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Nonlinear and Linear Motions of a Rectangular Barge In a Perfect Fluid R. Cointei, P. Geyer2, B. Kings, B. Molin2, M. Tramoni2 Assassin d'Essais des Carenes, France), (2Institut Fran~ais du Petrole, France) ~ Abstract The motion of a rectangular barge in beaten seas is studied within the framework of potential flow theory. A simulation technique based on tile Mixed Eulerian-Lagrangian ~netl~o`1 is de- scribed. It allows the simulation of tl~e how and tile resulting barge motions to lie performed wilds either linear or fully nonlinear boun`lary conditions on tile Idyll an`! on tile free surface. Efficient artificial bou ndary conditions are i~n- ple~nented tight allow nonlinear and linear si~n- ulations to be performed over a large number of wave periods. Results from linear frequency <1 - main theories are recovered and nonlinear pl~e- nomena are presented. 2 Introduction Nu~nerical remodels based on linear potential theory have proved the~nseIves to be efficient tools to predict tile seakeeping behavior of floating structures. As a matter of fact, very often they have been found to perform surpris- ingly well given all tile litnitations of the tI~e- oretical framework (sn~all ~notion., small wave stems, Coo viscous eire`..ts). A known exception is tile roll Notion Of ships ant] Urges at or near resonance, where (litrraction-ra(liatio~ co(les are known to over- predict the response. Addition of a supplemen- tary damping (accounting for tile vortex sI~ed- ding at the bilge corners) to the motion equa 85 tion usually notably improves the prediction. For barges however, it has been argued tight the nonlinearity of tile exciting forces ((rue to the large variations of the wetted part of tl~e Imp) should also be taken into account (e.g. see Denise [1~. The relative importance of these two factors (viscous damping and nonlinear ex- citation) leas lee} to seine controversy. In past years some attempts slave been Inane at modeling the vortex sI~ecI`ling from the bilge corners within linear potential flow ~nodels. Excellent agreement leas beets reported witty available experi~nenta1 data id], t3] . This success seems to rule out the possibil- ity of the nonlinearity of tile potential loacli~g playing an i~-nportant role Ail. It is our feel- ing however that the publicized numerical a.n(l experimental results are too scarce to entitle any general conclusions to be dra.w~. Obvi- ous~y, many parameters such as tile slope of tile bilge corners (sI~a.rp or rounded), tile beam over draft ratio, tile amount of potential da.rnp- ing present at resonance, etc., are to Inlay a role. (We are currently perform i Fig extensi; e exI,eri- ~nents to forge our own religions, by varying all these parameters). It seems therefore to be of interest to quantify the amount of nonlinearity present in tl~e excitation forces, ant! tills is tile aim of the numerics techie described in tile present paper. 'ago investigate taxis problem we make use of a time domain numerical model, Sin(l- ba(l, based on the so-called Mixed Eulerian- Lagrangian n~etho(l, which simulates a two

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dimensional wavetank. Sindbad I' as been val- idated on such problems as sloshing in tanks, wave generation by piston type wave makers, and wave diffraction over submerged cylinders, t5i, ted, [7~. Two difficulties had to be solved in order to extend its capabilities to the wave response of a freely floating body: a means must be devised so that the waves reflected by the body do not reflect again on tile wavemaker and pollute tile incident waves. In wave basins taxis problem is avoided by locating the body far enough from the wave maker, so that a steady state can be reached prior to contamination of tile incident waves. In the numerical mode! it is necessary to re- strict the length of tile tank in order to limit the size of the problem; the hydrodynamic loading on the hull needs to be calculates! carefully in such a way to permit a simultaneous integration of the body motion equations. Section 3 is devoted to a short descriptions of bile mode} and to the way these two problems have been solved. Even though Sindbad can be run to solve both the nonlinear and linear problems, it was felt that a proper assessment of tile validity of its results required some confrontation with more traditional linear frequency domain mod- els. Comparisons have been made with two such models, one based on matching of eigen- function expansions Or tile potential in tile three sub-domains limited by a rectangular barge (left, right, and underneatI~), and the artier on a Rankine source distribution on the hull and free surface. Both models are briefly described in section 4. The last part of tile paper (section 5) is devoted to tile presentation Or some nu~neri- cal results. First tile Midyear Otis Or Simian is checked by recovering, once a steady state Iran been reacts, tile frequency domain results to a satisfactory accuracy. Results from non- linear simulations are then presented for the diffraction problem (fixed barge) and for the diffraction-radiation problem. 3 Nonlinear and Linear Tran- sient Solutions 3.1 Introduction Since the work of Longuet- Higgins anti Cokelet [8] that pioneered the Mixed Eulerian- Lagrangian method (MEL), this method has been widely used for the simulation of two- dimensiona] free-surface flows; numerous i~n- plementations exist. Many authors slave been interested in the description of steep waves and the MEL has proven very successful in describ- ing the flow up to overturning, see for instance ted. Vinje and Brevig {10] first modelled waves interacting with rigid obstacles, either fixed, in forced motion, or in free motion. This prob- lem has since been addressed by several authors (e.g. [11i, [12i, [13i, [14~. The motions of a rectangular barge have been studied using act implementation of the MEL at St John's [15~. However, many of these studies were somewhat qualitative and it appears that little attention leas been given to the validation of tile resulting codes by proper comparisons with other tI~eo- ries and experiments. For instance, if computa- tions of the free motion of floating bodies have been performed, systematic comparisons witty results from linear theory are almost nonexis- tent. In our opinion, it is now time to validate the MEL and to use it to provide quantitative results. An implementation of tile MEL at the Institut Frangais du Petrole, and in cooper- ation with DGA, led to the development of the code Sindbac] (for Simulation nu~nerique d'un bassin de houle). Developments concern- ing this code are presently under way at both organisms. Its application to the simulation of two-dimensional flows in the presence of a submerge or surface-pierci~g body in force(l Otis Ails been `~scril~e(1 elsewhere ill solve letai! (e g., tbi, A) I~ tile next sections attention will focus on tile new developments tilat Ilave been necessary to (leal with a freely floating bo(ly of arbitrary shape. First some indications concerning tile principle of the simulation will be given. 86

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i''' = ~ fe ~ + In 71, (~1) v s rd ~ ~4 Wave maker & Floating Absorbing zone absorbing zone boa, Figure 1: Si~dbad wave basils geometry 3.2 Outline of tile MeLl~od lee attain idea Of tItc n't''~erica.t procedure is to caboose markers i''itia.lly at tl~e free surface I'm! t`, f~,ll`,w tl~e'', ill their '''`~li<~. We use ~ coortli''~te system (~,y). 'rile x-axis coiticitiefi witIt tile refere'tce l~osilion Or tile free surface a'td tile y-axis is oriented ver- tiC?~ly t~l~wa.r`~s -see figure 1 rear geometric `~e0~,iti`~fi. 1 I'~ Il''i`! is ass'''~! t<' I'n i'~- l~ressit,le a.~' tI'n Ilt~w irrt~tali~.' so 1~;`t tail velocity field v is given by: v= V+, (1) witty: A~ = 0. (2) 'I've coal io't is l,crf~,r''le(! ill t~ot'''4etI do''tai'~. Along rigid boundaries (x ~ i',,(1~), tI'~ '~,r''~.! Wily ill 1.~e {~'i(x). 'Ellis yields': I9t 2 (~ s + 2 In p + city) <3~) - IFor the sake ol simplicity, we use llHitS Sll(h tl~at the acceleration of gravity, .9, the specific mass of water, p, and the sleuth of the tank, h, are eq''al to 1. wipers D i8 used to implicate a 'bacterial deriva- tive, ~ and n are vectors tangent a''~1 normal to tile free surface, respectively, amuck ~ is an ar- bitrary constant. Taxis co'~sta''t specifies tile tangential Notion of tile remarkers: ~ 1 i(len- tifies marker as particles while - O yields a zero tangential Notion of tile remarkers. 'Levis last cI'oice allows a current to lie si'~'ulaterl in tile tank. For tile applications discussed I'ere, = 1. The pressure i8 assigned constant along tile free surface; it call tI'us be included ill tile function of time city. Witty an appropriate choice of tile velocity potential, taxis r~'ctio~ can be taken equal to zero. Tile kinematic co'~stra.i~t AI-O. associ- ate(l witty tile boun(la.ry co''`litio'~ oil 1',,, per- '~,its tile free surface l~o'''~la.ry co'~`litio'~s (3~- (~1) to be expresser] as all evolutions equation for (,x). 'finis stereos frolic tile fact tight if, at a givers instant I, ~ is knower along plot) anti ~' is known along l.~n~t), tilell 4>,, Call be co''' l,'' te OCR for page 85
curvilinear a6fiCifiSa fiO tint tile calculation of tile Iliatrix e~elllelit,8 is rattler side a''`! vec- torizes well. Special care I,as to be taken at corners of tl~e fluid I'oun~lary and Fore particularly at tile intersections between tile free surface and a piercing Wily. Billie '''.''''~rical treated limit is used tilers is hase`l on a local asy''~l~totic a''al- ysis in tile weakly `'o`~linea.r regime tight corre- Si)On(~8 to a small accelerations (relative to grav- ity) of tile bo`]y. 'l'llis n`~erical treatment is discussed in snore details in t6] acid tot. 3.3 Wave Generations aback Absorp- [io'~ 3.3.1 Generalities We are Oiler i'' lid tale ''~'ti`~s of ~ Il~.`t I'd I,`~ly i'' 1 lie and aces'', s~'it- ted to a givers incident wave field, over several wave l,eri`~. Si''`e tare l,`~.ry `,r tall c''- tire n~'i'! ~Iot'~ai'' liar to be discretize`l, artificial boundaries must be introduced. I7or tile nu- ~nerical procedure to be eiHtcie'~t, tile Quill dm '''~i'' sI~' l,~'as s''';~.~' an 1,~,~sil,l~'. 11 is 1~`r`~- fore 'necessary to generate tile incipient waves and to avoid their reflect iota oil tile artificial bou''`laries introduced. IJitticulties ill reacl~i'~g a steady-state, event for waves of spa. steep- ness, probably explain wily con~pariso~s witty linear res''lts I~aVt? ''fit t,~ ''lore syste',~ali- ca,lly perfor'''e(l. Wisest tile stea`ly-state li''ear sol.~ttio'~ is c.o'~1l,'110tl, all ;J\( ill wave is l,~escril,`~' allli a ra(liation condition is written tila.t fral~slllit- te`l e'' d reflected waves '''test satisfy. Wrili~g ~ proper radiations conditions for tile second-order problem llas Slid been a splatter or controversy. For tile tra.nsie'~t problem, tile proper loel~av- ior at infirmity can be acco~.~'te`! for by using ~r`,l,er emery I, bile (~'rec'' r''''`- tion. 'Ellis al'I'roacIt requires a convolutio'' ifs tinkle a''<] is o''ly 1~fisil'le if tile 1,rol,le'~' is li'~- ear (at least ill all outer do'~a.i'~ exte''`li~'g to i'~fi~ity). I~t tile 3,l)fi~ICO Or Rely '~'e'~atically sat- isfyi'~g answer yiel(li'~g perfectly transparent boundary conditions' we slave chosen a l~rag- '~atic solution sitar to that use`] for all ex 88 peri went i n a tank. 'l'lli8 approach 4OeS IlOt i ll- volve ally I'ypotI~esis co'~cer''i'~g tile steeliness of tile outgoing waves. Waves are generate by a piston-type wave~naker and tile tank is closed at tile other end by a vertical wall. Two ~la.~l,- ing zones, one at each end Of tile tank, are used for tile al~sorl~tio'` Or tile l~arasilic waVe8 that are generated in tile tank see figure 1. 'Ibis is equivalent to having a'' absorI,i'~g beacon at one e'`d and an absorbing wave''~aker at tile other en'l. 3.3.2 Absorbing beadle Tiff absorbing beacon is a dig zone, si~- ilar to that used in Al. Ill taxis zone, tile free surface boundary conditions are '~o~lif~e~l lay ad`li~'g a calming terser. We write: '~` 2 () l' + 2 'r~ we're) (-'e) (G) - - ~ /~ ~ + din 7t-V(Xe) (x-ale), (7) where tile s~ll)~ctil,' ~ (orresl)ol'(~s to tile r``rer- e'tce co'~fi'guratio'~ for tile llui~! inhere, tile fluid at rest). Lie pri'~ciI,le of flus (Ia.~,I,i'~g zone is to absorb tile incident wave energy before it call reacts the wall. It may be seers intuitively tI'at, if tile absorption is too weak, part or tile i'~ci- ~ent wave energy Will reacts tile wall a. be rellecte(l. Inversely, if tile a.bsorptio't is too st,rol~g, It Or tllP elIergy will lie rellectet! I,y tile Handling zone itself. 'i'lle choice of tile tla'~l~i'~g coetH~cie'~t tJ(X) is crucial to its efficiency. rrI'is coeib~cie~tt is equal to zero except ill tile Wing zone (*0 < x < x'). In taxis zone it is choose to be con- ti~'uous an(l continuously `lilTere~tiable, anti is "tilde." to a characteristic wave fre~l''e~'cy atoll a characteristic wave number k: I,(X)-tt~ [2 (x - xo)1 (8) 2~l3 no < x < x1 = To ~ k walers car ant] ,6 are ~lililelisioll~ess I,a.ralileters.

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3.3.3 Absorbi'~g Wave'~aker 'I'l~e al~sorbi'~g beacI' al~ows lo~'g silllu~atiOlis to be perfor~ned in tI`e nu''~erical tank wI~en tI'e generation a~d tIte propagation Of surrace waves are ti~e o''ly co'~cerns. A ~lilficulty al'- pears ror tI~e si''~ulation or tI`e rree '~otions Or a body i'~ tI~e ta~k. Waves are l~artly rellecte~! 1,y tIte bo~ly a,'~d tIte'' rellectetI back by tIte wave- '''aker. In a real ta,'tk, tI'is 1'rot,le'~' is overco''`e by locating tI'e tested body sufl~ciently far fro~n tI'e wave'''aker. i'ar tI'e nu'''erical ta'tk, tItis would i~ply too large a nun~ber of discretiza- tion points. It is tI'erefore cr~tcial to develop a, ~netI`o`! to avoi OCR for page 85
'i'llifi C(lIl~tit)~t ('RI' D.~) t)O Wlit,~,011 11Sil~g '~n,tri cia,' ''<,t,ati<~s O.fi: - _ M To= 1'. (12) Eater forces al,I,lie~! to tile body, I', are written as tile 8Ulll of I'ydro`lyna''~ic forces and ogler external forces. No ~listi'~ctio~ is ''table l~etwee'' I'y OCR for page 85
In e~l''at,io'' (18), the velocity `` is givers lay e~l~atio'ts (15) a''t! (1(~. 'I've a,ccelera,t,i`~' ax is given by: Hi - Ma-(p-YG) H-((X-AGO 82 (19) if - YG + ((X-EGO ~-(F-YG) ~ .(20) 48 a consequence of tile8e e`luatiol~8, fin ~lel~e''`ls li''early o'' ~,r,. As ~ rest'lt, I~y~lrm `lyna'''ic rorces a,l~l,lied to tI'e botly at a given insta,nt t can fi~'ally i~e writte't as: [7h = M XG + :o. (21) 'I'l~e vector [0 corresl~o'~Is to 1,l~e forces al~l,lie`! to tI'e I'o~ly wI~e'' it8 accelera.tio'' is e~lua.' to zero. 'l'l~e ~na,trix M' is si'~ila,r to a'~ a`~de`] IliaSfi ''~atrix. It,s coell~cie'~l,s ca.~ I,e evaluat,e~! by co'~p~ti'~g tI~e forces corresl~o'~di'~g to a, u'~it a.~< olera,t,i~ ~,r ~ '' `~egree `,[ free`~. It, sI~'l<1 l~e ''ote~l tl'~t i'~ order to e.sti- ''~a,te t\~e rigl~t l~an OCR for page 85
right Or the Charge (l,) I lert (~' --- see fi'g- '''~ ,l. 11 l'.~n aIr<~.~ly l,~t a1,l,li'`~' tt, tI' salute l~rol'le''', see for i''sta'~ce Mel a'~' llIack [lid. (Our '~atcJ~it~g l~roce`Jure `lilrers so~e- wl,at fry tllI?ir,fi, a,11~\ is lit try (Barrett jlBsi). 'lithe total Velocity l~ote''tial is writte'' ill tile classical for''': ~ I'=! where ~'-XG, x2-YG am ~3 = 49. 'flee ~liRraction ~,ote'~1 ial rev am ra~liatio' ~ote''tials ~`j ad'''it tl~e followi'~g ~leco'',l~osi- tio'~: Do~nait~ 1~: {B = bo ko Z eiko(~- b) cosll kolI 00 ~J~ ~ b~l COS k,,z ~-kn(~-b) tt=1 L)o't~ainC: 'Pc = cO - u e-ikO (2+b) cosIt k~, /! oo 1- ~ c,l cos k,~z ek1'(x+b) (24) n=1 wI~ere ko,kn are tI~e roots Or tI~e eq~ation: w2-gkO ta'tIt koII =-gkn tat' kulf D0~! ali I' A: ~A (~~~y'!) = (~' ~~~ ~! ~t R02 b (2~) c`,sI~ A',x ~t ~, ~nl - - COS ~nZ n-~ ct~sIi Anb - si I'It At! x ~t >~ 47r~2 -; ~ ---~ ~ ~ t,8 ~ Z wI'~?re A,, - ,''r//' a''<' ~' i.s ~ ''arlic''lar s<~- I''ti`~' wI'i~ I' is 7~, r`,r tI'~ `lilEra' ti`~' a'~' sway r;l`li.~ti~' l,~t'''Iin.~s, ;1~' is''`1~! f`, (Z2- .~2~/2~! for I~eave a.~d to (~/2/~) (772/3_ Z2) rOr rof I. I~t tI'e l~revious exl~ressio'~s JJ is tI'e to- tal water(leptI', b tI~e I,alf-wi`ItI~ of tI'e barge, z - y + ll, a''`' I' = I1 - `1 tI'e water(lel)tI' u'tUertteatI' tI~e I,arge, c! I,ei~'g tI'e drart. ,l'l~e coe[li' ie'~ts a,~ ~ ~ a,,2 ~ b,~, c,, a.re ~' ~ er- '''i t'e`! I,y tr'' 't~ ~ ti'~g 1 I'~ i'~ fi '' i 1~? s`'rit's .~.' st,'',~` fi~ite orders Na'Nb = NC, aI\~] ctl''ati'ig tI\e ~ote'~tial exl)a.~ 8iOllfi all `l t I~ci r x ~'eri vat i ves a t ~c-~ and ~--b. Con~si`leri'~g, rOr i'~sta'~ce, tI`e (lifFraction proble'~, e~luati'~g ~n i- ~' a''~] ~,~ i~ ~-b yiel~s, after taki'~g a~lva~`tage Or tI~e ortI~ogonati ty of tI~e cos ~n Z f'''~ ctio'~s over t0 /~] by i'~tegrating in z: _ ~ ~ _ A' + A2 - A 11 + 1 ~(26) (22) wI`ere Ai = 7 (aOi, , aNai ), B- T(bo,...,bNb)'a~] ~A is ~ Na Nb Illatrix. Si~nilarly equati~g ~c + ~ an~] ~ at x = -b ylel(ls: A1-A2-~ C ~ 1~1 e 2ikoi (27) siInilar consideratiolls on tile x derivatives ('23) in ~ = b an'' x--b, co'''t~i'~d witl~ i'~tegra tions in z over to II], yiel(1: L7=~(f~+~2~2~+ll2 (28) C = ~ (fiAl -[2A2)- Il2 e-2ikob (29) wilere 1; is a IV~ Na Inatrix, a~l~] [~ f2 are two rliago'~al rl~atrices. Some Inanipuiation ot tI~ese 4 vectoria.t _ ~ equatiolls yiei~s t~le filla' systeln ill 13 + C aIl(] ~ _ 13 - C : ~ _ [~-~1~4](Li+C) ~ ~ll~l(l + e 2ikob) + 172(1 _ e-2ikob) [l - ~ [2 '4](~ - C) = (31) [2171(1-e 2tkob) + 1~2(1 ~ ~-2'kob) TIle re8Oliltioll of tilis systell1 yiel(Is tlUtl ( ate(l series exl)~.llsiolls ror tile l~ot,elltin.ls il ea(.ll (lolll;lil1 a.II(1 a.1I('w.s tile 11y(lro(lylla111ic eilicielltS all(l tile (li~[ractioll [orces to be (Olil p11 te(l. 4.2 Lil1ear Il~tegral-equatioll rl'lle ra(liatiOn all(] (lilRractiol1 potetitialls are rOUll(l l~y tile solutioll Or all illtegral e(~uatio 92

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Isfx ~1 ~ ~ :~ ~ ~ ~ AL 7 SO / SR FiglI[e 4: (;~01~1Pt[lC dC1}I';tIO1'S R)[ 1.I'0 II1~[ il~trgrn.1 e~llln.1.1cl' ~- ~ ~ INS ~ ~ ~ 141CI`1.51.y. '1'1~P "1~1~)n(:I1 1IS2~1 ~II0WS ticket 0[ ~ ~J ~n~ [~1 WE ~ ~ (I(,IB1;~11 LIZ IR 1In(!(I R,r 1,II(! p1'()I)I{!IB1 S()I1I'.1~11; ~O r~ ~"d~n~ ~ ASHY ~ M~K -~ ~ Bug Bun ~ US ~ 1~ . . ~ a, . ~ , . . B@1)1)[O"@(:I' IS I[Ce (JI I[1~41I~r I[~(111~11C#~S ".11(' IS 1ISUf1Ii Air (:;1.147[1II (:;1.I(:1II;1.1~11 (A IIII(~;1[ I1y(I1-~1y n~K Pay. To 1I11k11~)W1' 1~1~7111~;1.I ~, ()r am 1.~.k(!S tiff! him: _ ~ ~(',) -1 - o 0~1 1 / ~-~o -1- / (~ (~} - (at) . S" 0" J" = /~ ~ An 0~1 (:12) WD~ am Gay ~n~- IS ~ GINO ma= -~t nI'`' I\, in 1.1Ill~l~l Ill 1~'ll cl`>se to 8 s il1 tl~e slln.[l' fllers t:nse@ 'l Ile l~nl~r w11.1' rt~lIll~lr~I `:q,['lr[R llnn n. 1~61~e r~l~lllIs `!~ll~n. ~ 1~ ~ ~d i~ n-~ ~1- in ~l is tn 7.5 8. Ilf~tlI 1.11e 1.[n.'lslr'`1, `:~,llll~lI1.n.1.1~,ll n.ll`I 1.llr Oi8fIl-[lI11~1.~11 ~XllA.llSif)11 n.[P 1~[R>rlllr`1 i1' A nl~ depth. In -rr ~ ~d Gnl~ ~Irpth rP ~cts1 ~ `1~1.1' o[ 15(} 1~' wns 1lse`I. 'I`lle illlen.r

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~ 150.0r i''l,'~gr;~.~-~'l'';~.~.i','' ''tt`1.~' in 1~'r~,r''''`~' ill illli- Z ''it'' 't<~1'll'. '1'0 ~('011~lt [tar 801110 Vi8t'0118 dig and to avoi<' very l'''~g tra'~sie''t l'l~e''o'''e''a i '' life '''~steatly co'''l,''ta.lio'', ~ li''car It cm ellicie''t e~lua.! to 5 1()5 kg..- (a','] corre '',''''i''~ 1~, It's ~ .1, '?.'r,% Or ~ ril i' a.' <~;~.~, 1'i'~g) WE ;~?~! tat Me e'1~.1.i''''~ ',[ I. .Si''' IlolI-%ero tIl~ri7,~.' drift rorce is eXlleCte~ [Or tl~e nonlinear co'''l,''tati~', a linear restraint i'' HWR.y WR.N D.lR(, ~. 1 llP ('t~rit'81~(.~(til~g 8tilr ''ees i~ e~l''7~.' t~, 5 1(~6 N''`-2 a''<' tI~e 'l;~1'i'~g coellicie''t is e~lun.' lo 5 1()3 kg.~-~-~. 'J'l~is Ien.~s to a. ''~t''ra.! 1~eri~' i', swny l;~.rger tI'a. li ve ti'''es l t'~ 'ta.l '' ra.' ',eri<~] i '' r`~. D.2 Li''en' C,'u'''~[ntio'' 'l'l'n li'ten.r verni<~' ',r.cli'~.~' w;~s ''s~' '~, <~- 1,''1~' 11~' 1 r.~ni<~1 .~'~I,~tf;t' ''r ;~. '<~' I ;~;~.r t'arge sui,'llillef~ to ;~.~' i'''i OCR for page 85
E t;~).0 at o 30.0 ,- ~0.o _1 ILL 30.0 -60.0 1.00 ) ,. 0.0 60.0 a. r-~-~ t-- ~-~- ~ ~ 1 120~0 180~0 TIME (s) I;'ig''re 5 Mo'''e''t V8. ti'''e (li''ear si'''ulatio'') 0.200 ' '- 1' -- ' ' ~1 240 0 300.0 F;1~11~.1 (rounds/ on) ~Sindbad {rounded corners) _.IttIeg~liq (!lil~lil, C<,lileis) ~_Eiger) Functions ( harp corners) E ~ 0.150 g(~.fi~) ~ _~: =01005 ~ A ~ ~ o.4o ~5 I `` / ~ ~0.075 I '] /" ~ {0.201 1 j0.050i / / '-BY v, ~1 0.0254 it/' / (),(~) 1' ' ' '1 '' it' ~ 6.0 7.~' 80 9~, 0OOO~ ~, PEItI()I' (A 6.0 7.0 8.0 9.0 PERIOD (s) Fig''re 8. Swa.y trall~fer [ulictio 2.00 E 1.75 E 1.50 _' w 1.25 ~ 1.00 Q :5 0.75 > 0.50 ~c 0.25 O.OC Figilre 10: Itoll tra.'l~rer [tinctioll Sindbad (rounded corners) _.lntegral Equot~c~n (rounded corners3 . ... Integral Equat~on (sharp corr~ers) _Eigen Functions (sharp corners) ': . ~ ~ , . . . 6.0 7.0 8.0 9.0 PERIOD (s) It'ig~lre 9: llea.ve lra.l~srer [ll~lct.iol' Figilres 8 to 1() fi~IOW t~le SWR.y, ~leR.Ve, a11~! r`~! tratl~rer r~lil'[i~,l's 1~,r. tI'e i,~.rge witIt ro''''tlet! cor''~re. 'l'lte trn''fiiet~t si'''''la.ti`~` is 1~Pr[Olllle(l Witll ,2~) Il(~8 Of, tI'e free st~rra.ce a.~' 8~} o'' tI'e \~.rge. '.~.~e Ie'tgl.~. of ll'~ t;~.~,k its a',l~roxi~nately e~l'tal to G wa.ve le'~gtI's a''~! ll~e barge is locate`] at ''~i'ltank. I'o''rly ti'''e stel~s ''er ~,eriot] were use~. }\ goo`! a.gree~,ent is obtainet1 between tl~e transie''t si'''ulatio'' ant! |,l~e rre~l~le'~cy-~'laitl renilil.. '1'l'~ ~olil,lilig I,~- 1,weell tile tilOtiOll Or tile 11 11i(l R.11(} t1l3.t Or tllP i,;ltgO i8 1)rOl~er~y 1~1(~(~(~. D.3 No~llillear Motiolis of ~ llectn~l- gular llarge 9s 'l'lle sa.~'le col~ll~lita.tiolis n.s berore l~a.ve beell l~errOrille(] ~lfiilig tile rUIly llolllillea.r versioll ~r

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Sickbays. Due to `li faculties related to tile sliarp corners3, oddly tile recta'~g~lar barge witty rounded corners will be considered there. Figure 11 shows the moment about the center of gravity as a function of time for the diffraction problem, the incident wave ampli- tude being equal to 2 m (the wave period is still equal to 8 fi and the saline discretization as in the linear case in used). As in the linear case, a steady state is reacI~ed rapidly. However, non- linear eRects are very strong and tile presence of Weigher harmonics is obvious. A Fourier anal- ysis of the signal reveals that even the ampli- tude of the first harmonic is not well predicted by linear theory and tint the second harmonic is twice as large in magnitude. Figure 12 and 13 slew the sway and roll displacements (XG and 8) as a function of time for a coating barge restrained in sway. The wave period is equal to 7.5 s (corresponding to the resonance) en c! Else wave amplitude is equal to 1 Gil. Tile saline discreti%ation as in tile li~- ear case is used. A graphical representation of the free-surface profiles during tile simulation is sinews ill figure 14. 'rI'ere are much less trigger I~ar''~o~ics Plan in tile diffraction case, the bo`ly presur~ably acting as a "filter". The response amplitude in roll is significantly modified, indi- cating tint tile nonlinear potential flow effects are important. 6 Conclusion A simulation technique based on tile Mixed Eulerian-Lagrangian method leas been de- scribed. It allows the simulation of tile now and tile resulting barge motions to be performed with either linear or fully nonlinear boundary conditions on the hull and on tile free sur- face. Efficient artificial boundary conditions have been implementer! tight allow long simu- lations of difEraction-radiation probes. Linear frequency domain results have been recovere(l using tile Mishear version of tile code. 3~s shown before, the convergence is very poor for the barge with sharp corners. Note moreover that the singularity for the complex potential is expected to be in Z2/3 yielding an infinite pressure due to the V. Vie term. Comparisons have been perfor~ne`] ',ot only for global quantities, such as transfer functions, but also for local quantities, sucks as pressures. This linear version appears to be well validated. Nonlinear simulations have been per- formed, tasting for a large number of wave pe- riods. Even though these nonlinear results are difficult to validate, the accuracy of the linear version of the code gives some confidence con- cerning tile validity of the model. References [1] Denise, UP., 1983, "On the Roll Motion of Barges," TRINA, Vol. 125. [2] Standing, R.G., Cozens, P.D., ant! Downie, M.J., 1988, "Prediction of Roll Damping and Response of Ships and Barges, Based on the Discrete Vortex method," Proc. Int. Conf. on Computer Mo(lelling in Ocean Eng., Venice. t3] Downie, M.J., Graham, ].M.R., and Zincing, X., 1990, "Tile Influence of Viscous Effects on the Motion of a Belly Floating in Waves," Proc. Nit Int. Chef. OllsI~ore Mech. and Arctic Eng., Houston. t4] Robinson, R.W., an(l Stod^dart, A.W., 1987, "An Engineering Assessment of tile Role of Non-linearities in Transportation Barge Roll Response," TRINA, pp. 65-79. Cointe, R., Molin, B., an(l Nays, P., 1988, "Nonlinear and Second-or(ler rl'ransient Waves in a Rectangular Tank," BOSS'88, 'Ron cll~ei m. i6] Cointe, R., 1989, "Quelques Aspects de la Simulation Nu~nerique d'un Canal a Iloule," These (le Doctorat de I'Ecole Na- tionale des Ponts et Chaussees, Paris (in French). [7] Cointe, R., 1990, "Numerical Si~nulation of a Wavetank," Engineering Analysis with Boundary Elements, special issue on "Nonlinear Wave Analysis", to appear. 8] Longuet-~liggins, M.S., and Cokelet, E:.D., 1976, "The Deformation of Steep Surface 96

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600.0 of 4()0.0 200.C 250.50 250.00 249.50 r~ 249.00 248.50 A) 248.00 247.50 C) I - - r ~r-~ - - I - 0 060~0 I - ~ -r ~I--- -r -- 120 0 180~0 TIME (s) . , ., . ~., , , . ~. . . .. . . . 240.0 30 ).0 I;'ig~lre 11: Mel vet ti '''e (~'o',li''ear Si'',''l?~ti`~) _.Slr~dbad linear _Si~dbod Nor~lir~ear 247.00 1 , 0.0 1 00.0 0.150 0.050 L_ I O.0()O J C) _ ().()5() -0.100 TIME (s) I;'igll re l'2: Sway V8. t illle ... Sludi~ad l.lr,~(lr _Sir~dbad Nor~lir~ec~r ---em ~r--- 200~0 300 0 0~ 1 50 ~1 ~-~-~~~~~- - - 1 1 0 0 1 00~0 200~0 300 0 TIME (s) I;'ig~l to l 3. It oh vs. ti Ille 97

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DISCUSSION J. Nicholas Newman Massachusetts Institute of Technology, USA This type of complete fully-nonlinear analysis seems to show very different results for the roll response compared to the simplified (hydrostatic) analyses described in the papers by Sanchez and Nayfeh or Francescutto and Nabergoj. Perhaps this is due to the large beam- draft ratio of this barge and its large damping? If so, it would be very interesting to apply the Sindbad program to a vessel with smaller roll damping. AUTHORS' REPLY The question of Prof. Newman is very interesting but we fear that we cannot give a satisfactory answer at this stage of our study. Our motivation for performing fully nonlinear simulations of the roll motion of barges certainly stems from our desire to get a better understanding of nonlinear phenomena involved. In particular, we would like to use such simulations to assess the validity of models using different levels of approximation. In this paper, however, we describe the method that has been devised to perform the simulation and we insist on the procedure that is used to validate it. We understand that is somehow frustrating to discuss more the tool than its results, but we strongly feel that nonlinear simulations can only be useful if carefully validated. The result we presented is for a barge having a large beam to draft ratio, a very small radiation damping and an important roll-sway coupling. It does not seem obvious to us why, in this particular case, it should be very different from that of a simplified analysis, as long as the roll-sway coupling is properly accounted for. The fully nonlinear computation is still quite time consuming (it took several hours on an Alliant FX80 for the example shown) and it cannot be used for a systematic investigation in the phase plane. It is our feeling that intermediate models might be better suited to check if the analyses performed on a simple single degree of freedom system apply to real situations. We are currently performing experiments and we will use the Sindbad program, as well as more less sophisticated simplified models, to make comparisons. We shall report the results in the future and we hope that we will then be able to answer this question more satisfactorily. DISCUSSION Ronald W. Yeung University of California at Berkeley, USA I find it puzzling how you could eliminate the reflected waves in the upstream zone by applying the damping layer to (-ok), ii being the incident wave potential. The difficulty, as I see it, is that the reflected wave from the body does not occupy the same fluid domain as the incident waves since the problem is nonlinear. You must have made additional assumptions in applying this procedure. Perhaps you can clarify this point. AUTHORS' REPLY We thank Prof. Yeung for his question that will allow us to clarify some points concerning the absorption mechanism we use. The method we propose is based on physical considerations rather than on a rigorous mathematical analysis. As a consequence, it is not clear to us what is meant by "additional assumptions.. We substitute the original problem with another problem with different boundary conditions. As for a real tank with an absorbing beach or an absorbing wavemaker, the only way to assess the efficiency of the method is to perform tests to quantify the reflection. As pointed out by Prof. Yeung, the reflected wave does not occupy the same fluid domain as the incident wave. The definition of the reflected wave that we give is therefore difficult to interpret is one refers to the Eulerian velocity potential in the fluid domain. However, as we use a Lagrangian specification for the time-stepping procedure, there is no ambiguity for the implementation of the method; we refer to the potentials attached to a marker labeled by its abscissa xc along the reference position of the free surface (that coincides with the x-axis). Similarly, the damping coefficient v(xe) is that attached to the same marker. 99

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