Deriving vegetation drag coefficients in combined wave-current flows by calibration and direct measurement methods

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Delft University of Technology

Deriving vegetation drag coefficients in combined wave-current flows by calibration and

direct measurement methods

Chen, Hui; Ni, Yan; Li, Yulong; Liu, Feng; Ou, Suying; Su, Min; Peng, Yisheng; Hu, Zhan; Uijttewaal, Wim;

Suzuki, Tomohiro

DOI

10.1016/j.advwatres.2018.10.008

Publication date

2018

Document Version

Final published version

Published in

Advances in Water Resources

Citation (APA)

Chen, H., Ni, Y., Li, Y., Liu, F., Ou, S., Su, M., Peng, Y., Hu, Z., Uijttewaal, W., & Suzuki, T. (2018). Deriving

vegetation drag coefficients in combined wave-current flows by calibration and direct measurement

methods. Advances in Water Resources, 122, 217-227. https://doi.org/10.1016/j.advwatres.2018.10.008

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AdvancesinWaterResources122(2018)217–227

ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

Deriving

vegetation

drag

coeﬃcients

in

combined

wave-current

ﬂows

by

calibration

and

direct

measurement

a Institute of Estuarine and Coastal Research, School of Marine Science, Sun Yat-sen University, Guangzhou 510275, China b State-Province Joint Engineering Laboratory of Estuarine Hydraulic Technology, Guangzhou 510275, China

c Guangdong Province Engineering Research Center of Coasts, Islands and Reefs, Guangzhou 510275, China d Shanghai Waterway Engineering Design and Consulting Co., Ltd., Shanghai 200120, China

e School of Environmental Science and Engineering/Research Center of Wetland Science, Sun Yat-Sen University, Guangzhou 501275, China f Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, Delft 2628 CN, the Netherlands

g Flanders Hydraulics Research, Berchemlei 115, Antwerp 2140, Belgium

a

r

t

i

c

l

e

i

n

f

o

Keywords: Wave dissipation Vegetation Drag coeﬃcient Wave-current interaction Keulegan-Carpenter number Flume experiment

a

b

s

t

r

a

c

t

Coastalvegetationisefficientindampingincidentwaveseveninstormevents,thusprovidingvaluableprotections tocoastalcommunities.However,largeuncertaintieslieindeterminingvegetationdragcoefficients(C_D),which aredirectlyrelatedtothewavedampingcapacityofacertainvegetatedarea.Onemajoruncertaintyisrelatedto thedifferentmethodsusedinderivingC_D.Currently,twomethodsareavailable,i.e.theconventionalcalibration approachandthenewdirectmeasurementapproach.Comparativestudiesofthesetwomethodsarelackingto revealtheirrespectivestrengthsandreducetheuncertainty.Additionaluncertaintystemsfromthedependence ofC_Donflowconditions(i.e.wave-onlyorwave-current)andindicativeparameters,i.e.Reynoldsnumber(Re) andKeulegan-Carpenternumber(KC).RecentstudieshaveobtainedC_D-Rerelationsforcombinedwave-current flows,whereasC_D-KCrelationsinsuchflowconditionremainunexplored.Thus,thisstudyconductsathorough comparisonbetweentwoexistingmethodsandexplorestheC_D-KCrelationsincombinedwave-currentflows.Bya uniquerevisitingprocedure,weshowthatC_Dderivedbythedirectmeasurementapproachhaveabetteroverall performanceinreproducingbothactingforceandtheresultingwavedissipation.Therefore,agenericC_D-KC

relationforbothwave-onlyandwave-currentﬂowsisproposedusingdirectmeasurementapproach.Finally,a detailedcomparisonofthesetwoapproachesaregiven.Thecomprehensivemethodcomparisonandtheobtained newC_D-KCrelationmayleadtoimprovedunderstandingandmodellingofwave-vegetationinteraction.

1. Introduction

Uprightvegetationincoastalwetlandscansignificantlyattenuate in-cidentwaveenergy,thusprovidingprotectionstocoastalhabitatsand structures(Andersonetal.,2011;Vuiketal.,2016,2018).Thewave dampingeffect issignificant eveninstorm conditions(Mölleret al., 2014).Additionally,naturalvegetationecosystemscanadjusttheirbed elevationtosealevelriseviaecogeomorphologicalfeedbacks,which enableslong-termsustainablecoastaldefensesolutions(Arkemaetal., 2013;D’AlpaosandMarani,2016;D’Alpaosetal.,2016;Temmerman andKirwan,2015).Withincreasingstorminessinthefuture(Donnelly etal.,2004;Youngetal.,2011),theprotectionofferedbycoastal veg-etationcanbeofgreaterimportance.

∗_{Corresponding}_author_at:_Institute_of_Estuarine_and_Coastal_Research,_School_of_Marine_Science,_Sun_Yat-sen_University,_No._135,_Xingang_Xi_Road,_Guangzhou 510275,China.

E-mailaddress:huzh9@mail.sysu.edu.cn(Z.Hu).

Waveenergydissipationbyvegetation(hereafterreferredasWDV)is affectedbyincidentwaveheight(H,MéndezandLosada,2004;Bradley andHouser,2009),waveperiod(T,Augustinetal.,2009;Suzukietal., 2012),theratioofwaterdepthtovegetationheightinthewater(h/hv, Ysebaert etal., 2011;Yangetal.,2012),dragcoefficient(C_D,Henry etal.,2015;Losadaetal.,2016a,b),stiffness(Boumaetal.,2005;Luhar etal.,2017;Pauletal.,2016)andstemfrontalareaofplantsperunit height(i.e.N∗_b

v,Nisthenumberofstemsperunitareaandbvisthestem diameter,Augustinetal.,2009;FonsecaandCahalan,1992;Nepf,2012, 1999;Ozeren etal.,2014).Thisknowledgehasalsobeenadaptedin differentnumericalmodels(e.g.Augustinetal.,2009;Caoetal.,2015; Mazaetal.,2013;Suzukietal.,2012).Recentexperimentalstudieshave alsoidentifiedWDVisaffectedbyco-existingcurrents(LiandYan,2007; Pauletal.,2012;Huetal.,2014;Losadaetal.,2016a,b).

https://doi.org/10.1016/j.advwatres.2018.10.008

Received12February2018;Receivedinrevisedform4October2018;Accepted11October2018 Availableonline16October2018

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Table1

AreviewofC_Drelationsinvegetation-waveinteractionandtheirderivingmethods.

Reference Mimic Type Flow condition CD relation Deriving method

Kobayashi et al. (1993) Flexible plastic strips Waves CD = 0.08 + (2200/ Re ) 2.4 Calibration method

2200 < Re < 18,000

Méndez et al. (1999) Flexible plastic strips Waves CD = 0.08 + (2200/ Re ) 2.2 Calibration method

2000 < Re < 15,500 (no swaying)

CD = 0.40 + (4600/ Re ) 2.9

2300 < Re < 20,000 (swaying)

Mendez and Losada (2004) Flexible real vegetation Waves CD = 0.47exp( − 0.052 KC ) Calibration method

R 2 = 0.76

3 ≤ KC ≤ 59

Bradley and Houser (2009) Flexible real vegetation Waves CD = 253.9 KC −3.0 Calibration method

R 2 = 0.95

0 < KC < 6 Field data

Calculated using the relative velocity of the seagrass blades

Ranjit S. Jadhav et al. (2013) Flexible real vegetation Waves CD = 70 KC −0.86 Calibration method

R 2 = 0.95

25 < KC < 135

Anderson and Smith (2014) Flexible plastic strips Waves CD = 1.10 + (27.4/ KC ) 3.08 Calibration method

R 2 = 0.88

26 < KC < 112

CD = 0.76 + (744.2/ Re )1 .27

R 2 = 0.94

533 < Re < 2296

Ozeren et al. (2014) b _{Rigid wooden cylinders} _Waves _C

D = 1.5 + (6.785/ KC ) 2.22 Calibration method

R 2 = 0.21

Nv = 156m −2 , h v = 0.63m

CD = 2.1 + (793/ Re ) 2.39

Flexible plastic strips CD = 0.683 + (12.07/ KC ) 2.25

Nv = 350m −2 , h v = 0.48m

Infantes et al. (2011) Flexible real vegetation Waves lg C D = − 0.6653 ∗ lg Re + 1.1886 Direct measurement

R 2 = 0.77 _method

Hu et al. (2014) Rigid wooden cylinders Wave + Current CD = 1.04 + (730/ Re ) 1.37 Direct measurement

R2 = 0.66 method

300 < Re < 4700

Losada et al. (2016a,b ) Flexible real vegetation Wave ± Current CD = 0.08 + (50,000/ Re ) 2.2 Calibration method

R 2 = 0.60 (regular waves) CD = 0.25 + (75,000/ Re ) 9

(regular waves + currents) C D = 0.50 + (50,000/ Re ) 9

(regular waves-currents)

WDVismainlyinducedbythedragforceprovidedbythevegetation actingonthewatermotion,whichcanbequantifiedbyMorison equa-tion(Dalrympleetal.,1984;Morisonetal.,1950).Thedragforce(Fd) isproportionaltothesquareofthevelocity,vegetationfrontalareaand vegetationdragcoefficient(CD).Thus,choosingsuitableCDvaluesisof vitalimportancetoaccurateWDVprediction.Theparameterizationof CDiscurrentlyoneofthemajordifficultiesinmodelingtheinteractions betweenvegetationandwatermotion(Suzukietal.,2012;Luharand Nepf,2013;Mazaetal.,2015a;Caoetal.,2015).Thus,determiningC_D hasbeenamainsubjectinnumerousexistingstudies(seeTable1).

CDistypicallydeterminedbyexperiments,eitherbycalibrationor directmeasurementapproach(Table1).Thecalibrationapproachisa conventionalmethod,whichdeterminesCDbycalibratingitsvaluein WDVmodelstofitthemeasuredwaveheightreduction(e.g.Mendez etal.,1999;Augustinetal.,2009).Previously,thismethodcouldnot beappliedinthecasesofvegetationincombinedwave-currentflows, aspreviousmodelsdidnottakeintoaccounttheinfluenceofco-existing currentonWDV(Dalrympleetal.,1984).Thislimitationhasrecently been relaxedby a new modelproposed by Losadaet al. (2016a,b), whichcanexplicitlyaccountforWDVincombinedwave-currentflows. Thus,thecalibrationmethodcannowbeappliedtoderiveCDinboth wave-onlyandcombinedwave-currentconditions.Comparedtothe cal-ibrationmethod,thedirectmeasurementmethod isanew approach (Huetal.,2014).ThisapproachisbasedontheoriginalMorison equa-tioninsteadof WDVmodels,anditrequiressynchronizedimpact ve-locityandforcedatatoderiveCDbyquantifyingtheworkdonebythe dragforceoveronewaveperiod.AstheMorisonequationholdsinboth

wave-onlyandwave-currentflows,thisapproachcanbeappliedto de-riveC_Dinbothflowconditions.Thus,avegetationdragcoefficientC_Din wave-onlyandwave-currentflowscanbeobtainedbybothcalibration anddirectmeasurementmethods.However,themeritsanddrawbacks ofthesetwomethodshavenotbeenexploredinparallel.Adetailed com-parisonofthesetwomethodscanbevaluableforfutureexperimental andnumericalstudies.

Manypreviousstudieshaveidentifiedthatvegetation drag coeffi-cientC_Dinoscillatoryflows(i.e.wave-onlyorcombinedwave-current) varieswithReynoldsnumber(Re)(seeTable1andFig.1a).The ob-tainedCD-RerelationsallshowthatCDdecreaseswithincreasingRe. Itissimilartothosederivedfromunidirectionalflowsconditions,but CD valueshave greaterrangeof variation(0.1to100)inoscillatory flows(Nepf,2012).MostofthepreviousCD-Rerelationsareobtained ineitherwave-onlyorcurrent-onlycondition.Itisonlyuntilrecently thatnewCD-Rerelationsareextendedtocombinedwave-currentflow conditions(Huetal.,2014;Losadaetal.,2016a,b).Suchanextension is ofimportanceasthecombinedwave-currentflowsarecommonin e.g.naturalwetlands.BesidesCD-Rerelations,CDinoscillatoryflows hasbeenfoundtobeafunctionofKeuglan-Carpenter(KC)number(see Table1andFig.1b).MendezandLosada,(2004)foundthatCD-KC rela-tionsaremoresuitableforoscillatoryflowscomparedtoC_D-Rerelations. However,CD-KCrelationshaveonlybeenexploredinwave-only condi-tionssofar(seeTable1andFig.1b).Thus,suchrelationsincombined current-waveflowsareyettobeexplored.

Inthisstudy,weaimtoprovide1)athoroughcomparisonbetween thecalibrationanddirectmeasurementmethods,and2)agenericCD

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-H. Chen et al. Advances in Water Resources 122 (2018) 217–227

Fig.1. SelectedC_D-Re(panela)andC_D-KC(panelb)relationsfrompreviousstudiesthatlistedintheTable1.

KCrelationforvariousoscillatoryflows,i.e.wave-onlyandcombined wave-currentflows. Toourknowledge,thecurrent studyis thefirst studytoprovideadetailedcomparisonbetweentwodifferentmethods inderivingCDforpurewaveandcombinedcurrent-waveflows.The obtainedinsightscanbevaluabletotheunderstandingandmodelling ofthewave-current-vegetationinteractions.Toachievethesegoals,we re-analyzethedatafromrecentlabexperimentsthatmeasuredWDVin bothwave-onlyorcombinedwave-currentconditions(Huetal.,2014; Jadhavetal.,2013;Losadaetal.,2016a,b;Ozerenetal.,2014).Both calibrationanddirectmeasurementmethodsareappliedfor compari-son.TocomparethedifferentC_Dderivingmethods,wecreateaunique re-visitingprocedure,whichevaluateshowwellthederivedCDcan re-producedthemeasuredwavereductionandactingforce.Finally,a syn-thesisofthesetwomethodsandagenericCD-KCrelationforboth wave-onlyandcombinedwave-currentflowsisprovided.

2. Methods 2.1. Datacollection

ToderiveCDviadiﬀerentmethodsandaCD-KCrelationincombined wave-currentﬂow,wecollectedthepublisheddataofarecent experi-ments(Huetal.,2014).ThedataofHuetal.(2014)areanalyzedin detailbecausevelocityandactingforcedataweremeasured simultane-ouslyat1000Hz,enablingthedirectmeasurementmethod.

TheexperimentinHuetal.,(2014)wasconductedinawaveflume witha6mlongmimickedvegetationpatchwasplacedinthemiddleof thewaveflume(Fig.2).Thevegetationmimicswerewoodencylinders withadiameterof10mm.Thebuiltvegetationcanopywas0.36mtall andthetestedwaterdepthswere0.25mand0.50m,respectively.The submergenceratio(h/hv=1–1.39)isrelativelysmall(Nepf,2004). Reg-ularwavesareusedinthistest.Thedataofwaveheight,velocityand actingforceinthispreviousstudyiscollectedinthecurrentstudyto deriveanewCD-KCrelation.Theimpactvelocitydataatlocations1–4 weremeasuredbyEMFs(electromagneticflowmeters).Atlocations1 and3,theactingforceonvegetationmimicswasmeasuredbyforce sen-sorsdevelopedatDeltares(formerDelftHydraulics,TheNetherlands). Atlocations2and4,theforcewasmeasuredbyloadcells(model300) developedbyUIILCELL.However,thecells atlocation2and4were notfunctioningproperlyduringtheexperiment.Thus,onlytheforce datameasuredatlocations1and3areincludedinthecurrentstudy. Huetal.(2014)testedtheconditionswhensteadycurrentsflowedin thesamedirectionaswavepropagation,i.e.followingcurrent condi-tion,whileLosadaetal.(2016a,b)testedconditionswithbothfollowing andopposingcurrents.Currentstudyisconstrainedtoconditionswith followingcurrentsonlyforparallelcomparison.

Besides the above-mentioned two previous studies on combined current-waveflows,C_D-KCrelationsderivedpreviouslyinJadhavetal. (2013)andOzerenetal.(2014)forwave-onlyconditionsarealso col-lectedtobecompared withthenewrelationsderivedinthecurrent study.Ozeren etal.(2014)use rigidcircularcylinders,witha diam-eter of0.0094m,astemdensityN=156stems/m2_and_a_stem_height hv=0.63m,whichiscomparabletotheexperimentalconditionofthis study.Jadhavetal.(2013)collectedfielddataofWDVduringatropical storm,andthetestedvegetationwasflexiblesaltmarshplants,Spartina alterniflora.TheaveragestemdensitywasN=422stems/m2_and_the_stem heightwash_v=0.22m,whiletotalplantheightis0.63mandthe aver-ageddiameterofcircularcylinderisdeterminedas0.008m.

2.2. Dataanalysis

2.2.1. DeﬁnitionofKCandRe

TheKeuleganCarpenternumberKCisdeﬁnedas:

𝐾𝐶=𝑈_𝑚𝑇∕𝑏_𝑣 (1)

where Um is the measured maximum horizontal velocity in the wave propagation direction at the half water depth in both wave-only and wave-current flows. This velocity is chosen following Huetal.(2014)becausevelocityatthehalfofthewaterdepthroughly equalstothedepth-averagevelocityinthevegetationcanopywhenthe submergenceratioissmall(e.g.h/h_v=1–1.39inHuetal.2014).Thus, itisagoodrepresentativeoftheactingvelocityonvegetationstemsfor conditionswithsmallsubmergenceratios,andasuitablecharacteristic velocityforKCandRedefinition.Tiswaveperiodandb_visdiameter ofthecircularcylinder,whichisthecommoncharacteristiclengthfor KC andReinvegetatedflow(Nepf, 2012).Forrealvegetationcases, themeasuredmeandiameterof vegetationstemscanbe usedasthe characteristiclength(Jadhavetal.,2013).Reisdefinedas:

𝑅𝑒=𝑈_𝑚𝑏_𝑣∕𝜈 (2)

𝜈 beingthekinematicviscosityoftheﬂuid. 2.2.2. Velocitydataanalysis

ConsideringDopplerEffect,thehorizontalflowvelocityincombined wavesandcurrentflowisgivenas:

𝑈𝑤𝑐=𝑈0+ 𝑔𝑘 2𝜎_𝑤𝑐𝐻

cosh𝑘(𝑧+ℎ)

cosh𝑘ℎ cos(𝑘𝑥−𝜎𝑡) (3)

whereU0isunidirectionalcurrentvelocity,gisthegravitational accel-eration,𝜎wcistheangularfrequencyassociatedwithcombinedwaves andcurrents(𝜎wc=𝜎 −U0k),𝜎 isangularfrequency,kisthewave num-ber,Hiswaveheightandhiswaterdepth.Thesubscriptwcindicatesthe caseofcombinedwavesandcurrents.Uwcisthemeasuredcombined ve-locityatthehalfwaterdepth,asitroughlyequalstothedepth-average velocityinthevegetationcanopy,i.e.theactingvelocityonvegetation.

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Fig.2. Experimentset-upofHuetal.(2014)tomeasuresynchronizedﬂowvelocity(U_wc)andactingforce(F)onwoodencylinders(mimicvegetation)atlocations 1–4.(a)isthetopviewoftheinstrumentsandthemimicvegetationdeployment.(b)isaphotooftheconstructedthemimicvegetation.(c)isaphotoofsynchronized forceandvelocitymeasurementtoobtainin-phasedata.Thewhitedashlinesindicatetwoinstrumentsareplacedatthesamecross-section.

2.2.3. Deriving𝐶_𝐷_𝑐𝑎𝑙incombinedwave-currentﬂowsbycalibration method

Thissectiondescribesthederivationof𝐶𝐷_𝑐𝑎𝑙incombined current-waveflowbycalibrationmethodfollowingLosadaetal.(2016a,b).Itis onlyuntilrecentlythecalibrationapproachhasbeenextendedto com-binedwave-currentflows,asmostpreviousmodelsdonotaccountfor theeffectofcurrentsonWDV.Losadaetal.(2016a,b)modifiedthe an-alyticalformulationofDalrympleetal.(1984)toincludetheeffectof currentsonWDV𝐶𝐷_𝑐𝑎𝑙incombinedwave-currentflowscanbederived as: 𝐶𝐷_𝑐𝑎𝑙= [ 𝑔(1+ 2𝑘ℎ sinh2𝑘ℎ )( 𝑔 𝑘tanh𝑘ℎ )1∕2 +𝑔𝑈0 ( 3+ 4𝑘ℎ sinh2𝑘ℎ ) +3𝑘𝑈02 ( 𝑔 𝑘coth𝑘ℎ )1∕2] 𝛽∕ [ 16 3𝜋𝑁ℎ𝑣𝑏𝑣 ( 𝑔𝑘 2𝜎_𝑤𝑐 )3 sinh3𝑘ℎ_𝑣+3sinh𝑘ℎ_𝑣 3𝑘cosh3_𝑘ℎ 𝐻0 ] (4)

where𝛽 isadampingcoeﬃcientstemmedfromrelativewaveheight (K_v)attenuationinDalrympleetal.(1984):

𝐾𝑣=_𝐻𝐻 0

= 1

1+𝛽𝑥 (5)

whereHisthewaveheightalongthevegetationmimicareaandH0is theinitialwaveheight.Whenspatialwaveheightdataareavailable,𝛽 canbeobtainedbyﬁttingtheEq.(5).Subsequently,theobtained𝛽 can besubstitutedinEq.(4)toderiveCD_cal.

2.2.4. Deriving𝐶𝐷_𝑑𝑖𝑟incombinedwave-currentﬂowsbydirect measurementmethod

Inbothpurewaveandcombinedwave-currentﬂows,forceona sin-glestemcanbeexpressedbyMorisonequation(Morisonetal.,1950): 𝐹=𝐹_𝐷+𝐹_𝑀=1 2𝜌𝐶𝐷_𝑑𝑖𝑟ℎ𝑣𝑏𝑣𝑈|𝑈| + 𝜋 4𝜌𝐶𝑀ℎ𝑣𝑏𝑣 2𝜕𝑈 𝜕𝑡 (6)

whereFisthetotalinlineforceonavegetationstem,FDisdragforce,FM isinertiaforce,𝜌 isthedensityoftheﬂuid,𝐶𝐷_𝑑𝑖𝑟andCMarethedrag

derivedbydirectmeasurementmethodandinertiacoeﬃcients respec-tively,hvistheheightofvegetationinwater,bvisthediameterof cir-cularcylinderandUishorizontalﬂowvelocity.

Thedirectmeasurementmethodderives𝐶_𝐷_𝑑𝑖𝑟fromtheperspective of theacting force on thevegetation cylinders.The period-averaged 𝐶𝐷_𝑑𝑖𝑟isderivedbycomputingtheworkdonebythetotalforceover oneperiod.ItisassumedthattheworkdonebyFMiszeroorcloseto zerooverafullwaveperiod,anditholdsforbothwave-onlyand com-binedwave-currentflows(Huetal.,2014).Therefore,theworkdone byFDisequaltotheworkdonebythetotalforce(Fwc).Thus,a period-averageddragcoefficientcanbederivedfromthefollowingequation (Huetal.,2014): 𝐶𝐷_𝑑𝑖𝑟= 2∫₀𝑇𝐹_𝐷𝑈_𝑤𝑐𝑑𝑡 ∫0𝑇𝜌ℎ𝑣𝑏𝑣𝑈𝑤𝑐2 ||𝑈𝑤𝑐||𝑑𝑡 = 2∫ 𝑇 0 𝐹𝑈𝑤𝑐𝑑𝑡 ∫0𝑇𝜌ℎ𝑣𝑏𝑣𝑈𝑤𝑐2 ||𝑈𝑤𝑐||𝑑𝑡 (7) wherethetotalforceFandUwc canbedirectlyobtainedfromactual measurementstoderive𝐶𝐷_𝑑𝑖𝑟.ByapplyingEq.(7),itisnotnecessaryto separateFDandFMwhenderivingperiod-averageddragcoefficient. Ac-curatelyseparatingthesetwoforcescanbedifficultbecausebothforces arerelatedtoanunknown coefficient,i.e.C_D andC_M, andhave dif-ferentphaserelationswiththevelocity.Additionally,thisequationis applicableinbothwave-onlyandcombinedwave-currentconditions.

TocheckifitisvalidtoneglecttheworkdonebyFMinEq.(7),we quantifyandcomparetheworkdonebyF_DandF_M.Thetime-varying workdoneisevaluatedasfollowing:

𝜀𝐷=𝐹_𝐷𝑈_𝑤𝑐 (8)

𝜀𝑀=𝐹𝑀𝑈𝑤𝑐 (9)

wherethetime-varyingFDandFMareobtainedbyseparatingthetotal measuredforce.Weassumetheinertiacoeﬃcient(CM)is2for cylin-ders(e.g.DeanandDalrymple,1991)andcalculatedtheF_Mfollowing Eq.(6).FDisthenderivedbysubtractingFMfromthetotalforce. Period-averagedworkdonebydragforce(𝜀_𝐷)andinertiaforce(𝜀_𝑀)canbe obtainedbyaveragingtheEq.(8)and(9)overafullwaveperiod.

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H. Chen et al. Advances in Water Resources 122 (2018) 217–227

Fig.3. Theworkflowofrevisiting(checking)thederiveddragcoefficientsby differentmethods.ThecalibrationmethodderivesC_{D_cal}fromtheperspective

ofwaveenergydissipation,whereasthedirectmeasurementmethodderives

C_{D_dir}fromtheperspectiveofactingforceonvegetation.Weexaminethe de-riveddragcoeﬃcientsbyrevisitingnotonlytheirdirectlylinkedquantities(i.e. energydissipationoractingforce,thesolidredarrows),butalsotheircounter parameters(thedashredarrows).(Forinterpretationofthereferencestocolour inthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

2.2.5. RevisitingthederivedC_D

ThecalibrationapproachderivesCD_calfromtheperspectiveofwave energydissipation,whereasthedirectmeasurementapproachderives C_D__dirfromtheperspectiveofactingforce.Inordertoprovidean ob-jectiveandquantitativeevaluationofthetwodifferent methods, we revisitthederiveddragcoefficientsfollowingtheprocedureshownin Fig.3.Thederiveddragcoefficientsbybothmethodsareusedto com-puteboththewaveenergydissipationandtheactingforce.Thus,the deriveddragcoefficientswerenotonlyexaminedbytheirownlinked quantity(energydissipationoractingforce)butalsotheircounter quan-tity,providingacross-checkofthetwodifferentmethods.

TocheckthevalidityofthederivedCD_calandCD_dirinreproducing WDV,theywereusedtocompute𝛽 byreversingEq.(4).Theobtained 𝛽 isthenusedinEq.(5)tocomputeKv, andsubsequentlycompared withthemeasuredKvforevaluation.ThereproducedKvisdenotedas Kv_calwhenCD_cal isused,andKv_dirwhenCD_dir isused.Similarly,to checkthevalidityofC_D__calandC_D__dirinreproducingactingforce,they areutilizedinEq.(6)toreproducebothtime-varyingandthemaximum totalforce,whicharesubsequentlycomparedwiththemeasurements. ThereproducedmaximumtotalforceisdenotedasF_{cal_}_maxandF_{dir_}_max, respectively.ItisexpectedthattheWDVcanbebetterreproducedby usingCD_calandactingforcecanbebetterreproducedbyusingCD_dir. Theserevisitingproceduresareconductedtosetacontextforthe cross-check:reproducingforcewithCD_calandreproducingWDVwithCD_dir. Bycombiningbothchecks,wecanevaluatewhichmethodcanderive dragcoeﬃcientsthathaveabetteroverallperformanceinreproducing bothforceandWDV.

3. Results

3.1. WDVandC_Dderivedviacalibrationmethod

ThecalibrationmethodderivesCD_calbasedonspatialwaveheight reductionpattern,whichcanbeinﬂuencedbyco-existingcurrents.The waveheightdata ina recentstudies(Huetal., 2014)areshown in Fig.4todemonstratetheinﬂuenceofco-existingcurrentsonWDVand toillustratehowtocalibrateCD_calvaluesfromthewaveheightdata.

TheexperimentofHuetal.(2014)showsthatWDVcanbeeither promotedorsuppressedbyafollowingcurrent,dependingonthe (rel-ative)magnitudeofthecurrentvelocity(Fig.4b).TheWDVvariations leadtodifferent𝛽 values,andeventuallyarereflectedindifferentC_{D_cal} values.Thetestedvegetationdensitywas139stems/m2_,_and_the_tested mimicswere0.36mtall(with0.5mwaterdepth).Theincidentwave wasregularwavewith0.08mwaveheightandthewaveperiodwas

Fig. 4. Thereductionof relativewave height(K_v)along tested vegetation

patches.ThedatawereobtainedinHuetal.(2014).TestC05Wmeansthe

wavewith0.05m/scurrentvelocity.

1.5s.Inwave-onlyconditions,𝛽 isﬁttedtobe0.059.Withasmall fol-lowingcurrent(0.05m/s),𝛽 (andWDV)isreducedtobe0.041,butwith largerfollowingcurrents(0.15–0.30m/s),𝛽 (andWDV)increasesfrom 0.067to0.132,whichishigherthanthatofthewave-onlycondition. Thereasonforthevariationin WDVwithdiﬀerentfollowingcurrent velocitymagnitudeisillustratedinHuetal.(2014).

3.2. 𝐶_𝐷_𝑑𝑖𝑟derivedviadirectmeasurementmethod

Thedirectmeasurementmethodderives𝐶𝐷_𝑑𝑖𝑟bycalculatingthe workdonebythetotalforceactingonvegetationincludingbothdrag force(F_D)andinertialforce(F_M)part(seeEq.7).Itisassumedthatthe workdonebyFMisclosetozerooverafullwaveperiodormuchsmaller comparingtothatofFD.Thus,itisnotnecessarytoseparatethemwhile estimatingtheperiod-averaged𝐶𝐷_𝑑𝑖𝑟.Therelativemagnitudeofwork donebyFDandFMisthereforeofimportancetosuchassumption.

TheworkdonebyFDandFMovertwowaveperiodareshownin Fig.5.Thewaterlevelrigidplantmimicdensityandwaveconditions arethesame,andthecurrentvelocityincreasesfromFig.5atod.The temporallyvarying𝜀Dand𝜀Mhaveclearcyclicbehaviors.𝜀Disalways positive,but𝜀Malternatesbetweenpositiveandnegativevalues.With theincreasedfollowingcurrentvelocity,𝜀Dandtheperiod-averaged𝜀𝐷 becomeslarger,whereasthenetvalueofperiod-averaged𝜀𝑀remains closetozero.Inall4cases,𝜀_𝐷issuﬃcientlyhigherthan𝜀_𝑀.Thus,the workdonebythetotalforce inEq.(7)isdominatedbyF_D,andthe inﬂuenceof𝜀𝑀islimitedoverafullwaveperiod.

Fig.6summarizesallthecasestestedinHuetal.(2014),andshows 𝜀𝐷isingeneralmuchlargercomparedto𝜀𝑀.Itisclearthatwiththe increaseofKC,𝜀𝐷increases,whereasEIremainsclosetozero.Theratio betweentheabsolute𝜀𝐷and𝜀𝑀issmallerwithsmallKCvalues(around 10).Thesmallestratiobetweenthemisabout3,implyingthatthe𝜀𝐷 isalwaysthebulkpartoftheworkdonebythetotalforce.Thus,itis consideredacceptabletoderive𝐶𝐷_𝑑𝑖𝑟viaEq.(7)withoutseparatingthe respectivecontributionofF_DandF_M.

3.3. CD-KCrelationsinwave-onlycondition

Wefirstderivedragcoefficientsinwave-onlyflowsthataretested inHuetal.(2014),asitistheconditioninvestigatedbymostprevious studies.ItisclearthatCD-KCrelationderivedbythedirect measure-mentmethodsharesthesamegeneralpatternasthosederivedbythe calibration method,i.e.CD decreaseswiththeincreasedKC(Fig.7). Comparingtothecalibrationmethod,theCD-KCrelationfromthe di-rectmeasurementmethodleadstolessscatteringamongdifferentmimic

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Fig.5. Workdonebyactingdragforce(𝜀D )andinertiaforce(𝜀M )indiﬀerenthydrodynamicconditions;PWrepresentswave-onlyconditionandC05W,C15Wand C20Wrepresentthewavewithunderlyingcurrent0.05m/s,0.15m/sand0.20m/srespectively.

0

10

20

30

40

50

60

70

80

90 100 110 120

KC

0.00

0.05

0.10 W

ork done by drag f

o

rce or inertia f

orce (J)

Fig.6. TherelationbetweenKCnumberandtheworkdonebydragforce(𝜀𝐷 )orinertiaforce(𝜀𝑀 )overawaveperiod.‘pw’standsforwave-onlyconditionsand ‘cw’standsforcurrentwaveconditions.

densitiesandsubmergenceconditions(Fig.7candd).Followingthe di-rectmeasurementapproach,theCD-KCrelationforpurewavecasesis:

𝐶𝐷=6.94∗𝐾𝐶(−0.72)+0.87(𝑅2=0.79) (10) ItisnotedthattheaboverelationhasmuchhigherR2_value com-paringtothatderivedbythecalibrationmethod(R2₌_0.21). Further-more,theaboverelationissimilartotheCD-KCrelationproposedin Ozerenetal.(2014)(Fig.7a),butdiﬀerentfromthatinJadhavetal. (2013)(Fig.7b).ItisnotedthattherelationinOzerenetal.(2014)is ap-plicablewhenKCisintherangebetween5and35,whereastherelation inJadhavetal.(2013)isapplicablewhenKCisintherangebetween 25and135.

3.4. C_D-KCrelationincombinedwave-currentﬂows

ByusingthenewmodelofLosadaetal.(2016a,b),dragcoefficients incombinedcurrent-waveflowscanalsobederivedbythecalibration method.Previously,theycouldonlybederivedbythedirect measure-mentmethod.InFig.8,wecomparetheCD-KC relationsderivedby bothmethods.Bothrelationsforcombinedwave-currentflowhavethe generalreductiontrendsimilartothatinwave-onlyconditions.Because ofthesuperimposedcurrentU0,thecombinedwave-currentconditions wereinherentlyassociatedwithhigherKC.AsKCvariesintherange between7and120,the𝐶𝐷_𝑐𝑎𝑙fromthecalibratedmethodreducesfrom 10.59to0.25.Itisclearthat𝐶𝐷_𝑐𝑎𝑙hasasubstantialdegreeof scatter-ingamongdifferentmimicstemdensitiesandwaterdepths.Thedegree of scatteringismuch reducedin therelationbetween𝐶𝐷_𝑑𝑖𝑟andKC

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Fig.7.RelationbetweenKCandC_Dbasedonvariousdatasource.(a)isbasedoncalibratedC_DinOzerenetal.(2014);(b)isbasedoncalibratedC_DinJadhavetal. (2013);(c)isbasedoncalibratedC_DinHuetal.(2014);(d)isbasedonC_DdatathatarederivedbythedirectmeasurementmethodinHuetal.(2014).

Fig.8. RelationbetweenKCandC_D:panelaisderivedbycalibrationmethodincombinedwave-currentﬂow;panelbisderivedbydirectmeasurementapproach

incombinedwave-currentﬂow.

(Fig.8).Theobtained𝐶𝐷_𝑑𝑖𝑟iswithintherangeof1to2.No appar-entdiﬀerencebetweendiﬀerentdensitiesandsubmergenceratiocanbe observed.

To obtain a generic relation for both wave-only conditions and combinedwave-currentconditions,wesummarizeallthe𝐶𝐷_𝑑𝑖𝑟from Huetal.(2014)in Fig.9.TheCD-KCrelationforbothﬂowscan be

expressedas:

𝐶𝐷=12.89∗𝐾𝐶(−1.25)+1.17(𝑅2=0.66) (11) The above relationshows that in bothwave-onlyconditions and combinedwave-currentconditions,CD-KCrelationhasasimilartrend, i.e.,withtheincreasingKC,CDgraduallydecreasesandapproachesto aconstantvalue(i.e.1.2).Furthermore,thisempiricalC_D-KCrelation

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0 10 20 30 40 50 60 70 80 90 100 110 120 0 1 2 3 4

KC

C

D VD1,emergent-pw VD2,emergent-pw VD3,emergent-pw VD1,submerged-pw VD2,submerged-pw VD3,submerged-pw VD1,emergent-cw VD2,emergent-cw VD3,emergent-cw VD1,submerged-cw VD2,submerged-cw VD3,submerged-cw fit line in this study fit line by Jadhav et al.,2013 fit line by Ozeren et al.,2014

Fig.9.C_D-KCrelationinbothwave-onlyandcombinedwavecurrentconditionsbasedondirectmeasurementmethod.Theredmarkersand“pw” standforthe resultsofwave-onlyconditions,whiletheblackmarkersand“cw” standsfortheresultsofcombinedwavecurrentconditions.theﬁtequationfromJadhavetal. (2013)andOzerenetal.(2014),seeTable1.

C20W

Fig.10. Measuredandreproducedtotalforceactingonﬁrstforcesensor(seeFig.1)overtwowaveperiods.

issimilartothatinOzerenetal.(2014),butdiﬀerenttothatinJadhav etal.(2013).

3.5. RevisitingthevalidityofthederivedCD_dirandCD_cal

Tofurthertesttheapplicabilityofthetwodiﬀerentmethods,the deriveddragcoeﬃcientswererevisitedbyusingthemtocomputeFas welltheWDV.Thecomputedforceandwavedecayaresubsequently comparedwiththemeasurements.Thecomputed(usingEq.(4))and measuredinstantaneoustotalactingforceisplottedinFig.10.Withno orsmallfollowingcurrents(PWandCW05cases),thetotalforce oscil-latesinbetweenpositiveandnegativedirections.Whenthefollowing currentsbecomeslarger(CW15andCW20),thetotalforcestaysinthe positivedirectionsforafullperiod.Itisclearthatthetemporal vari-ationofthetotalforce calculatedusing C_D__dir (F_dir)agreeswellwith

themeasuredtotalforce,whileasthetotalforcecalculatedusingC_D__cal (Fcal)overestimatesthetotalforcewhenthefollowingcurrentsisstrong. Theshowndataisfromthetestcasewith6cmwaveheightand1.2s waveperiodinHuetal.(2014).TheCD_dirforthePW,CW05,CW10 andCW20conditionare2.51,1.76,1.33and1.37,respectively. How-ever,theCD_calforthesameconditionsare3.94,3.82,3.53and4.46, which areconsiderablylarger thanCD_dir.Themaximumreproduced forceoverawaveperiod(F_{max_cal}andF_{max_dir})showninFig.10are fur-theranalyzedinthefollowingsection.

Fig.11showstheresultsofthecross-checkprocessasdemonstrated inFig.3.Itisnotsurprisingthatgoodagreementcanbeobtainedwhen check thederivedCD_dir andCD_cal withtheirownlinkedquantities. TheR2_value_is_0.98_between_F

dir_max(i.e.themaximumtotalforceover onewaveperiodcomputedusingC_D__dir)andthemeasuredmaximum totalforce(Fmea_max)(datanotshow).Similarly,theR2valueis0.99

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be-H. Chen et al. Advances in Water Resources 122 (2018) 217–227 0 0.1 0.2 0.3 0.4 0.5 0.6

F

mea_max

(N)

0 0.1 0.2 0.3 0.4 0.5 0.6

F

ca l_ m ax

(N)

F_mea-F_cal y=x 0 0.2 0.4 0.6 0.8 1

K

v_mea 0 0.2 0.4 0.6 0.8 1

K

v_d ir K_{V_mea}-K_{V_dir} y=x R2_=0.2619 R2_=0.5768

a

b

Fig.11. (a)comparingreproducedmaximumtotalforcebasedonC_D_cal(i.e.Fcal_ max)withmeasuredmaximumtotalforce(Fmea_ max),(b)comparingthereproduced relativewaveheightbasedonC_D_dir(i.e.Kv_dir )withthemeasuredrelativewaveheight(Kv_mea ).They=xlinesareplottedforreference.

tweenKv_calandKv_mea(datanotshow).Thecross-checkprocess, how-ever,providesuswithfutureinsights.ItisclearthatFcal_max(i.e.the maximumtotalforceoveronewaveperiodcomputedusingC_D__cal)does notmatchwithFmea_max.ThecorrespondingR2valueisonly0.26. Simi-larly,theagreementbetweentheKv_dir(i.e.WDVcomputedusingCD_dir) andKv_meaisalsonotideal.TheresultingR2valueis0.58.Insummary, itisclearthatthetotalforceandWDVderivedusingCD_calhavebetter overallagreementwithmeasurements.

4. Discussion

4.1. CD-KCrelationsinwave-onlyandcombinedwave-currentconditions Innaturalcoastalwetlands,combinedwave-currentflowsare com-monflowconditions.Toourknowledge,existingCD-KCrelationsareall forvegetationinwave-onlyconditions,suchrelationswhereasin com-binedwave-currentconditionshavenotyetbeenderived.By reanalyz-ingthedataofHuetal.(2014),wederivedanewoverallCD-KCrelation forbothwave-onlyandcombinedwave-currentflowconditionsinthe presentstudy.Thisnewrelationretainsthesamegeneralformas previ-ousstudieslistedinTable1.WithincreasingKCnumber,theCDvalues decreaseregardlessoftheflowconditionsandgraduallyapproach1.The derivedrelationisofvaluetotheunderstandingandmodellingWDV.

C_D hasgenerallybeenexpressedasfunctionsof Re,andprevious studieshavederivedCD-Rerelationsforpurecurrent,wave-onlyand combined wave-currentconditions (Huet al., 2014). Previous stud-ieshavesuggestedthatC_D-KCrelationsaremoresuitable for oscilla-toryﬂows (Augustinetal., 2009;MendezandLosada,2004;Ozeren etal.,2014).ComparedtoRethatdependssolelyonmaximum veloc-ity,KCnumberscontainadditionalinformationofwaveperiod.Thus, theyareexpectedtoresultinmoresuitablefunctionsindescribingCD dynamics.However,thenewCD-KCrelationobtainedhereshow other-wise.TheR2_value_of_the_derived_new_C

D-KCrelation(includingboth pure-waveandcombinedcurrent-wave) is0.66,whichislower com-paredtotheR2 _value_(0.89)_of _the_derived_overall_C

D-Rerelation in Huetal.(2014). Thismaybeattributedtothetestvegetation mim-icsinHuetal.(2014)wererigidsticks,whereaspreviousstudiesthat obtainedbetterCD-KCcorrelationsgenerallytestedﬂexiblevegetation mimics(e.g.Augustinetal.,2009;MendezandLosada,2004;Ozeren etal.,2014).Thisindicates thatthedependenceof C_D onKC (wave

period)is strongerwithﬂexible vegetation.Nonetheless,thederived CD-KCrelationisofvaluetointerpretingtheWDVprocess.Ourresults conﬁrmthatinbothwave-onlyandcombinedwave-currentconditions, thevariationofCDwithKCfollowsthesametrend,whichhasnotbeen reportedbefore.

4.2. ComparingtwodiﬀerentmethodsinderivingCD

Thecurrentstudyderivesthedragcoefficientsbytwodifferent ap-proaches, aimingtocomparethemandprovide guidelinesforfuture experimentalstudies.Suchacomparisonwasnotpossibleuntilthe re-centmodeldevelopmentthatincludestheeffectofcurrentintoWDV modelling(Losadaetal.,2016a,b).Adetailedcomparisonofthesetwo methodsisincludedintheTable2.

Thesetwomethodsarecomparedintermsoftheirmainequations, required data, flow conditions, applicable environments,the perfor-mancesinreproducingforceandWDV,aswellastheR2_value_of_the CD-KCrelations(Table.2).Thecalibrationapproachmakesuseofthe energydissipationequationandwaveheightdata,whereasthedirect measurementapproachreliesonMorrisonequationandsynchronized velocityandforcedata.Clearly,theCD-KCrelationderivedbythedirect measurementmethodhaveconsiderablyhigherR2_value_than_that de-rivedbythecalibrationmethod(Figs.7and8).However,thecalibration approachhasawiderrangeofapplication,asitcanbeappliedinboth labandfieldenvironments.Withthecurrentinstrumentation,thedirect measurementmethodisonlyapplicableinlabconditions,asitrequires synchronizedforceandvelocitydatawithhighaccuracy,whicharenot feasibletoobtaininfieldconditions.Additionally,thecalibration meth-odscanbereadilyusedinflexiblevegetationcases(Mazaetal.,2015b; Laraetal.,2016;I.J.Losadaetal.,2016a,b),whereasthedirect mea-surementmethodisnotyetabletodoso.Itisbecausesuchmethod re-quiresmeasurementofactingvelocityonvegetationstem,i.e.relative velocitybetweenwatermotionandvegetationstemmotion,whichis difficulttomeasurewiththecurrentsetup.However,itispossibleifthe videoanalysistechniqueisincludedforrelativevelocitymeasurements (LuharandNepf,2016).

ThecalibrationmethodderivesC_Dfromtheperspectiveofwave en-ergydissipation,whereasthedirectmeasurementmethodisfromthe perspectiveofvegetation-inducedforce.Inordertoevaluatethesetwo methodsobjectively,weusedthederivedC_Dfromthediﬀerentmethods

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Table2

ComparisonoftwoapproachesindeterminingC_D.

Calibration approach Direct measurement approach

References Dalrymple et al., 1984; Kobayashi et al., 1993; Losada et al.,

2016a,b; Mendez and Losada, 2004; Möller et al., 2014

Hu et al., 2014; Infantes et al., 2011

Main equation

Wave height reduction by vegetation a_:

𝐶 𝐷 = [ 𝑔 ( 1 + 2𝑘ℎ sinh2𝑘ℎ ) ( 𝑔 𝑘 tanh 𝑘ℎ ) 1∕2_{+ 𝑔 𝑈} 0 ( 3 + 4𝑘ℎ sinh2𝑘ℎ ) + 3 𝑘 𝑈 02( 𝑔_𝑘_{coth 𝑘ℎ )} 1∕2 ] 𝛽∕ [16 3𝜋𝑁 𝑏 𝑣 ( 2𝑔𝑘𝜎𝑤𝑐) 3sinh3_𝑘_ℎ 𝑣+3sinh𝑘ℎ𝑣 3𝑘cosh3_𝑘ℎ 𝐻 0 ] Morrison equation: 𝐶 𝐷 = 2∫ 𝑇 0𝐹𝑈𝑤𝑐𝑑𝑡 ∫𝑇 0𝜌ℎ𝑣𝑏𝑣𝑈2 𝑤𝑐|𝑈𝑤𝑐|𝑑𝑡 Required data

Wave height spatial distribution Synchronized impact ﬂow velocity ( U wc )

and acting force ( F ) on vegetation cylinders

Flow conditions

Wave-only and combined wave-current ﬂow Wave-only, pure current and combined wave-current ﬂow

Applicable environ-ment

Laboratory and ﬁeld Laboratory

Applicable vegetation

Rigid and ﬂexible vegetation Rigid vegetation

R2 value when revisiting acting Forceb 0.26 0.98 R2 value when revisiting Kvc 0.99 0.58 R2 value of CD - KC relationsd 0.19 ( C D = − 0.024 ∗ KC −1.05 + 3.26) 0.66 ( C D = 12.89 ∗ KC −1.25 + 1.17) a_The_listed_equation_is_the_recent_formulations_derived_in_Losada_et_al._(2016a,b₎_for_combined_current_and_wave_ﬂows._It_should_be_noted thatavarietyofequationsexistincalibratingC_D,dependingontheappliedwavedecaymodel.

b,c,d_The_comparison_is_based_on_the_data_in_Hu_et_al.₍₂₀₁₄₎_.

tocomputeboththeactingforceandtheWDV.Thus,providinga cross-examinationofthesetwomethods(Fig.3).WeshowthattheCDvalues derivedfromtheperspectiveofforcecanbetterreproducethemeasured force(R2₌_0.98)_compared_to_C

D_cal(R2=0.26).However,theCD val-uesofthedirectmeasurementmethodperformpoorerinreproducing WDV(R2₌_0.58)_compared_to_that_of_C

D_cal(R2=0.99).Therefore,the CD derivedfromeitherenergyorforceperspectivecanﬁtbetterwith theirownrespectivequantitybutnotthecounterquantityasshownin Fig.11.

ThereasonthatdifferentmethodsleadstodifferentCDvaluesis per-hapsthattheworkdonebythedragforceisnottheonlyprocess lead-ingtoWDV.Otherprocessesliketurbulenceaswellassurfacefriction ofvegetationmimicsandflumewallsalsocontributetoenergy dissipa-tion,buttheyarenotaccountedinthecurrentWDVmodels.Thus,the derivedCD_calisactuallyasynthesisforanumberofprocesses,whereas CD_dirisonlyresponsibleforactingforce.Becauseoftheinvolvementof theadditionalprocesses,theC_D__calandC_D__dirdonotprovidethesame performanceinthecross-checkasthecheckwiththeirownrespective measurements(seeFig.3).Overall,thecheckofCD_dir obtainsbetter agreementwithmeasuredactingforce(R2₌_0.98)_and_WDV₍_R2₌_0.58). Therevisitingprocedurealsoimplythatnumericalmodelsthatarebuilt uponmomentumconservationequationsshouldseektousetheCD_dir, whereasothermodelsthatrelyonenergyconservation(anddonot ex-plicitlyaccountforturbulenceandfrictioneffects)shoulduseC_D__calfor bettersimulationsofWDVprocess.

5. Conclusions

Byre-analyzingthepreviousdataofHuetal.(2014),thisstudyaims toreduce theuncertaintiesin thediﬀerentCDderivingmethodsand thedependenceofC_Donhydrodynamicparameters(i.e.ReandKC). ThetwoavailablemethodsinderivingCD,i.e.thedirectmeasurement methodandthecalibrationmethod,arecomparedintermsoftheirmain equations,requireddata,ﬂowconditions,applicableenvironments,and

theresultingC_D-KCrelations(Table2).Furthermore,wecreateaunique re-visitingprocedure,whichevaluateshowwellthederiveddrag coef-ﬁcientscanbeusedtoreproducethemeasuredwavereductionand act-ingforce.Toourknowledge,currentstudyistheﬁrststudyproviding athoroughcomparisonbetweenthesetwomethods,whichmayassist experimentdesignforfurtherinvestigationofCD.

Additionally,weformulateanewempiricalrelationbetweenCDand KCasanextensiontotheC_D-RerelationinHuetal.(2014).TheC_D -KC relation is basedon thedirectmeasurementmethod,andit isa genericrelationforbothwave-onlyandcombinedwave-current con-ditions.ThederivedC_D-KCrelationforbothwave-onlyandcombined wave-currentconditionshaveasimilardecreasingtrendasprevious re-lations forwave-onlycases,whichhasnot beenreportedpreviously. TheobtainedgenericC_D-KCrelationisexpectedtobeusefultofuture numericalmodelingstudies.

Acknowledgements

The authors thank two anonymous reviewers and an Editorial Boardmemberfortheircommentsthathelpedtoimprovethispaper. TheauthorsgratefullyacknowledgeﬁnancialsupportoftheNational NaturalScienceFoundation ofChina(No.51609269,51520105014), and the Joint Research Projects NSFC (No. 51761135022) – NWO (No. ALWSD.2016.026) – EPSRC (No. EP/R024537/1): Sustainable Deltas,andScienceandTechnologyProgramofGuangzhouCity,China (201806010143).

Supplementarymaterials

Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.advwatres.2018.10.008.

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