BREAKING BOW AND STERN WAVES: NUMERICAL SIMULATIONS

M. Landrini,INSEAN, The Italian Ship Model Basin

Via di Vallerano 139, 00128, Roma, Italy

Email: maulan@waves.insean.it

A. ColagrossiOcean Engineering Laboratory, UCSB

6740 Cortona Dr., Goleta 93117 (CA)

Email: andreac@catena.mee.ucsb.edu

M. P. TulinOcean Engineering Laboratory, UCSB

6740 Cortona Dr., Goleta 93117 (CA)

Email: mpt@engineering.ucsb.edu

Ship-generatedwaveshave alwaysfascinatedscientists,andplayed a key role in surface-shiphydrodynamicsfor con-tributing to hull resistance,generatingsoundsandradiatingvery longnarrow wakesremotelyvisible. Someof thesephe-nomenaoriginateabeamtheshipthroughextensivebreakingof divergingbow andsternwaves,forminga wake.

In this paperwe summarizeand extend our recentre-searchactivity aimed to understandthe complex fluid dy-namicsinvolved in bow- and stern-wave radiation, includ-ing wave breaking.Theanalysisis limited to practicalslen-der ships,with a sharpstem,for which basicinsight canbeachieved by an approximatequasithree-dimensionalmodelbasedon theideathatlongitudinalgradientsof relevantflowquantitiesare small comparedwith vertical and transversegradients. A historical recollectionof slender-body theoryfor shiphydrodynamicsis givenby Maruo(1989),Tulin andWu (1996),FontaineandTulin (2001).

In this framework, two methodshave beendevelopedatthe OEL. One basedon a potentialflow model, wherethevelocity field is given throughthe Laplaceequationsolvedby aBoundaryElementMethod(BEM), andtheevolution intime is obtainedby integrationof free-surfaceboundarycon-ditions. Specificdetailsof thelatestcodearedocumentedinLandriniandColagrossi(2001a). Themethodhastheadvan-tageof high resolution,sufficient to capturebreaking,andtotracejet overturningup to the impactagainsttheunderlyingfreesurface.

Post-breakingevolution is studiedby a gridlessmethod,calledSPHanddevelopedby Monaghanandco-authors(seee.g. Monaghan(1988)),which we appliedto breakingwavessincethe previous Workshop. In this case,Euler equationsfor a weakly compressiblefluid have beenconsidered.Fur-therdevelopmentsled to a codenamedSPlasH, appliedto avarietyof free-surfaceproblems,andpresentedin Tulin andLandrini (2000),Colagrossiet al. (2000).A detaileddescrip-tion of thealgorithmis givenin Landrini et al. (2001b)

The global picture by BEM In the 2D�

t method,calcu-lationsarecarriedout in two dimensions,verticalandtrans-verseto theship,andsuccessively in time. Fig. 1 shows thethree-dimensionalsteadywave patternarounda Wigley hullreconstructedby collectingsuccessive free-surfaceconfigu-rations.

Two key featuresof the flow are evidenced: first, the

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Figure 1. Wave pattern around a Wigley hull (B � L � 0 � 1,D � L � 0 � 1,Fr � 0 � 1) by 2D

�t (BEM) computations.

systemof diverging wavesradiatedat thebow asa resultofthecollapseof the”splash”,andsecondthe”roostertail” dueto the gravity reboundof the free surfacejust pastthe sternwhich, in turn, is the sourceof anothersystemof divergingwaves.

A thoroughanalysisof the genesisof diverging bowwaves is given by Tulin and Wu (1996). The collapseof

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Figure 2. Evolution of the splash at the bow of a Wigley hull (B � L �0� 2, D � L � 0 � 1, Fr� U �� gL � 0 � 4) by 2D

�t (BEM) computations.

splashat thebow is detailedthroughFig. 2: thefree-surfaceflow is notmuchdeceleratedbeforethestem,but uponreach-ing it, is deviatedsharplyupwards,riseson andeventuallylevels off andfalls down. An entirethin sheetis formedinthisprocessandappearsasasplashoneithersideof thehull.The relaxationof thesesplashesis the prime sourceof di-verging waves. In thepresentcase,radiatedwavesarelargeenoughto break,following thetypicalevolution: crest-rising,front-steepeningandjet formation.

Diverging breakingwavescanbe radiatedalsoby tran-somsterns.This is shown in thecaseof a Wigley hull, suit-ably taperedto have a deeptransomstern(cf. top-plot inFig. 3). The computationis performedasin previouscasesup to the last section. We thenassumethat the transomisdry and the potential is continuousacrossit, this providestheinitial conditionsfor thefollowing free-surfaceevolution.The resultingwave patternfor Fr=U �� gL 0 � 3 is shownin the center-plot. Besidethe alreadymentionedbow wavesystem,herewe observe a sharproostertail, surroundedbysteepbreakingdivergingwaves.Theanalysisof theflow fieldshows that the roostertail is causedby i) the inwardmotionof thefluid in proximity of theshipassociatedwith thecon-tractionof thehull crosssectionandeventuallycolliding aftertheendof theship,andii) thegravity reboundof thefreesur-face. Both featuresarealso detectedin the caseof Fig. 1,but for a dry transomsternthecavity left in thefreesurfacegivesrise to a strongerrebound.Thegrowth of sternwavesis bettershown in theperspectiveview in thebottomplot. Atfirst, the freesurfacemove inwardsandupwards,creatingasortof triangularhump.Later, themainbulk of fluid startstocollapsedown, leaving athinnerandthinnerjet, andacoupleof steepwavespropagatingoutwardsemerge,breakingsoonafter.

Post-breaking analysis through SPlasH In practicalcasesship-generatedwaves break, and further analysisby BEMwould requirespecialtreatmentof thefreesurface,oftenun-physicalor notpractical.Thereasonsfor shipwavesto breakcan be found in ship geometry(bow-flare, transomstern),higherspeedof advance,interactionwith ambientwaves.

The gridlesscodeSPlasH hasbeendevelopedto han-dle thesebreakingcases.The evolution of a breakingbore,shown in Fig. 4, gives an idea of the capabilitiesof themethod. The flow is generatedby a piston moving hori-zontally with constantspeed. After a while an energeticjet appears,impactingwith the underlyingfree surfaceand

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Figure 3. Wave pattern around a modified Wigley hull with a transom

stern by 2D�

t (BEM) computations. Top: ship cross sections; center:

wave pattern; bottom: detail of rooster tail and breaking waves past

the stern.

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Figure 4. Breaking bore and splash-up cycles forced by a piston

moving from left to right in finite depth water. Computations by SPH

method.

creatinga cavity andthereforecirculation(hereclockwise).The strongsplash-upI appears,center-plot, evolving into acounter-rotatingvorticalstructureandanotherforwardsplashup II, bottom-plot.Thestrengthof theplungingjet is crucialin determiningthatof thefollowing splash-upandtheentireresultingprocess,which for the breakingborehereconsid-eredis characterizedby several splash-upcycles,Tulin andLandrini (2000).

We have discovered similar featuresin breakingshipwavesby analyzingpicturesof modeltesting(picturestakenby PennaandGuerraat the INSEAN modelbasin). In par-ticular, Fig. 5 shows the bow wave generatedby a frigate(DDG51-type of the US Navy) with a strong flare. ForFr 0 � 41,thesplashreadilyevolvesinto aplungingjet, creat-ing thesplash-upshown in theenlargeddetailwhich revealsalsothesurfacescarassociatedwith thebackwardfacingjetto becomparedwith thebottom-plotin Fig. 4.

We appliedSPlasH in a 2D�

t fashionto studythepost-breakingbehavior of shipwaves.We first discussthecaseofa Wigley hull. The initial evolution hasbeenalreadyshownin Fig. 2 and, in this case,the computationsare initializedby thoseBEM results. This proceduresimplified the mod-eling, increasedthe efficiency andallowed an optimal reso-lution of the computation,with a high numberof particlesclusteredinto the jet. Thethree-dimensionalwave patternisreconstructedin Fig. 6, whereonly the upperlayerof parti-clesis represented.Only the portionaroundthe breakingis

Figure 5. Breaking bow waves in model testing. INSEAN Model

2412 of the US Navy DDG51 (Fr=0 � 41).

shown, with the ship centerplanelocatedat y 0, andthemid-ship crosssectionat x 0. The impactingjet gener-atesa splash-up,evolving into i) a vortical structureleft be-hind thecrest,andii) remainingparticlesriding on thecrestemerging after thebreakingwhich resemblethesteadyeddyusedby CointeandTulin (1994)to modeltwo-dimensionalsteadyspilling breakers. This flow patternis bettervisiblein thetwo-dimensionalbottomplot, andis remarkablysimi-lar to picturesof (two-dimensional)breakingwavesshown inMelville (1996).

On this ground,in breakingship-wave patterns,we candistinguisha TYPE I breaking,resultingfrom lessenergeticbreakingwaves,andresemblingthesteadyspilling breakingobservede.g. pasthydrofoilsat small incidence,anda TYPE

I I breaking,with strong(possiblymultiple) splash-up,caus-ing largerair entrapmentandvortex generation.

TYPE I I breakingis (usually) generatedat the bow of

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Figure 6. Top: perspective view of 2D�

t computation by SPlasHof the wave pattern generated by a Wigley hull (B � L � 0 � 2, D � L � 0 � 1,

Fr� 0 � 46); bottom: two-dimensional view of late evolution.

shipswith pronouncedflare,andthenumericalsimulationbySPH requiresthe modelingof arbitrary curved boundaries.Thisextensionof SPlasH hasbeenrecentlyaccomplishedbygeneralizingtheconceptof ”ghostparticles”,anda prelimi-nary resultis givenin Fig. 7 for a frigate-typeship (usedbyO’deaandWaldenin their analysisof deck wetness).Theflareis lesspronouncedthanin Model2412,andthecollapseof thesplash,topplot, causesaweakerbreakingthanthatob-served in Fig. 5. The resultingthree-dimensionalwave pat-ternis reconstructedin thebottomplot.

Resultsfor Model2412andfor thepost-breakingevolu-tion of sternwaveswill bediscussedat theWorkshop.

ACKNOWLEDGEMENTSThis work hasbeensupportedaspart of a programfor

the simulationof ship breakingwaves by the Ship Hydro-dynamicsProgramof ONR, initiated by Dr. Ed Roodandnow supervisedby Dr. Pat Purtell. The researchacitiv-ity of M.L. is also supportedby the Italian Ministero deiTrasporti e della Navigazione through INSEAN ResearchProgram2000-2002.

REFERENCESR. CointeR., M.P. Tulin ”A theoryof steadybreakers”,

J. Fluid Mech., Vol. 276,1994.A. Colagrossi,M. Landrini, M.P. Tulin ”Near Shore

Bore PropagationandSplashingProcesses:GridlessSimu-lations”,Monterey (CA), OEL Tech.Report00-224,2000.

E. Fontaine,M. Landrini, M.P. Tulin On modelingthepost-breakingphase:splashing.Proc.of 15th Int. Work. Wa-ter Waves Float. Bodies, Eds.T. Miloh andG. Zilman,2000.

Figure 7. 2D�

t computation by SPlasH of wave pattern around a

frigate. Top: the collapsing splash at the bow; bottom: perspective

view of the resulting wave pattern.

E. Fontaine, M.P. Tulin ”On the prediction of free-surfaceflows pastslenderhulls using the 2D+t theory: theevolutionof anidea”,Ship Tech. Res., 2001.

M. Landrini, A. Colagrossi”TwoDPT: a 2D+t codeforshiphydrodynamics”,OEL TechnicalReport,2001

M. Landrini,A. Colagrossi,M.P. Tulin ”Gridlesssimula-tion of sloshingflows in tank”, OEL TechnicalReport,2001

H. MaruoH. ”Evolutionof thetheoryof slenderShips”,Ship Tech. Res., Vol. 36,pp.107-133,1989.

W.K. Melville W.K. ”The role of surface-wavebreakingin air-seainteraction”,Ann. Rev. Fluid Mech., Vol. 28,1996,pp.279-321,1996.

MonaghanJ.J.,”An Introductionto SPH”, Comp. Phys.Comm. 48,pp.89-96,1988.

M. P. Tulin, M. Landrini ”Breaking wavesin the oceanandaroundships”Proc.of 23rd ONR Symp. on Naval Hydro-dynamics, Val deReuil,France,2000.

M.P. Tulin, M. Wu ”Di vergentbow waves” Proc.of 21st

ONR Symp. on Naval Hydrodynamics, Trondheim,Norway,NationalAcademyPress,Wash.D.C.,pp.99–117,1996.

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