IOP P UBLISHING UROPEAN OURNAL OF HYSICS Eur - PDF document

MDenny thedragonwingsandbody,respectively)andistheinduceddragcoefcient[].Liftmaybeconstant(equaltoforhorizontalight)butingeneralit,too,dependsuponairspeed.Thecoefcientsdependuponmorphologicalfactorsthatare,atleastinpart,underthebird’scontrol.Weusethisfacttosimplifyequations(4)byassumingthatouralbatrossmaintainsaconstantratioof;letusdenotethisratioby,andadoptaminimumvalueof1andamaximumvalueof25foralbatrossightFromequations(4)andourassumptionweobtain() ()()() () again,foratailwind.Equation()isrelativelyeasytoworkwith;intheremainderofthispaper,wesolvespecialcasesofinterestanalyticallyandinsodoingobtainanunderstandingofhowalbatrossescanylongdistanceswithoutexpendingmuchenergy.3.Specialcases:glideangle,efÞciencyandequilibriumspeedFirstlyletusassumethatouralbatrossisinstillair(sothat0).Foraglidealongastraighttrajectory(constant),equation()reducesto() ()andsoaconstantequilibriumspeedoccursforadiveanglegivenbytan.(Negativeadescendingtrajectory;positivemeansascending,asshowningure.)Thiswell-knownresult[]showsthatahigh(correspondingtoalarge)resultsinalongglidedistance;analbatrossglidingfromacertainheightwilltravelfurtherthanotherbirdsglidingfromthesameheight.Nowweconsiderarisingtrajectoryinstillairwithconstant,sothatequation(reducesto(t) (t) .Thisequationcanbesolvedanalytically(viaanintegratingfactorexp())toyieldfortheairspeed:v(t) ()exp kt+g (twheresin() Letussaythatthealbatrossascendsaheightandinsodoingitsspeeddropsfrom0tozeroat2.Theheightisgivenbytv(t)(t).Fromequations()and()andfromtheboundaryconditions,itisstraightforwardtocalculatetheefciencyofthisclimbingightmanoeuvre.Thebird’sinitialenergyis anditsnalenergyismgH.TheefciencyisThefractionallossofenergy1forthisclimbingmanoeuvreisplottedingureasafunctionofthelift-to-dragratio,.Notethat,becauseoftheirhighvalue,albatrosseslosemuchlessenergyduringaclimbthandobirdswithlowerlift-to-dragratios.Wecaninferfromthisobservationthatthereissignicantselectionpressureforalbatrossestoevolvehighlift-to-dragratios(becausethesebirdsaresodependentuponefcientight).Nowweconsidertheequilibrium(0)speedforthemoregeneralcase0.Againwerestrictattentiontolineartrajectorieswithaconstant.Equilibriumspeedisreadilyfoundfromequation()tobe  ()wheretan() Barnes[]suggestsamaximumvalueof27whereasAlexanderandVogel[]adoptasomewhatlowergure18.Thisrangeofalbatrossmaximumlift-to-dragratiosishighforbirds,butislessthancanbeattainedbymodernglidersandsailplanes. Dynamicsoaring:aerodynamicsforalbatrosses79 1Š 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure3.Fractionofalbatrossenergylostduringaclimb(describedinthetext)versusthelift-to-dragratioDoesthisspeedcorrespondtostableorunstableequilibrium?Stabilityrequiresthatthecoefcientofinequation()(theterminsquarebrackets)benegative.Itisnotdifculttoshowthatthisrequirement,combinedwiththerequirementthatspeedbepositive,meansthatforthetwocasesofpracticalinterest(divinginatailwindandclimbinginaheadwind)theequilibriumspeedisunstable.Soanalbatrossmustconstantlyadjustparametersinordertomaintainaconstantspeed.4.RayleighdynamicsoaringWhatisspecialaboutthetwotrajectoriesjustmentioned(divingwhilemovingdownwindandclimbingwhilemovingupwind)?Whencombinedintoasingleloopingtrajectory,theseconstitutedynamicsoaring(DS),familiartoeverygliderpilot.DSpermitsaglider(oralbatross)toextractenergyfromthewindwhenthereisaverticalvelocitygradient,suchastheonethatoccursintheleeofahillorontheboundarylayeradjacenttotheearth’ssurface.HerewewillanalyseasimpletrajectorythatshowshowanalbatrossusesDStomoveacrossthewind,withnonetenergyexpenditure.Rayleigh,in1883,wasthersttosuggestthatseabirdssuchasalbatrossesmightexploittheDStechnique[Considertheighttrajectorythatissketchedingure.Herewehavejoinedtogetherthreetwo-dimensionaltrajectories,andsowecanapplyequation()toeachsegmentseparately.Thealbatrossbeginsatthetopoftheboundarylayer(altitude20m)with12ms,whichwetaketobetheminimumairspeedthatthebirdrequirestomaintainlift[].Groundspeedis.Therstphaseofthetrajectoryconsistsofadiveataconstantanglealongthewinddirection;thesecondphaseisalongglideatzeroaltitude;thethirdphaseisaclimbataconstantangleintothewind,endingatanaltitudeNegativecorrespondstomovementtotheleft,ingure,whichmeansthatthewind(assumedinguretobeheadwind)isatailwind.Similarly,negativeinatailwindimpliesheadwind.So,mustbepositive. Dynamicsoaring:aerodynamicsforalbatrosses81 Wechoose1.Wealsoseethatclimbangleshouldbesmall,andthattheshouldbeaslargeaspossible.Wechoose25,andwendthataclimbangleofworks.Theequationsofmotionshowthatthetrajectoryofgureispossibleonlyforsteepdiveangles;herewechoosediveangleCalculationsshowthatguretrajectoriesarepossibleinmoderatetostrongwinds,withgradientsexceeding0.5s(whichcorrespondstowindspeedsabovetheboundarylayer10msor36kph).Thechangeinalbatrossenergyduringeachphaseis mgh h) 22Š1 mgh h) Notethatenergyisdeterminedinareferenceframethatisxedtotheearth’ssurface.Thetotalenergychangeoverthetrajectoryiszero,inthisframe,byconstruction.TheairapplyatthebeginningandendofphaseII,asshowningure.Thesespeedsarefoundtobequitehigh,varyingfrom29.760ms0.5to40.094ms0.9.Solutionsinstrongerwinds(larger)arepossible,atevenhigherspeeds.Albatrossesarecertainlycapableofsuchspeeds:measurementsofonegrey-headedalbatrossduringanAntarcticstormshowedthatthebirdtravelledfor9hatgroundspeedsofbetween110kph(30.5ms)and168kph(46.7mss14].Thechangesinenergyforthethreephasesoftheguretrajectoryareplottedingure.Energyisgainedduringthedive(phaseI),lostduringthezero-altitudeglide(phaseII)andlostalittleduringtheclimb(phaseIII).Thisbehaviouriseasytounderstandqualitatively.Thealbatrossispushedalongbythewindduringthedive,andsopicksupmorespeedthanwewouldexpectsimplyfromthechangeinaltitude.Energyislosttodragduringthehorizontalwave-topglide.Perhapsmoresurprisingistherelativelysmalllossofenergyduringtheclimbphase.Herethealbatrossisgainingheightandyingintothewind,andyetitpicksupairspeed.Ofcourse,thegroundspeedfallsasthebirdrises.Wealsoplotthealbatrosscross-windandupwindspeedsingure.Thesespeedsarecalculatedrelativetothesurface.So,forexample,analbatrossthatrepeatedlyiestheDStrajectoryofgureinastrongwindgradientof0.9swilldriftmoreorlessperpendiculartothewinddirectionataspeedofabout19ms.Inagentlergradientof0.6s,assumingthatthebirdadoptsthesamediveandclimbangles,thecross-windspeedwillbeabout9ms,whiletheupwindspeedis8ms.Bychangingdiveandclimbangles,analbatrosscanalteritsdriftdirection.Finally,wenotethatDSstilloccursifthealbatrossmaximumisreduced(toandtheminimumvelocityisincreased(to15ms);thetrajectoryspeedisincreasedbyabout3ms,assumingallotherparametersareunaltered,andDSrequiresstrongerwinds(aminimumvalueof0.65s).DSismoreseverelylimitediftheminimummustexceed0.85sisincreasedfrom1.0to1.5.Wemayconcludethat,forawiderangeofparameters,itispossibleforalbatrossestoexploittheRayleighDSinmoderatetostrongwindconditions.5.DSmodellimitationsThecentralassumptionsofourDSmodelare(1)constantduringeachphaseofthetrajectory,(2)linearwindspeedprole, Dynamicsoaring:aerodynamicsforalbatrosses83 oursimpleDSmodelistodemonstratethatanalbatrosscanusethewindenergytomove;itcantravellargedistanceswithoutexertingmuchinternalenergy.NeglectingtheturnbetweenightphaseIIIandightphaseIdoesnotinvalidatethismodelresult,weargue,becausetheturnwillincreasethealbatrossenergy,sothatitiseasierthanpredictedhereforthealbatrosstoutilizeDS.Thusoursimplemodelcannotcalculatetheturn,andsounderestimatesthebenetsgainedbyDS.Thefourthassumptionismadebecausemodellingthegroundeffect,andtheinuenceofsurfacewavesuponalbatrossight,iscomplicated.Thegroundeffect(disturbanceofwingtipvorticesbytheground,oroceansurface,resultinginadditionallift)isexploitedinseveralairplanedesigns.Ithasalsobeenshowntobenetcertainbirds(suchasskimmers)thatyclosetothewatersurface[].Theactionofwaveshasbeenshowntobenetalbatrossight:thebirdgainsthrustwhiledragisreduced,toanextentthatdependsinacomplicatedwayuponthesurfacewaveparameters[].Thus,onceagain,ignoringacomplicationleadstoourmodelunderestimatingtheabilityofalbatrossestoexploitDS.ThusoursimplemodelpredictsaDScapability.6.SummaryanddiscussionAlbatrossightdynamicsisofinterestbeyondthebiomechanicscommunity.Thereismuchresearchtodayintomicroairvehicles(MAVs)andtheaerodynamicsofsuchvehicleshasmuchincommonwiththoseofsoaringbirds.Thus,muchoftheanalysisassociatedwithMAVdevelopmenthasincorporatedalbatrossDSstudies[]orhasbeeninspiredbyalbatrossight[].Radio-controlledmechanicalmodelsofalbatrosseshavebeendemonstratedtoexploitDS.(Thesemodelsarehand-launchedandtheenergyforight,apartfromthisinitialboost,isextractedfromthewind.)Detailedmodelsofalbatrossight[]aretoocomplicatedtobetransparenttonon-specialists.HerewehavedevelopedamathematicalmodelofalbatrossDSthatissimpleandyetcapturesmuchoftheunderlyingphysics.ThealbatrossisassumedtoprogressbyrepeatingaDStrajectorythatisdividedintothreetwo-dimensionalphases.Theresultingequationsofmotionforeachphaseareverysimple.Complicationsthatareomitted(thegroundeffectandtheinuenceofwavesduringightphaseII,andthedynamicsoftheturnthatoccursbetweenphaseIIIandphaseI)willaidthealbatross,andsooursimplemodelunderestimatesthebenetsthatitgainsfromDS.Despitetheseoversimplications,ourmodelagreesquitewellwiththefewexperimentalresultsandobservationsthatareavailable.Thus,forexample,thebankangleduringturnsisreadilydeterminedfromthecentrifugalforceandweightvectors:tan(B)/gR;inourmodelitvariesbetween0.55–0.9.Observedbankanglesareestimatedtobeintherange606020].LoopingDStrajectoryinourmodelvariesinlengthscalefrom230mat0.55to2240mat0.9,whereasobservationsfromshipsindicatelengthscalesofafewhundredmetres[].Trajectorydurationsof5.0–12.7scomparewithobserveddurationsof9.6–10.9ss20].Clearlythesimplemodelisatleastcompatiblewithobservations.Theturnincreasesenergybecausethebird’scross-sectionalareaprojectedalongthewinddirectionwillincrease,reachingamaximumwhenthebirdismid-turn,andthisincreasedareawillincreasethewindforceactinguponthebird,sothatthedownwindspeedatthestartofphaseIwillexceedSee,forexample,theYouTubevideoathttp://www.youtube.comwatch?vThus,forexample,weseewhyalbatrossesloopanticlockwisewhenheadingnorthandclockwisewhenheadingsouth:theprevailingwindsarewesterlyandsotheRayleighDStechniquerequiressuchtrajectories. 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