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    The Aerodynamics of Hovering Insect Flight. I. The Quasi-Steady AnalysisAuthor(s): C. P. EllingtonSource: Philosophical Transactions of the Royal Society of London. Series B, BiologicalSciences, Vol. 305, No. 1122 (Feb. 24, 1984), pp. 1-15Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2396072 .

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    Phil. Trans. . Soc.Lond. 305, 1-15 (1984) [ 1 ]Printedn Great ritain

    THE AERODYNAMICS OF HOVERINGINSECT FLIGHT.I. THE QUASI-STEADY ANALYSISBY C. P. ELLINGTON

    Departmentf Zoology,Universityf Cambridge, owning treet, ambridge B2 3EJ, UK.(CommunicatedySir JamesLighthill, .R.S. - Received 8 March 1983)

    CONTENTSPAGE

    1. INTRODUCTION 22. THE BLADE-ELEMENT THEORY 33. KINEMATIC GROUPS 53.1. Horizontal strokeplane 53.2. Inclined strokeplane 63.3. Vertical strokeplane 84. THE PROOF-BY-CONTRADICTION 105. OUTLINE OF THE PRESENT STUDY 116. SYMBOL TABLE 12REFERENCES 14

    The conventional aerodynamic analysisofflappinganimal flight nvokesthe quasi-steady assumption' to reduce a problem in dynamics to a succession of staticconditions: it is assumed that the instantaneous forces on a flapping wing areequivalent to those for teadymotion at the same instantaneousvelocityand angleof ttack. The validity fthis ssumption nd the mportance funsteady erodynamiceffectshave long been controversialtopics. Weis-Fogh tested the assumption forhovering nimal flight,where unsteadyeffects re mostpronounced,and concludedthatmost nsects ndeed hover ccordingto theprinciples fquasi-steady erodynamics.The logical basis for his conclusion is reviewed in this paper, and it is shown thatthe available evidence remains ambiguous.The aerodynamics of hoveringinsectflight re re-examinedin this series of sixpapers, and a conclusion opposite to Weis-Fogh's is tentativelyreached. Newmorphologicaland kinematicdata for varietyof nsects re presented n papers IIand III, respectively.Paper IV offers n aerodynamic interpretation f the wingkinematics nd a discussion n thepossiblerolesofdifferenterodynamic mechanisms.A generalized vortextheoryofhovering flight s derived in paper V, and providesa methodof stimating he mean lift,nducedpowerand inducedvelocity orunsteadyas well as quasi-steady flightmechanisms. The new data, aerodynamicmechanismsand vortextheory re all combined in paper VI for n analysis of the lift nd powerrequirements nd other mechanical aspects of hovering flight.

    Vol. 305 B II22. I [Published 4 February984

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    2 C. P. ELLINGTONA large number of symbols are needed for the morphological, kinematic andaerodynamic analyses. Most of themappear in more than one paper of the series,and so a single comprehensive able defining he major symbolsfrom ll of thepapersis presented at the end of this paper.

    1. INTRODUCTIONResearch into the aerodynamic principles of flapping animal flighthas long been hamperedby a limited repertoire f investigative ools. Aerodynamics s primarily practical science,concentrating n the forcesexperiencedby a wing in steady motion. The effects f periodicvariations nwing attitude nd velocity uperimposed n a mean motionhave been investigatedin flutter nalysis, but such analytical treatments mploy linearizations that restrict hem tosmall amplitude oscillations. The scope fortheoretical nalyses ofanimal flight as thusbeenseverely imited,and fromnecessity t is generallyassumed that the instantaneous forcesona flapping wing are those corresponding o steadymotionat the same instantaneousvelocityand attitude, the quasi-steadyssumption.Whether or not this assumption is valid forlargeamplitude, high frequencymotionshas been the source of much controversy,nd a decisivetestcan only be accomplished by comparingmeasured instantaneouswing forceswith thosepredicted by the assumption. The aerodynamic characteristics f a wingin steadymotion areeasily determined,but a direct measurementofthecyclicforcesproduced byflappinganimalwings has only been achieved recently (Cloupeau et al. I979; Buckholtz I98I), and acomprehensive est f the assumptionusing thesedifficult echniques s still acking. Researchershave thereforebeen forced to adopt a debatable assumption which could not be testedexperimentally.The validityof thequasi-steady explanationofflappingflight an be tested heoretically nlyin a proof-by-contradiction. he mean forces generated by the wings during a cycle arecalculated according to thequasi-steadyassumption. f these forces o notsatisfy he net forcebalance of theflying nimal, then theassumptionmustbe false. fthe mean quasi-steadyforcesdo satisfy he balance, then the only logical conclusion is that the assumption cannot bediscounted. The range of forcesgenerated by wings is restricted y physical considerations:unconventionalaerodynamicmechanismsmay produce mean, and even instantaneous,forcesvery similar to the quasi-steady values. A satisfactory xplanation of flightbased on thequasi-steady assumptioncannot preclude alternativemechanisms.

    The early quantitative studies on the aerodynamicsofflapping flighthave been reviewedby Weis-Fogh & Jensen (I956). The available data on wing motion, the kinematics, ereincomplete and often naccurate, so it was usually necessaryto base the theorieson roughkinematic approximations.Some studies concluded that the wing forceswere not consistentwith a quasi-steady mechanism,but the sources of error were too large foran unqualifiedacceptance of theconclusions. n a classicseriesofpapers Weis-Fogh andJensen thenpresentedwhat is still the mostcomplete studyofflapping flight orthe desert ocustSchistocercaregaria.The kinematics ffast orwardflight nd theaerodynamiccharacteristics f thewings n steadymotion were measured (Weis-Fogh I956; Jensen I956). Jensen (I956) then combined thesedata in a quasi-steady analysis and found that the mean calculated wing forces greed wellwith those measured from ocustsflyingn a wind tunnel.This is a very persuasiveargumentfor the quasi-steady explanation of fastforwardflight, ut it cannot be considered as a proof.Indeed, the cyclic forcemeasurements f Cloupeau et l. ( I 979) on Schistocercandicate that thequasi-steady mechanism s not a sufficientxplanation in this case.

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 3It is generallybelieved that the quasi-steadyassumption s valid for ast orward light ecauseof the low value of the reduced frequencyparameter.This parameter,the ratio ofa flappingvelocityof thewingsto the forwardflight elocity,may be expressed n severalways and wasintroduced to animal flight tudiesbyWalker (1925). At low values, thesteady flight elocity

    dominates the flow over the wings and reduces the spatial derivatives of the fluctuatingaerodynamic parameters. Thus unsteady aerodynamic effectsare small compared withquasi-steadyones.Atslowerflight peeds the reducedfrequency arameter ncreases nd unsteadyeffectshouldbecome more important.Hovering flightpresents the extreme condition where the flightvelocity s zero and the reduced frequencybecomes infinite.Bennett (i 966) and Weis-Fogh(I972, I973) emphasized the ncreasedsignificance funsteadyeffects o be found nhoveringflight.The wingsaccelerate and decelerate in a distance ofonly a few chord lengths:virtualmass forces nd unsteadycirculatory ffectsmayno longerbe negligible.The amplitudeofwingrotationabout a longitudinal axis during pronationand supination exceeds a right ngle, andthe rotationalvelocity s comparable with the flapping velocity. Under these conditions theaerodynamicbehaviourof thewings maydifferppreciablyfrom he quasi-steadyassumption.Byexaggeratingthe unsteadyaerodynamic effects fflappingflight, overingthus presents nideal case for the investigation fflightmechanisms.In addition to its advantage in stressing nsteadyeffects, study of hovering s usefulfromotherconsiderations.The forceand momentbalance during flights simplified y the lack ofa net horizontalthrust, nd the effectsf body lift nd body drag can be ignored. The absenceof a linear flightvelocity acting in combination with the flapping velocity also reduces thecomplexityofthe mathematical treatment.Excluding briskmanoeuvres and climbing flight,the liftcoefficientsf the wings during hovering must be greaterthan other types of flightbecause the relativevelocityof thewings s not enhanced by theflight elocity.Thus hoveringdemands maximum lift oefficients, hich are particularlyusefulwhen comparing thewingperformancewithsteady-state onditions.Finally, the total power expenditure of the animalis greatest n hovering flight, rovidinga convenient case forphysiologicaldiscussions.Weis-Fogh (I972, I973) applied a quasi-steady analysis to hovering animal flight ndconcluded that 'most insectsperformnormal hoveringon the basis of the well-establishedprinciplesof steady-stateflow' (Weis-Fogh I973). The aerodynamics ofhovering flight rere-examined n this study using new data and a new theory, nd the opposite conclusion isstrongly ndicated. Before this work is presented,the previous approaches to the problemwill be reviewed.

    2. THE BLADE-ELEMENT THEORYThe usual aerodynamic treatmentof animal flight s based on the blade-elementheory fpropellersdeveloped by Drzewiecki. Osborne (I95i) derived thegeneral equations thatapplytoflappingflight. he fundamentalunitoftheanalysis s the blade element,or wing element,which is that portionofa wing between the radial distancesr and r+ dr fromthewing base(figure1a). The aerodynamic forceF' on a wing element can be resolvedinto a componentnormal to the flowvelocity,called the lift, nd a component parallel to the flow,called theprofiledrag (figure1b). The lift ' and profiledrag Dprofor wingsectionofspanwisewidthdr are

    L'= lpcUr CL, (1)D;ro = 1P CUr CD ,pro, (2)

    I1-2

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    4 C. P. ELLINGTONwhere p is the massdensity f air, c is the wing chord, and Ur s the relativevelocity omponentperpendicular to the longitudinalwing axis. Any spanwise component of the relativevelocityis assumed to have no effect n these forces.CL and CD,proare lift nd profile rag coefficients,which can be defined for unsteady as well as steady motion. According to the quasi-steadyassumption, though,these coefficients re functions nly of the Reynolds numberand angleof ttack of the relativewind forgiven profile haracteristics. quations (1) and (2) are resolvedinto vertical and horizontalcomponents, ntegrated long the wing length and averaged overa cycle. For hoveringflight henet forcebalance requires the mean vertical force o equal theweight of the animal, and the mean horizontalforceto be zero.

    L'* F'(b)

    Dproo

    FIGURE1. (a) The wingbeat uringhoverings approximatelyonfinedo a stroke lane, which s inclined t anangle/3 o thehorizontal. he aerodynamic arameters or wing lement fwidth r andchord are shownin a two-dimensionaliew n (b).

    The relative velocity s the vector sum ofthe flappingvelocityU and the induced velocitiesof bound and wake vorticity.The blade-element theory gnores the existence of vortices,however, nd so can reveal nothing fthe nduced velocity.FollowingOsborne (I 95 I), a meanvalue of the vertical induced velocityw0 s usually estimatedby the Rankine-Froude axialmomentum theory f propellers.One method of applying the blade-elementanalysis relies on successivestepwisesolutionsofequations (1) and (2). Complete kinematicdata are necessaryfor his pproach: the motionof the longitudinal wing axis, thegeometricalangle ofattack and the sectionprofilemustallbe knownas functions f timeand radial position.The angle of attack of therelativewind arcan then be calculated, and appropriate values of CL and CD,pro selected from xperimental

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 5measurements.This was the method used so effectively yJensen I956) inhisstudyofforwardflight n the desert locust, but it becomes increasingly unreliable as the hovering state isapproached, where momentumconsiderations how that thewake induced velocityreaches amaximum.For many hovering nimals there re periods whenthe oppositewings re separatedby a distance of ess than a chord length, o thatthe bound vorticity f theopposite wingwillalso contribute significantly o the relative velocity of a wing. This effect can only beapproximated and, when coupled with the crude estimate of the wake induced velocity,aconsiderable inaccuracy can be expected for the calculated value of ar. The lift oefficientsstrongly ependent on a, so the solution to equation (1) is prone to seriouserror.Osborne (I 95 I) introduced an alternativemethod of analysis whichsolvesformean valuesof the coefficients,L and CD,pro, satisfyinghe net forcebalance. This isequivalent toassumingthatthe force oefficientsre constant long thewingand throughout he cycle. The assumptionis notunreasonable, because thewingscould well be operating at some optimalangle of attack.For small insects ike Drosophila, L isnearlyconstant nywayover the range ofar foundduringflight Vogel I967). The mean lift oefficients particularly nteresting ecause it is also theminimumalue compatiblewithflight; fCL variesduringthewingbeat then ome nstantaneousvalues mustexceed CL. The kinematicdetail requiredforthismethod is greatlyreduced sinceonly the motion of the longitudinalwing axis is needed. The geometrical angle of attack andprofilesection of the wings are extremelydifficult o measure accurately, and one of theprincipal advantages of thismethodis itsneglect of them. The double integralsof radius andtime produced by manipulation of equations (1) and (2) can be separatedwhen the coefficientsare assumed to be constant;theresulting ingle ntegralsmaythenbe reduced tomorphologicaland kinematic parameters, allowing simple expressions to be derived for CL and CD, prosatisfying he integralconstraints.The theoretical nvestigations fhoveringflight ave usually reliedon the mean coefficientsmethod (Weis-Fogh I 972, I973; Ellington I975; R. A. Norberg I975; U. M. Norberg I975,I976). The animals studied so far can be divided into three functionalgroups, for whichdifferentpproximationscan be used in the calculations outlined above. During a cycle thewingsbeat roughly n a strokelane,which s nclined at an angle ,/withrespect o the horizontal(figure1a); it is the attitude of this strokeplane thatdistinguishes hegroups.

    3. KINEMATIC GROUPS3.1. HorizontaltrokelaneThe most commonly observed type of hovering is characterized by an approximatelyhorizontalstrokeplane (figure a). This groupwas extensively iscussedby Weis-Fogh (I972,I973), who called thepatternnormalovering,nd includes thehummingbirds Stolpe & ZimmerI939; Greenewalt I960) and most nsects Weis-Fogh I973). The amplitude ofwingrotationduring pronationand supination is large, enabling thewingsto operate at an angle ofattackfavourable for ift n both themorphologicaldownstroke nd theupstroke figure a, b). Thewake induced velocity s small withrespectto theflappingvelocityof thewingsfor hisgroup,

    resultingn a relativevelocityveryclose to thehorizontal (Ellington I978, I980). This ustifiesusing the horizontal flapping velocityforUr in equation (1) and neglectingany small dragcontributions o the netvertical force.A mean lift oefficients then easily found such thatthe

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    6 C. P. ELLINGTON

    (a)

    (b) A W >\

    FIGURE 2. (a). The wing tip pathof a hummingbirdhlorostilbonureoventrisiewedfrom heside, llustratinghorizontaltroke lane.Thewing ttitudes ndicated nthepath. b)Thewingbeat atternorMelanotrochilusfuscus iewed rom bove.Adaptedfrom tolpe& Zimmer I939).net liftgiven by the integralform of equation (1) balances the body mass. A mean dragcoefficient annot be determinedby the integral constraints,but once CL is calculated anappropriate CD,promay be selected from xperimentalmeasurements.

    In his comparative survey of normal hovering animals, Weis-Fogh (I973) used simpleanalytical expressionsforwing chord and flappingvelocity; a semi-ellipticalwing planformwas used,and theflappingvelocitywas takenas simpleharmonic motion.These are reasonableapproximations,permitting quick calculation ofCL, but increase the errorto an estimated30%. Table 1 presents ome ofWeis-Fogh's results nd a Reynolds numberRe forthewings;theReynoldsnumber s based on the chord and the mean flappingvelocity t 0.7 of thewinglength.CL for the small wasp Encarsiaformosas takenfromEllington (I975). In thiscase thewingshape and velocitywere not well represented y Weis-Fogh's approximations nd a moredetailed treatmentwas necessary.3.2. Inclined trokelane

    Smallpasserine irds Zimmer 94; Brown 963; U. M. Norberg 7), bats (EisentrautI936; U. M. Norberg 970, 1976) and the 'true' hover-flies f the subfamilySyrphinae

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 7(Weis-Fogh I973) hoverwith the strokeplane inclined at a value of , between 300 and 400(figure3). Dragonflies (Odonata) also hover in this manner, although , is about 600 (R. A.Norberg I975). These animals appear to generate relatively mallforces n theupstroke.ThedragonflyAeschnajunceatronglyupinates tswingson theupstroke,making the angle of attacknear zero (R. A.Norberg 975). Excluding thehummingbirds, ertebrate liers re anatomicallyunable to rotate theirwings to this extent.The birds and bats partiallyflextheirwings duringthe upstroke,however,and the individual primariesof birds are also rotatedto a negligibleangle of attack. Any lifton the upstroke of an inclined stroke plane would produce a largehorizontal thrust omponent, nd this s probably the explanationfor nsignificant pstroke ift.The mean forceon the downstroke s thus primarilyresponsibleformass support.

    (a) (c)

    (d) (e) (f)

    (g: (h X

    FIGURE 3. The wingbeat fthe ong-eared at Plecotusuritus,llustratingn inclined troke lane during overingflight. daptedfromU. M. Norberg1970).The relative velocity Ur on the downstroke hould not be approximated by the flappingvelocity; because of the inclinationof the strokeplane, the induced velocitywill reduce thevalue of Orwell below U2. An accurate estimateof the induced velocity s therefore ssentialin calculating the magnitude and directionof the relativevelocity. t is likelythat previousestimates f the induced velocityare substantially oo small (see papers V and VI), which ledto an incorrect onsiderationofdownstrokeprofiledrag by Ellington I980). From the resultsof R. A.Norberg (I975) and estimatesbased on papers V and VI, thevalue ofCL requiredforweight upporton thedownstroke fAeschnajunceahould ie between3 and 4. U. M. Norberg

    (I975) calculated thatCLmustequal 5.3 for henet verticalforce n the downstroke o balancethe body mass of the pied flycatcher icedulahypoleuca.n paper VI, I estimate that CL forFicedula s about 6.0, so a value between 5 and 6 can be expected. It should be noted that inboth cases the morphological and kinematic data were not from he same animals, and so anun nown inaccuracy iSpresent.

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    8 C. P. ELLINGTONThe effects f different ssumptionsabout upstrokeforces on the mean coefficientswereevaluated by U. M. Norberg (1976) in a thorough nvestigation fhovering n the bat Plecotusauritus.Her resultsdemonstratethat minimum coefficients ccur when no lift s generatedontheupstroke,which thenrequiresCL= 3.1 on the downstroke;my estimateof CL is about 4.3

    (paper VI), and so a value between 3 and 4 seemsprobable. She also commentedonWeis-Fogh's(I973) analysis of Plecotus, oting that his procedural simplifications ere invalid. This is alsotrueofhis calculations forother animals using an inclined or vertical strokeplane.

    (a) ol(b)

    (C) (d)

    FIGURE 4. The downstrokef vertical ake-offyPieris rassicae.he stroke lane svertical,nd thewingmotionis perpendicularo thechord.The vortex attern reatedbythe downstrokes indicated, nd theresultinglarge-coredortex ing s shownnvertical ectionn (d).

    3.3. VerticaltrokelaneIn additiontothe nsects overed n paper III, I havefilmed heLarge Cabbage WhitebutterflyPieris brassicae . in freeflight.A unique kinematic pattern,an approximatelyvertical strokeplane, is often een during take-off nd hovering.Figure 4 showsseveral stagesof the down-strokewhich nitiates vertical take-off rom smallplatform.The wingsare clapped togetherdorsallyat the startof thedownstroke, nd then fling' open (Weis-Fogh I973) as in figure a.The wingsmove with the chord perpendicular to their motion (figure4b, c) and nearly claptogether t the end ofthe downstroke figure4d). The body pitchesnose-up and the wingsstrongly upinate duringtheupstroke,which is not shownhere,producingan angle ofattacknear zero. Thus littleforce s generatedon theupstroke, s for the inclined strokeplane.The sustainingvertical force obviouslyresultsfrom he pressure drag on the wingsduringthe downstroke.A drag mechanism of flighthas been suggested several timesfor flight tReynoldsnumbers below about 100 (Horridge I956; R. A.Norberg 1972 a, b; Bennett 973),

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 9where thickboundary ayersreduce the steady-state ift oefficientsfwings. The drag is mainlydue to skinfriction t low Re, however, nd is not very ensitive oar. A drag mechanism basedon a differential elocityof the half-strokes as therefore roposed (Bennett 1973), such thatthe downstrokevelocityand drag were greaterthan the upstroke. However, the small waspEncarsiawas found to use a horizontal troke lane and a circulatory iftmechanism Weis-Fogh1973; Lighthill 973; Ellington 1975). I have also filmed hesmall fringe-wingednsect Thripsphysapus and the fruit fly Drosophila melanogasterEllington 1983); the kinematics andaerodynamics are very imilar to Encarsia. ronically the drag mechanism does not operate forsmall insects, orwhichit was predicted,but ratherfor large one at higherReynolds number(about 2800). The pressuredrag during the downstroke s much greater than the skin frictiondrag of theupstroke t high Re, and so theneed for velocitydifferentials eliminated.

    TABLE 1. RESULTS FROM THE QUASI-STEADY ANALYSIS(References:1, Weis-Fogh 1972); 2, Weis-Fogh I973); 3, Ellington I975); 4, R. A. Norberg(I975);5, U. M. Norberg1975); 6, U. M. Norberg 1976); 7, present tudy.)

    species CL Re ref.horizontal troke lane

    Amaziliafimbriatafluviatilis 1.8 6100 (1)Melolonthaulgaris 0.6 3900 (2)Manducaexta 1.2 5400 (2)Bombuserrestris 1.2 3600 (2)Apismellifera 0.8 1600 (2)Eristalis enax 0.9 1600 (2)Tipula p. 0.8 630 (2)Drosophilairilis 0.8 210 (1)Encarsiaformosa 1.6 23 (3)

    inclined troke lanePlecotusuritus 3-4 15000 (6,7)Ficedula ypoleuca 5-6 11000 (5,7)Aeschnajuncea 3-4 1900 (4,7)vertical troke lanePieris rassicae 0 2800 (7)

    Equation (2) of the blade-element analysis could be applied to the Large Cabbage Whitebutterfly, ut a constantvalue of CD pro on the downstroke s unlikelybecause of pressureinterferencerom heopposite wings.A differenterodynamic approach has been developed,therefore, ased on the vortexpatterncreated by the wing motion.Air rushes nto the gapbetween the wings as they flingopen at the beginningof the downstroke,producing a sinkmotion bounded by thevorticity n thewing surfaces nd the freevortex inesconnectingtheleading edges and the trailing edges (figure 4a). This flow pattern is maintained andstrengthened uring the downstroke,resulting n a large-corevortexring (figure 4d). Thereaction forceand kineticenergyassociated with this ring have been estimated from flowvisualizationphotographs.Althoughthisflightmechanism s conceptually quite simple, manysubtle modifications o the flow are effected y wing twisting nd flexibility, nd a detailedanalysis will be presentedelsewhere Ellington I983).

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    10 C. P. ELLINGTON4. THE PROOF-BY-CONTRADICTION

    The mean lift oefficient as been chosen n all ofthehovering nalyses to test hequasi-steadyassumption. If CL is greater than themaximum steady-state ift oefficientL, max' then theassumption s contradicted nd unsteadyflightmechanismsmust be invoked. WhenCL is aboutequal to CL, max the test s not conclusive: some variation of the lift oefficients likelyduringa cycle, and instantaneous values may then exceed the mean.Values ofCL,max are not known for the animals in table 1. For the insectswe may note avalue of 1.1 at Re about 2000 for the flatforewing f a desert ocust, which rises to 1.3 whenthe vannal region s depressed as a flap (Jensen1956). Nachtigall (I977) measured a smallerCL, max of0.85, however,for a crane-fly Tipula oleracea)wing at a similar Re of 1500. Vogel(i 967) found a value of 0.6 for flatDrosophila iriliswing at a lower Re of 200, which ncreasesto 0.8 fora cambered wing; his results re very similar to those of Thom & Swart (1940) fora flat-bottomed erofoil n this ow Re range. Of the insects n table 1, Encarsia and Aeschnaare the only examples where CL definitely xceeds CL,max forthe relevantRe, and unsteadyeffects re thereforendicated. The flightmechanismofPierishas already been outlined andmust also be considered as unsteady. Values of CL for he remaining nsects re uncomfortablyclose in general to the expected values ofCL max) and several lie in the region of uncertaintybetweenJensen's and Nachtigall's results.Because of this and the large margin of error nWeis-Fogh's (I973) analysis,no firm onclusions can be drawn concerning these nsects.The values ofCL,max for bird and bat wings are more controversial han those for nsects.The wings are thin and highlycambered, and generally operate at Re between 104 and 105.Experimentson similar aerofoil sections have foundCL,max up to 1.6; a very strongcamberor an effectivencrease n camber resulting rom eading edge and trailing dge flapscan raisethe value to about 2.5 (Schmitz 1957; Abbott & Doenhoff 1959; Mises 1959; Tucker & Parrott1970; Hoerner & Borst 1975). Which of these sections best respresents avian and bat wingsisnotclear. Primary eparationon thedownstroke fbirdsmayfunctions slotted r mul'ti-slottedflaps (Graham 1932), which can produce values ofCL,max in excess of 3. The alula of birdwings s commonlyraised in slow and hoveringflight, ncreasingthe lift ome 25% by actingas a Handley-Page leading edge slot (Nachtigall & Kempf 197 1). These comparisons ndicatethatCL,max may be about 1.6 forthewings,and possiblymuch more.Measurementson birdwingsand wingmodels suggestthat thesevalues are too high. Evenwith the alula raised, CL,max for the wing of a house sparrowPasserdomesticuss only 1.1 atRe 16000, and for a European blackbird Turdusmerulat is only0.8 at Re 24000 (Nachtigall& Kempf I971). Measurementsfrommanyother birdwingsover an Rerangeof10000-50 000are much the same: 0.7-1.2 (Withers 98I). There are problemsofcourse,tomounting wingin a realistic ttitude n a wind tunnel. Studieson wingmodels ofthepigeonColumbaivia howa similarrange ofvalues, however,from0.8 to 1.2 (Nachtigall 1979). For glidingbirds andbats values around 1.5 are found for the maximum lift coefficientbased on wing area(Pennycuick 968, 1971; Tucker & Parrott1970), butwhenthe tail area is includedas a liftingsurfaceCL max drops to about 1.3 for the pigeon (Pennycuick I968). The values of CL maxactually measured forbird wings and wing models are thus well below those thatmightbeexpected.For the hummingbird CL is 1.8, which Weis-Fogh (I973) accepted as possible underquasi-steady conditions.Judging by the results bove, I think hat this s unlikely nd unsteady

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 11effectswill then be significant. or Plecotus L is 3-4, which should be greaterthan CL,maxfor bat wing. The quasi-steady assumptionfails forFicedula s well.The resultsof the analysis can now be summarizedfor the kinematicgroups. The animalsusingan inclined or verticalstrokeplane relyon unsteady flightmechanisms, s concluded byWeis-Fogh (I973); the quasi-steadymechanism s insufficientven as an approximationto theaerodynamic forces. Of the animals using a horizontal strokeplane, Encarsiamust use anunsteadymechanism,and this s probably trueforthe hummingbird s well. The proof s notconclusive forthe remaining animals, but theirvalues ofCL are very close to CL,max,From such resultsWeis-Fogh (I973) concluded thatmost nimals hoveringwith a horizontalstrokeplane rely on thequasi-steadyflightmechanism.However, threepoints must be bornein mind when interpreting hese results. f CL does not exceed CL, max) then the proof-by-contradiction nly demonstrates hat the quasi-steadyassumption annot be discounted; it doesnot prove that the assumption is true. Furthermore, ome instantaneousvalues of the liftcoefficientre likely to be greater than the mean CL over the cycle.And finally, he mean liftin unsteadymotion can be substantially maller than thequasi-steady estimatebecause oftheWagner effect, o thewings cannot actually achieve the maximum liftduringhovering thatis predicted by CL max (paper IV). In the lightof these ast two points, the close agreementbetweenCL and CL,max could ndicate that thequasi-steadymechanism s not ble to providesufficientiftforhovering.The example of the hummingbird ould also lead to a conclusioncontrary o Weis-Fogh's. The kinematics fmost nsects re similar o thoseof he hummingbird,and it would be surprising f different lightmechanismswere involved; if the quasi-steadyexplanation is indeed inadequate forthehummingbird, hen tmightnot apply to the insectseither. Much more research is obviously required beforeWeis-Fogh's conclusion can besubstantiatedor disproved.

    5. OUTLINE OF THE PRESENT STUDYThe aerodynamicsofhovering nsectflight re re-examined nthis eries fpapers,to evaluatetheroles ofquasi-steadyand unsteadyaerodynamicmechanisms.Any aerodynamicstudymustbe based on accurate kinematic nd morphologicaldata, and the previous studieshave usuallybeen unsatisfactoryn thisrespect: the amplitude ofwing motion was oftendetermined onlyby visual estimation, ome of the data used by Weis-Fogh (1973) were from nsects n slow

    forwardflight, nd themorphologicaland kinematicdata were not from he same animals inthe analyses of Aeschna, icedula nd Amazilia.Papers II and III presentcompletedata sets fora varietyof nsects to be used in this study.Weis-Fogh (I973) proposed two unsteady flightmechanisms for animals where thequasi-steadyassumptionfails,based on an aerodynamic nterpretation f thewingkinematics.This approach is followed n paper IV to discuss the possible roles of differenterodynamicmechanisms.The flightmechanismscohsidered in paper IV cannot be directly ncorporated into theblade-elementtheory.A more generalized vortextheory s therefore eveloped in paper V,which calculates the mean liftforunsteady as well as quasi-steady mechanisms.The theoryalso provides more accurate estimatesfor the induced velocity and induced power thanpreviouslypossible.The new flight ata, aerodynamicmechanisms nd vortextheory re all combined in paper

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    12 C. P. ELLINGTONVI foran analysis of the flightmechanisms,theirpower requirements, nd other mechanicalconsiderations f hoveringflight.This study would not have been possible without the help, encouragement and guidance ofthree men: the late ProfessorT. Weis-Fogh, who inspired the initial lines of research, DrK. E. Machin, whose many talents nd enthusiastic upport were vital to the fruition f muchwork, and Mr G. G. Runnalls, whose photographic expertisewas invaluable. To these three,I extendmy warmest gratitude.

    6. SYMBOL TABLEThis tablecontains ll ofthemajor ymbolsn this eries fpapers.Exceptfor fewgeneral ymbols,hepaperwhere ach symbolsdefineds givenbelow.Some symbols re defined ormally y equations, nd this s ndicatedby XX.xx, whereXX is the papernumber nd xx is the equationnumberwithin hatpaper.A 'hat' denotes

    non-dimensionalorm f symbol. thermodificationso ymbols bey he ollowingules nless specialdefinitionis presented elow: subscriptsmax' and 'min' refer o maximum nd minimum alues, bar over the symbolindicates time-averaged ean value, prime enotes force er unit pan, nd an asteriskepresents echanicalpowerperunitbodymass.paperorsymbol equationno. definition

    a V axial separation fvortex ingsn thefarwakeA V cross-sectionalrea of thefarwakeAo V.10 area of the actuator iscAr V area of thevortex heetproduced y a half-strokeAl V rolled-uprea of thatvortex heet? 11.4 aspectratioof thewingpairb V radius fvortex ingsn thefarwakebo V radius fvortex ings t the ctuator iscc II chord f thewing11.5 mean chordc II c/cC V.34 self-inducedomponentf thevortex ingvelocityCD dragcoefficientCD,f IV.7 CD for flatplate parallel o theflowCD,pro profile ragcoefficientCL lift oefficientd 11.17 mean diameter fbody/bodyengthd/u III ratioofduration fdownstrokeoupstrokeD VI magnitude f the meanprofile ragvector1^ VI.32 D/mgDpro profiledragf V frequencyf iftmpulsesf VI f/wingbeat requencyF aerodynamicorce n a wingorF V theforce rom momentumetg gravitationalcceleration11.11 mean wingthickness/wingengthI V circulatoryift mpulsef V.5 non-dimensionalorm fV the mpulse equired orweight upportorI II moment f nertiaIb moment f nertia or nimalbodyIv ~~II moment f nertia or hevirtualmassof a wingpair1w II moment f nertia or hemassof a wingpair

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 13J 111.23 advanceratiok V.31 specificnducedpowerfactorK V.33 component f thevortex ing elocity ue to other ingsnthewakeI 11.18 distance rom nteriorndofbodyto centre fmass/bodyength11 II distance rom orewingase axisto centre fbodymass/bodyength12 11.20 radius fgyrationor hebody/bodyengthL II body ength? II body ength/wingengthorL liftL VI mean ift/mgm bodymassmw II massofwingpairrAw II mw/mmk 11.8 thekthmoment fwingmass bout thewingbasen wingbeat requencyPd V disc oadingPw wing oadingPa VI aerodynamic owerPace VI.39 powerrequired o acceleratemass and virtualmassofthewingsPind VI.20 induced powerPm VI mechanical oweroutput fmuscleVI Pmperunitweight fmusclePpro VI.29 profilepowerPRF VI.21 Rankine-Froudestimate f nducedpowerr radial position long thewingf r/wingengthfk(m) 11.9 non-dimensionaladiusofthekthmoment fwingmassfic(S) 11.7 non-dimensionaladiusofthekthmoment fwing reafkV) 11.16 non-dimensionaladiusofthekthmoment fvirtualmassR wing engthRe VI.28 Reynolds umbers V.36 spacingparameterS 11.3 wing rea

    Sk II~1.6 kthmoment fwing reat timet t/wingbeat eriodU flapping elocity f thewingUr V.8 relative elocity fthewingUt III flapping elocity f thewing ipv V axial velocityfvortex ingorv 11.13 virtualmassof wingpair0 11.14 non-dimensionalirtualmassof wingpairVk II.15 kthmoment fvirtualmassV III flight elocityV III V/nR number fwing engths ravelled er wingbeatw V axial wakevelocityn thefarwakewo V axial wakevelocity t the ctuator isc= inducedvelocityWRF V Rankine-Froude stimate fwWO,RF V Rankine-Froude stimate fwoX0 IV distance romeading dge to axis ofwingrotationxO IV xO/ca IV geometricalngleof ttackat IV engleof ncidenceao IV zero-liftngle of ttackar IV effectivengle of ttackar. IV effectivengleof ncidence,3 IV half-angle etween hords uring lingor/, stroke laneangleI3o III value ffiintrue'hovering

    ,dr ~~VI.4 relativetroke lane ngle

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    14 C. P. ELLINGTONy VI.13 rotationalift oefficientr circulationr v non-dimensionalirculation=r/ ..)Ar v meanofr over rea ofvortex heetj2 V mean ofr2 ver rea ofvortexheetFo V circulationf pulsedFroude ctuator isca VI.34 magnitude f themeanprofile rag/mean f themagnitude f theprofile rage IV. 14 downwashngleore V core radiusofvortex ingnfarwake60 V coreradius fvortex ing t actuator iscy1 III roll ngleVa VI.53 aerodynamicfficiency6 111.22 angleofelevation fwingwithrespect ostroke laneA IV distance ravelled ywing lement/chordA IV. 17 value ofA over half-strokeA IV. 18 meandistance ravelled y wing ver half-stroke/meanhordp kinematic iscosityf air6 III anglebetweenmeanflight ath and thehorizontalp massdensity f airPb II massdensityf thebodyPw II massdensityf thewinga' V.43 spatialcorrectionactor ornducedpowerX V.44 temporal orrectionactor ornducedpowerpositional ngleofwing nthe troke lane111.24 non-dimensionalorm fqJstrokenglex III anglebetweenongitudinal odyaxis and thehorizontalXo III free ody nglew III angularvelocityuring ronationnd supination6 III meanangularvelocity/wingbeatrequency

    REFERENCESAbbott, . H. & Doenhoff, . E. von 1959 Theoryfwing ections.ew York: DoverBennett, . I966 Insect erodynamics:erticalustainingorcennear-hoveringlight.cience,.Y.152,1263-1266.Bennett, . I973 Effectivenessnd flightf mall nsects. nn. nt. oc.Am. 6, 1187-1190.Brown,R. H. J. I963 The flight fbirds.Biol. Rev. 8, 460-489.Buckholtz, . H. I98 I Measurementsfunsteady eriodic orces enerated y theblowfly lyingna wind unnel.J. exp.Biol. 90, 163-173.Cloupeau, M., Devilliers, . F. & Devezeaux,D. 1979 Directmeasurementsf nstantaneousiftndesert ocust;comparison ithJensen's xperimentsn detachedwings.J. exp.Biol.80, 1-15.Eisentraut, . 1936 Beitrag ur Mechanikdes Fledermausfluges.. wiss. ool. 148, 159-188.Ellington, . P. 1975 Non-steady-stateerodynamicsf heflightfEncarsiaformosa.n SwimmingndflyingnNature(ed. T. Y. Wu, C.J. Brokaw& C. Brennen), ol.2, pp. 783-796. New York:PlenumPress.Ellington, . P. 1978 The aerodynamicsfnormal overinglight:hree pproaches.nComparativephysiologywater,ions nd luidmechanicsed. K. Schmidt-Nielsen,. Bolis & S. H. P. Maddrell), pp. 327-345. CambridgeUniversityress.Ellington, . P. I 980 Vorticesndhovering light.n nstationdreffekten chwingendenierfliigelned.W.Nachtigall),pp. 64-101. Wiesbaden:Franz Steiner.Ellington, . P. I983 Paper npreparation.Graham,R. R. 1932 Slots n thewings fbirds.JlR. aeronaut.oc.36, 598-600.Greenewalt, . H. I960 Hummingbirds.ew York: Doubleday.Hoerner, . F. & Borst,H. V. 1975 Fluid-dynamicift. ricktown, ewJersey:Hoerner luid Dynamics.Horridge, . A. 1956 The flight fvery mall nsects.Nature,ond. 78, 1334-1335.Jensen,M. 1956 Biology nd physics f ocustflight.II. The aerodynamicsf ocustflight. hil. Trans.R. Soc.Lond. 239, 511-552.Lighthill,M. J. 1973 On theWeis-Foghmechanism f ift eneration. . FluidMech. 0, 1-17.Mises,R. von 1959 Theoryfflight.ewYork: Dover.Nachtigall,W. 1977 Die aerodynamischeolaredes Tipula-Flugels nd eine Einrichtungur halbautomatischenPolarenaufnahme.n Thephysiologyf movement;iomechanicsed. W. Nachtigall),pp. 347-352. Stuttgart:Fischer.

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    THE QUASI-STEADY ANALYSIS OF HOVERING FLIGHT 15Nachtigall,W. 1979 Der Taubenflugeln Gleitflugstellung:eometrische enngrossen er Flugelprofile ndLuftkrafterzeugung.. Orn., pz. 120, 30-40.Nachtigall,W. & Kempf, . 1971 Vergleichende ntersuchungenur Flugbiologischenunktion es Daumen-fittichsAlula puria) eiVogeln.Z. vergl. hysiol.1, 326-341.Norberg, . A. 1972 a Flight haracteristicsf woplumemoths, lucita entadactyla. and Orneodesexadactyla.(Microlepidoptera).ool.Scr.1, 241-246.Norberg, . A 1972b Evolution fflight f nsects. ool.Scr.1, 247-250.Norberg, . A. 1975 Hovering light f hedragonfly eschnajuncea., kinematicsnd aerodynamics.n Swimmingandflyingn Natureed.T. Y. Wu, C. J. Brokaw& C. Brennen), ol. 2, pp. 763-781. New York: PlenumPress.Norberg, . M. 1970 Hovering lightfPlecotusuritus. Bijdr.Dierk 0, 62-66. (Proc.2nd int.Bat Res. Conf.)Norberg, . M. 1975 Hovering light f the pied flycatcherFicedula ypoleuca).n Swimmingnd lyingn Nature(ed. T. Y. Wu,C.J.Brokaw& C. Brennen), ol. 2, pp. 869-881.New York: PlenumPress.Norberg, . M. 1976 Aerodynamicsfhovering lightnthe ong-earedat Plecotusuritus..exp. iol. 5, 459-470.Osborne,M. F. M. 195I Aerodynamicsfflapping light ith pplication o nsects. . exp.Biol. 28, 221-245.Pennycuick, . J. I968 A wind-tunneltudy fgliding lightnthepigeonColumbaivia.J. exp. iol.49, 509-526.Pennycuick, . J. 197 I Gliding light f thedog-faced at Rousettusegyptiacusbservedn a wind-tunnel. . exp.Biol.55, 833-845.Schmitz, . W. 1957 AerodynamikesFlugmodells.uisburg:Carl Lange Verlag.Stolpe,M. & Zimmer, . 1939 Der SchwirrflugesKolibri mZeitlupenfilm.. Orn., pz. 87, 136-155.Thom,A. & Swart, . 1940 The forces n an aerofoil t veryowspeeds.JlR. aeronaut.oc.44, 761-770.Tucker,V. A. & Parrott, . C. 1970 Aerodynamicsfgliding lightn a falcon nd other irds.J. exp.Biol.52,345-367.Vogel, S. I967 FlightnDrosophila.II. Aerodynamicharacteristicsfflywings nd wingmodels.J. exp.Biol.46, 431-443.Walker,G. T. 1925 The flapping light fbirds.JIR. aeronaut.oc.29, 590-594.Weis-Fogh, . 1956 Biology nd physics f ocustflight.I. Flightperformancef thedesert ocust Schistocercagregaria). hil.Trans.R. Soc.Lond.B 239, 459-510.Weis-Fogh, . 1972 Energeticsfhovering lightnhummingbirdsnd inDrosophila.J. exp.Biol.56, 79-104.Weis-Fogh, . 1973 Quick estimates fflight itnessn hovering nimals, ncludingnovel mechanisms or iftproduction. . exp.Biol.59, 169-230.Weis-Fogh, . & Jensen,M. 1956 Biology ndphysics f ocust light.. Basicprinciplesn nsect light. criticalreview. hil. Trans. . Soc.Lond. 239,415-458.Withers, . C. I98I Anaerodynamic nalysis fbirdwings s fixed erofoils. . exp.Biol.90, 143-162.Zimmer, . 1943 Der FlugdesNektarvogelsCinnyris).. Orn., pz. 91, 371-387.