes are

avtuil;u.ltie.

Mutultil tosts wore cairieci omit, wit,!u a uurrutlar cyllndnr(l)ody I) next Lit a reet.auugmiiar box (body 2). For titis paper-cotuipiltations were car-rieti iusiuig t!to panel uuiodeis shown in iìguure i. The total uimuinber of iu;uiueis miii time cylinder aiu,oiujjl,od to 392 atud oui tite box this auutoün

IA 432. ( otuipuiteui and iiieasuu roil dala are given for tuo foiiowiiig

u ii iiuoiisii ums of tite 1)0(1 ics:

l"iguu u-o i: I 'autel uluuumi('is of (yiiutmler atud Iteetsuiguular Box

osti I l.s turo cotti pared for to d islaitces between lite 1)0(1 ics i.e.

1(10.11 ut and 150.0 'ui.

4.1

Discussion of Results

'lite ju'siulis of (0111 i,mtt.ttiku,is aro coot lutinaI vitim uiueas(ure(i dala atid 051111.5 mii m'oiiti)utatiouis carrioul tuul, by valu Oorl,uiuersseii iii figure 2

lii rmumugii ligiu ri' 5. Figit re 2 shows lite surge added uuuas.s coil pling

m;oef-hicioiul.s at au -niud a21 it for tite lOOM iii distatice. Figure 3-shows lite

stuìge tl;uuuupiuug u'ouipiiiig coeilìcieiuls b1 121 aitd b.21 ii for lite saune

(lis-l.a tito I"iguuro 4 atud ligure 5 show lije correspondiutg (lala for thelarger

miisitu tito tuf i 5O() uit.

lit lioso iiguuu-os Olily Olio rm)iiui)tute(i iiimeis sluowii for tite coefficiettts. 'i'iuit.ii4 diii' to lue Íiul lItai, the u'oiiipiited resuuil.s couuuplyaiuuiost exactly willu lito -syiiuuruetu-y relaiimuusluiiis givetu iiI (1uitt.tiOut( 12). 'lite resluil,s

slut mwiu- lut the ligmu res i uidkate a it exceiieuu I. agreeuuuelut lietwceii

eoiui-ittuteti aiid liiensui iou data.. 'l'ue iItOtt.Sli 10(1 (lutti, also i'iuiilii'uit , lii rouigh

tite goiuii ugru'otuietil. luutt,WeOlt lIte iutt'asti od coeiliciu'utl.s, tite' syiuu ttuel,rv a lit tush ills.

10x105 05x105 05x105 --1,0x10 0 D a-1121 S a-2111

-

DELFAAC-frequency i ris

Figii re 2: Sit rge added lisas coil pii tug cochiieie,tt.s for (iiSl.aiiCC 100.0 lut

\

frequency in ris

Figure 3: Surge daluiuittg coiu)lilug cooilirieuit.s for distatuce I 00.1) tut

5

Verification of the method of successive

ap-proxirnation

In order lo verify tite accuracy of tIte uuiel,itod of successive

approx-iuuuali011s sollte COilil)mlltil101IS were carried out. for 2 free-floatliug ,fuhl' loaded 200 kdwl tankers. Tijesel-lul) ofl,he tauukers is shown in figure 6. 'Flu is CorresiolidstypiCahly to tluerelal,ivo posil.iout Lita-L a Slulittie ta n kor uuiighuL assnulue wlìeui luloored- ill tandeiti to a. tanker-based perittalueuliy luloored -lloati hg prod iictioii , -storage aiud oliload i itg u ni t (Fi 'SO). lui

reality Olte of tile two vessels would be iii lite baiia.st coludiliolt whtile

lite otiler -is in tite futily or partially loa.did cohudilion. For Lue sake

of collI parison we have assululed tite fully-loaded coluditiout for liotit

vessels iii order to be able to give au inupl'essiòn oI'l,iue shuielditig nilect

of tite I I 15() rebtuve to tite tuiker huloored lo tito ski Il I hut Shuttle taut ker is.positioiicd witit iLs how 50M Itt ,asl.erii of llietFI'SO, lite cetuter

of gravity of tite vessel is 25M to the starboard side oF tite FPSO atiul tite iteading angle is 10.0 (logrees to l)ort relative lo the -FI'SO. Tite

uttain dimensiouts of bolli vessels are tus follows:

(,yiitider

I )itu iuumtl.er I )rtuft f158 lii :toO ti Box i .outgtlu i reaill.lt l)r,uft, 109.7 un 101.4 ui ;to.o n V;ttoriIept.11 220.0 uit .6 4

E U)

z

₅₄ C o' C E c'i 10x105 05x105 0 -05x105

ligure

l: Surge ailriuitl into's rout)iilig roellicietil.s for distance l50.() iii

50000

25000

trequency in ris

4

trequencyin ris

liguri 5: Stitge dauiuitig u:oii)liu)g u'O(lliri('uIlS foi disl,auiu'n 150_o iu

8

'l'ue rda Li ve posi Lb n (Jr Lite vessels is showit i tu ligo re G wit ile Lite

tiri iiiuuulrls al-e sluowut iii ugt re 7. For boLli vessels tite salite panel utuudel was uscii. 'the total tuiiiiihu'r of p;uiuels oit carli vessel alltOtiiile(l

lu :oxx. Ait iii i urtaitL 8S1)PCL of liii) liyd rorlynauiuiu: iriteracLion between

lue vEssels is i-u'laIe(l Lo Lite tuicati sevoiid or(ler wave (trill Forces. For

liii' sake of

brevity

we have restricted ou r cotti l)arisons to the titean suit-ge aialsway drill forres ni! Imiti vessels for (tite wavedirec.Lioii heiiig

lue lirai! sea u:u)it(iitiout for lite FI ISO. 'I'IiIs locatis that lue (!irectiO!i

utnul isLui rl sal waves relative to lite olftaking lait ker is IO degrees

oli Lite sLarluuuard how. lIte resijlls OF ioiuipiil_atioitsof lite iiuea.ui (Irifi. forres art' sluuwti if ligure M through ligure li. it is of iiul.ercsl, Lo iìot,e

luir tite uitnl.lioul u1 sutu:u:cssive approxitutatiotuottly .1 iteratlotus were

utuailc'il tu, i)ruuuluuu:e uouivu'rgeutrn. 'l'itt' u:uniipuulatioit tuile %aS reduiced

lv 45

rcia.tj ve Lut tite direct, solver.

Figuire 6: Relative position of FF'SO itii Shuttle Taiiker astern

Figut re 7: ['aiuci titodeis of F I 'SO tund Shit tile 'l'ait ker

5.1

Discussion of results

Tite results sitowuu iii Lite figures iuidicate that tite resuills:obtaiited b

the uitet!iO(l of successive approxiutiatiotu are identical to those- obtaiited

by tisi uug a direct solver. As itudical.eci previously, tite method of situ:-ce-salve approximal,iout will he eFfective when sl.andiutg wave ellcu:t.s are suitali. For tite give-li geometry this is hide-ed tite-case. Frouii -lite 1)Oittt, of view-of the hydrodynaunic -ituteraLioiu elfecl.s- theutuselves il, -rair iii'

itoteil l.liaL -theme-ti-n surge drill. forres titi Lite Siuuut.tie Latukeu

are.siguii-ficantiy lower Litait for Lite Fi 'SO. Eve-ti thought tite -FPSO is in lie-ad seto', a situait -sway unit force is predicl,ed ritte to Lite assyutuuuuetriu:

set-tip of Lite Shuuitile tanker. For a propel- siunuuiatioui of tite itiotiotus

oF Litt' Sit ut t,tle Laut ker it, a itueti thu it il, will he necessary to Lake thu

iiydrodyttauuuic iuutera.ciiotus itito arc outiL.

6

Conclusions

in this paper a review is given of tite (:ouuul)uitatiolu tuietliod liase-ti oit

:I-(liuuueutsioutandiulra(:tiotu tluuiory fon tietertutuututug tite itydrodyutauttit:

itt-t,eractiout of mutitiple bodies iii waves. it was-shiowit that tite

ittcrau:-tiouu roehliciettl,s for added tuta.ss and (latut pi ng at-c correctly piedit:tcd for siutipte body shape-s.

Tue solution titethiod oF successive approxiuttatioui was tirate-ci and it. was sliowit tluatfor a. pu'tu:l.icai ctuan of a Situittic tauuken iutooied aft of

aîi FF'SO yielded re-stilts which are i(leniicai Lo tite-se obtaiuued ilsiiug a direct solver. For this particular rase ottly 4 iteratiout steps were

requtired to achule-ve titis re-stilt. Couttputt,stioit- tutte- was reduiu:ed l)y 45

percent. Front the above it ce-tiM he couuu:lttuled Litai 3-diuiueiusiouial

diffraction methods are ade-(lulate for predictiiig iutteractiöit elfe-cl_s iii

waves. I-however, tue a.uutoui lut, of data. lo sut ittaut titile iii is uoiirlutsiout

is very limite-d. Elforl,s shioutlil be itutuhe Lo iuuake available nuore

ex-peri tite-ui tal resti u.s w lu iu:iu are releva lut lu ur u tu-adirai rases he-ft re a li conc.iuisiout can he re-ached. 'l'lue mitethto(i of successive appròxiuuial.iötus

ttìay i)e an i ntert'sti tug option for itium iti hotly u:u till Pt la timo' provided the geoutietry precludes sigitilicatit stattdiuig Wave ebfe-u:l,s.

tuttiiti tliltltltisit)tts I u'utgt.lu iOrt'tuuhl.hu i )rtull. I )bspltiut'uiueui t. Wtu.t,eru leptIt ;u i oo nu 47.17 tut IM_O) ut 231000.11 ni 82.5 tut

¡\N\C1 -Ct'i

Thj-.

uNt(r

a-1121 a-2111

- DELFRAC

-10x105 o 8

E -'i -200 -150

z

C -loo -50 o

Figui ri M: Mea ii sii rgn drift force FI >So in head waves (ISO (logrees)

-200

-150

-loo

-50

o

o---0 de. sol

n---° succ. appr.

o

G- dir. sol,

'-'V SUCC. appr.

.25 .50

Irequency iru ris

.75

o

1.00

1.00

-frequencyiruris

ligure Il: Mean swa.y drift force Sluul.tlu' Tanker iii head waves

Refereices

Engelen, Il. van(1995) "l?hoatiiug Breakwaters", Msc. Thesis, Civil Ei-giuieerng Dept. Dehit ti. of Technology, Dehit.

Ferreira, M. D. and Lee, C. II. (1991) "Corn putation of Second-order

M(Ll( Wavul?orcesaiud Moiuueiit.s i,i M uihl.iboi.hy I uiteractions"

F'rii-ceedings Boss'94, MIT

Løken, A. E (1981) "I lyd rodyivanuic

h nteractiouu betweciu Several

Floating Bodies of Arbitrary For iii iii Waves" , International Sy

iiiposiuini on I hyd rodynaiuuics in Ocean Engineering, Trondhueiuuu

-Newman, .1. N. (1985) "Algorithiuis For thieih?ree_suurface Green 'Func-tion" , ,ionrnal of Engineering MaI,lieiuuatius, 19,57-67.

Oortiuiers.sen, G. van (1979) "Ilydrodynaniic Interaction between Two Structures ,

floating in Waves", hiaper No. 26,. Proceedings

Uoss'79, London.

Oortuuicrssen, G. van '(1981) "Sorne I Iydrodyivauuuic Aspects of Multi-body Systems" Iii te r national Sy in 1rnsi il un on Il yd rody nain ics i n

Ocean Engineering, Trond heiw

F'inkster, J. A. (1980) "Low-frequency Second Order Wave Exciting Forces on Floating Structures", Publication No. 650, Netherlands Ship Model l3asin.

Wichuers, J. E W. W. (1988) "ASiniulatioui Model for a Single Point Moored Tanker", Piibhicatiout No. 797., Maritiiuie Research

Insti-tute Net,luerIauids

T .Ñ. P>

3t<STEt2

¿

75 60 E

z

₄₅ 'ç G, o o 30 15 O .4 .6 .8 1,0 frequency in r/s

l'igui re 9: Meaui surge (hilt force SIuuitili' 'I'aiiker iii head waves

.25 .50

frequency ruris