horikawa chapter 15 (pag. 217-233)

8/2/2019 Horikawa Chapter 15 (Pag. 217-233)

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CHAPTER1 5ASTUDYONWAVETRANSFORMATIONINSIDESURFZONE

KiyoshiHorxkawaAssociateProfessorofCivilEngineeringUniversityofTokyoTokyo,JapanandChin-Tong KuoLecturerofCivilEngineeringChengKungUniversityTainan,TaiwanRepublicofChinaABSTRACT

Thewavetransformationinsidesurfzoneistreatedanalyticallymthispaperundertheseveralappropriateassumptions. Thetheoreticalcurvescomputednumericallyhaveaconsistantagreementwiththeexperimentaldatainthecaseofwavetransformationonahorizontalbottom. Ontheotherhand,mthecaseofwavetransformationonauni-formlyslopingbeach,theanalyticaltreatmentseemstobeinadequatetoclarifytheactualphenomena. Besidesthemthenumerousdataonwaveheightattenuationandothersarepresentedmthegraphicalforms.

INTRODUCTIONThephenomenaofwavetransformationmthesurfzonehasbeenamatterofgreatinteresttothecoastalengineers,thereforethenumerousinvestigatorshavetreatedthesameproblemonthebasisoftheappropriateassumptions'.''Theassumptionsaresuchthatthewavehasitscriticalheightasaprogressivewaveateachdepthofwater,orthatthewaveheightdecreasesexponentiallywiththedistanceofwaveprop-

agationfromthebreakingpoint. Theseforegoingtreatmentsseemtobeinadequatetoclarifythephenomenaofwavetrans-formationinsidethesurfzone,thusmorereasonablemethodisrequiredtobeapplied. Theaimofthispaperistopresentanapproachtothestatedproblemonthebasisoftheanalyticalandexperimentaltreatments.THEORETICALANALYSIS

Intheanalysisontheattenuationofwaveheightmthesurfzone,thefollowingassumptionsareintroducedasthebasisoftheanalyticaltreatment:a) The2ndorderapproximationofsolitarywavetheoryintroducedbyLaitorT isadoptedtoexpressthefea-turesofthebroken wavesprogressingmthesurfzone. Thatis,thewaveprofile,wavecelerity,

217


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218OASTALNGINEERINGandhorizontalomponentfwaterparticlevelocity-aregivenyheollowingequationsespectively:

c = /$A(i+ H / f l . ) 2 )

where%ssurfaceelevationmeasuredfromthestillwaterlevel,Hwaveheight,H waterdepth, Cwavecelerity,u.orizontalcomponentofwaterparticlevelocity,zverticalaxistakingupwardfromthestillwaterlevel,andx. horizontalaxistakingalongthestillwaterlevel.

D) Thewaveisattenuatedbytheeffectsofturbulenceandbottomfriction. Theeffectofpercolationontheattenuationofwavesisnegligiblysmall.c )hefrictioncoefficienthasthesamevalueintheentireregionofsurfzone.d)heturbulenceisisotropicanddecreasesexponen-tiallyaccordingastheincreaseofshowewarddis-tancemeasuredfromthebreakingpoint.ENERGYDISSIPATIONDUETOBOTTOMFRICTION

Inanoscillatoryflowtheshearingstressatbottommaybeexpressedapproximatelybythefollowingequation.T=Cf(Sz)* 4 )

whereTcisshearingstressatbottom,Cf frictioncoefficientand it amplitudeoftheaveragevelocitymdepth,aTheenergydissipation bybottomfrictionperunitwidth,per unittime,JiE b/dt >isgivenby

where0C-/(3H/ftf ,X*+ H) 6 )

ENERGYDISSIPATIONDUETOTURBULENCEWhenthewavebreaksatacertainpoint,agreatamountofairbubbleisentrainedintothewater,causingalarge


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WAVERANSFORMATION19scaledxsturbancemflow. Suchkindofdisturbanceseemstotakethemamroleofenergydissipationatleastattheinitialstageofwavetransformationinthesurfzone. Bytheassumptionthattheturbulenceisstatisticallyisotropic,theenergydissipationduetoturbulenceperunitvolume,perunittime,isgivenby

W =15H - 7 )where V Vistherateofenergydissipationduetoturbulence,/^coefficientoffluidviscosity, &'fluctuationofhorizontalvelocitycomponent,and A.microscaleofturbulenceordissi-pationlength.

Thekineticenergyofturbulenceseemstobeinverselyproportionaltothedistancefromthebreakingpoint. There-foreitmaybepossibletoexpressthedecayofturbulenceasfollows:

U'xoc exp(-j8x/L) 8)where J3 indicatesadampingcoefficientofturbulence,Xdis-tancemeasuredfromthebreakingpointand Lwavelength.Thusthedissipationlengthmaybeexpressedbythefollowingrelation:

atxHereweassumethatthemixinglength,Jt mthePrandtle'shypothesisisproportionaltotheheightabovethebottom,Zt'=i-=x(Z + A ) - |10)where XistheKarman'suniversalconstant,ftaterdepth,Uhorizontalcomponentofparticlevelocityand Z .axistakingupwardfromthestillwaterlevel. Thereforeweobtainthefollowingexpressionsontherateofenergydissipationduetoturbulence,$ andthelossofenergyduetoturbulenceperunitwidth, oLEt/dt

W -Hrf e+ u'tiSf11)


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220 OASTALNGINEERING

4| = fj\Xlz = * *&(4f( I + 3 9 9 -oo -f t

+ 7.2f9*'(#)*0+%f1 3 )

T h etimerateofenergytransport,dts/dt i s asfollows:

WAVETRANSFORMATIONON AHORIZONTALBEDT h eruleofenergyconservationi sexpressedinthenextequation:

3=_(W+Zt"' 1 5 )Substituting Eqs.( 5 ) ,( 1 2 )and( 1 4 )into E q .( 1 5 ) ,wefindthefollowingdifferentialequation:

c C xowy^wy^smw*wn)(16)where

5 "

'1 7 )T h eintegrationoftheaboveequationcannotbedoneanalyti-cally,butbedonenumericallyasshownmP i g .1 . Here J3 i sselected t o beequal t o5 ,andtheeffecto f bottomfric-tion i sincludedma factor o fCfT/Jd/ft selectedas a param-eteroffamilyofcurves.WAVETRANSFORMATIONON A UNIFORMLYSLOPINGBED

Theslopeofthebottom i s definedby5 = -olk/dx, andthetimerateofenergytransportperunitwidthcanbe


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WAVERANSFORMATION 221

H 0.5

P = 5 Symb CfT/g/h

00.1250.250

5 ^* ?*^^-

2Fig.. Numericalntegrationurvesfq.16). (Horizontalbottom)

Fig.. Laboratorynstallation. (Firstetfxperiments)


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222OASTALNGINEERINGexpressedyhe followxng:

#-=(-*+#!&s)Takingintoconsiderationthenextrelationships,

#-$W(*)V)'* V ( 1 9 ) wemayobtainthefollowingdifferentialequation:

d&_

^'/^nri^rF^)+o^q(i+^rw)MI)- '^520)wherethefunctionso f F ( H / R . ) and^WR.) arethesameasgivenmEq.(17). Theintegrationoftheaboveequationcanbedonebythemethodofnumericalcomputation.

EXPERIMENTALANALYSISHORIZONTALBED

Theexperimentalstudiesbyusingahorizontalbedwereconductedforthepurposeofdeterminingthedampingcoeffi-cientofturbulence,^ whichwasdefinedmEq.(8). Itseemstobequitereasonabletoassumethattheturbulenceofflowinducedbybreakingofwavestakesthemostimportantroleonthewaveattenuationinthesurfzoneonasmoothhorizontalbedcomparingtothebottomfrictionandothers.Thewavechannelusedforthepresentstudiesis1 7mlong,0.7mwideand0.6mhigh. Atoneendofthechannel,anelevatedwooden horizontalbottomwasinstalledandcon-nectedtothechannelbottom withaslopeof1 /5asshowninPig.2 . Thesurfaceofthehorizontalbottommentionedabovewascoveredwithasmoothrubberplate. Wavesweregeneratedattheotherendofthechannelbyaflaptypewavegenerator. Theincidentwaveswereforcedtobreakthemselvesontheslopingbottomandthen propagatedtotheelevatedhorizontalregion. Amongthevariouskindsofwavesgener-ated,weselectedonlytheparticularoneswhichbrokejustatthecornerbetweentheelevatedhorizontalbottomandtheslopingbed. Thecharacteristicsoftheselectedwaves


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WAVERANSFORMATION23 aregiveninTable1 .

AsampleplottingoftheexperimentaldataispresentedinFig.3 . Herethetestedwaveshavethesameperiodof1sec,butthewaterdepthabovetheelevatedbottomhasadefferentvalueforeachrun. Figure4givesacomparisonofthelaboratorydatawiththeanalyticalcurvedeterminedbytaking ^8 =5". Fromthisfigureitmayberecognizedthatthewavesteepnessmdeepwaterofincomingwavesseemstohaveverylittleeffectonthewavetransformationmthesurfzone.

Summarizingtheresultsofourlaboratoryexperiments,weconcludethatthedampingcoefficientofturbulence,^canbetakenacertainvaluebetween4and5mthepresentexperiments.Inordertoinvestigatetheapplicabilityoftheabovetreatmenttothepracticalphenomenamfield,wetookthefielddataof waveheightmthesurfzonewhichwereobtainedbyIjimabymeansofstereophotographyontheNugataWestCoast. Thebottomslopeofbeachatthequestionedsiteissogentle,thereforethebeachslopeisassumedtobehori-zontal. Figure5showsthecomparisonofvariouscurvessuchas( l )laboratorycurve,( 2 )analyticalcurvecalculatedundertheassumptionof = I and. o . o 5 ~ (3) curvepro-posedbyIjimaempirically,and( 4 )meancurveoffielddata.Theagreementbetweentheanalyticalcurveandthecurveof

fielddataseemstobequiteconsistant. Buthereitisnecessarytoremarkthatthevalueoffimlaboratoryis4~5 ,whilethevaluemfieldis1 . Theabovefactsuggestsustheexistanceofscaleeffectofturbulencemthepresentproblem. Fromthispointofview morefieldworksarecer-tainlynecessarytobedone.UNIFORMLYSLOPINGBED

Anotherseriesofexperimentswerecarriedouttorevealtheinfluenceofbottomslopeonthewavetransformationinsidethesurfzone. Thefirstsetofexperimentsmwhichwetestedthebottomslopesofl/20andl/30wasconductedbyusingthesamechannelasmthepreviousexperiments. Thesecondsetofexperimentsinwhich wetestedthebottomslopeofl/65andl/80wasdonebyusinganotherchannelatChengKungUniversity,thesizeofwhichwas75m long,1.0 m wideand1 .2m deep. Theslopemthelattertwocaseswasmadeofconcrete. TheconditionsofbothsetsofexperimentaregivenmTable2 .

Thedimensionalanalysisintroducesthefollowingrela-tionshipamongthewavecharacteristics,waterdepth,andbottomslopeconditionmanon-dimensionalform.


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T-Osec h -5cmRun Symb Ho/Lo 60 006 X 0083

* 62 A 007363 + 006564 o 008S\+ e-5/A O +i/ +

+ A * XX X

Ao

X/T/gF

Fig.. Comparisonfhexperimentalesultswithheheoreticalurve.(Horizontalottom)

X/TTgiTFig.. RelationshipetweenH/hnd/T-y/gfiobtainedromvariousources.(Horizontalottom)


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226 COASTALNGINEERING

Table2 . Experimentalconditionsonslopingbottom

Bedcondition Wave characteristics Depthd(cm)Surface slope T(sec) H0(cm) o' 0Rubbersurface

1/20 2.0, 14.1 5.6 -6.9 0.008 .053 43.31/30 2.2, 14.1 4.7 7.2 0.007 0.052 39.0

Concretebed

1/65 1.56, 2.02.0, 1.8 5.9 4.5 0.009 0.065 78.0

1/80 1.6, 1.41.2 5.8 6.7 0.011 .072 75.0

H( c m )

28

24

20

1 6

1 2

Symt) H% I //'0065 65+ 0053 .**0043 0038 .


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or


4~t f f e #" f c , S)2 2 )wheresubscriptoandbdenotethevaluesmdeepwaterandatbreakingpointrespectively.Accordingtotheresultofdimensionalanalysistheexperimentaldatawereplottedasshownmthefollowingfigures. Figure6givesseveralexamplesofwavetransfor-mationontheslopingbottomofl/65>andindicatesthatthelimitingconditionofsolitarywave, H = < - 1 8 l l isnotsuita-bletoexpressthewaveheightinsidethesurfzone. InPig.7isshownthecorrelationbetweentherelativewave

height, H/Hb ,andtherelativewaterdepth,K / k j ,,foreachbottomslope. Scatterofdatamthesefiguresseemstobecausedmainlybytheinstabilityofwavesinsidesurfzone,butthesteepnessofincidentwavesmdeepwaterseemstohavesmallinfluenceonthestatedrelationship. ThereforetheeffectofbottomslopeonthewaveattenuationmthesurfzoneissummarizedmPig.8 ,fromwhichitmayberecog-nizedthatthegentlerthebottomslopebecomes,thesmallertherelativewaveheight,H/Hb becomesatthesamerelativewaterdepth,k / h . b Theabovefactisduetothatthedecaydistancefromthebreakingpointonagentleslopeislargerthanonasteepslope.

InthesamewayweplottedthedataonthefollowinggraphsasshownmPig.9inordertofindouttherelation-shipbetweentherelativewaveheightwithrespecttowaterdepth, H / h . ,andtherelativewaterdepth,h/hb ~ ?0T eachparticularbottomslope. Thereisalargescatterofdata,butitispossibletodraw meancurvethroughtheplotteddata.AfamilyofcurvesthusdeterminedisgivenmFig.10withtheparameterofbottomslope. Thefigureshowsthattherelativeheight, H / h . hasitsminimumontheconditionofK/hb= F 0 . 6 Theanalyticalresultsobtainedbytheintegra-tionofEq.(20)undertheconditionsof^5=4 and C f= 0.OZarealsoplottedmthesamefigurebydotsanddashes. Theagreementbetweenthecomputedandexperimentalresultsisnotfullysatisfactory,thereforeitisquitenecessarytotreatthepresentproblembymoreregorousapproach.Lastly,itwillbementionedherethattheBoussinesq'sexpressionforwavecelerityhasthebetteragreementwiththeexperimentalresultsasshowninPig.11. Theequationisasfollows:

C =/#t { I t ( M l ) } { / + to/au} ( 2 3 )


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I0090807

05040302

Symb "yi 1 0X AVD 0

AV+

000800080009001100100130013014010016017

-^o

X * A/S I+ + /V

A /A r*T=22ec

A* X % o*& J ff0

fit

X AS0 yyi*A

is

Fig. 7CorrelationetweenH/Hb andh/hbwith1/20,/30,nd/65bottomlopeespec-tively.

0 0 02 03 04 05 06 07 08 09 10 (a) /hb

II

I00908070605040302010

Symb " X * 1 oX A

*- - AV

00170016010 01400120100070009

""30

V V VT = 22sec', X*

V fr'9 | p e-^rf (A

? & A fo0 01 02 03 04 05 06 07 08 09 10 (b )

/hb


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II

I0090807

/Hb605040302

Symb HKo Q= 1 o X

4- V D 0V

006500530046003800320030002500180017

65

A /

vf1 5/T= l56sec X

V/ v f t 8V

a ni

o ^ I io

o V

Fig. 7 CorrelationetweenH/Hb andh/hj,with1/20,1/30,nd/65bottomlopeespec-tively.

(c)0 01 02 03 04 05 06 07 08 09 10

Vh b

lnC 0/

0 ,2 % ,Fig.. Effectfheottomlopenhewaveattenuationnsideurfone.


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230 COASTALNGINEERINGI1

Fig.CorrelationetweenH/handh/hbwith/20,/30,1/65,nd/80ottomslopeespectively

X

I

10

05 05 10(b)


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%

Fig.CorrelationetweenH/handh/hbwith/20,/30,1/65,nd/80ottomslopeespectively.

w (c) % ..

H

I2I

I0090807

'/Hh605040302O

Symb HVL9 1 oX+m \AV a0A V

00390043004700500054005600600061006500670072

" 80

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< A

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VVv X4#\-+

33

V"VV X

0 01 02 03 04 05 06 0.7 08 09 10 (d )


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1.5

0.5

1 \\\\ \1 \\ \\ \\ \\ \\ \\ \\ \

HV078 \ v ^^_-/20 --''

V ___^^Vo^^0.5 1 .0

Fig.0. Comparisonfheheoreticalandxperimentalesults.

20

s= y0 o T= 4secx T22sec

iX X X "?6o

*oo_-XOo xo xo x

0345 078Fig.1. ComparisonfheBoussinesq'sxpressionorwaveeleritywithheexperimentalesults.


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WAVERANSFORMATION33 whereC L , isthecrestheightabovethestillwaterlevel.

ACKNOWLEDGEMENTSTheexperimentalworkswereperformedattheCoastal

EngineeringLaboratory,UniversityofTokyoandattheHydrau-licsLaboratory,ChengKungUniversity. TheauthorsareindebtedtoDr.MasashiHom-ma,UniversityofTokyo,forhishelpfulguidance,andalsotothepersonnelatthebothlaboratories,whoassistedtheauthorsmcarryingoutthelaboratoryworkssuccessfully.REFERENCES

1)om-ma,M.andK.Horikawa Waveforcesagainstseawall,Proc.9thConf.onCoastalEngineering,1964.2 )jima,T . Wavecharacteristicsmthesurfzoneobservedbymeansofstereophotograph,ReportofTransportationTech.Res.Inst.,Rept.No.31,1958.3 )ajiura,K . Onthebottomfrictionin anoscillatorycurrent,Bull.EarthquakeRes.Inst.,Univ.ofTokyo,Vol.42,1964.4)aitone,E.V . Thesecondapproximationofcnoidal&solitarywave,Jour,ofFluidMechanics,Vol.9 ,1 9 6 1 , ,5 )ato,S . Designofseawall,Proc.1stConf.onCoastalEngineeringmJapan,1954.(inJapanese)6)awaragi,T . Effectofbottomroughnessonthewaveheightinsurfzone,Proc.9thConf.onCoastalEngineeringinJapan,1963-(inJapanese)

horikawa chapter 15 (pag. 217-233)

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