Numerical Solution of Water Entry Problem of 2- D wedges

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NumericalSo lution of Water Entry Probl em of 2-Dwedges

by

©Min Zhang St. John's,Newfoundland, Canada

A thesis submitted tothe School of GraduateStudies

in partialfulfillment of the requirement sforthedegree of

Master of Eng ineer ing

Facult y ofEng inee ring andAppliedScience MemorialUniversity of Newfoundl and

June 20 11

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Abstract

Thenumericalsolutionof nonlinear waterentry problemfortwo-dimensional (2-D) wedgesispresentedin thisthesis..The Boundary Element Method (BEM) wasusedfor solving theLaplace equation,and theMixed Eulerian Lagrangianscheme was employed to trackthenonlinearfree surface. The freesurface profileand thevelocitypotentialare representedbyCubic-Splines. Theforwardfourth-orderRunge-Kuttamethodwasused fortimemarching.A cut-offtreatmentwas applied tothethinjetto avoid computational instability.

Verification studies were carried outfor awavemaker with impulsivemotions.Pressure distributions,freesurface elevationsandhydrodynamic forceswere calculatedand comparedwithanalyticalsolutionsandother numericalresults. The developednumerical methodwasthenemployedtosolve the symmetric water entry of2-D wedgeswithvarious deadrise angles.Pressuredistributionsand free surfaceelevations werecomparedwith experimentalresults andsolutionsby other numericalmethods,suchasthe similarity method,BEM,and the constrainedinterpolationprofile(CIP)method.

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Acknowledgements

Iwishto express my sincere thankstomy supervisor Dr.Wei Qiu,whosuggested the research topic,forhis encouragement,guidanceandconstructivediscussionsd uring the development ofthisthesis. Also, Iappreciate the guidancefrom my co-supervisor Dr.

HeatherP eng.

I deeply appreciatethefinancialsupportofthe NaturalSciencesandEngineering Research Council(NSERC)of Canada,theMathematics ofInformation TechnologyandComplex Systems(MITACS)and the GraduateFellowshipfromMemorialUniversity.Without their support, this workcould nothave been possibly completed.

Manythanks are extendedtomy colleaguesat the Advanced MarineHydrodynamics Lab (AMHL),fortheusefuldiscussions andsuggestions throughoutmy work.Ialsowould like toacknowledge advicesfromDr.Hinchey.

Finally, Iwould liketo give my special thankstomy family.lamprofoundly gratefulfor theirlove,supportand understanding.

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List of figures

Figure I-ISlammingExamples....

Figure2-1Coordinatesystemand definitions (Sun,2007)...

.. 1

.. 8

Figure 2-2 Elements, mid-points on the boundary 15

Figure2-3 Procedure of regriddingfree surface with parametric Cubic Splines 21 Figure2-4C ut-off treatmento f the jet tlow(S un,2 007)...

Figure2-5 Flowchartof computationalprocedure . Figure3-1Snapshoto f impulsive wavemaker(Li n, 1983)....

.... 22 ...24 ...26

Figure3-2 Gridson thewavetankboundaries... . 27

Figure3-3 Velocity potential on thewavemaker at t=O+... .. 28 Figure3-4 Velocity potential on thewavemakerfordifferentRts.. . .. 29

Figure3-5Free surfaceelevationat t=0.025s.... .. 29

Figure 3-6Free surface elevationat t=0.050s.... . 30

Figure3-7Free surfaceelevationat t=0.105s... .. 30

Figure3-8Freesurfaceelevationatt=0.145s... .. 31

Figure3-9 Freesurfaceelevationat t=0.195s... .. 31

Figure3-10 Pressuredistribution on the wavemakera t t=0.05s.... ....32 Figure3 -11 Pressuredistribution on thewavemaker at t=O.IOs. .. 33 Figure3-12 Pressuredistribution on the wavemakera t t=0.25s... .. 33 Figure3- 13 Pressure distriblitiono n thewavemakeratt= OAOs.... .. 34 Figure3-14 Horizontalforceactin g on the wavemaker ... . 35 Figure3- 15Girdson thetluiddomainboundaries ... .. 36 Figure3-16Grids neartheintersectionpoint ... . 36 Figure3- 17 Pressuredistributionsfordifferent gridsizeson wedge ... . 37 Figure3-18Freesurfaceelevationsfordifferentlengths of equal-sizedgr ids 38

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Figure3- 19Freesurfaceelevationof thewed gewith deadri se an gIe of60°att=0.006s39 Figure3-20Freesurfaceeleva tionofthewed gewith deadr ise angle of600at t=0.012s 39 Figure3-21Freesurfaceelevationofthe wedgewith deadri se ang leof 60° at t=0.018s 40 Figure3-22Freesurfaceelevationofthe wedgewithdeadriseang Ie of 60°at t=0.024s 40 Figure3-23Freesurfaceelevationof the wedge with deadri se ang leof600a t t=0.03s..41 Figure3-24Sna pshotofwater entryof wedge with deadri se angleof 60°(Lin, 198 3).41 Figure3-25Press ure distributi on onwedgewith deadri se ange l0f600a t t=0.006s 42 Figure3-26Pressu redist ributi on onwedgewith deadri se ang leof600at t=0.012s 43 Figure3-27Pressur edistr ibuti on on wed gewithdeadr ise ang leof6 00at t=0.0 18s 43 Figure3-28 Pressur edistributi on on wedgewith deadri se ang leof600a t t=0 .024s 44 Figure3-29 Pressur edistributi on onwedgewith deadri se angleof6 00at t=0.03s 44 Figure3-30Freesurfaceeleva tio nofthewed gewith deadri se angIe of45°at t=0.006s 45 Figure3 -3 !Freesurfaceeleva tio nofthewed gewith deadri se angleof45°at t=0.0 12s46 Figure3-32 Freesurfaceelevationofthe wedge withdeadrise angIe of 45°att=0.0 18s46 Figure3-33 Freesurfaceelevationof the wedge withdeadr ise angleof 45°at t=0.024s47 Figure3-3 4Freesurfaceelevationof the wedgewith deadr ise ang Ie of 45°att=0 .03s.47 Figure3-35Sna pshotof water entryofwedge with dead rise angle0f45°(Lin,1983) .. 48 Figure3-36 Pressur edist ribution onwedge withdeadrise ang le0f45° att=0.006s 48 Figure3-37 Pressur edistribution onwedge withdeadrise ang leof45°at t=0.012s 49 Figure3-3 8 Pressur edistributi on onwedge withdeadri se ang leof45°att=0.0 18s 49 Figure3-39Pressur edistr ibution onwedge with deadri se angle0f45°at t=0.024s 50 Figure3-4 0 Pressur edistributi ononwedge withdeadri se angle0f45°att=0.0 3s 50 Figure3-41Freesurfaceelevationofthe wedge withdeadri se ang leof 30°at t=0.024s 51 Figure3-42Frees urfacee levationo f thewedgew ithdea d riseangleof300at t=0.03s..52 Figure3-43Freesurfaceelevat ionof the wedge withdeadr ise angleof3 00at t=0.039s 52 Figure3-44Frees urfacee levationo f the wedge w ithdea d riseangleof300at t=0.048s53

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Figur e 3-45 Free surface elevation of thewed gewith deadri se ang leof 30°at t=0 .057s 53 Figur e 3-46 Pressuredistributi on on thewedgewith deadrise ang leoDO°at t=0 .024s .. 54 Figur e 3-47 Pressur edistributi on on thewedgewith deadrise ang leoDO°at t=0.03s....54 Figur e 3-4 8 Pressuredi stributi on onthewedgewithdeadri sean gle of300att=0.039s..55 Figur e 3-49 Pressuredistributi on on thewedgewith deadriseangleof30°at t=0 .048 s..55 Figur e 3-50 Pressur edistributi on on the wedge with deadrise ang leof30°at t=0 .057s..56 Figure3-51 Free sur faceeleva tionof thewed gewith deadri se ang leof 8 1°a t t=0.045s 57 Figur e 3-52 Pressur edist ribution on thewedgewith deadriseangleo f 8 1°a t t=0.045s.. 57 Figur e 3-53 Free surfaceelevationofthewed ge with deadri se ang leof75° at t=0.045s 58 Figur e 3-54 Pressuredistr ibuti on on thewedgewith deadrise ang Ie of 75° at t=0 .045s..58 Figure3-55 Free surfaceelevationof thewed gewith deadri se ang leof70° at t=0.057s 59 Figur e 3-56 Pressur edistributi on on the wedge with deadriseangle of70°at t=0 .057s..59 Figure3-57 Free surf ace elev ation ofth ewed gewithdeadri se an gle of 65°att=0 .057s 60 Figure3-58Pressur edistributi on on the wedge withdeadrise ang leof65°att=0.057s..60 Figllre 3-59Free surfaceelevationofthe wedge with deadri se angleof55° att=0.024s 61 Figur e J-St) Pressur edistributi on on the wedge withdead rise an gleo f55°at t=0.024s .. 61 Figur e 3-61 Free surfaceelevationofthewed gewith deadri se ang leofStf utt=0.024s 62 Figur e 3-62Pressur edistributi on on the wedge with deadrise ang leof5 00att=0.024s..62 Figllre 3-63Free surfaceelevationof the wedgewith deadri se ang leof atratt=0.036s 63 Figure3-64 Pressur edistr ibuti on on thewedgewith deadri se ang leof 40°at t=0.036s..63 Figure3-65Free surfaceeleva tio nofthe wedgewith deadri se ang leoD5°at t=0.036s 64 Figur e 3-66 Pressur edistr ibuti on on thewedgewith deadriseangle oD5°at t=0.036s..64 Figure3-67 Free surfaceeleva tionof thewed gewithdeadri se ang leof25°at t=0.036s 65 Figure3-68 Pressur edistributi on on the wedge with deadrise ang leof25°a t t=0.036s.. 65 Figure3-69Free surfaceeleva tionofthe wedgewith deadri se ang leof 20°at t=0.036s 66 Figure3-70 Pressur edistributi on on thewed gewith deadrise ang Ie of 20°att=0.036s..66

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Figllre3-7 1 Free surfaceelevationof thewedgewith deadrise angleof15° att=0.0495s

Figure3-72 Pressuredistribution on thewedgewithdeadrise angleof I5°at t=0.0495s 67 Figllre3 -73 Free surface elevation ofthewedgewith deadrise angleof l OOatt= 0.0558s ... ... . ... ... ...68 Figure3-74 Pressuredistribution on thewedgewithdeadrise angleof 10°att=0.0558s 68

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Nomenclature

F hyd rod ynam ic force

G',i' H',i influence coeffic ient matrices

M lengthof eleme nts

unitnorm al vector

Po atmos phe r ic pressur e

hyd rod yn ami cpressu re

bounda ryofthefluid doma in

SH bott om surface

S; symmetr icsurface

Sf, free surface

Su right- end surface

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s;

wavemaker surface

I'll time step

U velocity of wavemaker

velocityof wedges

X(x,y) Cartesiancoordinates

Yo submergenceof the wedgeapex

deadrisea ngle

water density

free surfaceelevation

n fluiddomain

velocity potential

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Chapterl Introduction

Slamming is of conce rn in manymarine applicatio ns.The resultantforces are some timessufficie nt todeformthe struc tureofashiporanoffsho restructure. In addition,slamm ing is a sourceoffati guefor offsho rest ructures,whichmayresultin structura l damages over time.Examplesof shipslamm ing phenom ena aregive n in Fig.I-1.

Figure I-I SlammingExamples

Slamm ingloads, asdefin edbyFaltinse n(1990),are impulsiveloads with high pressu re peak s occurring duringimpact between abody and water. Thisis cha rac terize d bylarge hydr od yn am icload sina short period of time, suchaswetd eck slamm ing,green water andsloshinginship tank s.Slammin gisusually cause d bylargerelati ve vertical moti on sbetw eenwave and body,which willlead topromin entnonlin ear effec ts.The typi cal formsofthenonlinearit y areshow nasthinjetsandsprays,non-viscou sflow sepa ration,and wavebreakin g etc.Inaddition,compress ibilityof airandwater,air

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bubbles and hydroelasticitymaybeinvolved. Thecombinationofthe abovefactors makesthe slamming problems very complicated(Faltinsen,2004).

Given the complex nature of slamming,somesimplifiedslamming modelshave been investigatedbymanyresearchersbefore thecompletesimulationsof slammingare applied to ships.Oneof themost extensivelystudiedslamming phenomena isth e water entryof 2-D wedgesinto an initially calm free surface.Inthisthesis, anumerical methodhas been investigatedto solve thewater entryof 2-D wedgesinto calmwater.

1.1 Literature Review

Thestudiesof 2-Dwedgesenteringwaterwithan initially calm"freesurfaceare motivatedbythe slammingofshipbows orsterns in heavy seas.These difficult problems areu sually solvedb y simplifyingth em odel as a rigid body enteringin itially calmwaterwithafree surface.Due to the rapid entryofthewedgesinto water,viscous effectsare negligible.Theco mpressibilitya ndai r-cushioneffec tso nly matterina short period of timeafter impactbetweenthebody and thefree surface.

The pioneering contributionsinthisfield weremadeby VonKarman(1929).Von Karman useda flatp lateapproximation in thes imulation,which underestimated loads forsmall deadrise angles.ArmandandCointe(1986),Cointe(1991)and Howison etal.

(1991)extended Wagner' s theoryto analyze the waterentry problem usingmatched asymptoticexpansions. Thelimitation of abovestudieswasthatit requiredthe impactingbodytobenearlyparallelto thecalmfree surface,i.e.,withsmalldeadrise angles.

Another way of solving thewater entry problemwasfirstputforward by Dobrovol'skayatlvcv),whichprovided a similarity method for thesymmetricwater entryo fawedgeof infinite extentwithconstantvelocity.Fully-nonlinear freesurface

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conditions and the exact body boundary conditions were satisfied in hismethod.

However,the solutions werenot given inanexplicitformand limitedbythe assumptionsofsymmetric body shape,constant water entry speedand gravityw asn ot accounted for.

In additiontothe above two approaches,numericalmethodshave aIsobeen extensively developedto solve thenonlinearwater entry problem and havebecomethemainstream ofresearchin this area. The advantageof numericalmethodsisthatthere are theoreticallynorestrictionsforthebody shapeand thewater entry speedwhilegravity canalso beincluded.TheBoundary Element Method(BEM)based on potentialtheory and the CFO(ComputationalFluid Dynamics)methodsbased onNavier-Stokes equationsare the commonly usedto simulate thenonlinearwater entry problems.

For the CFO methods,manyresearchershave contributed to thesimulationsof slamming.Forexample, Kleefsman etal.(2005) usedthevolume of fluid(VOF) methodtoinvestigate adambreakproblem and water entry problems.Kimetal.(2007) usedthe smoothed particlehydrodynamics (SPI-I) methodto simulatethe water entry of asymmetric bodies.Yangand Qiu(2007)applied the constrained interpolation profile (CIP) methodto solve the slamming problem of2- 0 wedgesentering water.

CFO methods are moretime-consuming and moredemandingincomputercapacity compared toI3EM.Many efforts havebeenmadeto solve theproblem of2-0wedges enteringwater with an initially calmfreesurface based on theBEM. The first successful numerical simulationof2- 0steep water surface motions andoverturning waves wasmadeby Longuet-HigginsandCokelet (1976), in whichLaplace equation wassolved byBEMbased onGreen'sfunctionand thefree surface wasupdatedbythe Mixed Eulerian-Lagrangian method (MEL). Vinjeand Brevig (198 1) modifiedtheir methodbased onCauchy'scomplex integraltheorem.GreenhowandLin(1987) later usedthismethodto solve the waterentry problem.However,their results were only satisfactoryfo r deadrisea ngles larger than60 degrees, becauseof the instabilityof

3

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computationscaused bythethinjetalong the wedgesideforsmall deadrise angles.In addition,this method canonly be applied to 2-Dproblemsduetothenature ofthe complex integraltheorem.However, theexact description of thethinjet isnot necessarybecause the pressurein the jetisnearly atmosphericandith asli ttleinflu ence onother parts of thefluid.ZhaoandFaltinsen(1993,1996)developed a BEMwitha cut-off treatment of thejetflowto avoid thenumerical instability.Anew segment was introduced at theje trootand theupperpart ofjetregionwasremoved.They usedthe finitedifferencemethodto calculate the time derivative ofthe velocity potential, and then computed thehydrodynamicpressures.However,thismethodingeneralcan not givegoodaccuracyofthepressuredistributions.Inaddition, themethod for representingthefree surfaceateach time step wasnot convenientt ou se.

Based onZhaoandFaltinsen(1993)'swork, Linand Ho (1994) introducedlinear elements, instead ofthe constantelement,to evaluatethe free surfaceelevationsand potentialderivatives. The benefit of usinglinear elementswasthattheflowfieldnear theintersectionpoint can bedescribednumerically with physicalmeaning.However, thepressureswere stillcomputed using thefinitedifference method.Later,thefree surfacewasbetterrepresentedusing cubicsplinesor non-uniformrational B-splines (NURBS) toincreasethe accuracy.Battistinand lafrati (2003) introducedcubic splines todescribe and rediscretizethe freesurface when evaluating hydrodynamic loadsduringwater entryof 2-Daxisymmetric bodies.ChuangandZhu(2006) used NURBS torepresentthebody surfaceand thefree surface profilewhen solving the 2-Dwaterentry problem.

Toimprovethe accuracyof thepressure,twomethods arewidely used to giveabetter prediction ofpressuredistributionsinthe recent studies.Oneis theco mplexve locity potential method. Greenhowa nd Lin(1987) usedthismethodto evaluateslamming pressurefor2-Dwedgesentering water. Sun(2007) further developedZhao(1993, I996)'smethodto compute pressuredistributions.Wuet al.(20 10)also used the complexvelocity potentialto solve thewater entry problemin the stretched

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coordinatesystem for a symmetrical wedge, an asymmetrical wedge and twin wedges.

The other methodto compute pressuredistributionswasdevelopedby Tanizawa (1995),in whichthetimederivative of velocity potentialwasdirectly obtained from solvingaboundary integralequation, thenthepressure can be computed using Bernoulli'sequation.Qianand Wang (2005) usedthismethodto calculate pressures onshipsand floating structures.Kihara (2004,2008)applied the same methodto compute pressuredistributionsfor slamming problems.The lattermethodwas adoptedin thisthesis,duetoitslessrequirement forcomputing resourcesthanthe complexvelocity potentialmethod.

1.2 Summary of the Present Work

In thiswork,the numerical solutionfor2-D nonlinear waterentry problem was developedbased on Zhao'smethod (1993).Othercomputational techniqueswere adopted in thisworktoimproveboththe accuracyandstabilityofcomputationsin thisthesis.Theinitialdisturbanceis simulated byWagner's approximation(Sun, 2007).The forth-orderRunge-Kutta schemewasused for thetimeintegration anda five-point smoothingscheme was applied to eliminate numericalinstability. The parametriccubic-splineswereusedtorepresent and rediscretizethefree surfaceand velocity potential foreach timeinstant. A scheme based on thework ofKihara (2004, 2008) hasbeendevelopedforcomputingpressuredistributions.The constant elementswere usedin the computation.Thecut-off method was employed to deal with theintersectionbetweenth efree surface andth ebodysurface.lnthiswork,the velocity was assumedconstantandgravity wasneglected.

The numericalmethodwasfirstverifiedby applying it totheproblem of impulsive wavemaker.Validation studies werethen carriedouttothe symmetricwaterentryof 2-Dwedgeswithvarious deadrise angles.Pressures,free surfaceelevationsand hydrodynamicforceswere computedandcompared with experimental results,

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analyticalsolutions,andsolutions by other numericalmethods.

1.3 ThesisContents

This thesisis arrangedas follows.Chapter2describesthemathematicalformulation withregardstothenumericalmethod. Firstly, the governingequation, boundary conditionsand therelated initialconditionsare presented. Secondly, theBoundary Element Method (BEM)obtained fromtheLaplace equationandequationsfor hydrodynamicpressures are brieflydescribed. Thirdly, thenumericalimplementation of theBEMin thetimedomain isrevealed and theprocedurefortimemarchingis explained.Finally, the cut-off treatment ofthethinjetflow and the CubicSplines usedfor regriddingoffreesurface profiles andvelocity potentialsare introduced.In Chapter3, an impulsivewavemaker caseisfirst computed to verify thedeveloped method.The methodisthen employed to symmetric water entryof2 -0 wedgeswith various deadrise angles.Thefreesurfaceelevations, pressuredistributions and hydrodynamicforces aresubsequentlyobtained. Numericalresults are comparedwith experimental results, analyticalsolutionsandsolutions by other numericalmethods.

Conclusionsand recommendations are presented inChapter4.

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Chapter2 Mathematical Formulation

2.1 GoverningEqua tion

A Cartesiancoordinatesystemo-xy,fixed inspaceischosenwiththe originat the undisturbedwaterlevel,withoxrepresenting thehorizontaldirection andoythe verticaldirection,positiveupward,as showninFig.2-1.Itis assumedthat the fluidis inviscidand incompressible and flow isirrotational.Itis alsoassumedthatthere are no aircushionsand nonon-viscousflow separation.Inaddition,thefluidacceleration associated withinitial impactis generallymuchlargerthanthe gravitational acceleration,and the time duration that is investigated int his problem is muchsma ller compared to typicalwaveperiods.Therefore,theeffect of gravityisneglected.A velocitypotential¢(x,y,t)satisfiesthe Laplaceequation inthefluiddomainn.

(2.1)

The waterentryproblemcan be formulatedas aninitialboundary-valueproblem(BV?) for thevelocitypotential¢andsolved witheitherNeumann or Dirichlettype of boundaryconditions.

2.2 Boundary Conditions

For theproblem of2-Dwedgesenteringwater with aninitiallycalmsurface,only half ofthefluid domain(x>O)is studieddueto thesymmetricalpropertyoftheproblem aboutthey-axis,as shown inFig.2-1.

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S,. We dgeApex

Figur e 2- 1Coordinatesyste mand defini tion s (Sun,2007)

Fullynonlin earbound ar y cond itionsareimposedon the freesurfaceS, The dy na m ic

bound ar ycondition on thefree sur face in theLagran gianformis

(2.2)

satisfiedon the exac t free-sur face,whereI]=TJ(X ,t )istheinstant an eous free-s urface

eleva tionandPoisthepressur e on thefree-surf ace assumed tobe zero. Ifthe effec tof gravity is neg lected co mpa red to the large fluidacce leration, itcan be rew ritte nas

(2.3)

The kinem aticbound ary condi tionson the freesurfaceintheLag ran gianformare expresse das

(2.4)

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(2.5)

Alongthewed ge,akinemati cboundar y condition isimposed as

i}1=Von

an

^(2.6)

whereVisthevelocit y vecto r of wed ge andIIistheunit norm alvect or on thewedge point ing outof thefluid.

Theno-fl uxboundarycondit ionimposedon thebottornSj, on thefarfieldS"ando n

the symmetric bound aryS;is

i!1= a n 0

2.3 InitialConditions

Initially,thevelocit ypotentialis zero on theundisturb edfree surface.

¢(X,O)=0

(2.7)

(2.8)

Forthewater entry probl em,theinitialdisturb anc e onfreesurface profilecau sedby theimpactbetw eenwed ge and wat erhastobetak eninto account. Based on thework ofSun(2007),thefollowin gWagner ' s approx imatio n meth odto simulatethe initial free surfaceelevation is employed.

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1](x,o)= ~arcsin(~)-

^Y^o

c=~

2ta nu

(2.9)

(2.10)

whereYoisthe subme rge nceofwed ge apex relat ivetotheundisturb edfree surface, anda isthedeadrise ang le(see Fig.2-1).

2.4 Hydrodynamic pressure

Thehyd rod ynami cpressur e p canbe computed accordin gtotheBern oull i'sequation.

(2.11)

wherePais theatmospheric press ureandpisthewaterden sity.S inceth einflllen ce of gravitycan beneglect ed,the eq uatio ncan berewritt en as

P_ P"=-p (

~ +~IV¢12)

^(2.12)

Itcan beeasilyseen that thetimederi vative ofthe veloc itypote ntial8¢/8 tmustbe computed inorde rto obta in pressur e.Althoug h itcan be comp uted by applying the finitedifferen cemeth odtothe veloc itypote ntia lasin the workof Zhao(1993)andLin (1984), the computationa laccuracyofthepressur eisgenerallynot sufficie nt.For more accura te predi cti on,the timederi vat ive ofthe veloc ity pote ntial8¢/8tcan be

obtai nedby solv ing the Laplaceeq uat ion fo r8¢/ 8twiththe follow ing boun dary condi tio nsaccord ing tothework of Kihara (2004).

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¢, =_~I'V¢12 _gy

~

a n

=o

onSw

(2.13)

(2.14)

(2.15) (2.16)

where

a/as

isthe tangentialderivative along thebody surface,and x denotesthe local curvature of thebodycontour. Thegeneralform of thenormalderivative of¢, on thebodyin motion inEq.2.14 was simplified forthetranslationalmotionwith constantvelocity by Tanizawa(1995).

Theinitialcondition isnecessarytocompletetheproblems,whichis givenas

¢,=0at1=0 (2.17)

Theprocedure ofsolving for ¢,is similartothat of solving for velocity potential¢ . Since theinfluencematricesfor ¢, aresame asthose for¢ , noadditional matrixsetup or inversionisnecessarywhen solving theboundaryintegral equationfor¢,.Therefore, thismethodfor solvingfor ¢,willnot intensify thecomputations.

Integratingthepressure along the body surfaceS';will resultin theforceF

F=

L

^pnds ^(2.18)

wherepisthepressuredistribution, and1/istheunit normal vectoron the wedge surface.

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2.5 Boundary Element Method (BEM)

TheBoundaryElement Methodhasbeen very successfully usedinthe solving2-D and3-D wave-body interactions.Insuch problems,the method offersthegreat advantageofdescribingtheflowbyitsboundary valuesonly.Therefore,the dimensions ofproblems are reducesby one.Moreover, intheanalysis ofwave-body interaction,the only informationneededin the computationistheboundary geometry, velocitypotentialand itsfluxon theboundary. Applicationsof BEMtononlinear wave-body interaction problems essentially consist oftwo coupled parts.The firstisa solutionoftheLaplace equationata given time. Thesecond partis a forwardstepping tothenexttime instantatwhich the Laplaceequation is solved repeatedly.

In BEM,the Laplaceequation is converted toa BoundaryIntegral Equation(BIE) by introducing Green'sfunctionG(p,q)=-f:;-Inr_N,where rpq=jp-ql,pisthefield

point andqis thesource point. The BlEo btainedbyapplyingG reen'ssecond identity

to ¢(x,y,t)andG(p,q)is givenas

whereS representst he boundaries ofthe fluid domain.When thefield point pis onthe boundary S,Eq.2.19ca nberewritten as

Inthenumerical evaluationofEq.2.20,freesurfaceS; body surfaceS',,bottom

surface S';andfar fieldboundary SRaredivided into anumber of constantelements,

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on which¢ and

o ¢/on

areconstant. Eq.2.20can berewrittenin thediscretized form as

where Ni sth enumb er of elementsdi stributed onth eb oundaries.

Generallyspeaking,highresolutionisneededtodescribethefree surfaceclose tothe intersectionswith highcurvature.In theregion close totheintersections,fin er elements are usedwhile elements with larger sizeare distributed on theboundariesfarawayfrom the intersections.However,it isnotpreferablethattheratio of thelargest and the smallestelements is overly high, sincethe drop in computational accuracy mayleadto numericalinstability, especially in theregionwherethe velocitygradient ishigh.

AccordingtoKihara (2004),therati o ofl engths oftw o adjacent elementss hould be no morethan 3inorder tokeepthecomputation stable.The effectofdifferentratios of lengths of adjacent elementswillbepr esentedin Chapter 3.ln addition,inordert ou se the smoothing scheme, equal-size elementsareused onp art ofth efree surface nearthe intersections, whilelarger elementsare distributed on therest ofthefree surface.It should benotedthatth e control of spatialdi scretizationdurin g computationi s vitalt o boththe stabilityofthecomputation andaccuracyof theresults.

Introducing

Eq.2.2 1 becomes

H i ./= - -:};

t-/;;ln(rl"l)dS/

Gi

. /=--:};

Ijln(r'''l) dsJ

1 n- n

(o¢)

-2¢,+'L Hi.l¢/='LGi.J-;-

;=1 j=1 un}

(2.22)

(2.23)

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Eq.2.23can berewritt en as

" " (8¢)

L

H;,A =

L

G',J;;-

;::::1 I,d unJ

whereH;,1andG',Jare theintluence coefficientmatri ces.

H;,J=Ii., forioF-j

=ii,.J+

~

^forⁱ⁼^j

The logar ithmi c sing ularitycanarisewhe ni=j,i.e.,pcoinc ideswithq.Fori=j ,

duetothe orthogo na lityofrpqandn(Kythe,1995).

(2.24)

(2.25)

(2.26)

where I;isthelen gth ofthe eleme nti.Forr""=¢I;/2 ,r=0 at¢=0 and r=±I,/2 at¢=±I ,

For ie] ,theintegral foreachelemen t is eva luate dbythe 4-po intGaussian

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wherexp.YParethecoordinat es of midpoint oneleme ntp;X"I.Y"Iare the coordinates ofthefirstnod e ofeleme ntq;X,,2.Y,,2are the coordi natesofthe second nod e of eleme ntq,asshown in Fig. 2-2;c,areGauss ian quadratur e coefficients andWiare weight s;JistheJacobi an ofintegrationfrom globa lcoord inate tolocal coord inate;

and /" isthelen gth of the eleme ntq.

quadratur e.

H,.}

=-:};I -/;;

In(rl"l)ds}

= -:};Jt w, [n.(~

-xp)+n/v:--YI')]lrI"l2

~q = :t

_,0,^.!..[(I₂ ^{- C,)X}^q,⁺^(1+C^,)X^q2]

Y

q=t H(I- C,)Yq,+(I+C,)Yq2]

J=.!..' 2^q

1 r

(x", Y,,) (x"•.Y,,)

Figure2-2Eleme nts, mid-point s on thebound ary

(2.28)

(2.29)

(2.30)

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c,=-0.86 11363dO c2=-0.3399810dO cJ=0.33998 10dO c,=0.861l363dO

1V,=0.347 8549dO

1V₂=0.652 1452 dO

IVJ=0.6521452dO

1V,=0.3478 549dO

One can solve thismatrixequationnumeri callyusin g a sch em eforiinear syste m,such asthe Gauss ianelim ination meth od .Asforthepressur ecomputation ,theintegral eq uation fortimederiv ativ e of velocit ypotent ial can be solvedinthe sam eway.

Ateach timeinstant,bynumerically solv ing theEq.2.20,theunknowns,boththe veloc ity potenti aloneachelement, and thenorm alvelocit yoneach elem ent along the free surfacecan beobtain ed .Thenewfree surface profileand velocity potenti al on the freesurfaceforthenexttimeinstant can thenbe computed.

2.6 Time-M archingScheme

Whensolvingnonlinearfree-surf aceproblems,the exac t free-surfacebound ary cond ition hastobe satisfied on the exact free- su rfaceposition.The nonlin ear wave problemwas st udied by Schwartz(1982) usin gthe seriesexpa nsion techniqu eunder the Euler ian descripti ontoapproxim atevelocitypotential and freesurface elevatio n.

How ever,thedisadv ant age of usin gthe Eulerian descript ionwasthatboththe bound ar y cond it ions andtheboundarypositionhavetobe appr oximated.To avoidthi s probl em , an opti onisto employLagrangiandescripti on.lntheLagran giandescrip tion, thefree-surf acepositi onisknown,but thedisad vant age ofit isthatthe govern ing eq uationhas to be approx imated.A new numeric alscheme called Mixed Eulerian-Lagrang ian (MEL)meth od wasdevelop edbyLonguet andCokelet(1976), which canoverco me thedisadv ant agesof both the Eulerianand Lagrangian descripti ons.

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TheMELmethodwasfirstusedto simulate thetime-history ofsteepsurface waves.

Inthetimemarching,the surface wasrepresentedbymarked Lagrangian fluid particles, This methodrequiresthatthe integralequationsaresolvedateach time step in the Eulerianframe,with fullynonlinearboundaryconditions satisfiedon theinstant free surfaceand thebody surface.Thenewposition ofthefree surfacefor thenexttime instant can befoundbyintegrating Eq.2.3,Eq.2.4 andEq.2.5.Thex componentof velocity8¢/8x and theycomponentof velocitya¢/fJycan notbe obtained directly from the HEM.However,from thevelocity potential ¢ on the freesurface,onecancompute thetangential componentofvelocitya¢ / as,combinedwith thenormal componentof velocitya¢/ allsolved fromtheHEM,thexcomponentof velocitya¢/ax andthey componentof velocitya¢/ fJycan be expressedas

~=_a_x ~n_as_y+~n_._an

.r

(2.3 1)

wheren,andlIyarethehorizontal and vertical componentsof theunit normalvector 1I(1I"II,.)onthefree surface,respectively.Thetangentialderivative ofthevelocity potential8¢/as can be computed byusingthefinite difference method.

The time integrationof Eq.2.3, Eq.2.4 andEq.2.5isthenperformedby a forth-order Runge-Kutta (RK4)scheme,as shown below. The formulaeforRunge-Kuttamethod are given byBurden andFaires(2005).

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devldt=!(t,(O)

(2.32)

wheretocan be eitherx,yor¢.ev, and(0;+1arecorrespondingva luesat times tep n+l andn , andMisthetimeinterval

During the timemarchingprocess,the sawtooth-typeinstabilityofthe freesurface occurs naturally when theintegral equation is solved numerically.Longuet-Higgins andCokelet(1976)adoptedafive-point smoothingschemeto avoid thesawtooth phenomenon.However,the scheme isinapplicabletothe first two and lasttwo points onthesurface.A smoothingscheme derivedfromthe third-order least squares approximationover fivepoints wasintroduced in this work(Sun,2007).

Itshould benoted that the smoothingscheme is onlyapplied tothenearintersection regionon thefree surface wherethefree surface profile changes dramatically.This smoothingschemecanalso beusedtosmoothout the x,y coordinates andth ev elocity potential ¢.Thefir sttw o equations of Eq. 2.33 are usedforthefirst two pointsa nd the lasttwo equationsof Eq.2.33areemployed tothelasttwopoints.The thirdoneisused fortherest of thepoints.

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J;=-k (69Y1+4Y2-6Y3+4Y4- Ys) J;

= is(

^2YI⁺^27Y2⁺^12Y3-^8Y4⁺^2yJ

J;=

is(

^-3Yi_2⁺^l2Y,_1⁺^l7Yi⁺12Yi+1-3Y'+2) (2.33) IN-I

= is(

^2YN_^4-8YN_3⁺^12YN_2⁺27YN_I+2YN) IN=

-k( -

^YN-4+^4YN_3-^6YN_2+^4YN_I+^69YN)

whereYiare theo riginal values befores moothinga ndf,are values aftersmoothing, andN isthetotalnumber of points on the freesurface neededtobe smoothed out.

The time stepsize/11inthetimemarchingprocedure should be chosenwithcare, because alarge Mmayleadtonumericalinstability whilea small/11will take unnecessary computationalefforts.Based on the Courant-Friedrichs- Lewy condition (Roache,1972 and Kihara,2004),M wasdecided inconsiderationofthefollowing condition:

where\7¢,isthe velocityofthemidpoint on the elementi ,andSl,isthelength of theelementi.

2.7 Regridding by Parametric Cubic Splines

After updatingthefree surface bymovingmidpoints oneachelementateach time step, thenewmidpointsmaybecome overlyconcentratedorscarce,especially in theregion nearthe intersections, whichislikelyto causecomputational instability. To avoid this phenomenon,regridding ofthe freesurface ateach instantisnecessary.Parametric

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cubicsplines arechosen tointerpolatethenewnodepoints and midpoints ofeach element,aswellasthevelocitypotential on newmidpoints.However, given the special property of thefree surfacethat itcan turnoveron itself,p arametriccubic splines using polygonalarclengthisdevisedtoapproximatethefree surface.

For parametriccubic splines,eachsplinesegmentis expressed in terms ofan independent variablesas(Burdenand Faires,2005):

x,(s )=a,(s-so)'+b,(s - so/ +c,(s-so)+d, (2.35)

wheresisthepolygonal arclengthfrom thefirstpoint ofthe cubicsplinetothepoint of desire;Soisthepolygonalarc lengthfromthefirstpoint ofthe cubicsplineto the starting point ofanysegmenti;x,can be xcoordinates , y coordinatesorvelocity potentialtobe computed.Thecoefficients a.ib .,»,andd,arecalculated usingknown positions of theupdated midpoints andcontinuityofthefirst and the second derivatives ofx"In thiswork,thefree surface wasregriddedby equalarclengtha nd the distances were calculated fromthefirstpointon the splines tothe evenly-spacedpoints.Based on the arclength, acorresponding cubicspline ischosentocompute thex,ycoordinates andvelocity potential ofa desired point.

The wholeprocess can be summarized intothree steps,as shown inFig.2-3.Firstly,the midpoints A,B, C,0,E, F are updatedtothenewpositionAJ, B"CJ,OJ, E"F1from originalfrees urface S.Secondly,parametric cubics plinesa re formulated torepresent thenewfree surfaceSlusingknown positionsof theseupdatedmidpoints and continuityof thefirst and the second derivatives.Thirdly,based on the given length of new elements,newnodesA2,B2,C2,O2,E2on thefree surfaceSIcan be calculated,as wellasthemidpoints of these elementsand velocitypotential on them.Thenewnodes formthenew free surfaceS2,which providesupdatedboundary conditionsfor next

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time step.

AI~A2

⁶¹

~ 62

^C2

A~C IDIEI ~ 52

fl 51

6

C 5

Figure2-3Procedur e ofregridd ing free surfacewith param etri c CubicSplines

2.8 Cut- off treatment of the jet flow

Inthe st udiesofnonlin earwave-b od yinteract ions, spec ially non linear freesurface flowinducedbyimpulsiv emot ion,theloss of accuracyat the inte rsectio n point swill occure.Theso-ca lled intersectionpoint iswhere athi n jetand bod y surface meets on the free surface,suc has pointA showninFig.2-1

According tothe workof Sun(2007),a very thin jet will run upalong thebod y surface whe nthe body withimpulsive moti onimpactsthe wate rsurface,espec iallywhe n the contac tang le betw eenthebody surfaceand thewater surfaceissm all.For the cases of sma ll co ntac ta ng les,computationa lerro rscaneas ilyar ise in the area ofintersecti on point, in theform of some point s goingthro ugh bod y,whichisphysicall yimpossibl e.

This kind oferrorscauses thenumer ical codeto collapse.The theoreti cal explanatio n ofthe problemisthatthevelocitybecom es sing ularat theintersection point s,that is, thevelocityisinfinite.The refore,thethinjetflowmustbeprop erl ytreated.

Two aspectsare need edtobe considere d in thetreatm ent ofthe jet flow:one isthe

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computationala ccuracY,and theo ther isco mputationals tability.The two are usuallyin conflict withoneanother. Whenfiner elementsare distributednearthe intersections, more accurate resultscanbe obtained,h owever,the computationeasilygets unstable.

On the other hand,whenfewer elementsare used close totheintersectionpoints,the code becomes stable while thejetflow ishardtobe simulatedaccordingt oth ew ork of Lin(1984).Oneof the commonly usedmethodsto solve thisproblemisacut-off treatment of the thinjetflow.

Inthework ofZhao(1993),anew element normaltothewedge surfacewas employed to cutoff the jet whenthe contactangle betweenthe wedgesurfaceand thefree surface is smaller thanagiven value.Kihara (2004) also usedthecontact angleb y introducing anew element tocutoffthe jet, however,thenew element wasnotrequiredtobe normal tothewedge surface.Sun(2007)cutoff thejetflowbyintroducing athr eshold valueofdistancefromthepointnexttotheintersection pointon thefree surface tothe wedge surface. Aslightly differentmethodfromthe one thatKihara (2004)usedwas derivedin thiswork.Based on the factthatconstantelements were employed in this work,itgave theflexibilitytolocatetheposition ofintersectionpoints.Thecut-off treatmentisillustratedinFig.2-4 forawedge.

WedgeSurface

Figure2-4Cut-off treatmentof theje t flow (Sun,2007)

As showninFig.2-4,A,B, Cand Dare updatedmidpoints on the free surface.Usually,

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theinte rsec tion point istheintersecti onpointPI ofthe wedge surface and theline A B.

How ever,whenthe ang le

r

betw eenthewedge surfaceand thelineABis sma ller than

athresh old valueZijthelineBCisthenusedto calcu late theintersection pointPj.Ifthe

contactang le betw eenthewed ge surfaceand thelineBC is stillsma ller than /3,theline

CDis use di nstea d unti l thecontac tang le iss ma ller than/3.

Thecut-o ff treatm en t allows forthesimulationofthejet flow witho utcomprom ising thestability ofthenumeri cal code.An important factsho uld ben oted thattheupper jet flowhaslittleinfluen ce on thepressur e distr ibutionon thewed ge surface ortherest of the freesurface.Thepressur eisnearl y equalto atm osph er ic pressur e in theupperpart ofthe jet.That is why thethinjet can be cutoff indiffe rentways witho utaffecting result s suc hastheprofil e of therest ofthefree surface and pressured istributi on on the wedgesurface.

2.9 Summary of the Computational Procedure

Thecomputationa l procedu re can be summa rizedasthefoll ow ingflow cha rt.Starti ng with theinitialbou ndary cond itions,then usetheBEMto obt ain potentialve loc ity¢o n

the wedgesurfaceand itsflux8¢j8nonthefree surface.The tan genti alder ivative of

veloc ity potenti al8¢j 8s onthefree surfacecan be solved us ing the finitedifference meth odbased.ontheknownpotenti al veloc itydistributiononthefreesurface.

Trans for m ing8¢ j 8n and8¢j8sto8¢j8x ando¢j 8y ,the freesurface profil e and veloc ity potenti al on the freesur facecan beupdated for thenexttimeinstan t. Afte r updat ingthe free surface bymovin gthemidpo int oneacheleme nt,the grids maybe overly den se or scarce neartheintersecti onpoint on thenew freesurface,therefo re, parame tr icCubicSplinesareused to regrid thefree surfaceandobta inveloc ity

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potential on thenewmidpoints after theregridding.In order tomakethe computation stable, RK4integralmethod and five-point smoothingschemeareemployed in the time-steppingprocedure.Thenewfree surface profile andvelocitypotential on the free surface will thenbeusedastheupdatedboundary conditions forthen exttim ein stanl.

Figllre 2-5Fl ow chart ofc omputationalpro cedllre

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Chapter3 Numerical Results and Discussions

To verifythe numericalmethoddevelopedin thiswork,computationswerefirst conducted for an impulsive wavemaker withprescribedmotion.Itis similar to,but less complicated than,the cases ofsymmetric2-D wedges entering water. The present methodwasthen applied to symmetric2-D wedgeswith variousdeadrise angles. The timehistories of pressuredistributions,free surfaceelevations forboth cases are presentedandcompared withexperimental results,analytical solutionsandnumerical results obtained bythe similarity method,BEM (Zhao,1993),andthe CIP method (Yang,2007).

3.1 ImpulsiveWavcmakcr Motion

Animpulsive wavemaker starts tomove witha constant horizontal velocityUfroma state ofrest. Atr-O+,the velocity is a step function and the acceleration is infinite.

Analytically,alogarithmic singularity is expectedattheintersectionpoint ofthe free surfaceand thewavemaker. This phenomenon is confirmed bythe experiment performedbyLin(1983),shown inFig.3-1.Itwasnoticedthatthe waterroseup to a levelthatthewavemaker was almost paralleltoit before ajet ejec ted fromthe intersectionpoint. According toLin(1983), the ejectedjet quicklybrokeup underthe effectofsurface tensionand possiblyaircurrents generated bythewavemaker.

Therefore, only the veryshortperiodof time after theimpulsivewavemakermoved wasinvestigatedinthisthesis.

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Figure 3-1 Snapshoto fimpul sivew avemaker (Lin,1983)

3.1.1 Gr idgenerat ion inthechosendomain

To simulate themotion of theimpulsive wavemaker numerically,afluiddomain is chosentobea longrec tangular tank,asshown inFig.3-2.On theleft-endboundary,a piston wavemaker pcrforrnsprescribedimpulsivemotion and theright-endboundaryis arigid verticalwall.Thelengthof thetankis10m, which islong enoughthat influence ofthereflectionfromtheverticalwall can be avoided duringthe short periodof computation. Waterdepthis1111,and thehorizontal velocityofthe wavemakerislm/s, In the computation, theboundaries ofthedomainarediscretizedinto anumber of constantelements,onwhich¢a nd8¢ j 8naresetconstant.Thesize ofthe elements variesalong theboundaries, as shown in Fig.3-2.

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Figu re 3-2Gridson the wavetankbound aries

The number of eleme nts is NF=74on the free surface with theminimum size equa lto 0.04mclosetothe intersec tion point ofthe free sur faceandwavema ker, NW=25 on the wavemaker, NB=50 on thebottom and NL=5 on theright -end wallatt=O+.The sizeoftime step needstobe sufficie ntlysmall.Itwas chose nasdt=0.0Is accordi ng to Eq.2.37 .

Theveloci ty potent ialdistribu ted onthe wavema keratt=0+isplotted inFig.3-3, and compare d with theapproxim ate analytica lsolutionandnumerica lresults by Lin (1984).Theappro xima teana lytica lsolutio n is give n by Eq.3.1(Lin,1984).

k=(2n+ I):r

n 2d

'7=-3!!....log (tanh !!.!...)

:r 4d

(3.1)

The result sbythepresentmethod agree wellwithothersolutions.FromFig.3-3,it can be seen thatthe slopeofthe veloc ity potent ial curve aty=O isfairly stee p.This sugges ts thatthe vertica lvelocityat the intersectionis almos tinfiniteatt=0+,as pred ictedby the relatedtheory.

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Figure3-3Velocitypotential on thewavemaker at t=O+

Toinvestigatetheinfluence ofdifferentratios of lengths of adjacentelementson the velocitypotential, arange ofratios ofle ngthsare usedto solve forthevelocitypotential on the wavemakeratt=0+.Rtrepresents \he ratio oflengths ofadjacentelements.

WhentheratioRtis equal to 5,thelength of elementson thewavemakeris0.04m whilethelength of elementson thebottomis0.2m.WhenRtis equal to 2.5,the length ofelementson thewavemakeris0.04m while thelength ofelementson thebottomis 0.1m.WhenRtis equal to1.25,thelength ofelements on the wavemaker is0.04m whilethelength of elementson thebottom is0.05m. The resultsfortheseratios are plotted andshowninFig.3-4.Itcan be seen fromthisfigurethattheresultsnearthe intersectionpoint ofwavemaker and thebottom getclosertothe analytical solutionas the ratioapproaches I, which suggests that comparablesizesof adjacentelementscan leadtomore accurate results.

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Figure3 -4 Ve locity potentia lo n thewavema kerford ifferent Rts

3.1.2 Free surface elevations,pressuredistributionsand hydrodynamicforces

Forthe subsequent time steps,compariso nsof freesurfaceelevationso btai ned bythe prese nt method,theana lytica lsolutionandLin (1984)are presented at/=0 .025s, r-0.05s,/=0.105s,r-0.145sandr-0. 195s inFig.3-5to Fig.3-9,respectively.

Figure3-5 Freesurface elevatio natt=0.025s

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~ ^0.1

Figure3-6 Freesurfaceelevationat t=O.050s

Analyticlll- IJIl(I'Il"4-)Y

FigureS-?Freesurfaceelevationatt=O. I05s

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0 .6 1

04 ,

~

_:0.

....

~ ...~.-.-:1f--._

Figure3-8Freesurfaceelevationatt=O.145s

Anal}'tkuJ- Un(l')MJ X

Figure3-9 Freesurfacee leva tio natt=O.195s

Itcan be observed fromthesefiguresthattheresu lts offreesurface eleva tions bythe presen tmeth od agreewellwithresul tsfromthe analytica lsolutio nand Lin'smeth od.

Theonly rem ark abl edifferenceisthepositi on of the high est pointsintheplots,whi ch isalsothe pos itionoftheinte rsec tion po intof thefreesurfaceand thewave maker.The

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positions of intersectionpoints given bypresentmethod are lowerthan Lin'sresults.

This phenomenonbecomesmore obvious whenthejet along thewavemaker gets higher.Thereasonisthat acut-offtreatment ofthe jetflowis employed in present methodbased on thefact that thepressurein thethinjetal ongth ewavemakeri salm ost equal toatlllospheric pressure and does not contributes ignificantly to the total force actingo n the wavemaker. lna ddition,the positiono f the highest point is closely related tothedensity ofthe grids in the intersection region.AccordingtoLin(1984), the denser grids neartheintersectionwill resultin a veryhighintersectionpositionbut the rest ofthefree surfaceshape isnot affected.On theo ther hand, thelong and thinjet can easily cause computationalinstability.The advantagesofcurrent method over Lin's arefewergrids neededin thecomputationduetothe cut-off techniqueused,hence less computational time,and better stabilityof thenumerical codes.

Fig.3-10-Fig.3-13showcomparisonsof pressuredistributions given bythepresent method and Lin(1984)'s methodatt=0.05s, t=0. 10,t=0.25, /=0.40, respectively.

!

0.5

Figure 3-10 Pressuredistribution on thewavemaker at t=0.05s

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Figure3 -!!Pressuredistribution on thewavemaker at t=O.JOs

Figure 3-12 Pressuredistributionon the wavemaker at t=O.25s

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~. ^0.;';

Figure3- 13 Pressur edistribu tion onthewavemake ratt=OAOs

Itcanbe seen thattheresult s obtai ned bypresentmeth od and Lin'sme thodagree we ll ingene ral.Thesma ll discrepan ciescould be cause d bythe differe nt meth odsusedfor computingthetimeder ivativ e of veloc ity poten tial ,which is a significa ntcompo ne nt ofpressur e.Lin usedfinitedifferen cemeth odtoget thetime derivati veof veloc ity poten tial whi le thetimederivative ofveloc ity potentialwasdirec tlysolvedfrom the BEMinthiswork.

Theforce acti ngon thewavem ak er werecomputed byintegrat ingpressu re alongthe surfaceof wavemak er,andaresho wn inFig.3-14.Despite thevar yin g freesurface eleva tio nsateach ti me insta nt,itca n beseen that the force isi nse ns itive in timeinboth Lin'sresult s and presentresults.Duetothe fact thatthe pressure distribut ions pred ictedbypresentmeth od are slightly lessthanthoseby Lin'smeth od,hencethe resulta ntforces are also lessthan Lin's result s.

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Figure3-14 Hori zontal forceactingon thewavem ak er

3.2 Symmetric waterentryof2-DWedges

Forthe cases of symme tric2-D wed ges enter ingwate r, thecomput ation swerefirst carriedout forwed geswith deadri se an gles of3 0,45 and60 degrees.Thefree surface elevationsand pressur edistrib ution s onwedgesurface were computed andcompa red with result sbythe sim ilarity meth od ,BEM (Zhao, 1993) and the CIP method (Ya ng, 2007).

3.2.1 Gridgc nc rat ion in thcc hosc n do mai n

Afluiddom a inwith alengthof l Omanda depthof 1m was chose n for thenumerical simulation.No sepa ration was cons ide redin thisproblem .Duetothe symmetric propert y of thisproblem about the y-axis,only half ofthedom ain(x>O)was stud ied.

The do mai n bounda r iesa re div ided bystraightlinesegme nts,asshown inF ig.3 - 15

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r

Figure3- 15Girdson thefluid dom ainbound aries

Anenlargedviewof thelocal regionnearthe intersect ionpoint ofthefree surfaceand wedge is given inFig.3- 16.Itcan be seen thattheden ser gridswereuse d in the regio n neartheintersecti onpoint.Notethatthe sizesof eleme nts betw eentwo adjace nt sur facesare comparable sothatthe computationalsta bilitycan be achieve d.On the wed ge surface, the sizeof the eleme nts was eq ual.On thebott om and theright- end wallsurface,larger buteq ual-s izedeleme nts weredistribut ed.On thefree surface,the first 20eleme nts were of eq ualsizefor the smoo thingsche me tobeused,and the size ofeleme ntsgrad ua llyincreasesas farfrom thewed ge on therest ofthefree surface.

Figure3- 16Grids neartheintersecti onpoint

To facilitate the simulationof wed ges entering water, computation was sta rted initially with wed ges slightlysubme rged intowater.The apexof wedge waslocatedaty=

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-O.Olm at t=O+.Sincethereis arapidchangein the freesurface profileat theinitial water entrystage,especially fora wedgewitha small deadriseangle,Wagner' s approximation method (Sun,2007) was employed torepresenttheinitialfree surface elevation when gravity isnegligible.Theconstantvelocity of wedge entering water was set to Zrn/s,

3.2.2 Free surface elevations and pressure distributions

To investigate the convergenceofpressuredistributionfordifferent sizesof gridson wedge surface, threedifferent gridsizesare usedincomputation and results are compared forwedgewith thedeadrise angle of3 0 degrees.Thegridsizeon thefree surface neartheintersection area is setas0.005m.Thechosengridsizeson wedge are 0.00866m,0.00692m and 0.0052m.The results aregiven inFig.3-17.

Figure3-17 Pressuredistributionsfordifferent gridsizeson wedge

Itcan be seenthat theresultsagreewiththesimilarity resultsbest when the gridsize on wedge surfaceequals 0.0052,which is closest tothesize of gridson the free surface neartheintersectionpoint.Thisalsoprovesthat comparablesizeofgridson two adjace ntsurfaces isdesirableinthecomputationasindicatedin the wavemaker

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Asmenti oned above,eq ua l-s izedgridswereemploye d in theregionnearthe intersect ionpoint of free surfaceand wed ge surface,and lager size of grids used on therest of free surface.Toinvest igatethe influe nceofthelen gth ofthe eq ua l-s ized gridson thefree surfa ce profile,the eq ua l-s izedgrids weredistributedintheran ge of O.04m ,O.05mand O.06mfromthe intersecti onpoint at the startofcomputation, respective ly.Thesizeofeq ual-s izedgrids was setasO.002mfor all tests.Theresults aresho wninFig.3-18.

Figure3- 18Frees urfacee levations for di ffere nt lengthso fe q ual-sized grids

Itcan be seen thatthelen gth s ofequa l-s izedgridsare usedin the computatio n donot have a greatimpacton thefree surface profil e. The reasonfor employ ingeq ua l-s ized grids isthat the smoo thingsche mecanonly be applied to eve nly-s pace d point s.It sho uld benotedthatthis areagetsgreate rasthedeadrise ang les decrease.

The timehistories of freesurfaceeleva tio ns fo rthe wedge withdeadri se ang leof60 degreesfromO.006stoO.03s at an increm ent ofO.006sareshow n inFig.3-19toFig.

3-23.The result s are nond imension alizedby dividingVI,andVisthe veloc ityof wedgeentering water andIis the timeinstant. Theresultsarecompa red withsolutions

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bythe sim ilarity meth odandexpe rime nta l result s (Gree nho wand Lin,198 3),as showninFig.3-24.Itcanbe seen thattheagreem entbetweenthepresentresuitsand othe rsolutions isingene ra lgood.Inthebeginnin g stageof the simulation,sma ll discrepanci es around thejet spray root area areshown inFig.3- 19andFig.3-20.

However,thediscrep anciesdisapp earasthe computatio n becom es steadyas shown in Fig.3-21,Fig. 3-22and Fig. 3-23.

Figure3- 19Freesurfaceeleva tionof the wedgewith deadri se angleo f6 00at t=0.0 06s

~, u

Figllre3-2 0Frees urfacee levatio no f the wedge w ith dea d riseangIeof60°at t=0 .012s

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Figure3-2 1Free5lIrfacee levalionof lhewedgewilh deadr iseangleof600al 1=0.018s

Figllre3-22Free surfaceelevat ionof thewedgewith deadr ise angleof 60°at 1=0.0245

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s.

0

Figure3-23'Frees urfacee levationof the wedge with deadrise angle of60°at t=0.03s

Figure3-24 Sna pshotofwater entryof wed ge with deadri se ang leof 60°(Lin,1983)

In thepresentmeth od , only part of the thin jetflow was simulate d inthe computation andcompare dwith the sim ilaritysolution.

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Fig.3-25 toFig.3-29show thetimehistoriesof pressuredistributions on thewedge surface with deadrise angleof60 degreesfrom 0.006sto0.03s at an increment of 0.006s. The resultsarenondimensionalizedbydividing0.5pV²,wherepis the waterdensityandVisthevelocity of wedge entering water.The results arecompared with the solutions bythesimilarity methodin thefirstfourfigures,andcompared with other resultsby the CIP method(Yang, 2007) and BEM (Zhao, 1993)in Fig.3-29.

Goodagreement withothersolutionscan be observedin thosefigures. The resultsby thepresentmethod areslightlysmaller thanthe similarityso lutionsin Fig. 3-25 and Fig.3-28 whilethepresentresultsare slightly biggerthan the similaritysolutions in Fig.3-27and Fig.3-29. However,in theduration of the simulation, it was shown that theresultsweresteady withoutviolentfluctuations.Theresults agree wellwith similaritysolutionseven in the very beginning of the simulation, whichmay suggest thatth e smalldi screpanciesinfree surfaceprofil eh avelimitedimp acton thepressure distribution.Theslight variation of thepressuredistribution over theprocess of simulationcould becausedbyri gridding ofth efree surfaceateach time instance.

Figure3-25 Pressuredistribution on wedgewith deadrise angelof600at t=0.006s

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Figure3-26Pressuredistribution on wed gewith deadri se ang le0f600at t=0.0 12s

Figure3-27Pressuredistributi on on wedgewith deadrise angleof 60°at t=O.0 18s

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Figure 3-28 Pressur edistributionon wedgewith deadrise ang le0f600att=0.024s

Figure3-29 Pressuredistributi on on wedgewith deadri se ang le0f600att=0.03s

Thesamecomputat iona l procedur ewasappliedto awedgewith deadri seangle of45 degreesfrom 0.006sto0.03s atan increment of O.006s. Thetimehistories of free surfaceelevationsareshowninFig.3-30 toFig. 3-34. Theresult sare compared with

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solutions bythe similarity method andexperimental results (GreenhowandLin,1983), as shown inFig.3-35.Ingeneral, thepresentresults are ingoodagreementwith the similaritysolutions.The discrepancies around thejet sprayrootareashowninFig.

3-30almost disappear asthe computation becomes steady.Sameasinthelast case, the upper partofthe thinje tis cutoff,which doesnot affect therestofthe freesurface profile.

{ ()

Figure3-30 Free surfaceelevationofthe wedge with deadrise angIe of45°at t=0.006s

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~ II

Figure3-31Freesurfaceelevationofthewedgewith deadr ise angleof 45° att=0.012s

~. ^II

Figure3-32 Free surfaceelevationof thewedge with deadr ise ang leof 45° at t=0 .01 8s

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Figllre3-33Freesurfaceelevationof thewedgewith deadrise ang leof4 5°at t=0.024s

Figllre 3-34 Freesurfaceelevationof thewed gewithdeadri seangIe of 45° att=0.03s

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Figure3-35Sna pshotof water entryof wed gewithdead rise ang le0f45°(Lin ,1983)

Fig.3-36to Fig.3-40show thetimehistori es of pressur edistributi ons on thewed ge surface withdeadri seangle of4 5 degreesfrom0.006sto0.03s atan increm en t of 0.00 6s. The result s are compa red with the solutions by the sim ilarity meth od .

Figure3-36Pressur edistribut ion on wedgewith deadri seangle0f45°att=0.006s

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Figuref-S?Pressliredi striblit ion onwed ge withdeadr ise an gle of 45° at t=0.012s

FigureJ-J'SPressur edistributi on onwedge with deadri se ang leof45°at t=0.0 18s

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Figure3-39Pressuredistributi on on wed ge with deadri se ang leof45°at t=0.024s

Figure3-40 Pressur edistributi on on wedgewith deadri se ang leof45°att=0.03s

It can be seen thattheresult s ofpressur edistributi on sbythepresentmethod are in goodagree me ntwith the sim ilaritysolution. The simulatio n wasthen carriedout for a wedge with deadri se ang leoDOdegrees. The timehistori es offreesurfaceelevationsat

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t=0.024s, O.Q3s,0.039s,0.048 ,0.057s areshown inFig.3-4 1 to Fig.3-45.The results arecompare d withsolutio ns bythe sim ilar ity meth od.Itcan be seen that the agree ment betw eenthepresentresults andsim ilaritysolutions is,ingenera l,good.It sho uld benotedthatit tookmoretime ste ps in this caseto achie ve thesim ilarity result sthanthepreviou s case s.

~,⁰

Figure3-41 Free surfaceeleva tionof the wedge with deadriseangle of 30°att=0.024 s

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{ 0

Figure3-42 Freesurfaceelevationofthe wedge with deadrise allgleof 30°at t=0.03s

s..(l

Figure3-43 Freesurfaceelevationof the wedge withdeadrise angleof 30°at t=0 .039s

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Figure3-44 Freesurfaceelevationof the wedge withdeadri se ang Ie of30°at t=0 .048s

Figllre3 -45Freesurfaceelevationof thewed gewith deadri se ang leof 30° att=0.057s

Fig.3-46to Fig.3-50sho w thetil11ehi stori es o fpressliredistribliti on s onthe wed ge surface with deadri seangle of'Ju degrees at1=0.024s,0.03s,0.039s,0.048,0.057s . The result s arecompared withthe solutions bythe sim ilar ity meth odin thefirst four

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figures, and comparedwithother resultsbythe CIP method(Yang, 2007) and BEM (Zhao, 1993)in Fig.3-50.Ingeneral, thepresentresults showgoodagreement with the othersolutions.

Figure3-46Pressuredistribution on thewedgewith deadrise angleof300at t=0.024s

Figure3-47Pressuredistribution on thewedgewith deadrise angleof 30° at t=0.03s

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Figur e 3-4 8 Pressur edistribut ion on thewedgewith deadrise ang Ie of3 0⁰at t=0.039s

Figur e 3-49Pressur edistributi on on thewedgewith deadri se ang leofJO°at t=0.048 s

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Figure3-50 Press ure dist ributi on onthewedgewith dead rise ang leof30^oat t=0.057s

3.3 Symmetric water entry of 2-D wedges with various dead rise angles

Free surfaceeleva tio nsand pressu redistribut ion s onwedges withvario usdeadrise ang leswere alsocomputed by the presentmeth od. Thedeadriseang lesvariedfrom10 to 81 degrees.Duetothelimited access toresult s gene ratedby othe r methods , compar isonsof the prese nt results wereonly made with those fromthesimi larity method,BEM (zhao,1993) and the CIP meth od (Yang,2007) fordeadr ise angleof 40 degrees.Thefreesurfaceresultsare shown inFig.3-63whileFig. 3-64 presentsthe result s of pressuredistr ibutio ns.Inaddition,compa riso nswe re also made betwee n the presentresultsandsim ilaritysolutio nsfordeas r iseang les0f81, 25, 20,15, and 10 degrees,as show n inFig. 3-51,Fig.3-52 and Fig.3-67to Fig.3-7 4.

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Figure3-51Freesurfaceelevationof the wedgewith deadrise ang leof81°att=0.045s

Figure3-52Pressur edistributi on on the wedge with deadrise angleof8 1°at t=0.045 s

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Figure3-53Freesurfaceelevationof the wedge withdeadri se ang leof 75°att=O.045s

Figure3-54Pressur edistributi on on thewed ge with deadri se ang leof75°at t=O.045s

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0:: j l _

-o_~ 1

Figure3-55 Free surfaceelevationofthe wedge with dead rise ang leof700at t=0.057s

Figllre3-56 Pressu redistributi on on the wedge with deadri se ang Ie of 70°att=0.057s

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FigureJ-S?Freesurfaceelevationofthe wedge with deadrise ang leof65°at t=0 .057s

Figllre3 -58 Pressur edistribut ion on thewed gewithdeadri se angleof 65°at t=0.057s

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Figure3-59Freesurfaceeleva tionof thewed gewith deadrise ang leof55°at t=0.024s

Figure3-60 Pressuredistr ibuti on011thewedge withdeadr ise angleof55°a tt=0.024s

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Figure3-61Freesurfaceelevationof thewedge with dead rise ang leofStr att=0.024s

Figllre3-62 Pressuredistributi on on the wedge with dead rise angleof 50°at t=0.024s

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~.(J

Figur e 3-63Freesurfaceelevationof thewed gewithdeadr ise ang leof40°at t=0 .036s

Figure 3-64Pressuredistribut ion onthewedge with dead rise angle of 400att=0.036s

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Figure3-65Freesurfaceelevationofthewedge with deadrise ang leof35° att=0.036s

Figure3-66Pressuredistributi on on the wedge with deadrise angIe of 35° at t=0.036s

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{ 0

Figure3-67 Freesurfaceelevationofthewedgewithdeadrise angleof25°at t=O.036s

Figure3-68 Pressuredistribution on the wedge withdeadrise angIe of 25°at t=O.036s

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Figure3-69Freesurfaceelevationof thewedgewith deadri se angleof20°at t=0.036s

Figllre 3-70Pressuredist ribut ion on thewedgewith deadrise angleof20°at t=0.036s

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~, ^o

Figure3-71Freesurfaceelevat ionof thewedgewith deadr ise an gle of15° at t=0.0495s

~. )

L. --=~:o---=-/";/ .

\

Figure3-72 Pressur edistributi on on thewed gewith deadrise angleof I5°at t=0.0495s

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{ 0 ----.:--.:..-...:.--..:.-.----~--'--->..-_,4_

Figure3-73 Freesurfaceelevationof thewed gewithdead riseang leof10° at t=0 .0558s

Figure3-74 Pressu redistribution on thewed gewith deadri se ang Ie of10° att=0.055 8s

From the above figur es,itcanbe obse rve d thatthe agreeme nt betw eenthepresent result s (freesurfaceelevationsand pressur edistribut ion s) andothersolutionsis generallygood.Thelargestdifferences occur in thepredi ction ofthefreesurface

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profile close tothejetflow.Thiscould be associatedwith the jet flow cut-off techniqueimplementedin the computation. Asthedeadrise angle increases,the jet flowbecomesless apparentandthemaximum pressure sharply decreases. The positionofthemaximumpressure occurred closetothe jetroot regionfordeadrise angles lessthan 45degrees.Whenthedeadrise angleis equal to or largerthan 45 degrees, themaximumpressureis at the apexofthewedge.InFig.3-50,thepressure for deadrisea ngleo f 8! degrees,predictedbythepresentmethod,is smaller thanthat bythe similarity method.Thiscouldbe explained bythe singularityat the apexof wedgeswith largedeadriseangles,aspointed out byVim(1987).Thesingularity was avoided using constantelements.However,itstill hasinfluence on thedistribution of velocity potentialon thewedge surface, whichleadstoinaccuratepressureprediction.

Inaddition, it can bededucedthatthe effectofthe singularityat the apexofwedges becol11esgreater with the increaseof deadrisea ngles.

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Chapter4 Conclusions

Thenonlinearproblem ofsymmetric water entryof 2-D wedges governed bythe Laplaceequation was solved by anumericalmethodthatwasdevelopedbased on the BEM.Inthe computation, theLaplace equation wasnumerically solvedateach time instantusingtheBEM.The boundary conditions forthenexttime step wereupdatedby atimemarchingprocedure.Thefree surface was captured bythe MELmethod and smoothedout bythefive-point smoothingscheme to eliminateanysawtooth phenomenon.Thefree surface wasthenregridedbyparametric CubicSplines toa void overly dense orscarcesegments neartheintersection region. A cut-offtreatment was performedforthethin jetflowneartheintersectionpointstom aintaincomputational stability.For thepressurecalculation,thetimederivative ofvelocity potentialwith boundary conditionswas solved usingtheBEM and thepressure can thenbe obtained usingBernoulli' s equation.

Validation studies havebeencarried out for an impulsivewavemaker and symmetric wedges with variousdeadrise angles.For the case of impulsivewavemaker,pressure distributions, freesurface elevations and hydrodynamic forces were computedand comparedwith theanalytical solutionsand resultsbyLin (1983)' smethod. For the 2-D wedgesproblem,pressuredistributions andfreesurface elevations were computed andcompared with the similarity method,the CIP method (Yang,2007) and resultsby BEM(Zhao,1993).The results bythepresentmethod aregenerally ingoodagreement withothers.Ithasbeendemonstratedthatthenumericalmethoddeveloped in this work is able to solve thenonlinearbody-waterinteractionproblems with highly distortedfree surfaceand provide good predictions of freesurfaceelevationsand pressure distributions on wavemaker orw edgeswitha rangeofdeadrise angles.

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Forfuture work , stud ies canbe extende d totheproblem of symme tric wed ges entering water with sepa ratio n point s, asym me tricwed geswaterentryprobl em, and three-dimen sion alwater entry probl em ofbodi es of arbit rarygeomerry.