A MODEL :I"ORTHE , T W O DIMENSIONAL INTERACTION BETWEEN NON-LINEAR WAVES AND

"

A MODEL :I"ORTHE , T W O DIMENSIONAL INTERACTION BETWEEN NON-LINEAR WAVES AND

UNIFORM CURRENTS

( BY

@SH A OWENSONG

Athes is submlete dto'tlleSchoolof GraduateSt ud iesin,partia!

fuIn llmentoftherequ i;;e m ents.fc r the degreeof Mastersof

.Engineering .

" .

FACULTY OFE~GINEERING^{AN.nAPPLIEDSO{ENCE} MEMORIA l/UNIVERSITYOF NEWFOU NDLA.N D

St . John's

October IUBS

Newfoundland Canada

i._",

I..

"

~"

Per.1as1on~has~been gr a n t ed L'autorhat i o n"1tltd ae c ord'.

f xe the'lila tiona l Librar y of .A 1. BlbUoth~ql!-e-'naHonale C.,aqad a to .ierofll l1l tlJh' du. Canada',de-". i e rofi l. e r

theds'-and to le n d or se ll eeeee thi••··et.de prater au

c.op1eeof tbe f11ll1. ~~i:~ndred~. eUl\plairee du

'~ . \a~t~~r.~ c~~y:~~.h.t~O:~.er~.

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ez ten d ve eztrac"t•...,..;f roll it ~t:ll:tr a i t . de cel l e-ei he '

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Acknowled gements

~The aut ho r,wis~estoextend his~incere a~preciationfor the help and guidan« givenbyhi"11Uperv:isor:Dl':R:.~..Blldd5ul',d.uringthe worle.on this^{4 t h e s}is.

The author woulda1IoIikejcthank thetechnicalstaft' ofCPniputer·.

.

Servi~ ~d' ihe Facui~y ^of E~ginee~~

^{Center'for'CAE}

at.Memorial.u~i,

~er'sity

^{Eor thelr'}

b~lp,

^consultation

~d

^the

s~'.;.}l

;um- aro'und

tinie.~-

'In'

'clOSi~i.

^the

a~t~or

^wouldlike'

~a e~r~~ his grati(\l~e ~~

^:the

:S~~~l

~f GrBdn~t~ ~'tudie8

^&nd'

~he

^.Facultr1',f

~~gineep~g '~~rAP~~i~~~cien:ce :

.

Eo~ t~e ~~ci~

^.•

~·

^{in' the,}

fo.nn~

^oE

fel~OW~hi~ ~d .bUrsu~~(·:

,.

:~:. : ^/

0 '.

I

.i ·

~~

^{... ; ••}

~:.~'

^{••' .:}

~

^{I'•••· I.'." :: '•.• ,} .'. ",;;.,. ,'. <:..~.

. .

The intl'racti?lI or.•wate rwavetra.in~ ~itha uniform current nor- mal to the wave crests is considered.';l'Jiecombinedwav!'current'mo-

wave-currentproperties throughHo •L_o,do.and UoIs establishedby

. ' .

.

where6d=d -do,

au

⁼

u -

UO•The pre dictionof the combined tion resulting.rrolp the interactlon is assumedstable andIr~otational.

The'velocity.pote.ntlal,!fispersionrelation,theparticle kineinatics and pressuredistribution.~pto th.~ thi~d ord~laredeve~ped.Th~ conse~­

vati~n^{ofmean mass,'komentum}and energy of the current- freewave,

~;ve.free

^current

~nd combin~d wa.ve:curr~~t

^ftelds

bero~e

^and

A

th~ ~nteraction-are used, togetherwiththe.dispersionrelat-iOIi.'~nthe".

~;~'es~rface^to~,erive^{a:set of}~O};Ir non~ncar^equations,thro~g:^which the ftationshipbetweenH.,Ltd;Uand

s.,

L", do,

u,

~8eetabllshed, whereH~tLo ,d"arerespe ctively the current-freewave~eight.wave

I~ngth, me~n' w~ter 'depth,

^arid

o,

the

Wa.Ve-ff,~

^{current'speed,}

!I'

~"_ _ ._ _·~_~peetively-th~~gt~.,_w_a:t~"~d~.p~th"---'-;--c' current speed of the combined

~ave-currentfi.eld-irffirtMirilei;ac~ron.-r---­

Thisis a new approach to an

. · old.

problem.

.

A numericai method is

.

used to solve

the

system.ofnonUnearequationsto calculateR,"L,d, andUwhen given the values

~f

^Ho,\,Lo,",do"a.nd,uo.Numerlealre- eulte forthe changes'of wa'!e~eight,-length,waterdepth, and current speed in,the form offi/Ho•LjL o•l.i.d/do,.and.I.i.U/Uo are presen ted,

!,

usin",the ,:elocity

poten~i~

^{ofl'heC:;mbinled}

~a\~cu,re~;

fieW.and

. ,' .'

' . '

.

the numericalTault. of!I..L,d,ull U.Aco~p&lilO D.betweee the ex~rimen.talrftu,lt.o,(Thom.u(1981)I;lld'the

,ts,*,.

of thepresent ,JIItheory' ispreeeeted , udJurpri~ing.agreement

i.

^observed.

DLKullion.

o~

^the

rl~'t1oil'st~essi~d en~rgy

^{tra n. rerinthecom-}

-iuned

w1rVe-"tIl'~rent fi~d',

^andbtl

the'n~riae~ca1 ~eth~

IUId prec1510n,I'••'\.

tontrolare

gi~:en

^hithe

i1,p~~n~ite~~ , .~

^{..' \ . :..}

/~

^{..,' . - -: . :}

, ."

\

^~

" \ i ,

,"

,. .

iii

Contents

1..Introduction

2,Liter etureRev iew

2.1 ,Two DimensionalWaves and UniformCurrent Inter action. 2.2

~hte:e.~t~on o{'W8.v~

and Uniform'Current

'~t

^an

~ngl~· .

^{,. 10}

".?,3•

v.ia~~

^endShear'curr ent Intera tion. 12

? .

3 Combined we ve-curreneFi.e!~

3.1 Firs't order approximation

13 16 3.{.1 Potentialfun'ction linddispersio:-releticn 16 3.1.2 Properties of.the first order~ave.currentfie~d. . 3.2 Seeoi"id'order approxi:nation .. .

.

_3,2.1 Potential (unction enddiepersion

.

relation

19 22 22 3.2.2 Properties ofthe secondorderwave-currentfield 28 3,3 Third orderapproximat ion

3.3.1.Potentialfunctionand dispersionrelat ion 32 3.3.2 Proper~iesof thethird orderwave-currentfield 36

4)Cha nges in Wave an d~urre llt 40

4.1 C~nseryationequat ions: . ... .... .41

, .

^4.2.Meen mass, momentum,energy fluxes

4.2.1 Flu xes in a current-free W8.v.efield

iv

43 43

'

.. -

^:

'.,:'.

4.2.2 Fluxes inII.wave-freecurrentfield ... ,'.... . . 46

I ' , ..

.4.2:3 .F~uxes i~a wave-current field../ ... ...• 47

.4.3 Relationshipt betweenL,H,d,Uand

I(

H••de.U". ... Sl 4.4

eon:aput~tion~

^{eonsldera tlons}

and'-r~ul15

^.

~

^{.:.•..", 5;}

.- .!

5 Predictio~ ~ftheCombined Wa ve-C urrent Field'Pr oper.

. ' I

< . .

tie~ "'\ '" ~ .,. . ,- .. 03

"

,- . .

^" , ...

compariSbn!'Bet~een

^{P)esent'Theo; y}

~nd Ex~erini.e~t

^.71'

- ' .

~ ^{, '. ' "}

.7 .?O'lelUS,iOl1~l~ndReecrnmendeflcner~r~Fu~her Res~arch^.8.0,'

.

\ "

.

Referen ces . , , . . .. 81

.

^{.'. : '}

: 1 .

FurtherReadings 8j

!

Appendic es 93

103 100

110

"

)

94 A A Discu ssi on onth e RadiationStres.and Energy Transfer

in a

wave-cu~r~nt

^{Fietd. " \}

I · .

DNUl~e ri(:allonllider a ti o ns and Precls lonCOlltrol

C CGmp~terprOgram,

o

SomeNumericalResults

i I

l. ^{...1 I . ... .}

:~;~ "',;~~,~:.,;>,~

^. ..

.-:.:

) Norarions

Tirefollowingnotations!Lreused in this thesis:

C wavevelocit y.;' - waveamplitude;

a; - tho~izontal^part,ideacceJ.erati~n;

~'.. -~vert ical par,tideacceler ation; "

.

^'

water depth;

wave- free~urrentenergyfiux;

com.binedwav~currentfieldenergy

ftwc;

acceleration dueto'gravity;

waveheight;

T~lativewave velocity;

'Curre nt -freewa ve energyt1ux;

E.

H

[( wavenumber; wavelength ;

vi

_;M.

wave-free current-m?m~ntun1flux;

M", current-free wave momentum~u,x;

M",c combined wave current field momentumflux;

P ,pres s ure;

.Qc_. wave-free'cur rent~assflux;

Q",

\..t -

~I"rrent-fr~wav: massflux;

Q :"': -

/comb inedwaveeurrentfield massflux;

.- I "

T .

=-;

wave period; •

J:.

^{,time; " \}

/' u ,I

~W"rent^speed;

horizontal wave particle velocitrl verticalwaveparticle-velodty; . horizont~axis;

. /

j

! :'

verticalaxis;

wave ,frequency;

relative wave frequency;

wave profile;

vii

•

.r: '--,,;

_/ -

i>, . i>.

wave-freecurrentfield potential ; I

current-freewavefildlpotential;

com binedwave-currentfieldpoten t ial .

Parametersw}th (with out)eubscj-ipt"0" denote"'the val ue before(after) the interactionbetween the waves andth ecur reu!-

/,"

viii

Fig.4.1

Fig.4.3

ListofFigures

Wave

len~th

^{ratioL/L,va eurrentspeed ratid}VD/Co'for..

plane W8:ve parameter.s 0;= 0.75rn,Lo""100.~ , m,.do=25.0.~~.

Wavehei~tratio H/Ho.vscurIentspeed rat io Vo/Cofor. planewave.~arameteu^{0 0}= 0.75m,Lo",,1oo,0m,do=:,2~0.m:.

Current,,'cb'ang~' ~atio

^(U-Vol/Co

~s

^currentspeed

~: tio

Vo/Coforplane~ave.pararneter9ao~0.75·m.'Lo0:=100':6 m,

do";'25.0m. ' ~ . .

Fig.4.4 Depthdiani:ra't)o(d -'do)/v ?>

cu tTe~t ~peed

^{;a tid ('}

'f., I ·Uo1.Coforplen ewevep4rame're'rs00= 0.75 m,Lo=:lO(}.bm,, d..=25.0m. .

.'Fig.5:1 Maximum.:articlevelocity4istri!>uti~n^alo.ng~a.ter^depth•

.forVo/C o_-0.15. \. . ~

1\ . ' ,' .

Maximumpar ticlevelocity ·distribu tion alongwater dept h, forVo/Co=:-0.10.

!. •

Fig:5.3 Maximump'article'!VeIO'Citydistributionalon't.wa.terdep th;',

fo~Vo/Co=: -0.05. , •

"

--

Fig 54 Mexim usn.p eet iclevelocitydistri b ut ionalong waterdepth.

,forVolCo

=

^{0.00 . •}

F.ig.5.5 Maximumparticlevelocitydistributionalong waterdeptl~

forVo/C o⁼0.05.. .

Fig.5.6

Fig.. 5.7

~;g.• .9

,.

Fig .5.10

."..:.';

Maximumparticl~v10citydistributionalong water depth,

forV.I C. ""0.10.' . •

Maximu m particl evelocitydist ri bution alongwa terdepth, forV.le.~0.15 . ..

MflXimumparticl e accelcration -distributi o o along water depth,forV.IC.=-0.15 . .

MaxilUum .particle acceler a tion distri bu tio q alongwater depth,forV.IC.=-0.10.

Maximumparti~leacceler a tion'distri but io n alongw-ater .

depth,for

V .IC..

^{=-0.05. .}

Fig~5.11

t. :

Maximumparticle acceleration dist ribut io n along water

depth ,rorV.IC.=0.00. ' ~.'. _ .

Fig .5.1,2"Max imull\partiCIeacceler ationdistrib utionalongwater de,Ptb,~orV./(:•.=0.05. ' \ '

Fig.5.13 Maxi mum particl eacceleration

distributio~ a1o~g

^water

depth, for.U.I C.

=

0.10.

Fig.5.14 Maximumparticl eacceleration distributionalong water depth.r?r

u./e.

=0.15.

fig.

^5.15 Tra jectori es of partic'j;atz/ d=-G.2inth~wave-current

~ld.,forU.IC.=~O.02. .

Fig.5.16;I'rajector ies ofparticle atz/d""·0.2in the wave-current field,IcrU. /C..=-0.01.

::.-.l·., _'::

J

Fig.5.17

Fig.5.18

Fig.6.1

Fig.6.1

Fig.6~2

·J i g .6.3

Fig.6.4

Fig.6,5

Fig.6.6

Trajeetories"ofparticle.atz/d=-0.2in the. wave-current field,forUo/Co=0.01.

Trajectoriesof particle at;J.s;(=-0.2in thewave-cu rrent field;forUo/Co=0.02. .

Schematicsection of thewind-wave-current£lumin Hydraul icsLab o ra toryusedin Thomas experiments.

Comparisonbetween thepresenttheore ticalpred ictionsend . Thomas (1981) experimentalresults forwaveleng t hchanges.

Comparisonbetween thepresertt

theoretic~ predicti~ns and

Thom~(198 1)expe riment al results for wave heightchang es.

~~i:::~~~;~e:x~::t~:~~:~:~s:i~t~:~th:~~~~~:

horizon t al parti de velocity distributionalong water• depth,torUO,,;;-59.7 mm/B, , Co~pariJlon^between'the presenttheoreticalpred ictions and Thomas(1 981)experimen talresultsforthewavelike horizont alparticlevelocity distrib utionalongwater depth.forU,:='-116.2mm/s.

Comparisonbetween the present theoreticalpredictions and Thomas(1981).experim entalr~sult8fox the waveiike horizontal partic levelocity distribu tion along water depth,forUo

=

-159.8mm/ s.

Compariso nbetweenthe presenttheoret ical predictions andTho mas(1981 ) experimental rcsultsforthe wavelike .

~:~~t~.n;::1t:c~e2~;~~c:m~:tri~l!tio.n·al~ng

^{wa ter}

"

'; "".

Lis t of Tables

.::'

Tab.6.1 Comparisonbetweenthemeasured (T homas1981)and th e predicted.(present theory)wavelengthchllI1ges.

Tab . 6.2 CompwliOI1 betweenthe measured (T homas1981 )and thepredicted (prese ntth eory ) wave heightchanges.

Ta.b.6.3 Predictedchanges in currentspeed and mean waterdepth..

Tab.DJ Numerical resultsof wave lengthratioLIL"

Tab.D.2 Numer'i~al r,es~lt8 ofwa~e^heightratio... HIH"

Tab.

·0.3

,Nu merical results ofcurren t velocity chan ge ratio

(U7'"U.)X1O~4 • '

Tab.D.4 Numerical resultsof water depthchange ratio (d-d. ) x10-⁴

- ..

xii

,---... Chap}er 1

Introduction

. . ⁾

\

Accountin gfo.r thea~tionof environmentalloadsonstructu~e8i,s I\"typical tll{~^forc~astal^{andoceanengineers.}TJie theorie s for evaluating the loads.

in:duced"by~1I.~es,wi~dand~un.::nthave~lat.ivelY'b~enweit,developed . Ho~eve~,forthat aTtheecmblued effect

o f

waveand currenthavenot.At present, mos t offshore design ofdrillingplatformsend oilBtor.ag~tanksis dorie on the d csJgn wave concept,tha t~8,the forcespredictedon thestruc- tureareassocia ted with themaximum waveto beexperiencedbythe str uc- ture during'itslifetime .Quiteoften,th e presenceofocean ic or wind-driven clfrr ents isneglected'inthe design,o'rconsidere dsepara tely fromth~waves, while rare ly wouldthis quiescentcondition exist . In (ad ,thepresenceof ( cu r rent. willsignificantly influencethewave

8.ctio~

^on

st,rUl:;tu~elJ.

^Two

examples~ementionedhere,showingthesign ificant magnitudechanges of thewa~eandwave forceby thepresenceof acurren t .S~hu~ann(1974, 1975)reportsthatin theAgulhl;lAcurr entnearSou thAfrica whenthe fully

I

.developedscawaves~th^{heighhH- m}m~t~iththis currentthe result~. may begiant wavee'wit hwaveheightof18 m oreven more.R.A.Dalrymple (1~73)~~epo;lI^{th at}if th e' maximumhori~^{vd ocitydueto}the design·

waveis16£ps, thent~·prnen~of a 2fpscurrentwould increase thedrag a forceonthestruct ure by over 26%'.evenwhenthe currentvelocity issmall relat ivetothepa.rt~levelocity inducedby thewave motio n.Itillobvious, then,thatraUan a!oft'~hore'design must include the -effect ofcur~ents.

Evaluatingthe combi ned wave-current

r orce z d

nda on.thewellun-.

\ . \ ',

ders t an din g ort,~ewave-curr entinteraction,n Iy theuuders jan dingQ{

the.chenges of the waveand the~rren~,after thein~eractionan~them~el.

ing of thecombinedwave-cunent"fieldfOr1n~'bythe wave~dthecurr ent interact ion.,_~ Th eOries.for_. IIm~p~-tude_. -... w~vesinter actin,;^\ with. a~ur.·

.rentha....:eb~developed. see.forexamp~.Longuet-Hiumsand Stewart

(1960,1,961),Dalrymple(1973).Inthese theories the c.ha.ogell inthe'cur-

rent andthe.mean ,",ter depth inducedbyth~inte~actionare neglected.It isthe objec tiveof'thistbelluto~vestigatehigherorderwavesinteraction withacurrent.to model the bigbetorderweve-curreetfieldt.nd to findthe

" . ....

.changesbothin tbe wa veandin the eurrept,as well as in themean water

depth e:.ftertheinter~tion.

Int~is.thesis.t he interacti.onof twodimenslo.nalfiniteamplit udew~ves

and a uniform cu rrentis studied; i.e.finiteamplitudeperi odicalwav es on still water

p~paga't.ing Tn~~

^a

:-va~e-free.

iuotationalcurrent in the sameo"r

'::.

",

. .

,' ^...::.' ^'.j.

opposite direct.ionof wave prof\agation,isconsid ered.. The processoCthe interactionbetween thewaveandthe currentinthis situ ationisassumedto bedlvidedinto threesfuges.Instege'(1)the waveand thecurren tfieldMe

assumed to existseparately. Instage (2)waveandcurr ent encountereach other and theint eractio n betweenG the .waveandthee~rr~il;takee.plece.

This st ageisan unstab leone, since thewaveand curre ntcharacteristics change

~Hh

time. After the

i~terflC~i~n,

^{in'stage(3)"a}

8i~ble,

^{uniform;irroo}

tational.wave-cu rrentfield is'assumedto be formed.Itis forthiscombined wave-cu rrent-Hel dthat..the potential{unction,d~spersion'relation, particle kinema:tics,andpressure distribu tion uptothethi~.1orderar~developed.' The correspond ingphysicalsit uatio nand' possib leexperiment alset:ttpis

~ . .,' "

.

welldescribedin~home.s(1981)and-cit ed'in Chapter'6~{thi~ the~is.Th~

results of thefirstorder (see~.1of,thisthesis)developedb¥the approach"

ofthis thesis qualitati'velycompare well withthatof previousstudies,.!Iuch

.. "

^{" " .}

&.9Longuet-lti~nslandStewart (1961).Furlhermore,thisapproachcould

also be used to

.

fimi"the atreemiunctionsoltitionofa weveend shearcurrent field(S. Song endR..E.B..eddour 1987,R.E.Beddoran Song1088).

Therelationship

..

bet ween

.

thecurrent-freewa ve"'.lengt L.,."waveheight, H~,'wave-freecurrentspeedU".mean we t erd eptbd.befrethe interactio n andtheirtountez:t;afts,'denoted without_. subscrip t"_. Ii" ofthe com'bined_. wave-cu~rentfield'isest ab lished by using the:Iun d e menalcons~rvat~nr~

';lations for therrieen rateofm~s,momentumande,ngytransfer for the ,

consideredfluidflows. Conservation of wave crestsis also assurrred,which impliesthat the wave.periodT remains~onstant.The energy exchange

I .

betweenthewave and thecurre ntistake ninto account. Numerical results

' j

Jorthe variationinLI Lo,

JIll..,

6U/Uo!,.and6.d/d~arepresented.This is a new method for 80lvjnga welldocumented problem, Using this method

, .

^.

.notonlythech angesin the current endinthe water depthcould~eealcu- lated,butmore importantly" thecalculationof Ehee;ho.ng es in wave height and wavelengthis done by consideringthe changes inth-ecurr~ntand.in the depth,I'\hich~ho~ldbe,consideredwh'~consideringfinite.amplitude waves Irrteracjiorrwith'a current ..

Complltlsonbetweenthe experimental results of 'Ibomea(1981) andth e

res~lt~~ ~(th~ ~resent :theo~,.

^{for the}

wa.v~' le~&th ~~i~h.t r~tioJ' L i

£0 .'and.HIH..respecrlvely, andthewave particle velocity dis tributionare also

presente d , goodagreeme ntis generallyobserved.

Thethesisis organized)nto seven'~apters, fourappendice-s, alist ofreferences md

a

^{list of}furthe r readings,'FolloWingthis

i~troduction,

Chapter2isalit erature review on thewaye-eutre~tinteractionproblem, Chap~r3presen t sthe velocitypotenti~(unetio;, disperaioa.relationand

~ I . .' .. '

otherpropertiesofthe combined wave-currentfieldupto,tne third order in wave amplitude.Thesep~oper~iesarewr ittenip.~erFs.ofthe waveheigh~

H,wavele~gth

i;

~aterdepthd,and current speedUofthe combined".., w~~e1Jrre.~tf1.eId.In et;.apter4,th e relationshipsbetweenH,L,{i,V, lUld

: ....-,.' >

.

,

....

H.,L" . d., Uoareest a blishedoyusing the mean

m~s. m~~entum.

en- ergy conservation relati o ns,.lIS wellas.the dispersionrelation.Anumqikal methodis used tocalcul ateH. L,d,Uwhengiventhe values ofHOIL~,dOl andU", .Chap ter5con tains,the predictionofthecom b ined wave-current fieldpropertiesandtheresults~ftheircomputationfor arange ofval~e8 ofthe current

.

ratioU,,/C•.A negativevalue

.

for

." v.le"

indicates

-

B.curr en t opposin gthe'current-freewavepropagati on."Inchapter6,B.comparison between the expc;rimental;esults'ofTh om es(i9 S-1) an d theresultsof the

"p reeen e theoryis presented. Ch apter,7cOllf ain sconclusionsend eugges-:.

ucnef~rfU~Urework..AppendixAcontainsa discussion

on

the radiation stress

and

theene; gy

t~an~rer

ⁱⁿ

ih~

^combined

wave-current'fiel~.~Nu~er.

ical~'..~si?eratio~9^arediscussedjnApp endixB.Th~cOtl'!pulerpro~ra.m9

develop edfor thenumerical sol uti on of thf;nonli nearsystem ofequatlone , ..

~d

wa: e+currentfieldpropertiescalcula timl

tU'~

^listed

i~·Ap;endix.C-:-AP:~·---

^.:

-.pendix D givessomeofthenumericalresultson theehan geaorwavelength,

.waveheight,current sp eedand thewater depth.

.-

.. ...

~'itO- '..

Chapter 2 ·

Literature Review

..

The effects

0( ^.

a followingoropposing uniform currentonthe propagation of surfac'7

gra~ity

^Waveswere

fir~t~

discussed by

~n.ra

^{(1942).andSverdrup}

(1D44) andtheir

inve~tigations

w,:re

'basic~lY

on

fi~ding

thechanges in wave length andweveheight kinematic ally.

Johns~n

⁽¹⁹⁴⁷⁾found the effectson waves which enteraunifonn current at an

~e

^{end he}

,

sugges te d that

.

..

majoroceancurrent,suchasthe GulfStream,mayhave~app reciableef- feet onthe height,1entth,anddirection of waves approachingthe shoreand undersomecircumst an ces may causealmost completereflection .~Arthur (1950)investigatedtheco~~inedeffectof'nonuniformcurrentandbot t om topographyofshallow waterwavesand made an applicationt~wavesen- - terin.g~.intenseripc~rrent.Sarpkaya (1957) investiga t ed experimentally the height and length changes whenoscillatory gravitywaves propsgated intoflowing water.Ts8.0(1959)studiedthe wave interactionwitheshear current .

I'

...:,'

Si~ee 1960;tlu~_~p edsofthe in t er ac tio nbe tween gravit ywaves~da currentmotionhave~ceivedincreeai rigattention,fover il:lg awidespectrum ofproblem srM gin g",fro m studieson the co.r:nbinedw~ve- currentfield to

&a.ngCIIin wavea.a:ap~itud~and wave lengt h{ etc..'l'h~wo~ksof D,a1rym p le (19'13,1975),Longue~.Higgin8end SteWart(1960,196 1 ) ,.Whith~(1~62), Peregrine (1976),JOf\8on andSkovgall.l"~(1 9~8),.Bre~_i.kandAu(198 0), 'I'hom es (198 1)ton~e b~tafew,are already.elassic~.':themech anism is intim ately.co lUleete dwitl} theso-calledrad iationstr es s [Lcn gu et- Hlggi ne '..

and

Stew~ I~O, 1 964 1 '""

^action

(Breth~rton

^{and(;;a rret1960,}

JO~88on

1978 ,~ra~per1984 )Ill.!wellas meanenergy level(Jon~son1978,and Jon s so n, Br-ink-Kjaeeand Th?m u1978).

Inth\s chap ter-t he beist in gtheOriesarereviewed oyelass ify ing them I

und erthe following three topics:

1.Two~mensionalwave s~dunifbrmcUI'rc~tin te raction;

e.

Interactionofwavesand uniform currentat anangle;

~

,

.

3. Two dimensionalwavesand she ar current interaction.

2.1 Two'DimensionalWavesand Uniform Current Interaction.

..I

Wave inter actionwit h aunifo~current inthe~ameor op posi te directi on , i'.e.twodirnenaional-I'nterac tio n,is atypical probleminthe areaofwav~

andcu rren t interaction.Thepurpose of theexisting theor ies is ms.inly

-

^.

·-" '·'7

^':

··'··

,::· ::"":2

··\, ··' ·~·

.

^{', ' ,'}

I ,

,/

/

todevel~me thods,fordetermin!ngthechangesi~w&\.en&thandwave/

heightafterthewaveisaffectedby.th.ecu rrent.Unn.(194 2)el5tabli'hed,~

methodto

determi~e

the changesin

~ve

^{lengthand'waveheight by}

uI~g

. . ~. I

the dispersionrelationandtheenergyconservation equationgiven

b y

the

following two .qoatio", . ' • / .

c=c,+u. (2.1)

.ec; = E(C,;:

Uo) .~)

wherec.o ».eandUare-respectively

th e

wave ee.lerilY:'group velocity,. waveene~~^{density,andmean}current speed;

with

thesubscript"o"'ln di·

eating the parameterbefo rethew~\'etrai nand currentmeetandsub.script

"r^Windicating.th~relativevalueoftheparameter.

.Byfurther assumingthatCp==(* tanli 1f~t"&n9.E==lpgH~.:~th

ddenot i ngthe depth and

If

the'wave height.Thechanges in wave-length and wa veheiFht were given&oS

;

..

^-

i;

⁼⁽

¥;tMh·¥-d+U . ;/(J~tanh'¥;d) ,.

.. .

s. s ,

=l - !Q2C.

(2.3 )

(2.4)

(.

.I'.:

Longuet- HigglneandStewart (1960,1061,HJ64) showed tha tthere WAS 110couplingb~tw.eenthewaves andthe curre nt consideredinUnn~:9

, ·8.

r

."".'

...method.They foundthechangein,waveheightclynl'ln~allyby consider- - '''-9"""

d in g this coupling.For small amplitudewaveand current in teract ion ,the

B" '\ .:

ch an geinwaveheightwas givenas

f2.5) Some specialsituationshave.also been studiedsucheatheint~raction ofsmallamplitudewaveswithsurface-currents(Thomsonand West 1975), wave andcurrent int\lra.ctionsi.: shallowwater (Yecn end~ilipH186) etc.Brevik and Aas (1980)reviewandstudyexperimental lythe amplitude variation whenperlodic, waves,in it iallyonstillwater,propagatei~to11.,

knowncurrent ted frombelow. They identifya.set downor themean ,D^watersurfacebecauseof the' current, but the additionalcontribu~ionto the set.down because of thewavesis, asthe~ut it,·completely.negli~ible^• ...Theproblemof waves propagatingthro~aknownslowly-varyin g , depth independenthorizontalcurrent is also reviewed byCraik(1985).Ba ddour and Song (1988),invest ig a t ed the problembyconsidering thechangesin thecurrent as wellasint~ewater-depth,

Some'experimentsfor specialsit ¥ationsha7ealso been reportedbySerp- kaya (1957), HughesandStewart(1961),Huang, Chen andTung(1072), Br evik and Aee (1980a,198 0b),KempandSimons(1982). G. P.Thomas / (1981) stud ied the problemnumericallyand experimentally,and prcsentecl- results onwave height and wavelength , andthe we...cparticle velocity

J l

distr ibution in thewave-currentfield,

The obje"ctiveoftheprese ntstudy istoiD* \iga i et~e^{interactionbe-c..}

tweensteeperwaveandaun~cur rent and to develop a metho dof calculatingthechangesboth in,the wave~dcurrentaswellasinthe meamwater depth.

2.2 Int.eraction of Waves and Uniform Cur- rent at an Angle .

Whenwavesmovingthrough stillwaterencounteracurrentatanangle with thewavedirec tion,thewavesarerefracted ,undergoingchangesin length ,steep n ess and

dir~ction ,

^{Johnso n(1947)}

djs~u8sed

^{thisp; oblem by}

usingraytheory. Th edispersionrela.t ionWlI.8eseumecltobe,

S- ~ u. +s...

^(2.6)-'

smc sm/3

where ais theinitialangle between the inconiing wa ve and curren tdi', recttcne:fJistheangl e betweentb,e waveand'curren tdirectio ns~terthe interacti on,

Fromtheray theo ryit isknown that "-,

L L.

sin'"=sinO' (2.7)

Thusthe~,8.IlgCllinwavelen g t h,waveheight, andin the anglecouldbe fou,nd , Theyaregivenbythefo llowin g equations

10

"

^I

"II

\

)

,

":

. .

/

(2.8)

(2.9)

-.

~

..J .(2.10)

:,hereM==U-,,/Co.

SiInilady,

'Jom~~

^{andSkovgeerd(1978)l:lsing"thesame}thoo l'; stu die d the,problem

~~

^w:Ves

prOPll.~ating

^{frQIDcurren t'(hto.'}

cu~rent U ,

^{with M,}

.

.

angle.Th~re8,!111.1weregivenas

" -- . .

(2.11)

./

where

G,

=

2!( ld .sinh 2K1d . • 2/(Jd

c,

=:sinh2K~d"

-'.

(2.121

.'C2.l4)

...

"

...

wit~subscript s,1and2de~oteth e parameter!.rel~te.J^{toU1'and[;,resp ee-'}

tively. ). »

Longuet-HigginaandStew&lit(1060) also discussedthj~~roblem^I\San ext~nsionoftheirtheory.

11

,-.:.

\

2.3 Waves an d Shear Current Interat io n

Waveinteraction with a shear current is.:more

~pca.ted

problem, be- cause ofthencnpotentiel propertiesoftheshear~nt.Therefore,investi- gation of this probleminvolves considerationofthestre~functioninstead of thepotentialfunction,andonIrtwo dimensionalsituationhllBbeenstud- ied.Th~^{existingtheorieson thistopicmainly}coricentr~ted^{onfindi;' g}th~^. ,solutionofthe combinedwe ve-eheercurrentfield instead of finding the

changesinthewavelengthandwave height.The beh avior ofsurfacewaves on alinear ly~yingcurrent wasfirstworked-out byTllIO(195~).^He~a~e

.the stream function~dthe~ispersion·rel.ationof the'wave-lin~1U'current

field. Delrym ple P973)studied theinteract ionofwaves witha bilinear. eurreet,arid.~adean attempt toanalyse numericallywavesand-nonlinear cu~tinte ract ions;~n~andBaddour(1987)~addourand Song(1988b) investigated theproblem of~avesintera ctionwitha linearcurre nttothe second

o"~~.

^{Some.:ork has}alsobeen done

on:~re&ting

special situations.

Longue t-Higgins(1961)st udi edthe problemofwaves and current which. variesgradu~lyin thedirect ion~fwave prcpegeticn .Freeman andJohn-• son(1970)investig~tedthe~"ituation^of8hall~^{waterwaYell' on}shearflo~s.

Dennis (1973)enelreedthep~obl~mofnon-linear grav ity-capillarysurface

waves in a slowly varying current.

·.1

'""

..

.:

i

\.

\ .1 ·

,/": I~.- ~. -

.12

Chapter 3

Combined Wave-current Field

Aa.d iscussedin theintroduct ion,the processof the interact ion,betweena wavetrain and a current inthe same01'oppositedirectionofwave prope- gati.~~isassumedt~'rol1~wthree8ta~es.·InBt~geone orthe stageb~roreI, theinterac~ion^thecurrent-freewavesand wave-free curreqt areassumed toeiist.in s"taget~.waves propagate into thecurrent,~d^{hence thein-} .terectiontakesplace. The waves andthe currentproperties;as well asthe

"

. .

water depthkeepchanginguntil finally a stable"combinedwave-current field isassumed to beformed, whichishere'~alledthl;! thirdstiige.or the 'stage after theinter acti on'It is forthis combinedwnve-cu rrentfieldthat

the:potentialfunct~on.^dispe;sionrelation and.o therpropertie~^to,thethi~d .?rder in wave a:mplitudeare developedinthisch~pter.

The following assum ptionsare made:

1.The combinedwave-current field is a two dimensional, lrrc retlonal

( ,

13

2.The fluidis assumedto be invlscid,incompressibleand homogeneous"

3. TheSea bottom is assumed to behorizontaland impermeable.

Since thecombined wave-currentfieldis assumed to be irrotaeional,

a:

po- ⁽¹⁰ tential functionexistsand satisfiesLapl ace's equation,that is

(3.1) everywhereinthe fluiddomain where~_denotesthe potential function of

"

the combined wave-currentfield.

'"'Adop tin"g a system

~fcoor~inates,

with a verticaliy upwards,:c

hbr~.

h,~,'andth~origin onthe Undisturbed fr ee surface,·theb~tto~boundary

<liz=0 %-=-d

wheredis thenie~const":ll't water depthof the wave-current -1ield and subscriptsdenote partial differentiation.

The kinematicfreesurfaceboundary coedirionis

%='1(x,t)

( :rw

where'1is theprofile of the surfaceelevat ionof the combine dwave-current

field. ' "/ .. " (>

The'~icfree surfaceboundaryco~di.tiorican be expressedM 14

r

"

.~.".,I

for some fqnctionF(t)tobe determined.

I:,.:sreasonable to assume thatthe function'1(z,Odeseribln gth~Cree surface elevationoftheco~bined'wave-curr~tfield is periodicandex- pressedintheform

'>( 3

.s)

~somecons~antsli....to~e.evaluated,and wheretc==2trIListhewave number. 8.I!d^(J'is the frequency oft~,eperiodi calmoti~n.Tisthe period .hencefixing(J'all211'IT. "

From theperiodi~tyof~.i.tfollowsthat,~(z. z.t)could

b e

assumed of'. the form

<t-Ao{z):r+

~A

..{,;r).in n{K%-qt)

·~(~.6)

forthe horizontal bottom case.where

A.. .

n:0.l.2...•are functionaof

~only. Keepiag one•t...o, three termsof tjl e sene.intheexpansione

i::'

(3.5)and (3:6)areherecalled first,second or thirdorderapproximat iona.

,I

respectively.

I'

3.1 F ir st order ap p roximatio n

. ' c'

3.1.1 Pot entialfun ct ionanddispersion rel a t ion

FollQ':;rri,t;-'~I8SBica1

^{approach(see,forexample,}

L~b

1915) equations

\

(3.3)and(3.4) areexp~ded.inaTaylor serie~.abotite

=

O. Keeping onlythe first orderterms ,yi~ld8thekinemati c andthe. dynamicsurface con'ditioos

(3.7)

nnd

I" . •

. i.

+~,+ 2[(~')'+(~.)'J-F(t)·~O (3.8) tobe sati sfied onz

=

0

I Tothefirst order ofap proxim a tion,thefreesurfaceelevation(3.5)and the poten tial funct ion (3.6)arewrittenas

~d

r(=acos(l~,r-<7t ) .(3.9)

wherea

=

H/2 is the amplitudeof the wave'on thesu rface oft~e combin~d field,Ao(~)andA~z)are functions ofz only.

'\

16

)

( F>;.

Bysubst i,tuting'equation(3.10)ineq~ation^(3.1)and by satisfyingthe bott omboun dary condition(3,2),Ao(~)andAt (.:).':lUi'be found in the.

form

wh ereBoendB~^areconst an ts to be determine d.Equat ion(3:10) hence becomes-

Ao(z)=Bo (3.11)

(3.12).

" .

To determineBl,equations(3.9)-and{3.13)are~ubstitutedinto the free surfacecond it i6n(3.8)and keeping terms upto thenrat orderin0yields

I

BI

!'

si~h I(d!liAl(I(:t-ut)- -ousin(K.:z:"'"vt )+

/ ^.-

, "

+Boa](sin(I(x-: (Tt)=0

Fromequation(3.14)

1!f

is .foun d

to

^be

Ii':' Sin~I(iC -

^{B o)}

17

(3.14)

(3.1')

<.

whereC=(T/Kis thctab:solutevelocity of thesu rface disturbanceof the wave-currentfield.Thepotentialfunction41ishence offheform

4>

=

u«+a(C-

U)cos~~(;<; d)~sin(K!

^{-(Tt) (3.16)}

.~^{.., wh-;;re}U

:= s,

^is

'4F

aperiodiealpartof thecombinedwav~~t^motion.

Th e dispersion rela tionof.f.l;I.esurfacewaveisf~~ndbyassuminglJP/lJz= 0,on z=,'1.Differep.tiat in g par tia.!lywithres~cttoz, equation(3.8) gives

;J hq

'·a'~.

¹

a Q~~ ' 8~ ,

• .; 9a; +ar8t + :;ih !(ihJ' +(a;)]=0 (3.17 ) SuBUituting

~ua~ions·('3.~)'~~

(3.16) into(3.17)

~d

kee:tgthe first o~derterms'Yields

)

-gal<sin([(;i-O't)+

r+a~,:,," U)K.~ C'Oth Kd sin(Kr - O'~t)-• -Ua(C~ U)K2 coth ~d sin(Kz-O't) =0

...

^(3.18)

(3.19)-, Fromequation(3.18)tbe'followi~gquadrat ic,equa.tionCaDI' o~tained

k . , .

^.

.(C:" U )2 - f t an bKd =0

- .

Solving.equation(3.19) yieldsthedispersionrelati onof the combined Wave-currentfieldtobe

18

. ...

.:'".'".".."

. '

whereC~=[(g/J<) t anhJ(dJl/2=17,./^J(isthe celeri ty relativ eto thecurrent streamUand17.is the relat iverfuquency.Equatio n(3.20) is knownas the Doppl errelat ion .

F(t) .in~ernoulli'sequation (3.8)isfound ,by.subst itu.ting:or 11 andt i~equation~3,8),.~

c= u±c,

F(t)~~U'

(3.20)

(3.21)

3.1.2 Pro pert iesofthe firs~orderwav;cur re nt field'

Th e'p~opertiesofthe combined wave-curr ent fieldto thefirs torder

or

^ap-

proximationcan bederived from thefirs~orderpot~ntia1functionend-the, dispers ion rela ti on.Theyare listed.below.

1).The velocity potential is given by (3.16)

".

w=Ux + a(C._ V)cos~~~(;(;d) siQ(ICr _ D't)

^{_(3. 22)}

2).Th~.disper sion rela tionwas. found tobe

10

"

._.v"

where

,ando-isthe relative wavefrequency.

(3.24)

3). Particlevelocity A,

.The x

and

zcomponents~f^{particlevelocity}V

=

'Vel>i~·the^combined

fieldar~respectivelygiven by

4).Particleacceleration

Thexand z components ofparticle acceleration

DV . a - -

ii=m=(fjt+V.V)V in thecombined field could be obtained inthe form

a"

=

ga](COS~~(;(~^{d)sin(J( x _}

u ;)

20

\

5).Part iclepath

To find the particletr.ajectoryinthe wave-currentfield,~quations(3.25) and(3.26)at a certain point (Xi,Z;)areintegrated- withrespecttotimet toyiel~

'\sinh f( z;+d) ,

Z=Zi+a sirih I(d COS(I(Xi - Urt ) (3.31) Combining equations (3.30)end(3.3i')thepathoCtheparticlecanalso bewritte nas

where

(Z-Zi)a (:t-%;-Ul)'

~+

- - p- '- -

=1

-i -:

Q_ asinhf((zi+^d)

- sinhKd

{j==acoshf((z;

+

d) sin~l(d .

(3.32)

(3.33)

(3.34)

6).Pressuredistri~ution

Thepressure distribution ina.pc tentialflow is givenbyBern oulli's·

eq ua tion in'theCorm

21

r " ' . " "

P' ;-pg z

_p~._ ~(~)1 + (~)1~ +PF(t) (3.3~)

Substi tut in gF(t)and~from equations(3.21) and(3 .22)in(3.35), the

\ " "

pressu re distri butionin acombined wave-currentfield td' thefirst orderof approximation~uldth en be,obtainedin theform

p

~ -pgz+ psi;~iC ':- ,!)2eosbK(Z

^{+d}co.s<Kz-O't) (3.36)}

3.2 Second order approximaflo"n

3.2.1 Potentialfunction and dispersionrelat ion

Keepingthe secondorde r termsin expressions(3.5)and(3. 6)for

the

^{surfa ce}

elevationandthepoten tial fund.ion,gives thesecond app roximation to the wave-curr entfieldas

and•

of»=.~o{z)%

+

AI(z)~in(Kr-o f)

+

A2(Z)si,e2(Kr-ot) (3.38 )

\.

22

Substitutin~uB.tion(3.38) into thegoverninge~uation(3.1)the fol- lowingsystem ofordinarydifferentinl equationscouldbeobtained

",.

,

A~(z)

=

0 (3.30) _

(3.40)

(3.41)•

.~

I

Equations(3.39),(3~40)and (3.41)desc~ibe^II.system ofo~'dinarydiffer- entialequations, to be solved for Ao(z) ,A1(z )andA~(z) .A prime denotes diffrrentiation with respecttoe,

'The bottombo~ndarycondit ion(3.2) implies that

A;,( , )~O z=-d (3.42)

A;(z )=0 z =- d (3.43)

and

I

A;(z)= O z=-d (3.44)

which on solvingequations(3.39),( 3.40) and(3.41),su~cstthe for?, ofthe function sAo(z ),Adz ):A~(z)for anyconstan t/(,as

23

/.

(345

1

A,( z )=B,cosh2K(z

+

d)

.(3.46)

.)3.4 7) {orsom~^constantsHo, HI. B, tobedetermined. H:ncefor mally~^is,to. second order,written intheform

'2'>

=

Box+B.

COSh ; z r +

^d)sin(Kx-nt)

+

"-./' .

+B2cosh2K( +d)sin 2(lf x-ut ) (3.4~) Nowexpanding equation(3.3)ina Taylor's series ebout .e

=

0and keepinz;terms uptothe secondorder.yieldsthesecondord er kinematic surfacecon dition

. ^,

~ - ~ - ~~ ⁺ ~:;~ - :::z~~= O

tobesatis fied onz

= O .

\

(JA9) )

Suhs tit utingequations(3.37)and(3,48)for '1and~,in equ ation(3A9) yields

i

and

+02B \](2cosh[(d -4\ 8,](2c'?5h2I(d+

+2aIB2I(~ cos~-2'i(

^d,-

~a2BIJ(2

poshJ\"d-

-;:~a:BI'1(3'sinh I<'d

^:::0

2B2K sinh2I(d-202(1

+

2?2K Bo

+ A-~aIK2~1

^{cosh Kd'+}

~a'IBIJ~2 Ccish I(d

^:::0.

(3.50) L:-.:-.

C.

".-:' '.

(3.51)

- ,

. .

On solvingequa tio ns(3.50).and(3.51)thecoefficientsBIand,8 , are determi ne d tob~

B

I=41(C- Bo) • sinh KJ

8,

= ( ?-

B('d) [oZ-

.~a:I\

^{c:oth ; {df}

, smh 21 . .2 . ' .(3.53)

.. ...

! '

. .

. For thes~ereasonasfor thefirst ord: r, setting...B^Q

·=

Uan,dsubsti- tuting(3.52 ) and(3.53) backin equation{3A 8)the potentialfunction to

. . ,

the secondord er is expres sed as

25

' -- -

al(C -U) • . .

1'= Uz+ si nh J( d coshl(.t+d).sm(l(z-D't ) +

-cosh 2K (z+d)sin2( Kz -0'1)

..

- ^.(3.04)

-

\

...

Alsoe~rufdingequation{3.4}in aTaylor 's series~lit.z

=

~.toth8::'"

~econd orde~eldS

the dynamic freesurfaceconditio n inthe form

-,

I

,,+

~

_8t⁺

!I(~)'

_2...

ar .

⁺

(~)'I

8z

-

^F(t)+

. .

\ ~'lP40 M.a²⁴⁰ 8~a2<1> =0. (3 ) Jj.'tJta:rr+

a;

^8z.8z'1

+

£h.a;2'1 .55 to besatisfied on z=.O.

Oifferentiatin g"e<tualion"(3;5S)partiallywith respecttox t;ives.

,

"

. .

^:

.26

'; -,

:\.

I' ^, .

+?!

_o~

8'~ ~ _{oz' o::r;,}

^{=0 (3:56)}

.Onsubstitu ting equation(~.37)and(3.54) intoequation(3.56) thedis- persionrelationand thesecond

~rder

term amplitude

a,

could befo:unV 6 be

'.

I~

\

,- .,'"'..:

-:

,

where

c=

U±C,

\

c.

=

( 7<

^tanh[(d)I/2

(M 7)

(3.58)

v

wher eH

=

2Ulisthe waveheightforthe firstorderv.:ave amplitud e0\.

ltis worth noting that the disper sionrela t ion ofthesecondord erap- proxi mat ionisthe sa meallthefirstorderone .This is in agreementwith theStokes'ssecond,de rwave\heoryf~rthe special case whenthe current

speedUiszero. ./

Bernoulli'sequa tioncons.tantP(t )is foun d to be

(3.60)

3.2.2 Propertie softhe secondorderwave-curr ent field

For conveni enc e thepropertiesofthesecon dorder wav e currentfieldare listedbelow.Inthefollowinga, wit houtesubsc riptI,isused to denote the firs t orderamplit ude.

1).Potentialfunct ionis

<fl

=

Ux

+

a(e_U)cos~~~~(;(;^{d)sin(I{x _ut)}+ +[0.1-~a1J(cothI< d](C ':"" U)·

.cos:~~(~;{; d).,sin 2(I(Z

^{_O't) (31B)}

28

where

£ -,

^{a ' = -:rraL2(1+- - , -2$mh3/\ d.-)<othKd (3.62)}

¹

and a

=

^8/2,His thewave height of thecomb inedwe ve-currentji eld . l~ 2)Freesurfacepro~il

'1=dcos(Kz-.(Tt)

+

aJcos2(I(z-at) (3.63 ) (

3).Dispersionrelation~givenby

and

4).Partkle.velocity

c= u±c.

,

c.::::

(k t an h'/l d) I/ J

(3.64)

(3.65).

Thecom po n entsofparticle velocit yMepvenby goKcosbl« :-

+

d) . U

+ -;:-

coshKd cos(I(z~at)+ .+

4~~~~;~d c~h2I((Z

+d)cos2(l\x -at) (3.06)

29-

•~l.·

gaK,sinhK(z

It!) .

(l( )

~~sJn z-ut+

+ ti:1~;~d

^{'sinh2J(z+d)sin2(/( x- ut) (3.67 )}

.

..

.where therelativeIrequencyisdefinedby

.

^{-'" (3.68)}

t,

5).Parti~leacceleration '\ . " .

Thez

and

^zcomponentsofparticle accelerationare, tosecondorder.

giveb

oy

,,,... J

Q 6

= g~l~c09~~(;(;~)

^{sin(l( z-at) +}

3Q2J(u2

+(2 sinh4

K d

^cosh21((z+

cf1 -

.-

Si~:~dls1n2(l\Z

^{-at) (3.69)}

ga2

K2Ii~i:~~~~

^d)_

_

gaKsi~~~i)

^{cos(Kxrut )-}

- ~ ~2~q:d Si~h21«z + d) JOS 2(l(x- ut)

^(3.70)

28mhJ(

I .

,

30

6).Partide path

Thetrajelttory ofa particle,initiallylocatedat(Zi,:",)in'thec:o;nbined

.

,

wave-current field,is des c ribed to secondorderbytheparametric eqcencne

(3.71)

(3.72)

7)..Pr essuredistr ibuti on

;Thepressureclistribu t .ion inthe comb ined wave-currentfieldis, to the secondorder,given by

p= -; gz _

~pgJ(a2

^sinh12Kd [c09h2K (z +,d)-11+

+pgacos:~~(;(;

d)cOs(I< :t -ut)

+ +~pgl(a2sinhI2I<d[cos~~~;(;(;

^d)

'~I

co s2 (K:r;-0'1) (3.73)

31

3 .3 T hir d ord e r.ap p rox imation

3.3.1 Pot entialfu nction anddispers ionrelation

\

Inthe same""'y~inseetio~(3.1) arid (3.2),keepingth irdorderterms ~ in express ions(3.5 ) and (3.6),givesthethirdorde rappro xi mationof the sur faceeleva tion

aZd

the potentialfunctionofthe comb'ined

W&ve-,c~mnt

fieldinthe(orin

,,=

lJlcos (K:t-- ut)+atcos2(J(.r -at)+

+o3 c os3(K x':'O't)

t= Ao(z)l'+A.{z)sin(Kl'-QI)+

+A,{l }si n 2(K.:r-utl

+

A3{z)sin3(1\%- trl) (3.74)'

(3.75)

Throug hthe~arneprocedure as'(orthetinteqdsecondorderaproxi.,

-;ma tlons, Ao(:),Al( z),A,( z),~(~)in{3.75 )·ue found,on solvinga cor-

re s ponding system ofordinarydiffe rentialeqceccns,to be

.Ao(z)= Bo

Al(l) :=B1coshK(z+d)

32

<:

(3.76)

(3.77)

" .

(3.78)

(3.70) forsomeconstan.~sED,~I,82and 83•

Thus4'couldbewrittenintheform

4>= Box+B1CO~hJ((z.+d)sin(Kx-O"t )+ +B, cosh 2K (z

+

d)sin2(ICr":"'"at }+ +83cosh31((z+d)sin 3(Kx-O't) (3.50) Again,expandingthefr eesurfac e kinematicand dynamicconditionsin , Taylor's ex pansion aboutz=

0

upt athethird order termsgiYesrespectively

£! _ £!l. ....

£!.~+,Po1'1/- az, fJt. Ox ax' 8z²

(3.81) and

33

'.';~

801>8'w- 12 {j3~ lP<'l! 2 +&;,8 : 2 11+'21/[eta :2+(8$ {)z )+

,

+~8~8~2 + (::~12 ~ ~~z~)

⁼⁰

"

(3.82 ) ..

tobesat is fie<!on z=O.

Bysa ti sfy in gequ a t io'n s(3.81)and (3.8 2)toaIr con sideredordersthe coeffici ent!a""m=1,2,3;En,n=I,2 ,3;the dispersionrelat ion,andfu nd io n Fet)couldbefo und,in termsof thefirstord eramplitud ea" andun,ifonn .cur r ent- like speedU,to be

a,

=

~J(a2.(2_:i;;;hI:~l.d)^{cos hJ\"d (3.83 )}

(l3

= i4[(2

^a

3

¹~~~~s;;:\'d

( 3 .8 4)

80=U' ,(3.85)

~ 34

8^t=I<s~:hI<d ^(3.86)

c=ir±.~ ^{(3.80 )}

where

an d

(3.00)

(3 .91) The surfacewa ve heig h tHinthisceae isgive n by

(3 .02) Notethataisusedtodenotethe fir st orderamplitudeal.

35

3.3.2 Prop er t iesof the thirdorderwave -cu rre ntfield

As in sections (3.1.2) and{3.2.2) thepropert ies of thethird order wave- curren t field arelisted' belowfor com p leteness.Theexpress ionsarejust more complicat ed .

1).Potentialfunction

The potentialis,ecth~'^{thirdorder,givenby}

au ' f

Ux

+

J(sin~J(dcoshK(z

+

d) sin(Kx-,ut)

+

3 ~~ . '

+Ssinh4Kd cosh2K(z+d} sin 2(Kx-ee)

+ +~J("ra3(

¹¹

~i:~~~~~J(d)

c~h 3J{( z

+

^{d) sin3(I(x -eft)}

where"r is given by equation(3.90).

2).Free surface profile

The free surface profileto thirdorder is

(3.93) •

1]

=

acos( J(x-at)

+

a²1C,(2 +cosh21(d)coshICd . .

+4 sinh3Kd cos2 (I\,T-O"t)+

+~~(2a3(1 ~i~~~s:;:Cd)

^{cos3 (K x_O"t) (3.94)}

36

, u

r-".. ,

' ."

3).Dispersio n rela.tion

4).Part iclevelocit y

.Tot~irdorderth~,velocitycompo nents orn pnrt iclearc

u +

si::~dc~sh l«:+d)c os(J(x-l1t)

+

+ ~S~:;~dsinh21{(:

^Td) cos2 (!(x-l1t)+

+_~K1~ra3 11~i~~~~J{d

^'-

cosh 3-K(z

+

d)c~s3(I { ;r-11/) (3.'3G )

sina: ;( ds in h/«(z

+

d)sin( H x-l1t)+. .

+~ Si~:;~d'sinh 2]«(:

^{+d)sin2(K.:r -11 t)+}

+~J(1I7ra311 ~i~~~~~~J(d

'sinh 3J( 4

+

d)sin3(I ( x- (71) (3. '37)

5).Particle acceleration-

Also the compo nents of aecelerattoncanbe writ ten as

,

37

\ .

"1>

(l¥= {hl7.co~hJ({:+d)-

-~!lhI((SinhJ((z^td)sinh 2l\"( z+,d)

+

+cosh](l+d)_,.cosh 2(z_,.+d)l}sin{Kr - crt)_I

+

+12f,0'~

^{coeh2I\' (ztd)-}

~nJ(ISinr(J(r

^-0'1)+

+(3hO'~cosh3[( z +d)+ +~lihl([sinh[( zttl)sinh'lK(ztel)-

-cosh1\(:+d) c osh2l( z+.d)]}sin3(K r-0'1) (3.98 )

a, = I:Ksinb l\ (z +d) roshl\"(z +d)- -{!J17,.sinhli: (: +d)-

-~flf2l\{sinhJ(z+'0^co'sh2K(z+d)+ tcosh K(z+d ) sinh21\( ztd)J}cos(K r-at)- -2bl1.. sinh2K{z+d)cos2(Kl'- O"t)-

" ,

where

-(3hO'.sinh3J(ztll)-

- ~h~2J(lcos~J( z

td ) sinh 2K ( zttl)-

~

-sinhI\(z

+

d)cosh2 I«ztd))} cos3(Kx -at)

3S

(3.90)

(3.100J

6);Pressuredistributi o n Thepressure is givento third orderas

,(3.101)

~3,102 )

o.

p~ ^-pgz"":'

ipincos .h

^{2]( :}+d):"1]+ +p{hifcosh]{(z +d)+

+~hh(sinhJ(( l:+

d)sinh2J((z+d)+

+~sbJ(z+d)cosh2K(z

+

d)]}cos(f\.:r -".t)+

+p[h~

^{cosh2/« z}

+

d)

-.imcos

^2(Kx-u,t)+

+p{!J1Zcosh 3K (z +d)+ +~fd2J sinh l( z+.d)sinh2K(z

+

^d)

\

. • -cosh I«z+d)cos h2]{(zj)]}C093(Kx-Qt) (3.103 )

where

s. .

s;hRreexpressed abovebY(3.,lo),(3,:OI ),(3. 102)respectively:

39

.:i:'

:~

, Chapte r 4 "

Changes in Wave and Current

\ ~

Inthe previous chapter the solution'an dthe propertie!>of the combined .wave-cu rr entfieldhavebeendeveloped'intermsof thesurface.disturbance wave lengthL;.wave heightII,water depthd,and uniformcuuen.tspeed Uof the combinedwave-current field. S;~cethe current-freewaveand weve-free'Surrenr-cndergcchanges after the interaction,L,H,d,Uwill bedifferentfromthe length Lo,he igh tH6"water-dept hdo.and current speed Uo of the current..Ireewave and wave-freecurrent throughwhichthe

~ombin~d wnve-cur~ent

^{field is}

for~ed

^(a

s~bscript

"0"isused in this thesis todeno te theperemet ersbefore'

the

wave-cur rent inte ra ctio n ),It is

th e

objectiveofth:.prest:nt sectiont~find,the rela tionbetweenL,B,d, Uand Lo•Ho,do, Uoandto describeamethodto calculateL,B.d,Uknowingthe' valuesofLD,Ho,-do, andUo_,

'

^.

. , .

40

v

, ..

"

4.1 Conservation

eq u a t io ns

:'

..

^,

From11general point ofviewinII.fluid,the mess,momentumand energy ecnserva ric nrdl\~ion!loere respeerlve lyI!fven by

J.

^•

^{P(i' . n)d' + ~ J.}

^{.v• • •. . /}

^{pd, - " .}

^(4.1)

wherendenotes the~nitvector nor m alto and directed{rom insideto ,'outside ofthe fixed controlsurfa ces inthe fluid domainenclosin gII.volume.

rofflui4. With:axilvertically upwar ds.

V •fandP.denote respectivelythe velocityvector,the fluid body force per unit volume and surfaceforceperunit area. For an incompressible.

<invlscid

flui~

s:tea?y .flow undergravity:theabo ..·e'

~uattons

^takethe"ferml

lpFdr -,!.PfidJ-J.p(Vfi)iid.~= o ^(4.5)

"j.(p+.i pV2 _ P9Z)(V ii)dJ=O (4,0)

41

J

-, For th.ctwo dimensional wa ve curre ntinter actionconsideredhere and\

neglecting energy~issipation,lh~seequ a tions become simply

M"",+Ma>-M..,c '=O

(4.7)

(4.8)

./ . (4.9)

whereQ.}II,Erespectivelyrepresentthe~~an^{tnRSS,momentum,and} energy flux'trans fer acrossvertic alplanesnormaltothe xcoord inate and .of unit widt h..Thesubscri pts"wo" , "co"are usedres pect ively to denote the current-fre ewave andwfl-ve-free£7cntfi~~q~anti~es,and "we"isused todenotea combine dwave-curre nt field quantity."Aliflilxesinequations (4.7),(4.8 )I.UId(4.9)

ha~e

^{thesamedatum,}the horizontalJolidbed.

The three mean flu.xesQ,M,Eare givenbythe

foljow'i~g.eqUation;

(4.10)

(4.11)

42

. \

\

\

\ \

E ",,-L: (P+

~p(U2

^{+W2)+pgz)udz (4,12)}

whereu,Ware the horizontal and verticalparticle velocityrespect ively,P is the pressure,andthebar overtheintegrationrepresentstakingaverage over time.

4.2 M ean mass,

mo~entum,

energy flux es 4.2.t....,

Flux esirra current-freewave field

For the coordinat esystemwithz vertically upwards fromtheundisturbed fluidsu rface, the second order potential function andsurfac~profile ofn cu rrent-free wave field'are given by

'10"" aocos(I<o%-O'ot )

+.

+~~(oa~(l ,+ 2si~h~ J(o~o )

cothl( od. cos2 (J( 0a:-0'0t ) ,(4.14,)

43

f

·

.

The press uretotheeec:ond cederi~venas

t "'" =

-'pg:- p

a~",~

^_

~p[( O:;")i

^-:

(-a;;~ ),l

⁺

-fI~pgJ(~a~Sinh~J(.d..

^(4.15)

. .

Accordingto equat ion s(4.10),(4.11), and(4.12),fortheabovewave Heldthe mean mass,momentum,and enerD' flux couldbeob tai ned.They ll:I"einthefonn

.... I f.'·1~ ⁰⁰ Qwb

=

2;'I) _II.P0'; d:d8.

E_

= ~ 10

^2•

L :

^{P_

+ ~pl(a:; )2 +

+(0:;')2]+pgz}(0: ; >dzd8.

(4.16)

(4.17)

(4.18)

N~breaking the integra ti on from~d.to'1.intoaninte~ationCram -d.to 0plus anin;egrat!,nCram 0to'1.andtaking the followingepp rox- lmatlon

44

f

l8

^{0!(z)d z.}~^{zo!{ O)}

+

O(:tI~) whenZoisrelatively smal l,equ at ions(4.16),(4.17),(4.18)become

(4.10)

(4.' 0)

M.,

E. ,

i; L

^h

i:{PVJO

+

~pl(8:;),

⁺

( . D :;o),]

+ +pgz}

8~;

dzd60

+

+~ t"

^{1]o{PVJO(z, o.t} +}

~p[(a;;O(zI O,t))'

⁺

+(a: ;-(r,o,t))'1l

a:;' (r ,0,t )d8, (4.22)

\

Substitut ingP",o,!P",o.and1]0into the aboveequationsandperf orming the integrations yields,toO(J(a~)

45

(4.23)

(4.24)

(4.25)

4.2 .2 Fluxesinawav e-freecurren tfield

The potentialfunction ofawave-free uniform irrotationalcurrentfieldis givenby

(4.26)

for someunifo~treamUo.

Thepressuredist rib u tionalongaverti calsectionis

'e

Pc

=

-pgi - (4.27)

. .

.

.

Onusingequations(4.10),(4.11),and(4.12),the meanmess, mcmen - tum,andene rgyfluxesin awave-freecurrentfieldarethen

46

:,"•.,'~

(4.28)

(4.20)

Performing theintegr ationgives

Qro=pd.U. (4.31) .

~

Mct>

=

pdoU; (4.32)

s;""

~pdo·U:

^(4.33)

4.2.3 Fluxesin a wave- c urrentfield I

Forthecombi~edwave-currentfield the three meanfluxesacross an arbi- trar y ver ticalplaneintheflow of unitwidthcanbeexpressedas