NONLINEAR ENTRY NUMERICA

NUMERICAL SOLUTION OF 3D HIGHLY NONLINEAR WATER ENTRY PROBLEMS

by

@QingyongYang,B.Eng., lVI.Eng.

Athesissu b m it te dto the Schoolof Gradu a teStudies in par ti alfulfillmen t of the requirem en tsfor thedegr ee of

Doct or of Philosoph y

Facu lty ofEng inee ringan d AppliedScience Memori alUn iversity

Oct ob er 20 11 St.Joh n's,Newfoundland,Canada

Abstract

Ships or offshorestructuresmay experienceseverewat erimpactproblemsintheharsh environments,suchas cargosloshing,slammingandgreenwater on deck.These impactloads cancauseseriousst ructura ldam age andare ofconsidera bleconcern to the sta bilityandsurvivabilityofshipsoroffshorest ruc t ures.Alloftheseforces are associatedwith highlynonlinearfree surface flows.

Thethesispresentsthe numerical solutionofslamming problemsfor3D bodi es enteringcalmwatersymmetrically andasymmetrically,wit h prescrib ed entry velocitiesand free-fallmotions.Thehighlynonlinearslamming problemsnrc governed bytheNavier-Stokes equationsandaresolved bya Constra inedInterpolation Profile (CIP) -based finitedifferencemethodonafixed Cartesiangrid.The Clf?methodis employedfor the advect ioncalculationsandapressur e-based algorithm is applied forthenon-advect ioncalculat ions. For thepressur e computa tion,aPoisson-typ e equation is solvedateachtimestep bythe Conjngate Gradientiterati vemethod.

The solid body aIIIIfree surface interfaces arecapt ured bydensity functions. A panel-based meth odisdevelopedto capt uretheinterfaces of3D bodies.Themoti on of abod yisdescribedin terms of sixdegrees offreedo m.

Validation st udiesof thepresentmeth odwere carriedontfor several3D bod ies ente ringcalm water symmet rica llyandasymmet ricallywith prescribedvelocities and free-fallmotions.Waterent riesof3D bodieswithprescribed velocit ies werefirst studied. For thewat erentryofa3Dwedge,3DHow effects wereinvestigated . 3D flow effect stend tocauseareduct ion inslamming force.The compute dslamming forcesarcingood agreementwithexperimental resul ts .For the sphereenteringcalm waterobliquely,the computedvert icaland horizontal slamming forcesingeneralagree

wellwit hexpe ri mentalresults.Thc simula tionswerefurther carriedout foracouple ofbodieswithcomplexgcomctry.For the wat er entryofa 30shipsection, pressu res nea r theknuckleswereunder- predictedbythenumericalmeth od.The slamming for ceona :lO flaredbody wasalsocomputed bythepresentnumericalmeth od, and thcpredicted sla mmingforcesarcingoodagrccmcnt withtheexperi mentalresults.

Thcmaximum slamming force coefficientsofaplanin ghullwithdifferentpitchand rollangleswere compute dby thepresentnumericalmeth odandcompa redwiththese bythe20str ip theory.The 20 results areslightlygrcatcrtha n the 30solut ions.

The stud ieswerethcn exte nded to 30 bodies enteringcalm wat erwithfree-fall mot ions.Thepredict edmotion of thehalf-buoya nt cylinder withfree-fall motion agrees wellwith the expe riment aldat a .Fortheneut rall ybuoyant cylinder,reasoua hle agree mentisobtained, exceptat oneexperi mentalvaluewhich obviouslydevia tes frorntheother data .The complica ted free surface elevat ionsduringwater cnt ryof cylinderwere simulatedbythcpresent numerical method.They arc visuallyingood agreement withthephotographstakenfromthe ex periments.Thepresentnumerical meth od over- predicts thevelocity ratiosforwate rent ryof a cata mara n,especially forlar gedropheight s. Velocit y, acceleration,aswell asvertical and hori zont al hydr odynamicforces as afunctionoftime werepredicted bythepresen tnumerical mcthod fortheasymmetricwat er ent ryofashipsection.Asatisfact ory agreement withexperimentaldrop test resultsisdemonstra te d .

Acknowledgements

Iwouldliketo givemyforem ost ap preciat ionand thank stomy supe rvisor,Dr.Wei Qiu,whogreatlyenha nced myinterestinMa rin eHydrodyn amics,forhisconti nuous patien ce,kindness , encourageme nt,andsupportthroughoutmyworkonth isthesis . His ad viceandacadem icgu ida nce hav ebeeninva luableto thedevelop men t of this work.

I wouldliketoshowmy grat itudetoDr. DavidMolyn euxandDr.Hea ther Peng, whose guida nceandsuppo r ts enabled metodevelopanunderst anding of the subjec t.

Iwouldalso liketoacknowledgeadvice andcom me ntsfromDr. MahmoudR.IIadda ra and Dr. Serpil Kocabi yik.Many tha nks nrc ex tende d tomy colleagues andfriends at MemorialUniver sity fortheusefuldiscu ssionsandsuggest ionsthro ughoutvarious stagesof myresear ch .

Iwishto exp ress mydeepappreciationtothe Canadia nFoundati onfor In novation andtheNaturalSciences and EngineeringResearchCouncilofCanada. Iwould alsoliketo express myspecialthanksto Defen ceResearchandDevelopmentCanada Atlanticand Ocea n ic Consult ing Corpo ra t ionfor theirsupport thro ughapart nership program.Thisworkwould nothavebeen com plete d withouttheirfinan cialsup por t.

Iowemydeep estgra titude tomyfamily.Asformy wifeLeiLin,herlove,patien ce, andencourageme nthaveup held me allthetimc.Formypa ren ts ,X.L.Wang and E.Z .Ya ng,Iamprofoundly gratefulfortheir loves andsupport.Iwouldalsolike to give my spec ialthank stomy grand pare nts. whosekindness andunderstand inghave beenthe mostimportanttreasurein mylife.

iii

Contents

Abstra ct

Ack now ledgements

Lis t ofFigures

No mencla tures

Abbrevi a ti ons

1 Introd uction 1.1 Ba ck ground""

1.2 LiteratureRcview. 1.3 PresentWork"

1.4 Outli neoftheThesis"

Mat hemat icalFor mulat io n

iv

iii

vii

xv iii

xix

10

13

2.1 Governing Equ at ion s .

?? Discret ization Meth od

2.3 Free SurfaceModelingMeth od. . 2.4 CIPMethod.

13

16 24 27

2.4.1 Principleof thc CIPMeth od. 27

2.4.2 CIP For mula t io n inMulti ple Dimensional Cases. . . 32

2.4.3 Calcu latio nof Spat ial Der iva ti ves.

2.5 Mov ingBod yInterfaceXlod elingMethod.

2.5 .1 Gcnerat iouof Panels .

2.5.2 Calculat ionofDen sit yFuncti on .. .

2.6 Coupled Moti on ofRigid Body and Fluid.

2.6.1 Mot io nofRigidBod y

2.6 .2 Calculatio nof Hyd rod yn am icForces

2.7 Su m maryof theCou rp nt atioualMeth od

3 "Va ter En t ryof3DBod ie swit h Pres crib edVelo ci ties 3.1 Verification

3.2 Wa ter Entry of a 3DWedge.

3.3 Wat er Entryofa3 DShi pSection. 3.'1Water Entryofa Flar edBody .

41

42

·17

51

51

56

58

62 63

65

76

82

3.5 Wat er Entry ofa Sphere. 3.6 Water Entryof a:mPlaningHull .

4 WaterEntryof 3D Bodieswith Free-FallMotion 4.1 Water Entryof3DCylinde rs.

4.2 Water Entryofa:.lDCatama ra nModel. 4.3 AsymmetricWater Entr y ofa 3DShip Section.

5 Co n cl us io ns and FutureWo rk 5.1 Conclusions.. 5.2 Fu t ureWork.

Re fer en ces

Appendi ces

Append ixA: Co nj ug ate GradientMethodand Preconditioning

AppendixB:WaterEntryof a Free-F allLifeb oat

vi

86

90

98

98 104

109

117

117 120

128

130

132

List of Figures

I-I Violentfluid motion

2-1 Afixed Cartes ia ngridfor amultip lephasedom a in

2-2 Multi-p has e com p utatio na l dom ain

2-3 Prin cip leof theCIP meth od(Ya beet al.,2(01) .

2-4 Onedimen sionalgr id

2-5 A far upwindcell inone dimen sion al gr id

2-6 Twodimensi on al gr id

2-7 Upwindcub icgr idcell

o, Ca lcul ati on of densityfunctionfor a 20 wed ge. 2-9 Pointdistribut ion s onaline . 2-10 Quadrilat eraland trian gul arpanels .

2-11 Paneldistributionson a spher e

2-12 Paneldistri but ionsona 3Dlifeboat .

vii

15

24 28

29

30

33

35 41 43

44

46

46

2- 13Calcula t ionofthecont rib u t ion fact or .

2- 14 Effect ofgr id refinem en t onthegeome t rycap t ure : Top 80x120 x 120, bottom140x200x150 gr idcells.

2-153D bodi es embedde d infixedCartesia ngrids 2-1GPro cedu reof thcnu merica lmeth od .

3-1 Comp utedadde d massfor aspher e .

3-2 Awedgeused in droptest

3-3 Geom etry ofa3D wedge .

:l-4 Couiputat ionalmodel.

3-5 Exp er imcnt aldrop velocit y .

3-G Sen s iti vity st udies .

3-7 Vert ical slam m ingforcc.

3- 3D Aow cffect s . .

49

50

Gl

G4 G5

GG G7

G7

G9

71 3-9 Hydrod yn ami cpressure (Pa)distr ibu t io n onsectionsof a3D wedge . 72

3-10 Pressur e coe fficientsof testpoints . . . 74

3-11 Free sur faceelevat ion duringthcwat er ent ryof a 3Dwedge 75

3-12Geometryof a 3Dshipsectio ninthe drop test . 7G

3- 13Comp utationa l mod el of a 3Dsh ipsectio n .. 77

3-14Exp erim ent alvelocit y .

viii

3-15Vert icalslam miugforce . 78 3-16Numericalandexperimeutalhydr odynamicpressures (K Pa)forthe

shipsect ionenter ing calm water. 80

3-17 Freesurfacesim ula tionat t=0.058s. 82

3-18Geometryofaflaredbody . 84

3- 19Velocityofflar ed bodyinthecomp uta t.ion . 84 3-20 Timehistory ofthe sla mming for ces onaflaredbody. 85 3-2 1 Timehistory of the sla m ming forces ona sp here(Fn=2.2030) 87 :1-22Timehist ory of the slammingforcesona sphere(F n=:I.1156) 88 3-23Free sur faceelevation duringthewat er ent ryofa sphereent eringcalm

wateroblique ly (F n=2.20:l0). 3-24Velocit iesin the computa tion 3-25Geomet ryofa3 0 planin ghull. 3-26Computa t ional mod elofaSf) plan inghull . :l-2720stripandforce/velocity components.

89

91 92 os

3-2 Ma xinnunslamming force coefficient fordifferentrollandpitchangles 94 3-29Pressuredist ributi on(Pa)ouaplaninghullente ri ngcalm wat er

(pitch=5degrees and roll=O).

3-30 Pressur edist ribu tion(Pa)on a planinghullat5pitch anglesand different rollangles.

ix

95

96

4-1 Computationalmodelofa30 cylinder 4-2 Depthof penetrationduringwater ent ryofcylinders 4-3 Numeri calresult sbySun(2007).

4-4 Free sur face deformationduringwat er ent ryofahalf-bu oyant circ ular cy linder

99 100 101

101 4-5 Freesurfacedeformation duringwater ent ry of a neutrallybuoyant

cylinder 102

4-GEffectofgrid rcfinementonthejets(20) 103

4-7 Compa risonofcomputed jet s lO:l

4-8 Geome t ryofa:lOcata mara n model 104

4-9 Velocity ratio ofcatama ra n mod els lOG

4-10Comparisonof freesurface eleva t ionsat m'=().29,H/L=O. (Left:

comp uted, Right : expe riment al). 10

4-11Asynn nct ricwater ent ry ofa3D ship sect ion . 109 4-12Measur edandcalculate d hydrodynami cforce (ti lt 2.3°). 110 4-13 Xleasur ed andcalculatedaccelera t ion and velocity (ti lt:2.3⁰)•• • 111 4-14Measur ed andcalculatedhyd rodynamicforce (lilt:20.3"). 112 4-15 Measured and calculatedaccelerati onand velocity (ti lt:20.3°). 113 4-IGMeasured andcalculated hydrod ynamic force(tilt:20.3") . 114 4-17Measuredam!calculate daccelera tio n amivelocity(tilt:14.7°) . 115

8- 1Afree-fall lifeb oat ente r ingcalm wat er .. . ... .. . . 133

xi

List of Tab les

:3.1

Dat arelated to thc expe rime nta l dro p test ofa wedge. GG 3.2 Ri\ISE of thcnumer icalsolut ions bythe CIPmethod 73

3.3 Da t a related tot.he expe ri me n tal drop test of a sh ipscction. 77

3.4 Ri\ISEofthe numeri cal solut ions bythe ClPmethod 81

3.5 Offset. tab leofthefla redbod y 83

3.GRMSEof thenumericalsolutionsby theCIPmet hod 90

4.1 Ri\lSEofthcnumeri calsolut ionsbythe CIP meth od 102

4.2 Ri\lSEoft hcnum eri calsolutions bythe CIP meth od 105

xii

Nomenclatures

A

fl,b, c,d B B(l )

c,

matrix

proj ect edarea

coefficientsininterp ola ti onfuncti on bread th of wed ge

inst ant an eou s sub merge d depth colum nvecto r

sla mmi ng pressu re coefficient

sla nun iug forcc coefficient

velocity ratio

coefficient in interp olati onfuncti on soundspee d

diagon al vectorofpa nel compa risonerror

hydr od yn ami cforce Frondenumber

xiii

H

M

p"

resist an ce safety fact or

objec tive valuein interp olati onfuncti on bod yforcedueto gra vity dcri varives of objec t ive valu e gravityaccelera t ion drop height lengt hofhull diagonalma t rix bod ymass

added mass mass coefficient tot alnumber of' panels tot alnumber of nodes norm alvect or

pressu reintime stcpn

xiv

I"

pf(""~lIn'inlilllt,step11+I pressure ill atl\'t'diull phas..

radius of a sphere

coordinates

o r

central point oflllllllll'l coordinates ofccruer points of a !,i\lll'l

distance vector

time

bodvwlocitr

trauslarjcn velocrtyof1II;L"';('f'ut.'r

velocitveomponents

body vekx-iry components

veloeitycomponems ill limt"stt·pII

velocitycomponents ill time S\('P"+I velocitycomponents ill adn,'liOlll'ha,...·

velocity romponcntsill uou-advectiou ph'L""I

vertk-alvclocity

1I1lk 1lOWIl ('o 11l1ll1l\"('('Ior

c.m c.1 c.1f c.x,c.Y,c.z

coordinat es of massrent er coordinatesofcentro idof pan els coordinates inCartes iancoord inatessyste m par t icleposition of upwinddeparturepoint coord ina tesof nodes of pan els grid distributionfactor grid st retc hingfactor

mass ofaconiputat. ioua lcell time st ep

volumeofa computa t.ioua lcell gridspacings Relativ e erro r Kron ecker 'sdelt afunct ion pan el cont ribut ion fact or sma ll positivc cousta ntuumbcr velocitypotent ial densit y functio n

xvi

J19host

{lfJhost

P.o;oiit!

p"

p'

dynamicviscosity coefficient

dynamicviscosity coefficient for air

dynamic viscos itycoefficient for ghost point sinsid ethebod y

dynamic viscos itycoefficient forwat er

tcnsorof incrti aiuom ent

circnmferencerat io

massdensi ty

mas sdensityforair

massdensityforghostpointsinsidethebody

massdens ityfor solid body

massdensi tyforwat er

massdensi ty inlime ste pIl

mas sdensityin time ste p1l+I

massden sity afte rad vect ion phas e

tot al stress

contro llingfu nct ion forpan el concentration

ang ularvelocity

xvii

±

compu ta t ionaldom ain

cub ic polyn omi alinterp ola ti onfuncti on

der ivativ es ofcub ic polyn omi al inte rpo la t ion funct ion

tan gentfunct ion

positiv e,negativ einfinity

xviii

Abbreviations

I3E:--!

I3FC CF O CFL

CG

CIP EOS 1"O1\!

FPSO LNG LS RE R~!SE

SPO SPH VOF

10 20 3D

bou ndar yeleme nt meth od bou nd a ry-fi tt ed coord ina te computat iona l fluiddyn am ics Conra nt Fried richLcwy co nj uga tegrad ient constra ined interp ola tionprofile equat ionofstate finitedifferencemethod f1oatingprod uctionstorageall(loflloa d ing liquefiednaturalgas

level set Richard son extra po la t ion root-mea n-sq uareerro rs symmetric positivedefin it e

smoo t hedpar ticlehydrod yn a mics volum e of fluid

one- d ime nsional two-dimension al three-dimen sion al

xix

Chapter 1

Intro d uction

1.1 Backg ro u nd

Ships or offshorest ruc t ures may experiencesevere wat erimpactproblemsin theharsh environment,suchas slamming.greenwateron deck andsloshingin tanketc(See Fig.1-1).Theseimpactloads are associatedwith highlynonlinearfree sur face flows.

They cancauseseriousstruc t ura l damages, andareofconside ra bleconcern tothe sta bility andsurvivabilityofshipsoroffshorestruct ures.

Slaunniugisahighlynonlinear free surfaceflowphenomenoncausedbymotionsof aHoat ing structureina roughsea.When thehullemergesfrom thewat erdneto itsmoti on and re-ent ersthewater.the combinat ionofship motionand free sur face moti on canresultin largedynam icimpact loads.and subsequentdam agetothehull.

Slamming loadis oneof typicalissuesin thedyn amicload ana lysis. Forlarge ships, slamming force ana lysis is ofgrea tconcern instruct uraldesign.Forhigh speed marine cra fts,slammingimpactis aprimaryfactorin motionprediction.Inadd it ion,many offshore marineopera t ionsinvolvetheloweringofobjectsthro ughthefree surface.

Slamm ingforce can impactonthe operationsandcauselocaldamageof the objects.

(a}Slammilll'; (h) Gr....1lwatNond....k

Figure 1-1:ViolentHuidmotion {c)SI""hinl';

ToImprovethesafe find etrccuvcoperation ofsurface shtps anrl()lf~h()r('structures ill h",lvy"",L-;,it isnecessary toinfrl'ilsptheknowledgeof lIow behavior andwnve ilHlun'l lmotionof ships during water imparts. TIll'fOf("1'S and pressure that arc exertedhythelIuid impact on the structures are especiallyimportant.

~Il'myexperimental studies haveheencarried out011waterimpact.Gn-cuhowand Lin(I!.lKI)conducted "'JIm' 2Dslammingtt'Sts,<Iuriugwhich the snrfru-e of till' water WIlSdisplar-edn greatdealfromits undistributedposition. Pnrticnlar attentionWl<..<;

paid tothe point ofintersection ofthe free surface and a moving hody. Zhao"l al.

(19~Jfi)nlHil'{loutdropte-ts for a 2D"",'<11\"nmla l'eal shipsection.Hydrodynamic fo[(",'S am! pressure distributions wereobtained. :IlJlIow !'lft'lts were dist'us.<;l'llill theirwork. Troesch andKa ng (191\{j)runduru-damodol test ofa spln-re l'llll'ring r-almwaterwith bothvert iral amihorizontalvelocities. Experimentshaw beouthe primarysoun-eof pnl("tkal information. hut they an' expensive to conduct.

Nnmencelsimulations havealsobe-n lISI'I1to studythewaterimpact. The maindifficulty illtill'numerk-al method ad",'S from the treatmentofcomplir-nted hydrodynamicpln-nomonasuch 1<..<; high speed imJlIlt'tsOilstructures. breaking WIlVt'S.

jets.alld airbubbleentrainment. A fI'vipw of rarfior rr-soarr-h Oil water entry problemswas given byKoro hkinaudPukhnachov (I!lS8). Zhao amiFaltinsen(1!l~j;l) studied the water entry of a symmetrical wedge \Ising theboundaryelr-mcutmethod with constant clemente. Yangand Qill (2007):Nh'~'11tlit'20watt'r(·utry pwblcm

ofsymmet ricandasymmet r ic wedgeswith vari ousdeadrise anglesusing theCIP meth od . Asummaryofthenumer icalmeth ods onslamming is given inthe next sectio n.

1.2 Lit er atur eRe vi e w

Wat er entryis a complex hydr od ynam icproblem. Manydifferentmeth odshave been proposed . I3eforethedevelopm ent ofcomputer, thesewerepredomin an tly analyt icalmeth od sbasedon differentassumpti onsintend edtomaketheproblem mathemati callyman ageable.The theoret icalanalysisoftheimpactproblemwasfirst cond uct ed byvonKarm an(1929)and modi fiedbyWagner (19:32) toincludethe localuprise ofthewate r. Armand andCointe(1986), Cointe (1991)andHowison et al.(199 1)extendedWagner'stheorytoanalyze thewedge ent ry problem using matched asympto ticexpa nsions for wedgeswithsmall deadri sc angles.Furtherm ore, Dobr ovol'skaya (1969) develop edanana lyt ica lsolut ion in term s of anonlin ear singular int egral eq ua t ionfor theprobl em of symme t ricalent ryofawedgeinto calm water. Shiffman and Spencer (1951) develop ed genera lexpress ions for the pressur e distribut ionamislamming force ona cone.Milch (19 1)derivedthe added mass coefficients for adouble sphe ricalbowl with anana lyt icalsolutio n.For wedge-type bodies.these approxi matesolutionscanbe easily used to calculatethe slammi ng forces. However. therearelimita t ions when they are app lied tomore complex geomet ries.

Withthe genera lava ilabilityofcomputers,theemphasis in the trea t mentof free sur face problems is shift ing toward snumericalmeth ods.Manynumericalmeth od s havebeendevelopedtosolve thewat er ent ry problem on thebasis of pot ential flow theory.Basedonthework of Vinjeand Brevig (1981),Greenhow(1987)used Cauchy 'sformula to solvethewedgeentry probl em.In hiswork,bothgra vity ami

- -

nonlinearfree surfacecondit ions weretakeninto account. Zhao and Faltin sen (1993) studiedthe wat er ent ryof a symmetrica l wedgeusing the bound a ry clement meth od with consta ntcleme nts. Thejettip attheintersect ionpointof thebody surfaceand thefreesur face was cutandtwo sma llconsta ntcleme nts were distr ibute d. The gravity wasneglect ed .Zhao ct al,(1996) exte ndedthis meth odto generalasymmetricbodies withflow separation.When separationoccurs from a cont inuouslycurvedsurface,the sepa rationpoint s arcdet erm ined emp irica lly. There are numer icaldifficult iestotrace the wat erparticl es atthe intersect ion point .Linetnl.(194)present ed anapproach totreatthedifficulties.Intheirwork, the bound ar yintegralequat ion derived from Cauchy's formu lawasdiscretizcdusinglineareleme ntssothattheintersectionpoints canbeusedasthecolloca tion points.Chua ng eta!.(2006)developedabounda ry clement methodbasedonadesingular ized Cauchy'sIon nula.A numericalapproach was alsodevelop edto removethe corne rsingularityat theintersectionpointofthe bodysur face andthefreesurface.

Although grea tprogresshasbeenmadeinsolvingthehighlynonlinear free sur face problemwith meth odsbasedonthepotentialflowassumpt ion,there aredifficulties forthesemeth odsto treathighlydist ort ed or breakingfree surfaces.Thesedifliculrics can be overcomeby a comp utnt iona l fluiddyn am ics (CF D) meth od basedonsolving the Navier-Stokesequa tio ns. Free sur face mod elingmeth ods are essentia l inapplying CFD meth od sto solve the wate rentryprobl ems.Itisnecessary toresorttonumerical solutio nstofind theinterfacebetween airand wa ter. Calculati ngtheadvectionof free sur faceis atoughtask since the freesurface mayundergolar gedefor ma t ion and eventopo logicalcha nges. Formodelingthefree surfaces,severaltechnologies exist, and they cangenerally beclassifiedintoLagrangianand Eulerianmeth od s.

TheLagrangia nmethod, alsocalled themoving grid method,isused to const ructa computat ionalgridthatisfitted and moveswiththefluid bymean s ofabound ar y conformingcurvilineargrid,ablock-structured domaindecomp osit ion,orovers et

meshes(Fckkcn.2004).Itisa relatively sim ple met hodof defining and trackinga free sur face.Becau sethe grid andfluid move toge t her, the gridauto ma t icallytra cks free surfaces witho utsmearingtheinformationat theinter face.Hirtetal,(1970)used theLagrangianmeth odtosimulateatransient1I0w of viscousincomp ressib lefluids wit h free sur faces.Intheir work,theLagrangian methodgave accura t etrea tmeut of free sur faces.Theprincipallimitati onofLagrangianmeth od sisthat they cannot track breakingfree surfaces.

Aspec ia lcase of theLagrangian meth odistheSmooth edParticleHyd rodynamics (SPH)method, originallydevelop edbyGingoldand Mona ghan (1977).Ithasbecom e increasinglypopu lar,sinceit cantrea t largedeforminginterf aces andtopo logical changes.SP His ameshlesstechnique.Itdividesthefluid domainintoa finitenumber of mas s carrying particl es.Themovem entof theparticl esand pressur edistributi on inthefluidare obta ined through solving themomentumequat ionsandcont inuity equationwit h intheLagr an giandescr ipt ionofthemotion(Zhcng,2007) . Themethod wasusedbyMonagh an (1994)tosimulatethebrokendamproblem .Kimctal.

(2007) usedtheSP H method to simulatethewa t er entryof 20asym met ric bod ies.

A drawback ofSPH isitsinherentdifficult ywhenmod elingbound aries [Rogerset al.,2003).Anotherdrawbackisthat predictedpressuremaynot. bevery accurate duetotheuseofanart ificialpressureden sityrelationsh ip.

IntheEule rian method,alsocalledfixedgrid method,t.he comp ut.at.iona lmeshis treat edasafixedrefere nceframethro ughwhich thefluidmoves.Theinterfa ceis not explicitlytracked butis reconstructedfrom thefield va riab lesof thefixed grid.

Theinterfaceis ofafini tethicknessbutcanbe shar pe ned byvarious strategies.The Euleri anmethodis suitab leformod elin glargedeformation of free sur faces.However, theEulerianmetho d has someshor tcomingsassociat edwiththedet erm ina ti onof thefreesur facelocat ion.It isdifficultto applytheboundarycond itionsat the exact location of theboundary.Anotherdrawbackisthatsome interface accuracy

may belostwhendet ails of theinterfacecan notbe coveredby thegrid. Thefree surfacemod elingmeth oddiscussedbelow uses a fixedEuleriangridas the bas isfor computa t ion.

TheVolumeof Fluid(VO r) meth od ,developedbyNichols et al(1980)and Hirtand Nichols(1981),is afree sur facecapt ur ing method. In theVOF meth od , a volume fracti onfuncti onisintrodu cedwith valuesbetwe en zeroandone, rep resentin gthe fra ctional volumeofa cell thatis occupiedbyacertain fluid.The timeevolut ion of thevolumefra ctionfunct ionis obtainedbysolving an ad vectio nequa tion,and thevolumefraction functionis reconst r ucted inevery cell. Themostcommonly usedreconstructionmeth od s arcpiecewise consta nt rcconstru ctio n( HirtandNichols, 1981) and piecewiselinearreconstru cti on (Youngs,1982,19 7). In thepiecewise consta ntrecoustruct ion,theinterfaceisparall eltooneof the coordina teaxes, while the piecewiselinearreconstructionuses alinear approxima t ion with thcorientati on oftheinte rface with in eachcell. Thelinear approximat ionismore accura te,but a significant increaseinthealgorithmiccomp lexity isunavoidabl e. Based onthe reconstr uctedinterface, thevelocity fluxesarccomputed at cellfacesandthe fluid is moved inthe fixed grid . TheVOFmeth odhas been madetoworkwellby many successfulapplicat ions. Hirt andNichols(1981)applied thcVOF meth od to simulateseveral complicatedfree sur face flowproblems,i.e. ,brokendam,undu lar boreand breakingbore.Kleefsman et al.(2D05)solved the20slamming problems ofsynuuetricbodi esby thevor meth od. amiafinite volume discreti zati onwith a cut-ce ll meth odwas appliedon afixed Cartes iangrid.The vor meth odsusin g piecewiselinea r approx imation werehighly accura teand had no massconservat ion errors;however ,itsimplementationin3D wasdifficult(Yokoi,20(7).

TheLevel Set(LS) meth od , origina lly devisedbyOsh er and Sethi an (1988),has been proventobe successfulas afree sur facecapt uring meth od.In theLSmeth od , the deformationand movem entof thefrce surf acc canbe ca pt ured bya cont inuoussmooth

level setfunction, whichhas th e fea tur es of n signe d dist a ncefuncti on .At onephaseof flowthe level set functi onhas positi ve dist an cefrom thefre e su rface, negati vedist a nce at anot he r phase , andzerolevel corres po nds to the free surface, The L8 meth od ca ntrea thighl ydistortedinterfaces. Also ,thetopologycha ngesareincorpora ted au t oma tically (Fekk cn , 2004). Sussmanct al. (1994)solved incompressibletwo phas efluid probl em sbased on theL8 approac h. Walhorn etal.(2005)simula te d the2D fluid-stru ctur eintera cti onprobl em susin g the L8meth od. However ,mass conservationerro rsoccur re din thenumeri calpresen t a ti on oftheadvectioneq ua t ion for the det erm inati on of thelevel set funct ion(Suss ma nandFa t cm i,1999).

The Constra ine dInter polation Profile (CIP) meth od,develop ed byYab e et al,(200 1) based on thework of Tak ewaki et al,(1985),Yabe et al,(199 1)and Yab e (199 1),isa highorde rupwindscheme for ca p t ur ingfree sur faces. The CIP method uses afixed , Euleri an gridasthebasi sfor com p uta t ions.Italsoem ploysaLagran gian solut ion todet erminethefunctionvalu efittheupst re amdep ar tur e pointatthe new time ste p , dep end ing onan interp ola ti on fu nct io nof theinitialprofile.The CIPmetho d usesboth the advect ionfunct ionand its spatialderivat ivesto const ructthe high order inte r po lat ion functi on wit h inonegr idcellsotha t the interf aceprofile ins ide thegrid isret rieved.The CIP meth od , as an int er face cap t ur ing meth od,doesnot need anadapti vegr idsyste mand ther efor eremov estheprobl em s ofgr id dist orti on causedbyinterfacebreakupandtopology change.Furthermor e,the schemeca n treat multi-phas eprob lem s and itcan hand lelargediscontin uiti esor large gra dientsat the interf ace of differ entphas esbecau se of its compac tsche mecharacterist ics(Ynbc et al..2001).A pressur e-bas ed algorit h mcoup led with Cl P has been prov ento be stableandrobus t insolvingthe2Dslamm iug pro bl em (J-Iuand Kash iwagi, 2004).

Thevalidationandver ificatio nof 2DCIP meth od werepresented byVes tb osta d ct al,(2007),andthe numericalresul t showe dthat the CIP meth odwas both ro bu st andaccura te for cap t ur ing violen tfree sur face flows.21met al.(2005)stud ied the waterent ryandthe ex it ofahor izon talcircularcylinde r withthe CIPalgor it h m

in thc20com p uta t ionaldomain. Yangand Qiu(2007)solved the20 wat er entry probl em s ofsym me t ric andasy m me t r ic wed geswit h variou sdead rise ang les usin gthe CIP meth od.The effectofcomp ress ib leair for20 wed geswithsma ll deadrise ang les wer e st ud ied(Yangand Qiu,200).

From the literaturerevi ew ,itcan be seen thatman y expe rime nta l,ana lyt ica l,and numeri cal st ud iesofsla m ming 1I<I\'ebeen carr iedout. Thcvastmaj orityof thcwork is ,however,restri ct edtotwo-d imens ional bodi es am isimp leax isy m me t r ic bodi es.

Relativelyfew at te mpts havebeenmad etorigor ou sly solve impactproblem s of30 bod ies .Someresea rch esall30 impactprobl em swer e cond uc te d by Troeschand Kan g(19 6) .Theynumericallystudie d impactload s onthree-dimension al bodies usingtheboundarycleme nt meth od .In thethree-dim ension al comp utat ions,nor mal dipoledisr.ributionsamianeq ui-po te nt ia l freesur face wereused , and theresults werecompa re d withexpe r imenta l results .Faltinsen andChezhia n(2005)presented a numeri cal meth od for three- di me nsio na lslam m ingprobl em sbased onthegeneralized Wagner method .Thorodd sen etal,(200-1) studiedtheinitialstage of theimpact ofa solidsp he re into a wat er sur face usin g anovelultra-high-sp eedvideo camera.

Ther eis aneedtofurtherdevelopthrec-dim cn sion almuthod sto solve the sla nu u ing probl em s since most of water ent ry phen om en aarcthree-dim en sion al.

1.3 Pr esen t Work

Presentresear chworkfocuses on thedevelopm en t ofa numeri cal simula t io n toolfor highl ynonlin earwat erimpactprobl ems.Inpar ti cul ar ,thephen om en onofslamm ing on 30ship hulls,whichhas ca ug ht theinterest sof man ymarine eng inee rs and naval archite ct s for alongtime,is an important mot iva t ion. Mo reprecisely,themain ob ject iveof thisthesis istodevelop amet hod tosolve 30sla m mingproblemsby ad d rcss ing thclollowin g uspcc ts:

Three-dim ensionalHoweffect s. Thephysics ofslam ming problemsis ext remely complexsince it involvesbreakin gwaves and jet s,etc. Duetothe challengesof the realist icphysicalproblem and thenumericaltechn ology,thevast maj ority of this workhasbeenrestri ct edto20orsimple wedge-t yp ebod ies.Rclat ively few at te mpts havebeenmad etorigorou sly solve impactprobl ems of 3D bod ies .Thiswork aimsat developin g anumericaltechni qu etoinvestigat e3D nonlin earfree sur face How effects in theslammin gproblem .Inthe resear ch ,the 20Const ra inedInterpolat ion Profile (CIP) meth od (Yang,2007)isfurtherdevelopedto simula te the3D nonlin earfree sur face problems.

Coupledmotionsimulat ion. In order tosimulate theslam mingproblemmore realisti cally,thecoupledmotionofa 3D bod y and fluidhasto be solvedaswell.

Therefore, ameth odhastobedeveloped that is capableofhandlingfreelymoving bod ies.The coupling motion simulation posesmore cha llengingwork. Thebody velocity isunknown befor ethesolut ion for thefluid isfou nd.Tileaccuracyof tile pressure willbefedback thro ugh thefluid flow via the calculation of thebodymot ion, Inaccur a tepressur e canca use fals ebody resp onse.One of tileaimsof thisworkis to overcome thesedifficulti es andaccura telypredi ctthebod ymoti on and theviolclit Howphenomenaduringwat er ent ry.In thisresear ch. sla mm ing forces and moment s are obtainedfrom the Navier-St okes equa t ions whicharc solved bya CIP-base d finite differencemeth od.Themoti on ofasolidbodyispred ict edby numer icalintegrati on of differential eq ua t ions ofmoti ons conside ring the compute dslamming forces as exte rna l forces.Thefluid velociti esduetothebod ymoti on arccalculated.and they arethenusedforthefree sur facesimulat ion usingtheCIP meth od.By solvin gthe body and fluid moti ons simult a neously. the coupled moti ons arccompute d in thetime domain.

Three-d imensionalmovingbodyinterfacetreatm en t.. Todistinguish the flow and solidgeometryonafixedCartesiangrid,the volumefractionofsolidbod yin

each Cartesian gridcellmustbe calcula ted accur at ely,whichis importantfor the comput a t ionof fluid-stru ctur eintera cti onproblemsusin gtheEulerian grid meth od . However , allowingarbit ra ry 3Dgeome t ry(i.e..arealisti c ship).the computat ionof volum efracti on can becomevery complex. Esp ecially,consider ing not only tran slatin g,butalso rotat ingbodi es,the situa t ion becom es somewha t mar c complicate d. Therefore.it will be impo rta nt todevelop a geometry reconstru cti on meth odthat willcompute the volumefractionaccura te ly. Inthis resear ch , a pan el-bas ednumericalmeth odisdevelopedto capt ure the arbitra rysolidgeomet ries.

Thebody sur face isrepr esentedby aset of panels.Sincethebodyisassumed to be rigid,thepanels canbeusedtoupdate thebod yposit ionwithaLagran gianmethod ineach timeste p.Thecont ributionofeach panelis estima ted bythecont ribution fact or ineachcomputa t iona lgridcell.Thenthedensityfunctionfor asolidbody is obtai ned,and the solid phaseismodeledinafixedcomputa tionalgr id.

1.4 Ou tline of the Th esi s

The thesispresent sthenumerical solut ionof slammin g for3D bodies ente ringcalm water.Thehighlynonlin earwat er ent ry probl ems arc governed bytheNavier-St okcs equa t ionsamiarcsolvedby a Constra inedInterpolat ion Profile (C IP)-based finite differencemeth od onafixedCart esian grid. The CIP meth odis employed for the advect ioncalculat ionsandapressur e-based algorit hm is applied forthenon-adv ection calculat ions. The solid bod y and free sur face interfa ces arccapt ure d bydensity functi ons .Themoti on ofabod yis descr ibedin term s ofsix degrees of freedom .For thepressure computa tion,aPoisson-type equat ionis solvedateach time stepbythe Conjuga teGradientiterativemethod.

Chapter2introdu cesthecomputat ionalmeth od.TheNavierSto kesequat ionsarc describ edasthegoveru ingequat ions, Thefractional step meth odisintro d uced,

10

and the govern ingequationsarcsplitint othree-differentphas es.The CIPmeth odis discussedindetails andthe upwindcubicinterpolat ion functions nrc given. A Poisson equationof pressur eisderiv edbas ed onapressure-b asedmet hod, anda Conj uga te Gradientit erat ivemethod and Jacobipreconditioning arcintroduced .Free sur face and movin gbodyinterfacemodelingmeth ods arcex plained. Thenumericalmethod for coupled moti ou ofa solid hod y andfluid isdescrib ed .The computa t ionalmeth ods forhydr od yn ami cforces andmoment s areprovided .

InCha pte r3, a verificationofthe numericalmeth odis presente d. The added mass of a spherewas computedandcompa redwiththe ana lytica lsolutio n based onpot enti al flowtheory. Convergencestudiesofgridspac ingand time ste p were cond ucte d.

Valid a tion stud iesare describ ed forseveral3Dsolid hodi es enteri ngcalmwate r with prescrib eddropvelocit ies. The computat ions were carr iedout for a 3D wedgeente ring calmwater witha vertica lvelocity. The slammingforce was calculate dandcompa red withexperimentalresults.Threc-dimcusionalIlow effectswerediscussed. Thepresent meth od was alsoappliedtoa3Dshipsectionente ringcalmwate rvert ically.The pred ict ed slammingloads were compa redwithexpe rimentalresult s. Theoblique waterent ryofa spherewas st udied . The vert icalamihorizont al slamming forces were calcula te d fordifferentdropvelocit iesandwerecompa redwit hexpe rimenta l results andother numericalsolutions.Studies were alsoexte nded to thewate rent ry ofa3D planinghullat variouspitch am]rollangles.The 3D results werecompa red wit hthesolutio ns based onthest rip theory and2DCIP method .

Three-dim ensionalbodi es enteringcalmwater withfree-fallmotionsare st udiedin Cha pte r 4.The wate rent ryofa:3O hal f-buoyant cylinderand a3D neut rallybuoyan t cylinder were carriedout. Thepredi ct edmotions ofthecylinders were compa red with exper imentaldat a . Thewater ent ryofa 3Dcata ma ra n modelwit h free-fallmoti on was alsosimulate d. The variat ionsoftheratio ofhullvelocitieswere compute d amicompa redwithexperimenta lresults.Theasymmetricwa t er ent ryof a 3Dship

11

sect ion was simulated ,andthe computed accelerations and velocitieswere compare d wit hexperimental results.

InChapte r5,thisthesis endswit ha summa ryandconclusions. Some future perspectives arealsogiven.

12

Chapter 2

Mathematical Formulation

2.1 Gover n ing Eq ua t io ns

An Earth-fixed('"n,-slllll eoonlinatc SrS!l'JllisI\p]lIi,~ I. :-a.'( ~spciutingupward ..,,~' Fig. 2-1.TIll',lilf"n-ntialt"llllatiOlLsgo verning tllO'unsteadvlllotionofa"L';("()ILSfluid are given asfol k",l' "

8p 8p i} ll,

iii+

lI,or, "" -Pa;:

^{(2. 1)}

(2.2)

wher eti.sth ..rime:r,(I".1.2.3)an' Ih.-rou n liu., t,.,. ill II Cartesian coordinate S~'SI"IlI:pislh.·

Ill ''''''

deusitv:u,ar,'tht"veloeityn"lll"'l"'ul~:f,·LSthegra,·it~·fort'..

F'oraXewtouinnllll id ,tl",1II1a]strf'>'..SrailI..•writt enILS

1:1

U'J=-/,,)'J+2115,)-21IJ;jSu/³

S;j =~ (~ +~)

whereIIistill'dyn am ic viscosityroellk-ientandO,jisKrourc-kerdeltnfunctkm Theequation of stale(EOS)iswrittena.~p=f(p).Applyingthr-EOStoFAj, (2.1), the Pf(':'SIlWequation can bcohtainoda.~

~_D/+IIi!E.=_'fh, _~~_{'Dr ;} (2.')

lf turbuk-nt Hews are considered.tIH' ('1lna t ions ca n lH' avPnl l';''11 overthe thnc ,,,,ai,' of turbuk-ur-e amiadditionul''(I"atiollsdt',wri!Jingtla'Reynoldsstn~",,:,II''('(!to ],(' iutrodur-cd.Sinceslammingisa transientproblem.till'significant turbulent df,'Cts donothavetimt'to develop duringtherapidimpacts (Falt inseuetal..2()(J-l).The laminar flow is assumed inthestudiesfor till'slammingproblem basedOil till' work

"f Zhll dal. (2005)

A fix,,,! l'l"'lall Jl:lll<lr''<>!lIpUl.ati'l((al,l"lrlail! is ll"'''! in 11ll' prest'II1computueion,"',,' Fill:.2-1.An exterior Iktiti011S{lIl"{'dlla.wrt",,;t,.h sid""ftlj('I'ltysit"ill domain is added to allow impositiou ofdiscreteboundaryconditions, TIlt'pre-sure'1longthe top ofthecomputationaldomainis s"ltozero.Xo-slip wallLoundaryconditionsun- appliedtotln- domainsides,Furitwau-rentry prohk-m,ifthe domainhouudark-s an farenoughfromthebody,theexistcure ofthe boundarywillnotgin'significant influence dnriug therapid water entryproc ee- Ccmputarioualdonmin including t.hcgeomet ry of a movingbody,air and wutr-r,isdividedintoregular cubicgr ids.

TIlt'solidbody boundary isimmersedill the fixed Cartesiangrid with II fractional volume or an'a reprc-emauoutechnique(S,,,'Section2))).TIl('solidbody is st'l up

J.J

=0

Fictitious

Figllf('Z"l:A fixed Cartesiangridfor amultiplephasedo ma in

as emcving objectwitha dropvelocity and its contour isdefuu'daHano-slip wall cumhtiou (Sf""F,(I. 2.100).This approachallowsfo r a complexgeometrywithoutIt

limpconsuming gridgenerution asforItboundary-fittedcoordinate (Bj'C}method Itiswell known thatBf'C grids,"lUIh,>diilkultto ronstrurtandit tak •.,; a significant amountof tunetoestablishItworkable andwell-behavedgrid even with good gnd generators.TIl(' pn-sent nnnu-rk-al methodisdt~<nih,,<1illdetail..int.hofollowing

15

2.2 Dis cr e ti za t ion M etho d

Applviugthr-fractional su-papproachto Eqs 2.1to2.:\!('i\(!Sto tlwnumerical solntkms ofgowr umg equations illthreest p]", asfo llows(HuandKaxhiwagi,2tKl-I ).

2.Xon-advcrticu phiL""}

P'~/" + 11~tf

⁼⁰

1I:~I1i~ + lj~'~

⁼⁽⁾

(2,4)

(2.:i)

(26)

(2.7)

11~+~~

^u;'=

-f, iJ:;:.~1 l'''+~~

^p'=

_p.{'~ a~~~1

(2.l1)

(2,9)

(2.10)

W1Jt'IPtlwsllp,'rS{'ript~•and**in,U.-.at,>the iutcrmodiutc vahlO'suftert.lu-cak-uhrtious

"Iudve ctiou phase and nou-udvcction IpIHL~'"The superscript 1l+1 indk-atcsth" filial valuesatthenew timesu-p.The fractkmnl steps ill thePH's('1lI numerir-almerhodure arr ung cd ill the order of udvcctiou!,hasl'.uon-udvection phiL....·I.and nou-ndvcction

phaseII

The equation"illtill' edvectiouphasean'computed byII... CIPmethod whkh ...iII be uuroducedin &'-lio1l2..1

FOfanilK'OlllllC.~ihlt,fluMI.11M'uon-advectioupba...•I.r....l. 2.7.ranbeexpaeded es folio".."

u"- u' piflu' iflu' iJlu'

~

=;(

^all+[Jy2+iJ=2) (2.11)

11''';, II"

⁼

~(~~'~.

⁺

i;;~'

⁺

~;;~. )

^{_,Q (2,1;1)}

AuEuler cxplk-It Sdll'lllt'i,. 1Ist'<1forthe tim., inl' ''ItraI MJIlfur tl,,·I..ft-han d -s kl..terms ofEqs, 2,1110 2,1:\an,1a N"nITa!finite ditf''r''nct'SCh"lll('isIl",-,I10di"CWIVR the termson tilt' riJi:ht-halld-,.HIt'oftbese equations.TIl<'\'f'ltlCilirliill non-advectkmIrail bt'obtain ...!a.~f"IIt...·s

u ..-u·+

7(~;2·

⁺

~.

⁺

~:U2' )

^{(2. 1-1)}

17

These second-orderdcrivutives in Eqs. 2.14to2.lfiCIIIlbe appro ximated by

iPll' (u:+I._J•k-11:.}.kl/.l.r-(u:.},t-1I:_I.}.k)/~.r

i)xl= .l.r

iPv'_(lI:+l._{J •}k-I't.j,ll/A.r- (v;'ol,'

-1':

lol,. l/A .r

0.r1- A,r

iJ111" (u':+I.).k-u't,j,.l /iJ.r -(wi.)..-u';_lol,kl/A,r

D.r¹ Sr

iPll" (u·:.}+u -u':ol.kl/.ly -(n':.),.-IV:olul/Ay

iJy1 A,I/

IX

FortheIloll -ad\'t'('tiemphiL';o:II,iua g"Ill'r al 1m .\',lilt'd"lI~ it ~·Pis solvedfin-t,end thenthr-P"'·SMlf.. ('ali I.... raknlatedba-ed on Ill<'d"IL~i t ~·.Since thespeed ofsound, c.=,fiiPJiJP.i.~wrylargeforsolid01"liqu id"ha.w.;,11small dcusity errorcan lead10

"la rge prt'S>'urepulse. and ittendsto cause a di."j"1'>'ionI'rror inutnnerj calsinrulariou .

•".pressure-b....,,-...l algorit hm[Yabe.19'JI)kl'lllp~"{'(1to OWn"OlII('this problem.In thepr('Oo;,~llll'-tliL"'l...1 method. aPoisso n-ty pe prf'>'Sl,r(' {'(Ilialiem canheobtained~' lakingrh..dh1,"f'l"II\'" ofUI.2.9 ami substrtutjngj)'l,/iJr,uuo Eel. 2.10.

(2.17)

Eel. 2.17showsrhat'il/I! pis routiuuous at shilrpdiscontinuities .Intbis cuseif tIlt' d"lIs ilydl<lIl~"Sby S!'\1'rlllurders ofmngnitnde at TlIO'bUlllldilry,forexamph.. IWl w""1l liquidandgw<.tln-pn's.sIITt'grndk-nt canhI'{'llklllal .~II«'{'\lrall'lyenoughto "IlStlrt' tbccontinuouschangc.TIll"":lllali ollis\'t'f.\'robust"\""11withadensityratio lurgor than WOOalltlt llt' lIltllt i-p h,L"'I'w m p utiltiouSfftn 1" · f ltrri ,...10111.

For ape rfec till('(lll lp n ...ihl..filii,\.w..canassumer.=DO,ItsimplerPo is."Oneqnation canheobtai nedas Ix·I",,-:

o

IfJrI' · ' luu"

-(- -)~--,-'-

ur,p. ur, ~,irr, ^(2.11'l)

In th .. oom!,llllllinu,11K"""Iocil)'iTL"ielf'rhe bodyisforo....LtoIX'equaltotheri,;io;I bodyn.·ltX·it~·befon-soh-iugEq.2_11')_ The s"hlli"uofF"'I. 2_1!l pw\'iel,'Sthepr ...ure distr ibutionillthe....h"I"oolllJlutat ionaldomain.XoteIhal IIi<'pres suredi.~tr ibut iun obtainedinsi,lf'tfu-",-,lidbody is aficfitious0111',....hh-hSltlisti, ... thedivergr-nrefro...

eonditiouoftit., wloeit yti.,M(1111amiK,,,,,hiwa gi,24)(17).amiis {'OTlSiStt'1I1 withtI...

pH'S.SUfl' 011tit"IXltl)' sllr(<u'l'. Th{N'points insi,I, 'Ih.-Ixxl}'an'denoteda,,, glll"' t points.Thvreasonfordomgthisis from l\l'mcti"al('flIlllltltllliullltl]x ,iutofvir ...

\\"il h thist.reutmcnt,tI","ollnd a r)' conditionforpn'SSllrcal the intl'rfa{'eIw lw f'f'1I

1!}

differentph,L...·s isnotrequired.

Till'rentrnltillite dilf"rt'IH'"methodisusedfor thevelocityderivat ives 011 the right-hand-sideof E'I_2.18.

i~;;O⁼

iJ;;o

+i~~O+iJ;~o⁼!l;~ IJ,;~~I;:IJ,k+L':j+l.;~.~';j_l ,k+U':j.k+ ~~:L':j'k_1 (2,19) Thepress uretenuson the left-hand-siderunhl're-writtenas'

\\"1'start toapp lytheceutrnl finite difrl' lTlll'"lud hnd to tIlt' firs tterm 011the right-hand-sideofEq. 2.20.Thiscan ],Pdonebyfirstapproximuting7~at pointshalfwaylx-twecuthe gridpoint s ,usinga eentcredapproximat ion,

(2,21)

andthenapplying anothercentralfinit e dilfPTelH'"method toap proximatetilt' derivativesof this quantity,

Applyingth,·samenu-thod tothe SI'I:olHlundth irdtermson theright-hand-sideof

21J

Eq. 2.201('1lI1~luthefollowingcquations.

..z.(~Dpn+l)=...!..(_1_ 1':~+11.1- 1':~;:_ _ 1_Jl:7t - II;~+~ l,k )

Oyp. Dy t:,.yP;,j+l/2,k :iy P;,j_l/2,k D.y

=

i [p;,j+1112/):~+11"

^{-(P;,Hll/2'k+p;,j_l}l/2'k

h'~;~

⁺

p;,j_11/2,.II~,J+~1.kl

^(2.2;1)

and

.Jz(;' a~~~+l)

⁼

-b(P:J':+l/21'~;~+~~ P~;,:

^-f!:,J.•

l_l/21)::'~~ ~~~J~i-,)

=

~[P:,J.:+1//~;:+l ~

^(P:J':+lll+

P;,j.:_1/2)J!:~;~

^{+P;,j,:_I// 'J Ll(2.2.1)}

Suhsr.itutingEqs. 2.22to2. 2 ~intoEq.2.11',yields,

BI"rcar rall /(ingEq.2.25,thcfinal differcntial equation forpresslln'equerionca lltll' ohtaineda.~follows

21

~ (f.D~:~.:I )

⁼

-p~ti [~( Pi+l~2.i'k

⁺

Pi-I~2J'k )+~ (Pi.i:l/2'k

⁺

Pi'i-ll/2'k)+~ (Pi'i':+1/2

^{+PiJ.: -If2)]}

+P?: iJ,k(

~Pi+I~2.j.k)

^+P;'!/i,k(

~Pi-I~2'i'k)

⁺

]J~t~l.k(~

^PiJ+1lf2.k ) +

P~t~l.k(~ PiJ~I/2.k)

^+p;'J.i+1

(~ Pi,j':+1/2)

⁺

P;~t.Ll( ~Pi.j':-1/2)

^(2.26)

wherePi+l/2,i,k =~(pi,j,k+Pi+l,i,k),pi-l/2,j.k =~(pi,i,k+Pi-l ,j,k)'pi.i+I/2,k=Hpi,j,k+

Pi,i +l,k)'pi,i-I/2.k·=~(pi,i,k+Pi,i-I ,k)'Pi,j ,k+I/2=~(pi,i,k+Pi,i,HI )'andPi,i ,k-l /2=

~(pi,i,k+Pi,i,k-I)'

Alinear syste mofequations,Ax= b,can beobtain edfromEq.2.2(j awlEq.

2.18,in which xisanunknowncolum n vectorof pressur e,bisa knownvector of veloc itygra d ientcompo nents,andAisaknown,sq ua re,sym met ric,positi ve-d efinite mat rix. AConj ugat eGradient(CG)itera t ivemeth odisem ployed tosolve the linearequations,andaJacobipreconditi onerisap p lied toim provethe com put a t ion efficiency.A brief discussionon theCGalgorithmandprecond itioningisgiven in Appendi xA.

After thepressurefield isobtainedby solving Eq . 2.18,thevelocitiesin thenewtime ste pcan be calcu la tedbysolvingEq.2.9inthewhole comp utat iona ldomain,

U"+I-u·· 1p?: i,j ,k-P;'!i.i ,k -~-t- =-P: --2-~;-1'-

11,,+1-11" =_2.- ]J;~j~l,k-]J;~t~l,k'

~t p' 2~y

22

(2,2 7)

(2.28)

(2.29)

Tilt, wlol"iti"s innew thne st"p r-an 1>p otnatucd,I~~follows,lJi~~,'(1011 allEuler explicit scheme

u nH=I'"

_ 7fJCi.j.;~ ;C'~'k

vn+l=1'.'_~!J)~J+1l.t

-

P~.J+~1,1

p. 2t.y

(2,30)

(2.31)

Xoto that nuder tho inromprcssiblelIuid'L~sllmptioJl,the densitycun be dett'rlllim'(l bydensityfunctionsinsteadofsolvingEq. 2.1,which willIwintroducediunext S,'ctiol)

Inthe whole solution procedure for the governingequat.ioux, the«nnpuuitkm ofudvect.ionphiL'«' with theell' method has no restrictionon the CFL (Co ur a nt - Frie dr k-h-Lew y ](S('('Section2.4.1).The apphcatiouoftheuxplk-ittinu- lntcgrationscheme,i.e.Eulerscheme,results ill aCFLrcstricuon fortllP tinu-st,'!, size,1t6.I/ur .:s;I.Till' Euh-r explicit iutograt.ion yields first-orderarc-nnwyilltime iutegruriouofthcgoverningequations.Ceutrnlfiuitediff"n'Il(""giw ss''('OI"I-onlt'r at'("UrH,-yillspareil1lt'gratioll.llow,'vl'r,tlwfractiollalstl'pllll'thodrailrnusr-splitt.ing

"nor,tln-rcforcthe resulting methodwilllot'of first-order accuracy (Leveque. 2(/07)

2.3 Fre e Su rface

M odeling Metho d

F'igure2-2:Xlulti-phusccompur.atiounl domain

InIInmlt.i-pluw-romputution.in order10identifywhich partis occupiedhywater.

solidbodyor air in thecomputationaldomain,Itdensityfunction¢",isintroduced.

whir-h~ati.,til'S

wheren,o'III=I,2, and 3,denotesthedomain occupiedbyliquid,solid and air, rcspectivclv. Thedensity Iuur-tions have I!I"valuerang.,ofOm=[0,0,l.OI, If¢I=l.o ,

the(,l'JIis completely suunn'rgcd ill wutr-r;ifOJ=0,no wateris ill thecoll: if

o<¢l<I.II!" ('I'llispartially submergedinwater For "'\l'hrell.thedPllsily funrt.ionstuustflLlfillt ht'foJlowillgrto(juirPJIIPlll:

(2,:13)

TlwH' an' twotyp"s ofilll<'rfllt" ,~tlia t uecd toI",modeledin til" rnmputurion.S",' Fi!!;2-2. Uueis interfucr- hr-rweenairand water[thefr<" surface}.andtheotheri~

2-1

Ill., ill!t'rfan' lx-twrx-nsohdbott~·IUIlIwater(or air).TIl" lwhaviol'sot Ih,'St·t\\l't~1)('!i of illlt'rf;u"t' ar,'.litftort·1I1.and differentmunericalun-thods an'11",,,1 for I'lu·hofthr-m Tlu-deusitv fUUt'l iollfor wateris adver-ted with flow and th.·CII'un-th...l is thusIN..I 10i'roi',.~alt·01with timt'(St,· St, ·tiulJ2.-1).T!l{'IJOlI~',kllsit.\·fuut"lion02is 111.. lal",1 with adir...·1method underthe ""'S1l1Ui'tiullut a rigi,IIJOlly(5<....St'("tiull2.5).TIlt' tI"ll.;il\"functionfor aircan thushi'found hy unhzingEq 2.3.1.&'('lI.u>-(' Ihl' airand Ih.· Wltl.,. pluN'!ian'mo deleda..;asingle fluid ...ith \·ac.,.illli\1>ft'I){11il'li(L.·. d,·ll.;ity.

\·i..;cosity.etc.],thereis no11<....1tospecifv dynamic('onditiullsill h'ml'; of"Irt_'l;

al lilt' iUl."ffan' Iwt'A"t"ll airand'A"ah'r TIle dywunic roll<litioliSart· ItlllOlUllt icai ly

Aft.·rtlw,lt·llsityfllllction fort'a("hpha... i..;dMfOfllliIlM.lh,·l'hnit'lllprol){11it"'.SII<'h as \·i..;n)Sily..utitkll.;il~·.Callhi'calculateda..;f"lIollo'Sfor ...O('hcOIILlmtilliOllalct'lll..L...t 011Ih.,iucoenpn-ssiblef1oWa.>;><lIlllptiun

(2.3-1)

(2·"1

wh"rl'P-t..alltlP...act'the tklll'itie.,,;of water and air:1'_ ••awl II.. ,aceI h.·dynamit'

\i.;n)Sitit"l' ofW"I ..ranti air;andP~aud1'"...ar ...the art ilit-i"I, I.'n.;ilyand \·i..;ro,;ity r..r lh.'S<'ghllNIIN.iuls ill.;itlethe solidbody. IIItilt"r-omputunou.PrJ-'-tJ-..and

(2.:11;)

TIl<' I..fl-hand -sid"ofEq.2.:16i.~til<' materialderivariw.Eq,2.:Uiill,li,.all'Sthatifw"

folio".. a tluid part ide, the prOIM'rt)·of the tluidpartwh-dol's nul l'hauge ... ith timc.

The finite di!f'''''I1('1,' scbcmes ba...-dontill.' Eulerianreprt'N'!ltaliuu tend to prod uce Imnlt"l"ira.lditflL~ioll,Il,'hi<-hwillsmeal"tbeillitialsIMrl)fl~oflilt"interface. In Ihi...

work.a lallll:"lIt fUII('tion(Xiao,19'19)is.L...I to IralL"foclll the d"lIsityfunction into a ncwIunctjon.

(2.37)

... h..refis a»mallposiuvecon-t ..UI.TIL., Iactor ,I-I , ('ullhh-s IL';to get around-00 fnr Ol=Ilaud();Jfor~1^:zI, and to 11llWfortI~imlst''('l'llI'>i!Softill'transition luvcr TIll'parametereIH....b I" hl' ('!lOSl'U IUlihl"ia ll)'hd"n'r-ulculutiou.IutheI'ro'Sl.'1I1 romputution.fz=11.02.AC('<Jrdj ll ~tll thetalllo\"lIl.fuul"tiu ll, a sllla lll'reresultsin Il uum-ncallyshaq Jt'r siopl'll(·ru:.;..; litetransitionluwr.

iJJ;;l)+u,~~I)_O (2.:J..'l)

Eq.2.:ki ('all I ,;01\"1...1 "r Ih" CIPmerbod.The pfl""l'nl Ilurk fOC1L_ 00 3D bodies entermg ealm atcr , Therefore. the ,nih'"i~at rest initiallr b"fofe the body intf'fl"-·t!i witbthcwah-r. Sinn'11ll'compact SC'h,'uII'of CIPIIwlhodanditsSlIb('1,'IIn-solutjcu fe..uu re.theinilia.llihat)IIIl'M of the imf'rlll('t' canbemaintained\~.wellintbe-Il,'hol., ('1,)IIIVlllat,nnpr~.

(2.J~J)

TIlt' deusitvInurtjonofwntr-rthenranI" , "hlah "..1 hyEll . 2.:19 TIll'value of 411 r-haugesrontinnouslyln-tweenthevalues ufair'll" lwalt '!'III lin'fu--surfan' dll!'illR

21,

thesimulation.

2.4 en- Method

2.4.1 Principleof theelP Method

Using an upwind differ en cetechniqueforthe advec t ion term of theNavi er-St okcs eq ua t ionscan leadtoasta blecomp uta t ion inCF D meth od s. However ,the useof upwindsche mes willintroduce excess ive numeri caldiffusion andassociate d inaccuracies.In thepresent comp utation,thead vectionequa t ions of Eqs.2.5and2.6 for velocities and pressur e,andthead vect ioneq na t io nof Eq.2.38forfreesurface, are solvedbythe CIPmeth od to reducethenumer icaldiffusionandimprove the accuracyofsolut ion.

TheCon strainedInter pola t ion Profile (CIP) methoddevelop edbyYab e etal.(200 1) bas ed on thework of Tak ewaki etal.(19 5),Yabe etal.(199 1)ami Yabe (199 1),is ahigh orde r upwind sch em efor solving theadvecti on eq ua t ion.

TheCIPmeth od ca n alsobetreat edas a kindofsemi-Lag ra ng ia n meth od .Ittra ces hackalongthe cha rac t er ist ics in a fixedCart esian grid,andaninterpolationof the profileis requiredto determinethe valueat theupstr eamdeparturepoint. Thekey aspect in theCIrmethodisthattheinterp olati onfuncti onis cons t ruc te d byboth thefuncti onitself and its spat ial derivatives .The stra tegyof theCIr meth od can be exp la ine d by usin g aIDadvect ioneq uat ionas follows:

ot ot

at

+11 &=0 ^(2.40)

When thevelocityis constant, the solut ionof the eq uationgives asimp letransla ti on

27

Figun-'2-:\:Principk-oftill'CIP1l1l'Ihod('{all('"IaI., 2(01)

)JJoti,mofprolilt ,~withavr-hxitvu,A~ ~h0WIlinFi",'2-:1,thr-hurl ..1profileIUOI'P"like II,liL"h,..IIiILt,ill^1\1'''lIlillIlHlI~reprc-cut anon.au, ltlu-solntiou atgridpoiut.s dcuotr..1 bythetill, ..1cird .",isrhesameustheexact "olulioll.However.iftlw,Ia."h, ..1iiII" is eliminat...tas showninFilt. 2-3 (h),the informatioll"ftilt'protileillSidl:thegndr-ell is lost.Wh..u tJ,"l'l't>lilt,i,"('(JII"tnwtl..1b~'a liuearinlt'fIK,latiull, itis naturaltog"1It profilelik.· that shl""'11h~'Ih.· solid liue ill Fig. 2-3 (c), and uumerkal ditfu...ion ari,,,,,,, TiM'reason whvthe..solution becomes ....u~'is thaIIIll' bt'llit\'ior ofthesolutjcuiIL"iel.' grid('("IIb 1tf1tlertf't1.Ifth ..protilt" is approximated ....·ilb aII'...·1IU1l1t..-ic'a1 ",,1"'1I1t'to maketht'"~tiufunllaliouinsiek'ayid('("IIrt1rlt>n'tl.Ih ..nUlllt."l'K-aisclutiouwill become better.

DifferentiatingF...!_2_-10withrt'Sl>('('ttothespalial\'lU'iahll'r, "-r-canobtain.

(2.,11)

wl..'re,f.=iJfliJ.rslan ds 101'thospatialderivativeofl-Forsimplk-iry,It,'OJl,,,t all t udvectiouvelocity'Iisa"sulIll"!. Eq 2.·11 l""Im"" 'llt "till'I'rullili/;al iunofspa li a l

,1l'ri\"ll.lin·witha('(J1l~lali twlccitv u.l""illgElF'. 2.10awl2_-11. till'tirue l'\"ol uliu ll ofIandI.(,llilI...trl4N-rl.Sincethe "palialderivauvr-~~ l~....-dto aPI,wx ima l"th..

pr opag a rlou,Ih..pro ll'" alt"onestepislimitedto a "li't'{'itil"I>rOOk>. ('\'t'11 ius),.l..Ih..

gridcell.Th,' ,;ollll ion lll'COlIll-'; Iillld , CJ,oserl otlll'rl"a1 so l11 l ioll. a." " hOVl"ll inFig.2-3

Figur t'2·-!:On..dicn-nsiounlgnd

A high-order iutcrpolutkmfunctiou callhe',>onslnl('It~1ill nn upwind grid usillgth,- values ofIalHl/. ,"'~-Fig.2·4

a, ""f:";;;.0+1+2(.t:'.:i.-/ +I) b,⁼3(f,".1 -f,") _21:.,+1:""

.:i..r~ ~1"

Co=g~

FurI.l>O.II,.. su l_· ript1+ Lueeds10IX',h aIlK,...1intoI -1,and.it=-.it.

TIl<'profileal th,'n+1s"' p r-antheu beohlui lll~ 1h~·shirt iliK tilt' profilelIyadisl alll~'

"fll .:i.t

J;'+1=,/,(.T-lIC::. t )/ dx

(2.'12)

(2A3)

where (;=-ItC::. t, nmlC::.Iisthetimeste\' _Note that the right-hand-side of Eq.2,41 musthe calculated for a variablevelocirv,whichwill h,· discussed inSection2..1.3

The ell' method hasln-enpw w'<: ltohavethtrd-ordcr accuracyin tinm andspaceby Taylor expansion (Uts u mictal.. 19'J6). T'ln- constr uct.ion of Eqs.2..12alH!2,·l:\uwr a m-ighboringrell oftheinteres ted gridIimplies CFL

:s

1.In(JT<lt-1'to allow tht' Ils<'ofa h,rg.'r tim!'st<,plikeothersemi-Lagrangian methods. Eqs. 2_-12and 2.·t:! arc IJludifi(~1hyapply ingt!lt's" tothe farlll'wiudgrid('(,II(S.'t' the cdlmillFill;. 2-5) fromwhich theLagrangian jl<lftil'!f' startstothe pre-em posi tionufcouo-rn[Yahe.

:'!OOO),

Figure 2-;)Afarupwind 'TII in uno dimensional grid

Eqs. 2.-12ami2,43are modified to 1)('

30

(2..t-l)

(2.-13)

whefe'"is the grillpoint.dctcrmiurd be.'J'm<

I,.

^{<Xm+lforIt:S0 amIXm_l<}

.Tp<^.f",for ">U, "",Ixpi"the partid" 1,,-,,;it;OIlof upwiudd,'partur"pOiUI whid,i~

r-ak-ulatrd by

fp=.r,+!'

wit '+il ' wm-re (is til" distanrc lu-twecntl]('st~two points:

II",

=f).,n

:.;]'.",+1

+2(J~";I~;:'+I ) II",=3(f'':.;I,r~I;:,}_2/;',,,,

:,:;',,,,+1

\\'" theng,'tasemi-Lagrangian schomo lhalpermitsalargt>thue stepfn.'Iromthe n-strk-tionCFL:S1.

With the special treatment of tit .. spatial rh-rivat.ivc, onlytwopointsan>nt'('(!t'dfor coustmcuugthehigh·orderIntcrpclation approximationinOIH'gridcell. Theell' lllcthud achieves<I('OIllI',KtfO[1ll <lIlUprovides~uhecll rt'~ol"tioll.Th" k,'y po;nt" of

theel l'merhodcanI...·~lmUl..l.fi"A't1a:.Iollcws:

2.:\.ruba-inh'f"polati..ufllll("tioll is constructed .Illll iIllugh-orderSdK'lIM' is

3.Sinn'110n"-IIM...hingl"aklilatioui>;required.til<'l"olllput atiouMlilltl'('allusually

-t,Th.. ell'1l1l'lilod dOl'O<notinvolveillW iUI..rfan·NlI~~Ir\ll'liunprocedure andi.~

quit ..,'("<'lIomkalillromputationalcomplexity.''S!lI't'iallrfur3Dapp!i"',,!ions

2.4.2 CIP FormulationillMul ttplcDlmcnsloua lCases A.2DCIP1o.1",1o",J

TiI.- gowrlliUK{'tlll;<tiullSforfirst-orderspatiald..rh1Ith,.,.o(tlll'interpchuionfu nd io ..

innrultj-diun-usious art',l'''fiwo:lby r1i!f,'f{'lllialiugII...;\lh''t·t ioll ''q l~al ionwithft'SJM.'ct toI bt· spatial ("OOCd iua l...Tilt' gt' IK'I"al(orm""f2D",h'f'«"!iolll"(lllat iol~~can hol- wruten

iJ..!.L+ll~+J'~{Jt O-r tJy_() 32

(2.·Ui)

(2.·1,)

(2.-I~)

1'111'tcrtusOlLthe right-hand-sidesof Eqs. :lA7and 2..11'1un- indud"t!illth,' uou-ndvectiouphiL';"calculationofafractionalstepnpprouc hII:" shown ill SKlioll 2.U

;'

Figure2-6 :Twodill\<"l~siollalgrid

Sevr-rnl(" rillso(20r-uhicpolynom ialhawbeenproposed.Til"simpl.'!'tIll\<" I'Wll()I«'Il I~'Ya h,· (·1ill.(2001) is giw.·nasfoUows. seeFig .2-6.InIbisup...indcell."'1' _ llll\<"

lJ<O.v<0.

I(X. }")=CJljX3+ C2IX2}"+CI2.\"y2+Cm t'3

-t("20.\ "2+ clIS )"+ ('021"2+('lOX+ ("OIl"+'00 (2.1!1)

wln-reS=-lilli, }'=-1.'6.1.There aI"{' tenunknowncot'flki\,t1ts,Con",whichwill Ill' determinedi\.~follows bythe values of I, Iz,andI~atgrnlpoillts(i, j) ,(i+l,j) , (i,j+1),and the value of1at the grid point(i+Lj+I).These n,d!icknts are gin'lI usfollows

too=/(i, j )

clO= IAi, j)

C:ro=[b..r(fz(i+1.))+Iz (i ,j )) -2(f(i+I,j ) -j(i,j ))J/ti.r³

(;211=[-ll .r(f,( i+l.j}+2fr (i, j))+3(f(i+Lj)-j(i, j ))I/ti I¹

("0:'=[lly(f~(i,j+1)+f. (i,j ))-2(f(i,j+1) -f(i, j ))I!6.l

CO'l=[-A.II(f~(i,j+I}+'2fy(Lj ))+:l(/(i,j+1) -j(i,j )))/6.1

('12=[j~(i+1.j+I) -f~(i+1, j ) -f. (i,j+I)+j. (i. j )I!('211d..II) ("11=[I(i+l.j+I)-f(i+l,j) -j(i.j+I)+j(i.j)I /(6.r6.y ) -c~lll:r-Cl~.6.,~

Thefollowingchanges nre needed for11~l)au,ll!~lJ:i+ I=>i-Iandu.l"=>-ll I . j+1=>j-1ami

uy

=>

-uy.

:.II

n.3Dell' Method

TIll" .i:'·lwTlIl f"TIIIofthTl~'·<Iiu)('Ij.,i"llalndvectiou('(lllaliulLs calli...ritt.'1l1\.'rullows

i.!l+ /.!l.. +/!.!.. +

u,i}.!..=0

ill i)r i)y i);

'!!.!.+U~+l·r.!.!.!.+U·i!!!.=O

_{Of i); (}y i);}

'!.!..J. +

u'!!!. +I'i!!l. +U.'.!!..!.

=0

ill Dr {Jy {};

'!.!.:..+/:!.!..:.. +t·!!!..:..+ u,i.!!..:.. _o

ill [Jr iJy iJ;

(2.52)

(2.53)

(2.a-a)

(2.5[,)

TIl<'It'l"II\.'on the right-band-sidesof ECis. 2_52 102.~an-iuducinJ ill th c nUlHllh ...-non ph ..., calculationof a fractional step approacha>i~ho"'nin 11ll" IwXI S...·ti"n

.'~~,Jo" ' "

;(1 I.:J.

;)~, .-=r)"'" .

•"-. - - -."'J-' Figure 2· i :Upwnul"llhic grid cell

:\ r-uble-polynomialinterpolationfunct.icnis "(lustrul'l,..!innll upwindcd l(Fig.'1.7)

h;~~,..lontl...workof Yabe(19!J1).IIIthis upwindt-ell.it is;~"" 1lI 11 (..1that'j<0,

3'j

v<Uall<lW<U

1(1', ,1/,: )=[(CIS+C2l'+ClZ+(4)X+(5)'+f, (i,),k )].\' +lk~}'+C7Z+C"X+(';J)Y+CIOZ+f~(i ,),qp' +[(cliZ+cnX+cn)·+C14)Z+'·IS.\"+f,(i,), k)]Z

whe-reX=-uti t , }'=- rbol aut!Z=-IJ'bol.

(2,56)

Thr-W uuknowu C()('!fiCiPllls an'(kltTlIlim~1fromIll<'valuesoff, fr'f~andl,atgr id poinl s(i+I, j ,k ),(ioj+I,k)and(i,j, k+I)and that(Iffatpoints(i+1,)+J.k}, (i.)+I,k+I),(i+I.).k+1)and(i+I, )+l,k+1).1'1"",,' nwllici,'nt sarcgiven iL~follows :

C4=[-3D.-iJA f ;+I,j,J.'+2f' ,J,k)U.r]/ti x1

an

''''WH'D,_1'.I"Jr-1,,,Jr.DJ=1.,,+lJr- I,,,.t.D, ."1,,,Jr +1-I,,,.t.r...zz-1,,,Jr+ 1'.I"Jr+1,,,.lJr-1•••".lJr '^{Cf}= -}1."Jr+1'.I"Jr+1."Jr •• -1•••".1<+.,....,zz-1,,,.1<+ 1,,,+lJr+1.".1< +1 -1,,,+lJr.I'

o.'I,,·,ulinKolJthe "igll."of1.1.l'all<l1.1',thefolJov,'ingrhallll.",1rail 1H'Il\,;llI,':1+1~i-I aml.1r=--.1rfor1.1~O,j +l

=-

j- IalKI.1y

=-

-.1.r"furI'~O aml;+1=-; - 1

>l.ml.1:=--.1: fur1.1:~O

TIl" interpcluriouIunctious for tht,.;e spauald,'rimlh..'Sar,'11"'11ohtaim'll

+f. ^{(V i 7)}

ry=[3C6Y+2(C7X+CsZ+CU)]Y +(CI:lZ+ClfiX+CIO)Z +(C2X+C5)X +Iy

rz=[:lcuz+2(C1 2X+CI3Y+CI4)]Z +(C3X+cIG Y+CI5)X +(CgY+CIO)Y +Iz

2.4.3 Calc ulat io n of SpatialDeri vatives

(2.58)

(2.59)

IntheCIP meth od ,not onlythefuncti onofI,butalso its spa t ialder ivati ves,ha ve tobeupdatedatevery time ste p.

Eqs. 2.5,2.G and Eq.2.38 can bewrittenina genera l 3D formasfollows. Note tha t IDallll2D cas es cHnf ollowthc sam eprocedur c.

wherel/denot esthctot al termsonthc right-h and- sid cs of Eqs. 2.5and 2.G. Differentiatingit with resp ect to1:gives

(2.GO)

Denotin gjjfasL,

%t

asI,and~asL,the aboveeq ua t ioncan be rewrit te nas

Th,' I,·ft-Iuilltl-sidt' ofthis cquatjouis a 3DiUl\Tctitlll '~IIMlitlufurI~. The righl-hand-sid.. oflhi" ''lIUilli"ni..;a source term

..\p pl)·iugthe ffiU·tionalsh'p1II,.'t hudto the aho\'e "'tuatiou."i1...1"10

I;~,r:⁺u·~⁺t'·~⁺It.·~

_

0 (2.G-1)

/""-1"

- .1- ,-

^{=II (U N)}

I;·~~/; = ~ -(f;~ + I;~-t- /;~)

^(2.116)

Eo!".2.r~1and 2.64 are s<lln",1bythe CIPmethodd",;nih.'l! illSt",·tiun2A.2,;lW!

r

!tlldI;ar,' "blain.",!h)'F.o:f'. 2.fi,liaud 2.57'.

(2,(;7')

:I!)

Subst itutingEq.2.67into Eq. 2.66.thefinalequationfor1;,+1can be obtained,

It-I,j,k

and

1;+1=I;+Ir"'~~j,k-1:~~~j.~~:t+l.j'k^{+It-I J.•}·

_su];/l:~IJ'~~:l~L:'-l,j.k+I;V:~I,j.~~:f_I,).k+I;lU:~I·j'~~:l~Vf-l,j'k) (2.69)

Similarl y,I~'+Iand1;,+1can he obta inedasfollows:

1;+1=I;+I:.j"+\k-1::f-'I,;~:tJ+I'k+ItJ-l,k

_su];1I:~+I.;~~I:~)_l.k+I;VfJ+I.;~~':~)-I'k+I;lUfJ+I';~yW:'.j-l.k) (2.70)

ir:

=I;+J:.j:~+1-1,~j:L~~:,:j'k+1+ItJ.k-1

_su];1I:~j.k+~~;;II:~j'k-1+I;V:~j.k+ ~~::~j,k_ 1+I;W:~j'k+~~:V:J.k-l) (2.71)

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