Étudeexpérimentaledelaréponsed’uneplaqueflexibleàunforçage harmoniquedansunécoulement
Florine Paraz, Christophe Eloy, Lionel Schouveiler∗
Aix–MarseilleUniversité,CNRS,CentraleMarseille,IRPHEUMR7342,13384Marseillecedex13,France
a rt i c l e i n f o a b s t ra c t
Articlehistory:
Received14April2014 Accepted10June2014 Availableonline16July2014
Keywords: Fluid–structureinteraction Flexibleplate Harmonicforcing Resonance Non-linearity Propulsion Mots-clés : Interactionfluide–structure Plaqueflexible Forçageharmonique Résonance Non-linéarité Propulsion
Mostaquaticanimalspropelthemselvesbyflappingflexible appendages.To gaininsight intotheeffectofflexibilityontheswimmingperformance,wehavestudiedexperimentally anidealized system. It consists ofa flexible platewhose leading edgeis forced into a harmonicheavemotion,andwhichisimmersedinauniformflow.Astheforcingfrequency isgraduallyincreased,resonancepeaksareevidencedontheplate’sresponse.Inaddition totheforcingfrequency,theReynoldsnumber,theplaterigidityandtheforcingamplitude havealsobeenvaried.Intherangeofparametersstudied,themaineffectontheresonance isdue tothe forcing amplitude, whichreveals thatnon-linearities are essentialinthis problem.
©2014Académiedessciences.PublishedbyElsevierMassonSAS.All rights reserved.
ré s u m é
La plupart des animaux aquatiques se propulsent grâce au battement d’appendices flexibles. Afin d’avoir une meilleure compréhension de l’effet de la flexibilité sur la performance de la nage, nous avons étudié expérimentalement un système idéalisé. Il consiste en une plaque flexible, immergée dans un écoulement uniforme, dont le bord d’attaque est forcé en un mouvement harmonique transverse à l’écoulement. En augmentantgraduellementlafréquencedeforçage,despicsderésonanceontétémisen évidence.Outrelafréquencedeforçage,onaégalementfaitvarierlenombredeReynolds, larigiditédelaplaqueetl’amplitudeduforçage.Dansledomainedeparamètresétudié, leprincipaleffetsurlarésonanceest dûàl’amplitudeduforçage,cequirévèlequeles non-linéaritéssontessentiellesdansceproblème.
©2014Académiedessciences.PublishedbyElsevierMassonSAS.All rights reserved.
*Correspondingauthor.
E-mailaddresses:florine.paraz@irphe.univ-mrs.fr(F. Paraz),christophe.eloy@irphe.univ-mrs.fr(C. Eloy),lionel.schouveiler@irphe.univ-mrs.fr
1. Introduction
Animalswimminghasattracteda lotofattention fromhydrodynamicistsinthe last decades.This interestwas partly motivatedbythedesignofbioinspiredaquaticpropulsiondevices.Numerousreviewscanbefoundonthedifferentaquatic propulsion modes, such as the monographs of Lighthill [1] and Childress [2] or the articles of Lighthill [3], Sfakiotakis et al.[4]and Triantafyllou etal. [5]. Propulsion usingflapping appendages iscommon forlarge aquatic animals and, in particular, mostof the fastswimmers such assharks, tunaand dolphins flaptheir caudalfins to propelthemselves. An observationoftheseappendagesshowsthatthey aregenerallyflexiblealong thechord.It hasoftenbeenargued (e.g.[6]
and[7]), that thisflexibilityenhancesthe swimmingperformance. However, theoriginofthisperformance enhancement hasprovedelusivesofar.
Inspiredbythiscaudal-finswimmingmode,differentstudieshavebeencarriedouttoquantifythepropulsionefficiency of a flapping foil. Propulsion by rigid foils has been first studied theoretically by Lighthill [8] and Wu [9]. It has also beentheobjectofexperimentsby Triantafyllouetal.[10],Andersonetal.[11],Schouveileretal.[12],andBuchholzand Smits[13],among others.These experimental studies haveall reported that propulsionperformance ismaximized fora flapping frequency f corresponding toa Strouhalnumber St≈0.3, where St=2 f ATE/U , with ATE theamplitudeofthe trailingedgemotionand U the swimmingspeed.Thesamevalueof St≈0.3 canbeobservedinnatureforswimmerswith moderateaspectratios(cetaceans,sharks,salmons,troutsandscombrids)asseeninthedatarecentlycompiledin[14].
ThenumericalsimulationsofKatz andWeihs[15] haveshownthat,usingaflappingfoilflexiblealongitschordrather than a rigid one, yields an increase of the propulsive efficiency. In this case, because of the foil flexibility, its natural structural frequencies are introduced in the system, in addition to the flapping frequency. These structural frequencies depend not only onthe geometryand thematerial ofthe foil butalsoon the surroundingflow because ofadded-mass effects.Two-dimensional numerical simulations ofAlben[16] andMichelin andLlewellynSmith[17] have later revealed peaksintheamplitudeoftheplateresponse,whentheflappingandstructuralfrequenciesareresonant.Theyshowedthat efficiencycanbemaximizedattheseresonancepeaks.
Gain in propulsive performance through the chordwise flexibility has been confirmed by the experimental works of Prempraneerach et al. [18], Heathcote andGursul [19], or Marais etal. [20]. Moreover, Dewey etal. [21] for a pitching flexibleplate,Albenetal.[22]andQuinnetal.[23] foraheavingflexible platehaveclearlyevidencedthephenomenonof resonanceexperimentally,astheforcingfrequencyisvaried,andconsidereditslinkwithpropulsiveperformance,butnone ofthesestudieshasconsideredtheeffectofchangingtheforcingamplitude.
Inthis study,to havea betterunderstanding ofthe dynamicsof aflexible fin, we have examinedthe response ofan elasticplateimmersedina uniformflowandforcedintoaharmonicheave motionatitsleading edge.Theobjective has beentoquantifytheinfluenceofthedifferentexperimentalparameters:theamplitude andfrequencyofforcing,theflow velocity, and the plate bending rigidity. This problem involves complex fluid–structure interactions since the flow load deforms the plate, whose motion in turnaffects the flow. We believe that the full complexity of these interactions can only beaddressed withexperimental studiesbecause, atthepresenttime, accurate numericalsimulations are limitedto moderateReynoldsnumbersandtheoreticalmodelsusuallyassumelinearproblems,i.e.small-amplitudepropulsion.
This paper is organized as follows: first the experimental set-up and methods are presented. Then the response of flexible plates to harmonicheave forcing is analyzed andeffects ofthe differentexperimental parameters are discussed. Finally,someconclusionsaredrawnanddiscussedinthecontextofswimming.
2. Experimentalset-up
Experimentshavebeenconductedwiththeset-upschematicallyillustratedinFig. 1(a).Itconsistsofahorizontalflexible platemoldedoutofpolysiloxanewitharoundedleadingedgeandataperedtrailingedge(Fig. 1(b)).Theplatesusedhave athickness e =0.004 m,span s =0.12 m,andchord c=0.12 m,givinganaspectratio s/c=1.Duringmolding,arigidaxis isinserted insidetheplateattheleadingedge; thisaxishassome roughnesstoprevent therotationoftheplatearound it.Thisset-upreducestheflexiblelengthofthechordtoabout0.115 m.Theplateisthenimmersedinauniformflow, of velocity U ,ofafreesurfacewaterchannel.Thetestsectionis0.38 mwide,withadepthofwateratrestof0.45 m.
AsshowninFig. 1(a),theaxisisattachedtoaninvertedU-framesuchthattheplate,attheleadingedge,ismaintained parallelto channelflow. ThisinvertedU-frameissetintovertical motionby acomputer-controlledlinearactuator,which allows to impose an arbitraryheave motionto the leading edge.In the presentexperiments,the leading edge hasbeen forcedintoa harmonicheavemotion: ALEcos(2πf t).Theplateisconfinedbetweentwo verticalwallsseparatedfromthe platebylessthan1 mm.The roleoftheseconfinementwallsis,first,to minimizetheflow aroundtheside edgesofthe plateand,second,theinvertedU-framebeingoutsideofthesewalls,toavoidtheperturbationsfromthewakeoftheframe. Inaddition,freesurface effectshavebeenpreventedby arigidhorizontalwall placedabovetheexperiment.Channeland confinementwallshavebeenmadeoftransparentmaterialtofacilitatevisualizations.
Toinvestigatethe response oftheplatewhen forcedinto heavemotion,visualizations havebeen carriedout through the side wall of thewater channel andrecorded witha video camera. The successive shapesandpositions of the plate centerlinehavethenbeenextractedbyimageanalysisoftherecordings.Aquantitativecharacterizationoftheplateresponse
534 F. Paraz et al. / C. R. Mecanique 342 (2014) 532–538
Fig. 1. Sketch of the experimental set-up (a) and of the flexible plate (b).
Fig. 2. Responseofaplatetoanimpulseperturbation.Thedeflectionofthetrailingedgeisplottedasfunctionoftimefortheplateofbendingrigidity
B=0.053 N minwateratrest.
In thepresentstudy,the variationsof fourexperimental parametershavebeen considered.First,the frequency f and amplitude ALEoftheforcinghavebeenvariedintheranges f=0.2–8 Hz and ALE=0.004–0.014 m.Then,theflowvelocity hasbeenincreasedupto U=0.1 m s−1.ThisvelocityisexpressedindimensionlessformusingtheReynoldsnumberbased onthechord: Re=U c/ν,where ν isthekinematicviscosityofwater.Finally,threeplaterigiditieshavebeentestedusing differentmaterials:B =0.018,0.028,0.053 N m.Forthesethreeplates,thedensityrelativetothewaterisabout 1.2.
A typical experimentconsists, fora plateofgivenbending rigidity, infixing theReynolds number Re and theforcing amplitude ALE andinrecordingtheplateresponseasthe forcingfrequency f is gradually varied.Experimentshavebeen repeatedchanging ALE, Re, andusingplatesofdifferentbendingrigidity.
3. Results
Prior totheexperiments withforcing,we haveconsideredthe responseofthe plateto animpulse perturbationofits trailing edge, inwater atrest. The dampedresponse to such a perturbation is shownin Fig. 2 forthe plateof bending rigidity B =0.053 N m.Fromthissignal,wededucethelowestnaturalfrequency f0oftheplateinwaterthatwillserveas areferencefrequencyinthefollowing.Forthethreeplatestestedinthepresentstudy,wefindthenaturalfrequencies f0= 0.75,0.99,1.30 Hzforbendingrigidities B =0.018,0.028,0.053 N m,respectively.Notethattheratiosofthesefrequencies scalelikethesquarerootofthebendingrigidityratios(asexpectedfromthevibrationanalysisofacantileverbeam)with anagreementbetterthan 6%.
Theresponseoftheflexibleplatestotheharmonicforcing, ALEcos(2πf t),imposedattheirleadingedgeisthen consid-ered.Wefirstnotethatforallthecasestestedtheplatedeformsmainlyalongthechordanditsdisplacementisharmonic
Fig. 3. Responseoftheplatetoaharmonicheaveforcing.Therelativeamplitudeofthetrailingedgedisplacement,ATE/ALE,andthecorrespondingphase shiftφareplottedasafunctionofthenormalizedforcingfrequency f/f0,forALE=0.004 m,Re=6000 andB=0.018 N m.
Fig. 4. Modeshapes,atthefirst f/f0=1 (a),(c),andsecondresonancepeak, f/f0=6.3 (b),(d).Thesedeformationsareshowninthelaboratoryframe
(a),(b)andintheframeattachedtotheleadingedge(c),(d),ALE=0.004 m,Re=6000 (flowfromlefttoright)andB=0.018 N m.
The plateresponse ischaracterized bymeasuring theevolution oftherelative amplitude ATE/ALE andthephase φ as theforcingfrequency f is varied. ThisresponseisillustratedinFig. 3forarepresentativecase: ALE=0.004 m, Re=6000 and B =0.018 N m.Inthisplot, f is madedimensionlessusingthenaturalfrequency f0oftheplate.
Remarkably,theamplitudecurveexhibitstwodistinctpeaks.Thefirstpeakissharpandoccursatafrequency f close to thenaturalfrequency f0,withamaximumofthetrailingedgeamplitudemorethan2.5timestheforcing amplitude.The second peak,ata frequency f/f0 between6.0and6.5, isflatterandlower inamplitude. Itisrecalledthatthethreefirst naturalvibrationmodesofaclampedplatehavedimensionlesswavenumbers k0c=1.875, k1c=4.694,and k2c=7.855 and thattheratiosofthecorrespondingnaturalfrequencies f0, f1 and f2 scalelikethesquareofwavenumberratios.Itresults that f1/f0= (k1/k0)2≈6.3.Wecanthus concludethat thetwopeaksobserved at f≈ f0 and f1 inFig. 3correspondto resonancesofthe forcingwiththefirsttwo naturalstructuralmodes. Accordingto thisanalysisthethird resonancepeak wouldbeexpectedatafrequency f≈ f2 with f2/f0= (k2/k0)2≈17.6,whichisoutsideofthefrequencyrangeexplored inthepresentexperiments.
The evolution ofthe trailing edge phase φ relative tothe imposed leading edge motionis also shownin Fig. 3.The phaseappearstobecloseto−π farfromthetworesonancepeaksandapproaches−π/2 atthefirstpeakwhichiswhat isexpectedfroma simpledampedoscillatormodel.Incontrast,itisnotpossibletodistinguishacleartrendforφ atthe secondpeak,whereitcontinuouslyvariesbetween−3 and−5.
InFig. 4,thedeformation oftheplateat thetwo resonantpeaksis illustrated byplottingsuperimposed viewsofthe platecenterlineduringoneforcingcycle.Itshouldbenotedthatdirectvisualizationsoftheplatedonotrevealsignificative deformationsalongthespan,suchthattheplatedeflectioniswell representedbythecenterlinedisplacement.Itcanalso beremarkedthataweakup-and-downasymmetryinthemodeshapeisapparent,whichisduetotheplatematerialbeing slightlydenserthanwater.
Figs. 4(a)and(b)show themodeshapesatthetworesonancepeaks f/f0=1 and6.3,respectively,inthelaboratory frame.Contrarytothefirstmode,thesecondmodeexhibitsaneckclosetoitstrailingedgeshowingthathigherstructural modesareinvolvedathighfrequencies.Thesamemodesarerepresentedintheframeattachedtotheplateleadingedgein
Figs. 4(c)and(d).Theseviewsrevealtheirclosesimilaritieswiththeflutterinstabilitymodesofaclampedplateimmersed in a uniform flow investigated by Eloy et al. [24], among others. Note that these deformations are different from the
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Fig. 5. Effectsoftheexperimentalparametersonthefrequencyresponse.Responsecurvesasafunctionoftheforcingfrequencyf (a)andofthenormalized
frequency f/f0(b)for ALE=0.004 m,Re=6000 andthreevaluesofthebendingrigidityB.Frequencyresponsefor ALE=0.004 m,B=0.028 N m and threevaluesoftheReynoldsnumberRe (c)andforRe=6000,B=0.018 N m andthreevaluesoftheforcingamplitudeALE(d).
thesystemplate+surroundingfluidandnottheplatealone,asemphasizedbyMichelinandLlewellynSmith[17]intheir numericalstudy.
So far,we havedescribed thefrequencyresponse ofaflexible plateforfixed valuesoftheforcing amplitude ALE,the Reynolds number Re, and the bending rigidity B. We will examine now how variations of these parameters affect the frequency response. To do so, the sameprotocol is used: ALE, Re and B are fixed andthe relative response ATE/ALE is plottedasfunctionofthenormalizedfrequency f/f0(Fig. 5).
3.1. Effect of the plate rigidity
InFig. 5(a),theresponsecurvesareplottedasafunctionofthedimensionalforcingfrequency f for threevaluesofthe platerigidity B. Wenote thatthe amplitude maximaarenot affectedbychanging B and thattheresonance peaksmove towardhigherfrequencyas B is increased.However,thenaturalfrequencies f0 increasingwiththeplaterigidity,whenthe responsecurvesareplottedusingthedimensionlessfrequency f/f0,theyallcollapseonasinglecurveasseeninFig. 5(b). In conclusion,in thelimit of thepresentstudy,which considers plateswithbending rigiditiesvarying by a factor3,we couldnotdetectanysignificanteffectoftheplaterigidity,aslongasthefrequencyisproperlynormalizedwiththenatural frequency f0.
3.2. Effect of the Reynolds number
TheeffectoftheReynoldsnumber Re=U c/νontheplateresponseisillustratedinFig. 5(c)representingthefrequency responseforthreevaluesof Re. Thefirstvalueis Re=0 correspondingtowateratrest,thetwoothersare6000and12,000, corresponding to U=0.05 and0.1 m s−1,respectively.We observebotha decreaseoftheamplitude maximaandashift of theresonance peakstowardhigher frequencies, asRe is increased.This latterobservationis dueto thenormalization performedusingthenaturalfrequency f0oftheplateinwateratrestwhilethenaturalvibrationmodesaremodifiedbya
Fig. 6. ResponsecurvesobtainedwithEq.(3)fordifferentvaluesoftheforcing γ.Thefollowingparametershavebeenused: f0=1/2π, μ =0.1, ν=3, and γ=0.04,0.1,or0.14.
takenbyafluidparticletotravelalongthechordtotheforcingperiod.Forsmallervaluesof fr,onewouldexpectthewake tobecomemoreimportantandtoaffectthefrequencyresponsemoresignificantly.
3.3. Effect of the forcing amplitude
Contrary tothe other parameters tested, variations ofthe forcing amplitude ALE havea stronginfluence onthe plate response.ThisinfluenceisshowninFig. 5(d),wherethefrequencyresponseisplottedforthreevaluesof ALE.Anincrease oftheforcingamplitude byafactorof3.5(from0.004to0.014 m)leads toadecreaseoftherelativeresponseamplitude of the trailing edge by more than 30% and a slight decrease of the normalized frequency at the resonance peaks.This importanteffectoftheforcingamplitudeisasignatureofnon-lineareffects.
Thesenon-linearitieslikelyoriginatefromthelarge-amplitudeplatedeformations, whichintroducenon-lineartermsin both the plateandflow dynamical equations, through geometrical anddamping terms.Forinstance, the impermeability condition,whichensuresthecouplingbetweentheplateandthefluid,hastobeappliedonadisplacedinterface,yielding terms that depend non-linearly on themotion amplitude. These terms correspond tocubic non-linearities [25]. Another sourceofnon-linearityisthe dragforce exerted ontheplatenormallytoits surface asit moves,also calledthe resistive force,whichcorrespondstoquadraticnon-linearities[26].
Calling x the amplitudeofthefirstbendingmodeoftheplate,weexpectaweaklynon-lineardynamicalequationofthe form
¨
x+ (2πf0)2x+μx˙+ν|˙x|˙x+ δx3=γcos(2πf t) (1)
wherethefirst twoterms describethebending modeoftheplateasaharmonic oscillator(witheigenfrequency f0),the thirdtermisrelatedtotheinternaldampingoftheplate(μ>0),thefourthtermisthequadraticnon-linearityduetothe dragnormaltotheplate(ν>0),andthefifthtermgathersallthecubicnon-linearitiesoriginatingfromthefluid–structure interaction, the platedynamics, andthe fluid load (in reality there should also be terms ofthe form x2x and ¨ xx˙2). The right-handsidecorrespondstotheforcingduetotheheavingmotion,where γ>0 withoutlossofgenerality.Thisequation is the classic Duffing oscillatorwith an additional quadratic term, ν|˙x|˙x. It is beyondthe scope ofthe present paperto calculatethecoefficients μ, ν,andδappearinginthisequation,however,itcanbeformallysolvedbyassuming
x(t)=a cos(2πf t+ φ) +h.o.t. (2)
where h.o.t. stands for “higher-order terms” andrefers here to higher harmonics that can be neglected nearresonance. Neglectingalsothecubicnon-linearitiesforsimplicity,thedynamicalequation(1)canbeprojectedontothemainharmonics togiveanimplicitequationfortheamplitude a of thebendingmodeoscillations:
(2π)4 f02−f22 a2+ (2π)2 μf+16 3 νf2a 2 a2=γ2 (3)
Solving this implicit equation shows, in particular, that at the resonance (i.e. f = f0), the relative amplitude a/γ is a monotonicdecreasingfunctionof γ.Thisisconformtotheexperimentalobservations,wheretherelativeamplitude ATE/ALE
isalsoadecreasingfunctionoftheforcingamplitude ALE.InFig. 6,therelativeamplitudes a/γ obtainedwithEq.(3)for differentvaluesoftheforcingparameter γ areplottedasafunctionofthenormalizedforcingfrequency f/f0.Itshowsa goodqualitativeagreementwiththeexperimentalmeasurementsreportedinFig. 5(d)nearthefirstresonance.Inparticular,
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second peak observed inFig. 5(d)around f/f0≈6 isnot reproduced in the model(Fig. 6) since it only accountsfor a singlenaturalmodeofvibrationwhereastheflexibleplatehasaninfinitenumberofstructuralmodes.
TheexperimentalresultspresentedinFig. 5(d)andtheabovediscussionontheDuffingequationshowthatanaccurate descriptionoftheplateresponseatresonancemustincludenon-lineareffects.Itisimportanttostressthatthesenon-linear effectsarerelevanteveniftherelativeamplitudeofthedeformation, ATE/c, andtheanglebetweentheplateandtheflow