Cont ents

...

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AN EXPE RIMENTALSTUDY ONHIGH ER ORDE RWAVES AND HYDRODYNAMICLOADING ON VERTICA L

SURFACE PIERC INGCYLINDE RS

@KRlSHNAPRASADTHIAGARAJAN,D.Tcch

A thesis submittedtothe Schoolof Grll(\natc Studies inpartialfulfill ment ofthe degree ofMnstorofEll,ltilleerillll;

Faculty ofEngineeringand Applied Sciences Memor ialUniversity of Newfoundla nd

February 1080

St.John's Newfoundland

Abstract

Estirnationof second order"'1l\'1'loading011 olfs hon' slrudl1rt'S hils ilLInwt l'd recentattention in fI'."f'ard l.Thisthesis prcsouts therl'slllt"of lin t'XI"'l'i, mentalstud ycarriedout\0quantify second orderwavefort-esOila vortir-nl cy lin d er in1\wave tank. Theinitialpart ofthis tlwsisisdir..d.t,,1towards formulating andanalyzingLllewave field prosr-nt int.ll('lalJUra tory,TIII'sl' res u ltsfor medpart ofthoinputfor further slmlil'S nilWill'" [o rn 'S,])llt'l u thephysica llimitation sof the wave tank, itwasfOillidIhntS"\'1'1'1I] W,\\" ' S I',,·

exist withtheprogressiveI\',WCofinterestillt.Ill'rnnk. 1\11'lI1i\ly.~ispr<J('I~l ll r"

was developed usinga FilsiFourierTransfonu ll'l'huiq lll''lIId

.1

Il'ast"'Pli\1"~

cur ve fittingmet ho d to separatethe11'1\\' 1'ofillll'n'stIromtlu-sid"df"I,tsun.l idc n tifyits pr incipalparameters.Se\'l'f<llsi, I,·dfl'l'lsI\'!'f(''lllilllt ilil't1 illtIll' pro ccss,Woveforce s on^i\I'erlkal cyJin, I"r 10 "''''IIlIlI 01',1,,1'1\"'I'1·1.!l,",wtind ly fo r m ul a ted fromaliterat ureSU1'\'I'Y,Analysisof thl' lllc'asllfl,r]wnve f"rn's ill the wave lankinvolvedusing a [;1st fouril'ftrnns fcnn toi,II'lIti(\'tIll'lirst .111, 1 secondorderforcecompone n ts . Towar dstile <-'uJufthe study, it wase"'JlII I tha t the proposedfor mulat io ns forthe tota lWIl \,(!fipldillIlil' lank Mid till'

W;(VCforceson acylinderwere adequate .There,~" b tmanypllP lIC'llWl 1iIill

11wave lankwhoseeffecls onmeasure ment s eequlresFurtln-r reseanh. It.is emphasizedthatsystematicexperimcn talstudiesofhighl~rorlll:!'W/H"!Sfor m animpor tant part of offsho re resea rch flclldo pmellt,

To myparen t"

IIIho .lefoot8tc pJformed myllllld ~ r

i i

Aeknowledgcmcuts

Theneed for second orderexpcrhueutnl stu diesilltheJln~I 'lIttrendI,f resear chWIISfirst point ed10nil'by lily research super viso rrl'Ofl~~c'fR.E.

Dnddourof the Facult y ofEnglucc ring,~IUN.Jnm exl.rcuwly grntl'ful In himfor theencourngc m ont,active sup erv ision lind cno p l'l"lltinllII!'ll!f" fl 'd tomeduringthisstudy.

Thisresear chWf\.Sfinancially su pport ed by n grnduntc rcsenreh Iellow- ship from theSchoo l ofgrnduntcst udies nt.~IrlllOl'inlUnlverslty.Mytbnuks aredue totheDeanof GraduateStUlliesundhis sluff forthe'iflwl]Jurul supp or t. IamIndebted in grn li ludetotile officesofthe DeunuudA~ s{)·

elateDean of Engineeringfor eoo rdinu tiugIllyecuvlucsdurlugthe!lI'l"ioci ofstud y toward sanefficient ou t come.

The cooper at ionexte nded tome duringtheexperimentsbythewave tank crew andthesta ffofthestructureslabarcrCllll'll1lll'I"l·l!ntthisnp- porhmity.Japp reciate tilehelpofferedto me bytill:stnff oftheCCAEill prep ar ingthisdocum ent.

OnefinalkudostonllNewfou ndlande rs ,their hosllitnlit yand warmth formakingmy stay memorable at St.Jolm's.

x.r.

iii

Cont ents

1 In tro d uct ion

WavePropagation:Theo ryand Experiments Backgroundand Lit er atu re Review

3 Theoretical Develo p m ent :1.1 TII(> houur!Ary\"alileproblem. :t.:! S,'N)1lf1OT(!I 'rSrokestheeey .

"I.;I Tlu'tlrie,;for pisto ntyp eWAn"mak er

:U TIl('h,tillW,,\~·field. ... . . ... . ...

·1 Experimentalsetupand procedure

!> Disc ussionofresults

."1.1 First ord er resul ts .

rl.:! Secondorderresults .'"1.:1 t\lIl dllsio ns..

15 I~ 1$.»

:!6 30 41 43 16

"

II \Vav eForc es: Theoretical Fo r mula tionand Ex-

pe riments 64

o Da ckground andLit er aturereview The o reticaldevelopment i. 1 Fnrr eequat louinarealHow. i,:! [nertia forc es onarvlinn er.

,,;1 Dra gror n'1l<>l1ilcyl;nnl·r .

65 71 71 j.';

so

8 Experimentsand Anal ysis 9 Discussionofres u lts 10Concl uding remarks Listof Refer ences

Ap p en d ices A ANAL Y S·Sou r cecode B The FastFouri erTrans for m C Descriptionof thesub routine LSSIN D Harmonic components ofUlill

ee

03 If.!

liS

122 rue 13' lao

List of Figures

2.1 Ilnngeufvalidi ty of vari ou s\WW Ctheorie s 14 3.1 Ilepr cscutntivediagram of a two . dimensional progressive

WI\VC. 20

3.2 n"'pr('.~,..nt,al i \'cdiagra m oftwo . dimensionalwavegeneration

bya piston typewnvemilker . . 29

-l.I C'ullfig lH'lltio lLofthe pistontype wavemilker 39

4.2 D"lwh confi gu rnt.ion 39

·1.3 Iufurmnrion flowchartfor wave experi men ts. 40

·1.4\V1~"'Cproboculibrut iondiagra m: DateFeb12. 40 -1.5Fir~torderunalysiaHowchart for wave experiments 41 5.1 SampleWIlV,..formgraphs:{=l.OHz.H/ gT2

=

0.001 ;Lo-

cation zero (20m awayfromthe wave board meanposition) 52 :'.2 Samplewaveform grap hs:f=l.OHz,H/ gT¹

=

0.001 ;Lo-

rution one (O.2mfrom loca t ionzero ). 52

!j.3 Satlll'l(~wnve for mgra p hs:f=l.OHz ,H/yTl

=

O.OOLLo-

futioll two(O.4m fromloca tion zero) 53

5A SlImpleWl\W 'formgra phs:r

=

1.0Hz,H/yTl

=

0.001:Lo- ('nt ionthree(0.6m fromlocation zero) .... 53 5.5 Suurpk-wave formgraphs: f=1.0Hz,Hl aT1=0.001;Lo-

r-ationfour (O.Smfrom location zero). . 54 5.6 First orderampli tud efit: f

=

1.0Hz,Hl gT"l

=

0.001 55 .5.7 Secondordernmplitudefit:f

=

1.0fh:,1JIgTl

=

0.001 55

5.8 Relativedepth:theoryvsexperimen t 56

.5.0 Wn\'C makeramplituderatiovskh ... 56

5.HI Firstorderreflec tioncoefficientV.!Ikh 57

5.11Firstorderrt'!l"ctedwave phase angle\'5kh 57 5.1~Stok esnmplltuderatio vsH;H/gT2=O.0007 58

vi

5.13 Stokes amplitude ratio vskh;H/g7"1=O.OOI 55 5.14Stokesamplituderatio\' SH,;H/gT' =O.003., 59 5.15Stokes am plitude rAtio vskh:H/ gTl=O.OOO.. 59

5.16Freewavenumber ratio\"Sl·h. GO

5.1iFreewaveamplit ude rnrlo\'51-h . 61

5.15Free wavephaseangle\' 5l:h ...• ..•.. .. G~

5.19 Secondorde rreflect ion coe fficient\'S«h: G2 5.20Second orderreflectedwavephase nngle\'1\KlI. • 03 i.1 Rep resem a rive diagramof thewaveflumewith tllcWI\\'Cgl·n·

erat or,beach andcoordinatesystem ... 54 8.1 Configu ra t ion ofthetestcylind er in theWII\"Ctunk .. 90

5.2 Configura tion of theforcetran sducer. 01

8.3 Calibrnt io n chart for the forcetran sducer 01

8,4 Schemati c diagrnm of wavefor ce 1I1lnlYH1M 02

0.1 Sample wave forccgrnph:Run 01,D

=

0.lli Z5m,f=0.9Hz,

H/gT2

=

0.003,[\'C=2.12. 103

9.2 Sam pleWM'eforcegra ph:Ru n~2.D

=

0.Q.1i!lm.f

=

0.4H;t, H/ gT"

=

0.005.[':C

=

20.56 . . . . .. . ... 1113 9.3 \Vave forcespectru m forrunaI,D

=

O.Oi2i.im .. 104 9.4 waveforce spect ru m forrun~2,D

=

O.().Iiflm .. .,.., 104 9.5 Inertia coefficient(C...l\-$Kf'Ulegnn Carp<'lltcrnumber(/':C)105 9.6 Dra gcoefficient(CoilVIKelllt>gallCarpentr-rnumber(A' C ) lOG 0.7 Ratio ofsecondtofirst ord erinertiaforces\" 'nIUSSnrpkn Yll'H

parameter (6).. • ' ..•., . ... . .•.. •. lOi 9.8 Ratioofsecondto firs tord er drll.gfor ces\"'rSltSSarpknyll·S

par amet er(P) •• , , •••• '. ..,••, .• •• •, . , . 107 9.9 Comparison bet ween ealcull\tedend measured SI"'CIlIIII ordl or

inertia forceI\tlow steep ness(H/gT²=:O.oooi,0.001).'. lOS 9.10Comparisonbetween calcula tedand meM1Jf"l'iIIlN'f1nl!order

in ertia forceI\t highsteepness(H/ yT2

=

0.003.O.IKJS). 1fI8 9.11 Compari sonbet ween calculatedan dmcnsuredsccond(m lcr

dragfor ce atlow steepne ss(H/ yT²=0.0007,0.001). 10~

9.12Comparisonbetweencalculated andmeasured"'~I:lI11rJorder drag for ceAthigh steep ness(H/gT²

=

0.003,n.OOS) 10!) 9.13Compari son betwee nmeasurednnr]cnlculn tr.II!i/'cOililorder

ine rtiaforcefunctionvskh;H/gT2

=

O.1l03iD

=

1l.1I47f1m. 11ll

!:l.14 Comparison betweenmeasured andcalculatedsecond order iller tiaforcefunctio nvskh;H/ !JTl==0.005; D

=

^{n.0470m. III}

!:l.W Comparison between measur ed and calculatedsecondorder iner ti aforcefunct ion vskh;H, .,T²==0.003 ;D=0.0 72:im. 112 O.lGComparisonbetweenmeasuredand calculatedsecon d order

ilU"'rtin fo r'"efuner ion vskh;H/!JT²==0.005;D=D.0 725m. 113

C.l Flow('!lllrtforthe su broutineLSSIN. 135

lJ.1 CosineJunctionsath=0.2/1•,. 140

D,2Har mo n iccompon entsofIlf lih⁼0.1/1. 141 D.3 Harmoniccomponentsof11/ 1;h == 0.211.. 142

viii

List of Tables

4.1 System570spcdfications 36

4.2 Recordofwaveprobecelibrajionconstants 37

5.1 Parameters of expe r imental runs 51

7.1 Formulation ofthe second orderiner tiafo rceInuctjonj(1.:1I) 84 9.1 Experiment alWIW Cfor cesnndrelatedpnrnmctcrs... 101

ix

List of Symbols

'Ihefollowingnotntions nrc usedinthis thesis:

fll Firstorderamplitude of theincidentwave Secon d orderamplitude of theincidentwave a2~ Freewave amplitude

Firstorderreflected wave amplitude

ann Secondorder reflectedwave amplitude C,I Coefficient ofdrag force C,,, Coefficien tofinertia force

D Diameter ofcylinder

f Hrst orderWI\'''Cfrequency f Elementarywave force onII.cylinder F Tota lwave forceon a cylinder F/ Totaldragforceon acylinder F". Total inertiaforce on acylinder

Accelerationdue to gravity Stillwaterdepth H Firstorderincident wave height

~. Firstorder wave numberof the incidentWR"C 1\C Kculcgnn - Carpenternumber

flc Reynold'snumber of flow 5 Strokelength ofthewavegenerator

t Timeaxle

T First orderper iod of the incidentwave Horizontalcomponent ofwaterparticlevelocity Verticalcomponentof waterparticlevelocity (r . :;) Two-dimensionalCoordinatesystem

Phasedifference between reflect ed wnve andinciden twave J Snrp kn.YI\'s periodparameter

Symbols(contd .)

Phase differencebetweensecond orderrcflt'd cd\\"1\\'1:amiindo dent wave

Phase differencebetweenfree waveand iuridcutwnve Free surfacedls p luccruc nt

Free wave number

>. First order wave length

~ Wave generatordisplacement p Mass density ofwater

¢ Irrot u t.io na l flow velocity potential Firstorderangular frequencyof th" incidentWl lVC

xi

Chapter 1 Introduction

Esttnnuion oflIuilllnlldi lll;isunint ermedia testepin en offshorestructural df·sip;n.The finnl result isusuallytheresponseof nrthestressesindu cedin tl...·structure.On theasp ectof waveload i ng,presen ce of randomseasand i\xpcctrmuofwave heights and lengthscnIls fornprob abili stic.eetim...te

"fthe designWIWCconditio ns.Theforces arisint; due tothe designwave

nrcmudd ledb)"empiric alorsemi-e mpiricalformulae,where the wavesand

wavefUf C ('Snrc generally consideredlinear innature.

All/Itt from design wave conditions,lowprobability ext reme wavecon- diliunll occur.whicharc importan tforthe designof structuralfound ations 1\11()alsod.~if!;nagAinststructural overloading.Mod elling

or

extremeor 111('('1)wn..:csWI\Sdone usin g1\llonli neuwave theory,but theloadingwas neverthelessestimatedbythe sam e empiricalformulae. Though it ep- peurerlconceptually erroneous, lackof availebility of betterforce models

!lUllIt,thisIIIlt'ngim-'('ring practice.With a factor of snfet}'of30%included

in thedesign,most offshorestructures su rvivedbeyondIlu-ir~liJl\lI,lll'tl[if."

time. Searchfor betterforcemodelsintherecent pastha~1'''11towe-ll- developed theories for wave forces011offshore structures. DI,\",' I'll)llI"ul.ill theoreticaland analytical studiesliashoensubstanlj;III'.1hy veryfew expe-r- imental studies.

SimulatingrealSCIISin a lab or-atoryis inll('rf'n llycol1lpli";lh',1h.y1\1t' var i- abilityand randomnessof theseas andJ bythe scalingburrierwh kh,'xislll between the viscous flow conditio nspre\' ai!inginrealit y ami thoseatt.aill.~d in thelaborato ry.Empirir"~1correctionsareemployee!for this RowdHferc lLec and experimentalresultsare extrapolatedfor desig n.Apart fromthc1lt!fun- damen t al problems, seve ralsid eeffectsarise inIIwavetan k whichCiUIaired the resultsof experiments" Ma n y of thes eeffects arc usually ronsidf'[l,d111.'15"

ligible duringafirst order waveforcees t imation , becauseof thei r relat ivel y small magnitude, At the secondorder of analysis, two major problem;~[('as

I, Second order effectsduetot.hoequipmem.sllsl~l(geI Wl';\ttlr,111'ad l d,"_l have to be quantifiedand this areaof research is stillnot wi'll,l,'w lol,, ',l "

2, Many sideeffects neglectedpreviouslyrnigllL beSi~lIilic;ll lt.111~'alls'~

second order meas urcmc r-ts aresmall qua u t it i,:s.

The presentstudy began withasurvey oflilf~r;Ltllrl~for;111)"ne-w .t,:vd"I'- rncnts in thetwo areaslistedabove.:\If)rldlj ll~of the\\';tI"!'licld forllwd

a pre-requi site for nstudy ofwave forceson offshorestructures.Regu lar

\\'l~VI~Swithrelutivcdept h of0.137<hI>.<0.!l71 andwavesteepness of

O.CJ()44<H/>.<0.0474 wereconsidered in the experiments. Firstandsec- ond orderwave flcldain11.wave ta nk werefor mula ted and theamplitudes of variouswavecompone nt s wereestimat edusinga fastfourier tr ansform unrlaIcns t squnreslma lys is.Severa lsecon dorder effectswerequantified.

The (orn's onn ver ticalcylindrical pilewas conside redinthe secon d part ofthe exp...riments. Second orderwave forceswereformul a tedbasedon alif.ornturesurvey.Experi ments were conductedfortwo cylinder dlame- tcrswiththediffr actio n parame ter(D/ >.)varying over 0.0066to 0.0704.

Rosnltswer efilt ed intothe formulated model andanalysed.

Shortc omingsofthest udy Wer e unnlysed inthe lightof availableresults (otUlI!intill'Ij t,~ra t ll re(orsystemat icsecondorder experimental stu d ies.It wasexpect ed tha tthepresent resea rch woul dserve toward sthe following:

•Formulat enstmt cgy forseparati ngfirs t andsecondorder effectsin a laho rn tnrr nudwhereverpossib le,iden tifythe indi vidual com p o ne nts .

•Highlighton difficu lt iesand providesuggest ions(or futuresecond ordcrcxpcelmcutalstud ica.

Thisthesis has beenorganisedinto two parts.The firstpartexplains ti lt'wuvo(01"111experiment s, the litera turereview and theor iesthat formed t-lll'lmsiaforthe proced ureof experimentsand for thediscussionof re-

»nlts. Thesecon d JllU't iscompo sed ofthe waveforce form ulationbacked

bytheli t erature review.Experiments andeualyeisofrt.~t11bat efnllnwt'd bydiscus sionsandconclu sions.Ad ditionaldetailsare pr(,Sl'lltt'ttinfOllr IIp- pendicesfoUOI\ingthelastmapt e rongcncrnl ccncheionsand~u~(':jl ioll~

forfutureresearc h.

Part I

Wave Propagation: Theory

and Experiments

Chapter 2

Background and Literature Review

\r.:lH'Sgent'ra tedina lahoratory/lTCcoutamln a tcdbysc·\'!'til.]wavesrn'aln]

dueto thephysicilll imitlltionosil1\'oh'Cflintill'"PIWlrdllll 11,...,1. Thn;('W1\\"''':I togetherformacomb i ned wavefeldwhee("h ara{"lo~r i ,.t i'lIvary<I"wnti ll' flume. To elim inate these sidedrl'Cb,ernecan(·it ill" PUH"IHl''flCl n'arC-IIT'II,·

and refined apparatusOrseparatetilesidcdfl'C lsfromtlll'lulalwan 'lif' ]'l.

Thelatterroutecan beimplem entedby the-rol lowin~1)ITR'sl,·ps:

I.r\nticipatin lll hc significa ntddccffccu,cre-ateam...I.~u{1Itr· t" t,,1waw- field.

2.Measure thelolal wavefield,wh ich\IoTlulcl invul\'c'llll"ilSllri.11!;11ll'vnri- ation ofwavech aractcristicsdowntheIblllll~.

crredsandalso explaintI Ll~lli~rr{'paflcil'SifallY.

Thispar tof the thesis presentsalitera t ure sunerandthe theoretic a l I,a, :kllmun dr1~ qll ir,, [for creatingamodelof the wavefield uptosecond orde r.

Tl ll~expr-rhnerus cMril'(] out al t.hcMUNwavetan k facil ity tome asure the ,,,t;1!wan'fi"I,1 an'then (!''I, lailledfollowe dhyadiscussio nof the results of filotillllilleIIll'asun:,1 values into tIlederclo ped model.

Must oftill'rp~"'''rrhin wa\'"hYflroll)'nl'lmk~or iginallyhcganwith1\pure I'l"h1!,n'!isi v(' 11'111"'.'I'll('P1"o';r1~ssi\"cWilVI'theoriesarc solutionstotheboun d- aryvililu-prohll'muf01gravlty wave travelling in an idealfluid domain,sub- jI'l"!tu""rl l1iniuit.ialauclboundaryconditions. Each theoryis distinctin

il.s1'11111;"uf i1IJplinllJililyillteems of wate rdepth, waves tecpncss etc.The

dils sklllW;\I','theo ryofStokes(18·17)involve dtheperturbat ionapproach.

IIIr.his111('[ ll(ul,sol111~JIIsto auonler of accuracyarcobtainedby using th e rosultsofllll'jlrr'c pd inj\crdor.Tilefirst ord ersolutio ninwavestee pnessis ob- lailll'dLyfonnnlatlugthe problem withJinoanzcdboundary condi tionsvta ., II."[inonr wavnLlu-ory,Thesolut jouupto the secondord e r inwavesteep'

''' 'SSi~ol.l l1illl',1hy a man'precis e deflnit.ionof tIl,'probb-mandutilizing

tIlt'Hrstonl,'fsofution,(S lo!«.'S.1847;seealsoDe a n tlildDalrym p le,198·1.) Fo ril.,i1l1ls o i,lillrf(~surfac e, this theory predicts that the velocitypotential would be sinusoidalwithtimeand horizontaldistance andwo u ld decay hypt'rlJolic allywit h,!,-,pt h.Furtherdevelop ments of the th eory ha sbeen in I'Xh' lIdillg theso lution uptohig herorders, Skje lbrelaand Hendrickson

(1960)_Stokesform al ism wasfound tobemoreepplieeblcindL'l'1lwa ters thuscreatin ganec essit)"forshallow "..aterWIWCth eories, The shallow wa terwavetheory developedb)'Kcr tewegamiDeVri('ll (1595)l'XJlrt'!'so", the Wa\'C profileintermsof1\JacobianellipticintegrN'CII',TheSt l'fi\UI functi on theorybyDCM(106.5)isbasedonanumericaliterationtl'chllilln~

to get the bestfittotheboundar)"conditions.Furtherdetnils on these mill othertheo riescanbefoundinDeanandDalrymple (1!IS4),Snrpknyn and Isaacson(1981) an dthe Shore ProtectionManual(l9S4).

Chakrabar ti(19801» con ductedaseries ofexperime ntsove r n widerillLKe ofwater depthsan d wavestccpnesscs inanefforlto evnluntutherdativl!

val idityofvarious theories.Whilethelinea rtheorynndthirdcnl ..r Stukl'lI theory gave betterfits 10themeasuredwavelcngtilli, till.'irregulur strl'lIm functiontheory gaveII.bctter 6t tothemeasured.tweJlrofik'li. Donn (I!)74) wasinvo lvedinestablishingabasisiorselectionorl\wavetlil'tJry,gi \'f~1I1\

set or wan 'conditions,Anal )1.ical studiesre\'clllcd tlmtIIl'Cpwaterwnv.'!!

werewellre presentedby higherorderStok esthcori~.slllllluwwaterWIlV~

by the6rst ordercnoidal theoryandthe st ream{unction Ihmryprnvi,lNI lUIoverallgood\-alidity. Thestrcam fun ct ionlh(!OryalsosIIUWl-'l1J;uu.1 cor relationwith pub lished data,thus Dea n(1974)wlI~hll!lodthatstrcur u funct ionth eoryWASbestsu ited:ordesignpurposes.

The rangeofapplicabili ty ofvariousth eories lull'!hellllwelldisCllSSI:r!

in theShore Prote ctionManual(19 84 ),Fig , (2.1)f!'prmhu :clifromthl:

IIlnnllltli~I~plOloftwo par ameters,the depthpar am e t erhlgT 7andthe sl('('pllf'~.~parameterIl/9T~ ,TIl<'applica bilityorStokes theoryisrestricted bytheso-called Ursell parame ter{cf3.2),Blandan UrseJlvalueof26,the 111l~Jl"di("ill Slokl.~formdevelops asecondarywavecrest due 10thelargene ss of tile second or-der ter mcomp a redto the first order termthus restr icting till'ill'plin1Jioll(Jfthislh<'~lrytotransiti onal anddeepwaters,

l.idml""tor ygl'lwndiullofprngrl'Ssivf'wnvesisdifferentasoppose dtothe pun 'I'ro,c; n 's....ivl'lI"a\'('problem IlI'CnllSC ofthe viduity of thegenerat in g smf;I(·('.[\l)Ilow"("fOS .'"thl'1!;" lIn atin gsurface isAlladdit ionalcondition th e 11,,\\· ha,s I"sil1is f.\'inlhispmhl l'lIl,Havelock (1!l2 !l)was concer nedwith th e 1'1""lJlI'lll llf fun' l'l l\\'n\'I'S lilli'It)i~sinusoidally oscillating forcing sur fa ce, the pistuuIYI}(',l!,l ' tll't ntfll',lIissolution,basedon linear wave theory showedan

with.li~tnl1n'fromI,IIO~wavegenerator. Ursclletal.(I!lliO)haveobtained

"xpn 'ssi"nsIortill'ilillp litlllic ralio ofbot hpisto n andIll\lhl1l' typ egene ra tors.

1l.'SllllsofI'xpl ' ril1l<'llls~1"" I1l("(1to correspond well withthetheo retic al ...-suhs.1l1O\\", '\·..rforslfl'lwr\\';1\'1'1',theIhr....)r~·wasfoundtoove r-predict the r('Snll shyilS muchasIO'iL Exp('rilllf'nt~on plllngl'r typl' WR\'emakers

"'lIHhll'l,~1hyEllix,1n.1Arulllll,Lt;llll(I!)::!·I)andalsoEllix(l9B.J)showedth at 1111'I'xislill,Lt IllI'orr overestimated byas milch as20%.Poorcorrela-

lion at lawn freq uencies IlIftSattributed to leaka gearoundthe\\·t'tlg.~t)'pt' plunger,Chen(Uli8) hasdoneexperimentsn.ttheMUNWfw erankfllt"ilily toverifythelin ear wave generatortheoryfo rthe Ilisto lltnte~1'I,'rlltt>r installedin thetank,Two waterdepthsof3.0ftnlltl4.5ftinwnverondi- tions 0.059<hIA<O.69i7 nod0.0012<HI _\<0.1wereeousldcred.TllI~

resultsshowedaheavy scnttc r totheorderof50%,Theexplnnnt.ious giVl'1I weretoo generalnnd didnot serveto expl ainfullynsto whythegenr-ruto r

<lidnot followthe lincnt-generatortheoryC\'('IIinthellnenrWl\\'Ctll11ges.Iii thiscontext,la t e rwork b)'MuggeridgefindMurrny(lOS1)«hewedlhutthe sam e genera torfollowed the lineartheory for "widetlIu l,l;Cuf experhueutnl conditions,0.0 58<hlA<O.M 8nnd0.0012<HI ..\<V.11lt l\cCllIsla nlwn- terdepth of1 m.The ap pl\tentdiscre panciesRl nonll;thf"Sl'ltwotl~·nr,.j ll'rl<·

resultswere inexplicable.

Secondord erwavegenerat ortheorieshl\w~l)C'l'n prop'lS\'tlfnUnwin l';

Stokessecond order theory bysomercsoar el....ra,[Funtnnet,WG1;Mall - sen,

u no

Hli!;Daugaard,lOT.!;F1.i<-kandGn z al:>SO)IlIIt nenuILllli IW': 1I widelyaccepted.Solvin gforthe generator bou n d ar)'C'OIlllitinnatt111~sec - ondorder,thesetheoriesindicatedthe gencrarlouofII."':CUIICl lllltlll",,:c:rtl~~

waveattwicethewavefrequency,This wavewas cn/lsid c rcel pnr ll.'iitic:1.'1it travelledover thesecondorderincidentwave.Ex p ressio nsfurth.~frr'!'wave amplitudewere varied amongthese the ories.De taileddi~ Clls s i olls011th,~

secondorder wave generatortheo riesaredoneins,-,,~ t ioll3.3ofth i"t,llI'sis .

10

Eep orl mental work basedonthesetheorieshave been reporte d. Duhr Hunscn awlSvendsen(HI74) have analyse dthe beatingphenom enon be-

tWj~~1lsecondord e rincidentwnvcandth e free wa vegene ratedbythewave

maker. ExpenmcnrsIHWCbeencurriedouttome asure the amp litudeof t.h,!frc'.~wave atvnrimlSW1W'~steop nessesand rela ti vede pths,Recordwas t.nkoufromIIwnve probe travellingdown the flumeand a bandpassfll- ll""WlL."usol to remove th...first order signal from the ou tput.Resultsfor t.Ilf'fn'('W1\V.!nrrrplitudcwere inpnrt ial agreemen twit htheoriesofMad- se-n,Fun t/lIlt't lInd Daug ourd.Oneimportantoutco me of Buh rHan sen an d SVI'udsf!USworkwusthat theauthors were successfulin reducing thefree

W/W'!lllllplit1l1le bynnonsinusoidalmotionto the generator,basedonth e

f/w tt.ILnttil!'freeware was n linea rwave withrespectto its amplitude,as

ll/ullH'(,Il:;llggl~tcdh;rMadsen (IOil) . Ellix and Arum ug nm(1984)and

all'll)Elli x (1(184) were concerned with the secondordergenerato rtheory f" ra plUlII;I'rL.l"pegcnerutc r.Froma waveprobe placed atdistinctloca - ti<lllsdowntheHu me,recordsweretnkcnandanalysedus i n g aFas tFourier Trnusform(FF'T) and n le astsqua rescurvefitt ingtechnique.The author s sh owed tluuthe freewaveamplit ude was as muchasth r eetofourtimes th eseco n d orderStokeswaveamplitude.Notheor e tical com pariso nswere ma de because notheorieswere avail ablefor plunge rt~'pewave mak ers.

11

Wavesgeneratedatone endof a wave tankhns to beabsorbedhyall

energydissipatingbeachatthe otherend.All beachesabsorbmost of the energyand reflectthe rest.Beach rcfleeuonis an intrig uing nren of research and todate notheories are availableto estimatethe reflectedwaveallll)li tllll,~

andits dependency on various incidentwaveparameters.Eadl wavetunk has tobecalibratedto estimate the amoun tof reflectionfrom thebeach.

Thisis oftenrepresentedby the reflection coefficientwhich is a ratio ofthe amplitudesofthereflectedwaveand the incldcutwave. (lodaandSuzuki (1976)have presenteda techniqueforestimating the inciden tand reflected wave heightsfrom twotimerecords takensimultaneo uslyfromtwo wave probeslocated at lessthan a wavelength apart. Experimentsweredone both for regularand irregularwaves. The effectsof location and distan ce betweenprcbeewere discussed. EllixandArumugam(1984)andalso EIliK (1984)as partofthe ir exp e riments haveestimatedtill![irston ler rellectjon coefficient of the beach in theirwaveflume tobeabout1)%orIl ~ss.lIm\l(~ VN, the reflect ioncoefficient atsecond order wasa.~highas,IO-I)O'XI,whichWM puzzlingto the authors.Chen(1978 )hasusedseven waveIlro lll'SNfl,wed 2ft apa rtto meas urethe beat patt erndue tordleetedandin,:idl~lllW'l\lf'S. Iii!

noted lhat the reflect ioncoefficientat the~'I1J Nwaveta nkfac ilitywasilltill' orderof 10% at 3ft and L'5flwater depths.

wavestravellinginthe viciuity ofstatio nar y~lIrflln~sexperienceIlJ~sf)f

12

ene rgyrluctofrictionatthe se sur fac es. This is called wave attenuat ion.

Attenu atio noftwo- dimens ionalwav esin achannel offinit ewidthwas firs t treat ed hy Ilicscl(HI49)andlater by Hun t (HI52). They definedan attenuationcoefficient which\'IIU!composed oftwo com pone n ts,onedue tn bottomfric tio nand anotherdueto side wall frictio n.Chen (1978)as pnrt ofhisexp erime ntsfoundthat theattenuation fa cto rs at the MUN wuvotank werein the orderof 10-⁵thuscausing negli giblereduction to the wave amplit udes .

Theabov eliterat ur e survey wasfocussed on und erstand ingthe wave fiddll existingillIIlaborato ry.Secondorder Stokestheo rywasfound suit- ablefor theexperiments asmos t ofthe second order gene rat ortheories were validilltlmtrange.Followingany one par ticul ar generatortheo rywas not pussib le.Butcertai ncomm onresu ltsofthese theori eswer e used . Beach

!"!'flection was anine vitab le par tofth eseexpe rime nts .Itwasproposed to split thereflectioneffectsin to firs tandsecon dorder.Othersideeffe cts in till 'tnnk were genera llycons idere dsmalland neglected.

13

,

/ iF=O.071l2

DttpWll.te!f

tF=O.OOI~

j.:cSh:::al.:clo:...w_w:::,:::".:..,

I T-__ T ::: '=~=';:.:: ti=on ::al~ w :,: ,:::"::,,

^{_ _}

~~~~:":~

0.0.

2!....O.OOI

g7'2

0.0001

Stollet 'lVord er

,

I ,Stolle.'lIIOfM r I

,

I I I ISlob.'11Qrdff

LineuthCOf7

Figu re2.1Range ofvalidityof variousW8\' etheories(Shorepro tection manual , 1984)

14

Chapter 3

Theoretical D evelopment

Allwavesand associatedphenomenacan be obtained as solutionsto a basicbounda ryvalueproblemwithgeneralboundary and/or init ial con- ditions.'Vav!!nndwave generator theories are solutionstothis problem , withthe appropriateboundaryconditionsapplied. Webeginthisch apter bypH'sc n tingthe fumlnmentnl problem. Secondorder Stokes theoryis in.

tI'O(\lII'(.'<\IUlnsolution to this problem.Firstand secondordergenerator theories nrcsuccessively discussed. The chapter concludeswitha second orderformulationforthetotalwave fieldinalaboratory.

3 .1 The boundary valu e prob lem

The hound nry valueproblemofinterestis atwo-dimensionalgravitywave trnvclllng011thefree surface of a fluid (water),involving the following n"l<lImptioul<:

•TheHnid [wntr-r] is incompressibleandinviscid 15

•Zer osurfacetension

•The flow isirrotational and acyclic

•No underlyingcurrentexists

•weveeareofpermanent form

•Thepressure is constantand assumed tobe equal to zero onthefree sur face

•Gravityis the only bodyfor ce

•Thebottom is smoothandhorizont al.

Therepresentative.:iagr'1Illof theflowwit h thechosen axesarc shown ill Fig.(3.1). Theorigin of the coordinateaxesis onthe freesurface,with positive xinthe directionofwave propagation andposit ive zpointing upwards. Thewa ter depthisdenotedbyh.

Assumi ngthe validity oCthe aboveassumptions[Lamb,1932;^WdLaUSCll andLaito ne, 1960),the velocityofBowcan be consi dered to he the gnulimit ofa scalar namely the velo citypotential,p(x,z,t).This po ten tial satisfies the two-d imensionalLaplaceequation

(3.1)

inthefluiddomain . Theboundaryconditio nseppllcubleover this domain are(Deanand Dalrymple,1984;Sarpkaya andIsaucson,1081)

10

1.Noflow acrossany solid boundar y(S~)

ii.v+~=O

^{onSb (3.2)}

whereitisthe IInitvecto rnormalto the solid surface,iiis thevelocity ofthebonndnry nnd{)1J/{J/Iisthevelocity componentof the flow normalto thebody.

2.Thespatialperiodicityis therequire m entthatthepoten tialatany location~repeatsitself atz

+

A,whe re ,\ isthe wavelengt h.Thus

orr"~, ,)

=

o(r +A" ,')

3.Period icitywit h respect totime means tP(x.z ,t) = tP(z, z,t + T) whereTisthewave period.

(3.3)

(3.4)

-l.The kinemnticfreesurfa ce boundaryconditionstates thatthe water particleonce inthefree surfacecontinuetoremain with the surface.

Thismoansthat

(3.5) must be sntisflod011thefreesurfacerepresented.by z

=

T/(x,t)which is notknownIiprioriand where 11 is afunctionof zandt only.

5.Thedyn amic free surfaceboundary conditionarisesfrom thelaw of ronscrvnrionof momentum, utilizingthe fact that thepressure is

17

constant at the surface,This leadsto

00

^{1 [}

(00)' ( 00)']

at + '2 a; ⁺ a;

+9'/

=

Jet) ^(3.0)

tobesatisfiedon :='I(r, t) . f(t)is a{UIICtit>OoflimeUIII) 'ami whichisusuallyincluded int/J.{Sarpkayannd Isnacson,1981) 6.Aradiationcon dit ionimplying that the waveay etcrngenerated hyI~

bodyis an olltgoingone.

Since the free surfaceisIl l !impermeableuoundnry,thekiuemnr ic freesur- Ieee condition canbeincludedin theno flowcondtao u,E(I. (3.2). For('1I..'Iy com put ation , thekine ma tic and dynamicfreeeurfuccconditions are mmnlly comb in edto give a sing le (ree surface boundary condition.(Sarpka)'" and Isaacso n. 198 1).

3.2 Se cond or de r Stoke s theor y

Progr ess ive wave theories are solutionstotheboundaryvnlueprohl,~mwit h thebotto m as theonlyimper-meablesurface.The~prt.'ti4.'ntl\ti ve(lia g r fllll ofthe coord lna te sj-st-tmisgiven inFig. (3.1). Gnc cfthed:ls.o;icrumctho&, usedto solvethetwo-di mensio nal progressive waveproblemill thepert..r- barionapproach where the varia bles of flow arc develo ped asIIpowerSt~rics of a perturb atio nparam eter. FollowingStokes(1847), thispertnrlmtiolL paramete r(f) is thewave steepness.Accordingly,~andIIme writtenlL'I

(3.7)

18

(3.8) for some function s?j(x,::,/),i=1,2. ·· ·and'7i(r, t), i=1,2,",The free sur fucc boundary con di tionsco nta in non-linearterms which are evaluated Iltn vnrjublelim it,::=q.In orderto be applicableatz

=

0,they are expandedin theform of aTaylor'sser iesabou t::==O. Say given

on::=fl,

Taylor's expa nsion ofIf:)nbo ut :=0impliesthat for small'7we can writ e

j( z )

+ '1~+ '"

^{= 0 onz=0}

Suhstitut ingtheaboveexpansionsin thefundament alequationsof Hew, lIuclcollect ingter ms ofordere,the fir s torder problemisdescrib ed by

8¹11 +^{{)1t/Jl~ 0} iJ.r'

u:

¹

O.

~ 0 on :=-h

a;

IF¢l+

0.,

_{~ 0 onz}

_{= a}

-{)t2

^{g -}_O,

(3.10 )

(3.11)

__,I

(0fJt .')

'71

=

on z=0 (3.12)

E'1_(3.10 )isthe no flow conditionfortheprogressive wave, theflat bottom Iwing theonlysolidboundaryinthe viciu ityof theflow,

19

Collecti ngtheterms o(orderf2,the secondord er problemi" dcscrilx"ll by

onz=-h

(3.\3)

(3.14)

on:= 0(3.15)

on:

=

0(3, 16 ) Expr essions(or the velocitypoten tialandthefree sur facetosc!'Ollf\ll order ofa progressivewavepropagatingin the positi vex-dircctionareob- tained by solving theabove problems,Ass umingflosinusoidal freesurface, theyaregivenin dimensionalfonnsby,(Sarpknyaandlsuacson,1981;DI~nn and Dalrym ple,1984:Stokes,1847)

q,

= ~ cos~o:~~: z )

^{sin(kz _wt)}

⁺

+ ~ (T)

2W

CO:~::!hk~

^{z) sin2(kz_wI) (3.17)}

20

'1 :=

f cOS (~'x -wt)+

whereH,thewave heightis twice thefirst order amp litu dealandhisthe waterdep th .k the first orderwave numb eris equalto 27r/>', wher e).is tllf) firs t orde rwnvetength . w is the angular velocityof thewaveandis relutcd t.o thefirst. orderwave period(T) as 2rr/T.The linea r dispers ion relation s hip'iotdsgood uptothe secondorderandisgiven by

w~:=Oktan h kh (3.19)

R<-'Sults of Stokestheoryuptofirs t orde rcoincideswith the classica l lineartheory (Dea nandDalry mple,1984).The secondord ertermsare Inncrions ofH²andarc usuall y oneorder ofmagnitudeless thanil:iefirst orderterms.Limitations rega rdingth evali dity of thistheoryarises dne to:

1.Convergence ofthepower seriesused

2.Develo pment ofa secondarywa ve crest in shauowwa ters.

Itcanbeshown[Dean andDalrym ple,1084)that nn Ursellparame ter defined hy

>.'H

lTr= V

21

determinestheran geof validityof Stoke!theory.InordertoSR.tisfythe abovementionedconsiderations. it is required thnt(OeMI\udDalrymple.

1054)

Ur<~1l"' ::: 26

^(3.20)

The straightline withUr

=

26 is showninFig.(2.1)lindlimits the applicabilityof thistheory to tra nsit ional and deep waterranges.

3.3 The oriesforpiston type wavemaker

Apist on type wavemak erisIt.verticalflapthattrnOl/lntellnormal1.0its surface.Activa tedh~·drl\ulicl\lIyor pneu ma ticnlly, thewuvcmnkerf\isphu'l~

water while in motioncausingwaves.For theboundaryvalue pmhlemof two-d ime nsionalwave genera t ion. Eqs.(3.1).(3.6)holdgood.The no Row condit ion has to besa t isfied atthe bottom.Eqs. (3.10)and(3. 14) and nn thegeneratorsurface. If the equationofmotionofthe genera torisgiveu by

~(t)=~o sin ""t

theno-Bowconditionon thegeneratorsurfaceta kestheform (3.21)

""~oCOll""t- ~= o

^{onx=( (3.22)}

s =

2~ois theso-calledstrokelength of thewave mnker,The rcp resentutl ve diagra mfor thetwodimensionalwave generationproblemillgivenillFig.

[3.2).

22

A linearsolu tion totheabove problem uptofirstorderwas proposed as a combinationof asingle progresive wave and an infinitenumber of standingwaves, (Havelock, Hl2D;seealsoUrsel!ct ai,1960).According to this theo ry,the velocitypotentialis givenas:

<b

=

bucoshk(lJ+z)sin(kx-wt)+ (3.23)

+ eoswt~b"c-J:~'"

^{eOllk,,(h+z) (3.24)}

wherethe wavenumbersknndk"~atisfythe relationships

w²=gktanhkh (3.25)

(3.26)

(3.27)

(3.28) Theumplltudesholind b"arcob tained as:

/ .

0~owcoshk(h +z)J.:

flo

= ,,-"'.;;;-- - - -

k 1:

^cosh2k(Ii

+

z)dz b"

= L:~owcos k,,(h + ':)d:

1.:,,1:

cos²k,,(h

+

.:)d:

Tilerclutlonbetween the height oftheprogressivewave and 110is obta ined

hrevaluatingthe Creesur faceelevationfarCra m thegeneratorsu rface as

~cosh_g k1t = !!..₂

The ratioof theW(W Cheight to stroke lengthforthe pistontypegenerator

WIL'IgivenbyUrsel!et al(1960)as:

~

=

taohkll

eo

⁰¹

23

(3.29)

where n\ is the ratioofgroupspeed to celerityof theprogressiveWIWC,;iVl'lI

nl=

~ [1+ 2 kll_]

2 sinh2k11

The exponential amplitude standing waves decayrapidlywithdistnJl\'c Irom the wave maker.It hasbeenshownthatwithintwo orthr eewaterdepths away from the generating surface, most of thestanding wavetermsarc negligible, (Ursellor al,1060;Deanand Dalrymple,HIS4).

Secondorder theoriesfor thewave generation problemhaveheenpro- posed,hut none has been accepted widely.Atl1t.'orybyFontanet ,( HlGl ) is applicable to piston -typewavemakeea,butWIUIfoundto heeumborsome.

Madsen,(IDi O,I D71)employed an expansionsimilartothe Stukes pertur - bation techniqueto developasimple theoryforpistontypewavemnkcrs.

The authorhas used a perturbationexpansion for the piston motionillthe form,

(3.30)

where{Iand6representedthe first and secondorderpistonruotioue re- epectlve ly.The first order solutionofthistheoryis similartothe classical solutio ndiscusse dpreviou sly,In secondorder,thistheoryreveals, similar to Fontanet(1961), the existenceof a second har monicfreewaveillnddi, tion to the Stokessecond orderprogressivewave.Thenmpiitudeofthefree wave is givenby this theoryas

1

(H )'

1 ( 3

n ,)

^{tanh /'!.h}

a

^{2 2}

=2 '2

hta nhkh 4sinh²kh-2"

-n- ,-

24

(3.31)

where02is analogous to11\of Eq. (3.20)and isgiven by

(3.32)

,. isthewavenumb...r ofthefreewave andsatisfies the dispersion rela tion [Mmlson , Hl70jHl71)

(3.33)

Limitat ionsontheapplicabilityofMadsen'stheoryfollowstheusuallim- ito.tions ofsecondord er Stokestheory.A drawbackofMadsen 's theory is tha t thevalueof(In rapidly dropsto zerowithincreasingkh becauseof the llcKat ivetcrru intheexpression.

The theory byFlick and Guza,(HlSO)wasdeveloped to include a variety ufwave makerconfigurations.This theory resorts toamore exactsolution by includin gthelowestorderstanding waves into the secondorde rsolution.

Till: amp litudeofthefree waveis givenas

when:D"andDo are integralsoverdep th ofcer tainalgebraicallycompli- outedfunctions.

A few result s commo ntoallthese theori es, whichhavebeen assumed inthecourseof theprese ntwork areas follows:

•Thesecond orderwavefield consistsofa Secondorder Stokes pro- gressivewave and a secondharmonicfree wave.

•TheIreewavebehaveslikealinear wave and trnveleattwicethe first order frequency,ItsatisfiesthedispersionrelationgivenbyEq.

(3.33).

•ThestandingWfI,\'"arevirt ual lynegligiblenIe w wntcrdepthsewny fromthe genera tor,

3.4 The total wave fie ld

The total wavefield inthelabora torywasexpectedtoconsistapartfromthe desiredprogressive wave,cont ribut ions duetoreflection"fromthe beach andduetofree waveeffects. Effects ofthebenchwereuottbcorcricully quantified,butitwas expectedthatawave arisi ng line to thereflectionof theincidentwavewill bepresentin thetank.Thiswavewouldbeofthe sameIrequeneyas the incident waveandtravelin11directionoppositeto it.Yet anotherreflectedwaveofsimilarqnA.1iticswasfoundto ariseclueto thereflectionofthe freewev e.IThetotalwnvcfieldinnwnvc BurneWf\.'l

thusassumedasalinearsuperpositi o nof thefollowing wnves

•Firstorder Stokeswavewith amplitude a,andwave num berk

•Secondorder Stokes wave withamplitude^U1andwavenumber2k

•Seeondorderfreewave wit h amplitudeanandwaveunmbcr~ lit was initiallyassumedthatthisreflectedwavewculd benegligible.OurinAth,'"Ourllfl orexperim ents,a beatpaUerncorresp ondingtothisWII.Y<lWMfoundHll p~r i m p~II)V':r otherbeatpeu eene.lienee are-Icrrn uleticnhadtcbeelreeted,

26

•Firstorderreflectedwave withamplitudeURandwave numberk

• Second order reflectedwavewithamplitudeUURandwave number,..

Thisis representedmathematicallyas,

lhot:tl

=

1l11:os(kx-wl)ta2cos2(kx-wl)+

tn:22COS( ';:X-2wt

+

6)

+

aRcos(kz +loIt+a)t

(3.34) whereo.~,'Ylire the phases of the correspondingwaves withre spectto the first orderStokeswave,Rewrit ing the total wave field as:

\1'1'ohtniuthe followingexpress ions.

(3.35)

27

9, [

4lsin b-4ASin(b : +O) ]

arctan I'Ilcosl.:z+a n cos(kx+ a) (3.37)

Doththe firstandsecondorde ramplit udes.'/Innd'/2canbe eccutobe functions of the horizo ntaldistance,%downtheRurne.Th isis refe-r-redto u the beatpatternarisiugdue to a superposit ion oftwo ormorewaves.

25

Figu re3. 1Reprcscnm tivediagramofatwo-dimension al progressivewave.

,

1-( ( . )

.(x,')

Wave generator"

Beach

Figure3.2Represent ative diagramof two-dimensional wave generationbyapis- tontypewavemaker.

29

Chapter 4

Experimental setup and procedure

Thewave tanklocated attheFacultyofEngineering,MUNisII..~1"1·1re-in- fo rced structure with inne rdimensionsof58.27mx·Ui711lxa,U,IIlI.Ow' oftheside wallsisglasspanelledatvari ous depthsforvie willg pllrpOSf's.1\

pist on type wavegenerator is ins ta lled atone clIdof thetau k ussho wnill Fig.(,Ll).Thegeneratorisdriven hyahydraulicactuatorwithit"lll'ld ,ilily of·IS.SKi':force over aa.25mstroke.Electroniccout rolforlll'~wi.~·d>Ollr,1 i~

pro vid ed from....contro l roomth ro ugh anJI,f'l'S c10Sl'f1 loopsPf vn -/"OIllrulll ..1 systemwith errordetectionand compen sationappliedtIlrougha J.VDTf.-,..[,

back loop. Theboardhas a maximumspan ofO.SSmand isexpect edto re- spondwell to command signalsinll,frequency rangeof O.35Hz • 1.3 Hz. Both thefreque ncy andspa nca n be set manually by means(IfClIl ll l t c'rswhichllilvl ~ anaccuracy upto thesecond decimal.The waveb car d ha sII waterti,ltht

teflonsealalon gitssides andbottomandno leakage behin dthe boardhad bee n observedor repor ted to date. \Vavefilterplates affixedtothe frontof thebo ar d servetoredu cethe cross oscillat ionsinthetank.

Attheothe r endof the ta nkis located a para bolic beach. Thebea ch iscon s truct ed out of threemodu les as shownin Fig. (4.2). Thebasic struct u re issteel,top ped bywooden gridsand th reelayers of net to ab sor b energyefficient ly.A gap of one foot betw een the beach and the bottomof tankallowsn freeflow ofwater under the bench for mainte na nce purp oses.

A towing enn-iageruus on rails parallelto the length ofthetank. The carriagecontro lsystem emits10,000pulsesfor every1.0rn travelled. The horizon taldistance moved by the carri age canbe estimatedby tracking thenumbe rof pulses emitt ed usingafrequencycounter installedfor th is purpose.

Aschematicdiagramof thesetup used for these experiments is shownin Fig.(4.3).A resistance ty peWIWCprob e wasused du ringthe experiments to measure ins tanta neouswave surfaceelevations.Thisprob eop erat edon theprinciple thatvar iati on ofthe con ductivityof the wires would linearly ,1"p"1H1nil thelevel ofwa ter inbetwee nthem,providedtheamb ienttem- pera ture remained constant.The wavemonitor hostedan amplifierand a Wheatstone net work.The probewas con necte d to the Whea tstonebri dge illt.hemonito r. Sincethe vnrint ions inthebridgevolta ge dueto varia.tions inthe proberesistanc e were small, it was necessaryfor the sign al to be

31

amp lified. Th e amplified signalWASthe nrecordedon mllA:netictn~by1111

8chan nelFMrecorder.

On-line dataprocessingwasnotpreferredforthepresentst\l(lyliSthm- wasaneed to store analog datafor re-analysispurposes.Digitisntion IIf theanalogdatawasperformedusin g a workstationdnt..acquisiti onI\lId con t r ol devicecalled KeithleySystem570.Specific"tion~ofthissystemnf(' show nin Table(4,1).Therequiredin put parametersforthecligitisntillll programwer e:nameoftheout putfile,sampli ngrete ,lengthof recordnnd the numbe rof channelsof input.Thelength of recordwnsu:m nUy twocy- desandthesamp lingratewasdetermined basedonthe requirementof G4 samplesper waveperiod.IComman d s were issuedthrough1\desktopr-em- puterandth edatast ored asfilesintheVAX/VMS$530computerthrour;h theRemote accessfa cility(RAF)ava ilableontheVA X/VMSsystem.

Cali bra t iontestsforthe wavepro be wereconductedbefore theexpcr- ime n t stoevaluatethe relation betwee nthewater leveland thett5istnncc of th eprobe.Thesetestswerecondu ctedbymanuallyrailiinJ; «ndlow ceing theprobeandnotingthecorrespon di ngoutputsfro mthewavemorritnr.

Sinc e theprobesweresensitive tote m perat u reftllctllntion~,itwasnf..'CI'!I- SAry to calibratetheprobesevery dayof theexperim en ts.A recordoftltl! calib r a tionconstants thusestimate dis shownin Tahli!(4.2)for alltill!(Ilty~

when expe rimentswereconducted. Fig.(4,4)shows the cnllb mtiondntu Irorefficientfunctioningof theFFTalgorit bm,itw;ure~uireJtlllilthel"L,,1"a ml,I"..

beapo~rof 2.h wudeeldedthal64sample. perWIl.V'lperi"uwOIJ],Ilmoptimalforthe present purpose.

32

for a part icularday anJaleastsquaresstraight line throughthe data .The

~loJl I~oftheline gavethevalueofthe calibrationconstant,

Theproblem1!.lhandwnsto estim atethe individualwave parameters, giventhetotal wavefield. The approa ch adop t edinthepresentstudywas toobtninwave surface data atdiscretelo cations down the tank.From these tilil"si~lIals,tlll'freque ncy componentsat firstandsecondorderwereanal - y""11. Cllr\'l'St:Orr~pomlillgtoEllS,(3.3,')) ami(~.36)were fitted tothese l'UIII[)(Jl U'ntsand,the amplitudesorinte rest were discerned .Theexperimcn- 1;,1pro n~ llIrl'fortilisapproachwas:

•St'l l.lll'Irrqueucyand span

or

thewavegenerato r.

•Sl'!I'rl1l11111hl' rofloearionsanddis ta nce betweenthem

•Allowsulllclenttimeferthewavefield intheta n ktoattainsteady sta h'.

• Pla,:ctilt'probeattheHrstlocatic n, reco rddataover threecycles ,

• :\ltl\'l~ontothenextlocation,

• Cll'<I ralllocations ,prepare fornex trun.

Till'd'ltathu sobtainedwasrecorded.digitise d andstoredseq uentially as

1"llIl [ltlh'Tli lt'~,I';arhlllc was l'ollct!bya four digitnumber, thefirst two

digits ind ic a ting th erunnumberandthe last twosh owedthe locn tionIrotu which thedata wastaken . Thenumberand distancebetweenlocations depended onthe wavelength softhebeat patternspresent.Ilwns ensured thatatleas tfiveloc at ions were presentwithinthesmallestbeatwavelength.

Thefirstlo cation{designatedaslocationzero) wns fixedat 20mawny from the meanpositionofthe genera tor .

Analys isofth e datawasperfor med by aprogr a m calledANALYS,lie- velopedfor thispu rp ose.A flowch artofthis program is sho wnin Fig.(4.5) and thesourcecod eis provided in AppendixA.Inputstothe program were the sampleddata at every locatio nand theparam eters char acterisin gthe run like th e frequency , distance betweenlocation s ,numberof lorntkms, inter valwithinwhichthe wavenumbersof bent pa tt ernslie etc.Couvcr- sia nof thesampleddatain to the frequency dom a in was perform edbyR.

fast fouriertransfa r msuhro utinecal led F2TnFoftheIMSL/ MATHlihrllry availablewithintheVAX/ VMS system.Thisprogram utilisedthe data for one wavecycleand evaluated theamplit ude sof various eosineandsine com ponen tsof the data.Amplitudes of firs tandsecond orderwere thus obt a ined atevery location.Appendix Dex p lainsth e workingof a stnndurd fastfouriertransform.A subroutineLSSIN wasusedto fit nh:l\.'itS(llmrl:S sinecurvestotheseamplit udes, withthe wavenumberbeinglUIunknown.

Basicallyit performedthe followingoperations:

34

•From the variation of nmplitudes down theflume, LSSIN identified the wave number of the beat patterns within a user-specified interval byanoptimization procedure to minimisethe least squared error in the fitted curve.

•At the optimalwave numberacurve wasfitted and the amplitudes ami mean values ofthe beats wereevaluated.

•Fromthese values, the amplitudes of individual waves were estimated using Eqs.(3.35)nnd(3.36).

Appendix C explains the working ofthe subroutineLSSIN.In the equa- tlon for the totalfirstorderamplitude (3.35),there existedonly onebent putturn. So theworkingof the program was straightforwa rd. In the second order-equ ation (3.36)however, there existed three beat patterns.

The procedure followedinANALYS in this casewes to fit a curve to the predom inantpattern (the curve with numbcrx - 24-).Thiswas thensub- t.ructedfrom the amplitudes to obtaindata for theremainingpatterns. To minimise distortions caused due to the subtraction,the amplitudeswere iutcrpolntedulSillgsplinesto get a smooth curve.Subtractionof the fitted curvefrom the interp olatedcurvewas theneffecti ve.The data a..tailable aftrreu btrncrionwes fitted withth enext predominantcurve.

Essentiallytheavailable dataVIIlSfitted into the formulatedmodel to ovnluatcthe<!csirc(!quanti ties.These werethen compared with available theories to check thevalidity of themodel.

35

Table4.1 System570 Specifications

General Supplied software Host computerconfigu rat ion

Standardchannelcapacity Analog input Analog output Digital input Digit aloutp ut Powerrequirements Analog input Instrumenta.tionamplifier Programmablegain amplifier AIDconverterfull scale ranges Resolu tion

Max.samplerate Digi t al Outp ut Channelcapac ity Output ran ge Drive capability Accessoriesand optio ns AIM6

DaDiSPI DaD iSPII

36

SOFT500extensions toBASICA IBMpc,XT,AT COMPACpc,COMPAC Deekpro 286 with atleast256K bytes of RAM

32single-endedor16differential 2

16 16

5Vat2amp max. fromhost computer.

xi, XI0,XlOO,switchselectable Xl,X2,X5,XlO,softwar eselectable

±lOV,±5V,±2.5V,0tolOV,Oto5V, switch selectable

12bits(1part in4006) 31.4Ksamplesper sec.

16 non-iso lated TTLcompatible,lowtrue 10TTLloads,20m Asin k{j0.5V Straingageand RTD analoginput module DataAcquisitionand SignalProcessing software package

Extend edDigita l Signal Processing softwarepackage

Tab le4.2 Record ofwave probecali bratio n factors

Dateof Calib ration Experiment Factor V/m

Feb12 5.99

Feb 15 6.30

Feb 16 6.41

Mar 02 6.29

Mar 03 6.06

Mar 04 6.13

Mar 07 6.S2

MarOS 7.06

Mar 09 6.98

37

Figure(4.1) Configuration ofthe piston type wave maker

38

Input Amplified Inp ut

Sampled du.

Memor y

PC

m

~

ci

~

~ .;

! ;:;

.fr

~ ::l

~

0

e-

~

~

Figure4.3 Infor mationflowch artforwave experiments

V

-: V

/ ^V

-6.25 -3.25 -0.25 2.75

Displacem ent

(om)

5. 7 5 8.75

Figur e 4.4Wa veprobe calibr a tiondia gram;Date:Feb12 Slopeof leastsquaresline=5.99Vim

• -),Ieasu redpoin t8 39

8

Figure 4.5 Fir st orderanalysisflowchartforwave experlrnenta '0

Chapter 5

Discussion of results

TIll' /·xlu'rillll'lll.sW,'rI'Jc~igllcdto coverarangeof parameterswhere Stokes

~<','<mcl"TIlt'r tli''flry W01l1,1hevalid.ItW1\.<;alsoimpor tantthat third order ,·rr'-"lllI". lowillthis runge, asno thirdorderwavegeneratortheorywas a\·i1iJidJI....Furtln-r,tIlerall~Cwas alsosubject10lim itatio nsof the apparatus IIS" '1.'I'll"piston t)"p.,generatorwasstrokelimitedat lower frequenciesand I'''W''I"lilllii''(] llLIlighc rIrcqncn clcs.Itwas foundto performsa tisfa ct or ilyfor r""'lll" Ill'i"Hofll.:l,")-1.:1][;-;.Consideringallthesefact o rs, a set of28point s Il'pn ':-J11l.~1"1illFig. (2.1).Therunswe recomposedofsevenfrequ encies

nudtourH!l'( 'plll' SS l'liatevery frequency.Theseva luesandthose ofcert a in

rhaTiWh' risiligparamet ers are givonin Table(.''i.!).A constant wa ter depth of1.0mwas maintain ed throug hout.

Samp lewavefor ms(limehistor yof thefreesurfa ceele vat io n ] aresho wn ill Fi~s.{.'i.l)10 (5.5) as a plotoflime vs surfaceelevation .Figuresare

II

arranged accord ing to theorderof locntions from where the darnwas taken.

The timescale used dependedon the samp ling rate,whichinturnWIlS

depend enton the frequency ofth e wave. Free sur face elevation1I.'lshown is uncalibrnted.Thetota lwave amplitudecan be seento vary/lIangthe llllIlW.

Thevariationof the first andsecondorderamplitude"isshownillFigs.

(5.6)and (5,,7)wh ich areinterm ediate outcomes oftheprogram ANALYS.

First andsecond ordermean square amplitudes at variouslocut jons nre evaluated usingII,FFT andplotted as pointsin thesefigures.Lcns t squares fits,corresponding to Eqs. (3.35) and (3.36)are alsoplotte.l.Th ..pm ximi ty ofthe points to the curveshowsthat the modelascdforthc wave ti..·l<l matches well withthe actualfield.Correlatio nwas gCIll'1"llllygood forall the runseven at the second orde r, wher eLI",measured,p\lllltitit'~arc stunl l (for examplethemean squaredamplitude inFig.(5.7)is of theorderof 0.025sq.c m.]Discrepan cies stillexistedwhichwere Jlossihly(hlf'to

•Other unknownand unacco unte deffects

•Practical difficultiesin exactly positioning a location .AsW;L>;111<'11'

tioned before, movementfrom onelocation tothenext wasbyIIca r- riage, withthehelpofa frequencycounter,Precise1II0VCIIlt'ut to11

loca tionwould be pract ica llyimpossible thoughdfortsweredifl~dl~"

towerdaachievingthat.This effectcould be siJ;lLificlIul.011WhVl'Sof smallerwavelength.

42

5.1 First or de r results

Thofirstorderres ults fromthe programANALYS consisted of the first urderwave1I1l111l.lCr ,amplit udes ofthefirstorde rStokes wav e and the first 01'<1('1'reflect ed\\1:\ \' 1.'findthe phaseofthe reflected wavewithrespectto the illl'id.m tWIl VC.Result sarenon climcnsionaJised andpresentedinFigs . (5 .S)lhro1\gh {5.Il}.Fig.(5.8)is a plotofthe relative dep th parameter I.:hexpcr-irnent\'('1'5115theory. Thegraph is inte nde dto showtheefficiency oft,llI~,l!;oldf'Oser-fionnpt.imiantlontechniqu eto estimatethe optimalWI\ \'C

1l111I1h ,-1'whichwouldsuitthedntn.Fig.(5.8) shows thattheexperim ent al

HI<li/fl'fsfrruuthellll'orct ir.al~'hh)'amargi nof10%.Furtherim pr ove ment ofthisteeluriqueispossib le.To checkwhether thewavegenera torfollowed thl!lincnr gcm-mtor theory, thewaveheig ht(lf)tostrokelen gth (S )ratio wasplottedlIgninstkhin Fig.(5.0).Thecurv eofFig .(5.0)is a plotofEq.

(3.20)ami thcdot:- arctheres ultsfromANALYS.Ascatterto the order

of2[)'i(,is ,'vill"ut.Th i~isnot unprecedented.Chen(lOiS)repo rte dofa.

siuulur"I'atl"r at approximatelythesame wat erde pth.His explanations fo rthis«rnt.tr-rwere genorul. IIIthepresent study,the aut horexpectstwo ptll'sih ll~sourcesofthisscut tcr.

•Pn'S" II<'('of sl'<;'l1 lLdllr,yreflect edwaves at thefirstorder .

•lmpcrfevtiousillthe wave gene ratorcon tro lsystem,

H the reflecti on fro mabeach is high , second a ryrcn~ tl'l lW<lWSan' en'atl'tl duetothe reflection oftheprimaryreflectedwaves illthegl'lwl'alorIIlLrf a ,"', This effect is usuallysmallifthe prima ryreflectedW:\\'(,11fromI,ht'lu-arh;HI' small,Ifa seconda ryreflected wave we re10hepresent , itwouldheimpollsihl,' to separateit out fromtheprima ryincidentwaveIlt't~:l.llSf'hoth of1,11l'1lIwould beofthe same Iroquency. De pen din g011thephasetElfl'n'lIn ',lilt's,'WitW S inte r fereco nst ructively ordestructively.Comparison ofIh,' results ()fFi~.

(.'i9)with the reflection coefficients plot ted in Fig. (tJ.l lI)s htlws no ill'l'il r. 'nl correlat ion,There arc cases where Ill<' reflectionm,'ITid" nii~Iliglll.n tH,'aU,'r inFig, (5.9 ) loll'andvice versa, OnthelII'('oll tlilSIlf'd.

o r

till'wuvegen,'r at llr con trolsystem, fourpossible so urces oferror s are:

•lruper fect ionsinknobcontrol

•voltage se tt ing for maximumstrokeofthe board

•varlatlon of generator efficiency wit hue pl h

•temp oralvaela rlc n of boa rdmovement dul' topoo r Sf"llilll; ofthe l,,.jloll seal.

Manu al errorsillknob control rorspa nsdlill~totlu-"SI"lltIIr,,'y.,nn- possi b le, The volt a ge scuingfor maxhunmstrokeof lIwW'lIl'r;.lurisIIIL initialisingpararllO'tr.'rrorlll"~roll" rolsystmu.S!II;11tl"il<llll!;''''illIhisIMr ' lllJl'l."r can occur over till'rour-snnf thm-,ifHilldW"k ,"Il"'ri"rlin, lIy, TIl1'

"

amountof erro rwillhowever,besmall.The generator was designed foran optima ldept hof108m. Lowerdepth can resultinabigger strokethanre- qnirrxl becau se ofreduced hydrosta ticload . Improperseatingofthe teflon

~elllaroundthebonedWIISnoticedin theear-lystages oftheexperiment lind roetifled.The scatter inFig.(5.0)couldhave had cont ributio nsfrom all these four sources.

Firs torderreflectioncoefficients defin edby R=?!!)( 100%

a

an'plo ttl ,.1inFig.(5,10)versus therelativedep thparamet erkh. Bea ch mechuuisrusnrustillasubjectofstudy and henceitis possib lethatR CtJnlddl.'pl'lld011nnyparameter. Thegraph showsa considerablescatter varying over10• GO%.The main sou rceofthis high reflectio nwas the

"IH~footgnpbetweenthebeach and the tankHoor.The aut horwasalso informedtbutthe beachwas designedfor an optimaldepthof LSm,So atItwat er depthof1.0m, the reflectio ncoefficientsarelikely to behighe r thnnop timum. More intriguing thanthe reflection coefficientisthephase litwhichIIwavoisreflect edfro mabeach. Fig.(5.11)shows a plot ofth e phase differencebetweenthereflected wavean dthe incid ent wave ,ctvskh.

Resultsarcwidesprea dfrom-2:rtoO.Infillcases,Q'isnegative,indicating thatlht"reflect edwavelags behin d theincidentwave.Apartfromthat ,r. o

"nlldn~ionssee m possib le.

5.2 Second order results

Expected results for secondorder wer e:

•Free wavenumber,/'i

•Second order Stokes wave amplitude,(11

• Second harmonic free wave amplitude,(111

• Phase differencebetweentheIrcc wavenIH!incidentwave,Ii

• Second order reflected wave,a22R

• Phase difference between the secondorder rcflcoteclwave nnd th ...

incident wave,-r

Figs.(5.12)through (5.15)are plots of the Stokes amplitude ratio,(I.d"l

versuskh.This would be e teston thevalidity of Stokes theoryoverthe range of testing. Each plot is for one ofthe fourdiff(~relltSll'I'PIWSS('S ,rhur- ucterisedbytheparameter,H/ gT², Fig.(5.12) shows that att1wJOWI'st steepness, whereG2is typically of theorder of 0.01111 'experimental(('Suits are quite dose to theory.This was thefirst indicationorthe dfici"ney ..r

the algorithm behind ANALYS.Though the nutniued vall u-sor(IIwere con- siderablydifferent from expected (sec Fig. (5.9)),it did not seetu ttl hllv(' affected the amplitude ratios. Results are higher than theoryillFi~.(5.12) but the differencena r rows downwith increasingsteepness.This is beeuuse

46