461The Journal of Experimental Biology 201, 461–477 (1998)Printed in Great Britain © The Company of Biologists Limited 1998JEB1201

lands Japan

A COMPUTATIONAL FLUID DYNAMIC STUDY OF HAWKMOTH HOVERING

HAO LIU1,*, CHARLES P. ELLINGTON2, KEIJI KAWACHI3, COEN VAN DEN BERG4

AND ALEXANDER P. WILLMOTT1

1Kawachi Millibioflight Project, ERATO, JST, Park Building 3F, 4-7-6 Komaba, Meguro, Tokyo 153, Japan,2Department of Zoology, Downing Street, University of Cambridge, Cambridge CB2 3EJ, UK, 3Research Centre forAdvanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153, Japan and 4Facultyof Human Movement Science, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT, Amsterdam, the Nether*Present address: Department of Mechanical Engineering, Nagoya Institute of Technology (NIT), Gokiso-cho, Showa-ku Nagoya 466,

(e-mail: liu@pfsl.mech.nitech.ac.jp)

Accepted 2 December 1997: published on WWW 26 January 1998

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A computational fluid dynamic (CFD) modellingapproach is used to study the unsteady aerodynamics of theflapping wing of a hovering hawkmoth. We use thegeometry of a Manduca sexta-based robotic wing to definethe shape of a three-dimensional ‘virtual’ wing model and‘hover’ this wing, mimicking accurately the three-dimensional movements of the wing of a hoveringhawkmoth. Our CFD analysis has established an overallunderstanding of the viscous and unsteady flow around theflapping wing and of the time course of instantaneous forceproduction, which reveals that hovering flight is dominatedby the unsteady aerodynamics of both the instantaneousdynamics and also the past history of the wing.

A coherent leading-edge vortex with axial flow wasdetected during translational motions of both the up- anddownstrokes. The attached leading-edge vortex causes anegative pressure region and, hence, is responsible forenhancing lift production. The axial flow, which is derivedfrom the spanwise pressure gradient, stabilises the vortexand gives it a characteristic spiral conical shape.

The leading-edge vortex created during previoustranslational motion remains attached during therotational motions of pronation and supination. Thisvortex, however, is substantially deformed due to couplingbetween the translational and rotational motions, developsinto a complex structure, and is eventually shed before thesubsequent translational motion.

Estimation of the forces during one complete flappingcycle shows that lift is produced mainly during thedownstroke and the latter half of the upstroke, with littleforce generated during pronation and supination. Thestroke plane angle that satisfies the horizontal forcebalance of hovering is 23.6 °, which shows excellentagreement with observed angles of approximately 20–25 °.The time-averaged vertical force is 40 % greater than thatneeded to support the weight of the hawkmoth.

Key words: CFD, computational fluid dynamics, leading-edgvortex, dynamic stall, insect flight, hovering, hawkmoth, Manducasexta.

Summary

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Flapping flight is employed by most flying insect(approximately 750 000 species), birds and bats (nearly 8species). Animal flight covers a wide range of Reynolnumbers, from approximately 10 for the smallest insects wfringed wings to approximately 104 for large insects such ascertain beetles, moths and butterflies, and up to approxima105 for most birds and bats.

The question of how flapping animals fly can be translafrom the aerodynamic point of view to the basic problem the interaction between the wings and the surrounding air. Tway the wings are put in motion, the characteristics of thmotion and the main features of the wing–airflow interactiform the essence of the flight mechanism. Flying animals,general, differ from conventional aircraft in beating their wingrapidly to hover and dart, in performing manoeuvres with rap

Introduction

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accelerations and decelerations, and in quick plunging apitching motions. Therefore, we need to understand taerodynamics of the flapping wings undergoing such highthree-dimensional and unsteady flapping motions, i.e. of tcomplicated three-dimensional, unsteady and viscous floaround wings with a variable geometry.

The goal of understanding the production of aerodynamforces can be approached from either of two directions: characterising the structure of the resultant wake and/or near-field flow or by studying the time course of forcgeneration by the flapping wings. The approach that visualisthe resultant wake, combined with energy conservatioassumptions, allows the force production of the wings to calculated from the flows past them. The key point here is hoaccurately to characterise and model the unsteady wake pat

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H. LIU AND OTHERS

behind the flapping wings, which affects the advance ratio, strength and the location of the wake and, hence, the forcethe body. Rayner (1979) proposed a vortex theory of animflight based on the wake being modelled as a series of vorings periodically shed into the wake. However, flovisualisation of the wake behind a jackdaw Corvus monedula(Spedding, 1986) shows that this model is not quite realisthe momentum of the vortex rings produced less than 35 %the momentum necessary to support the weight of the bMaxworthy (1979) succeeded in his pioneering experimenstudy in visualising the near-field flow around a rigid thredimensional model of the ‘fling’ motion (Weis-Fogh, 1972and found a leading-edge vortex with spanwise flow. Usismoke-visualisation of Manduca sexta (Willmott andEllington, 1997) and of a hawkmoth-based robotic wing (Vaden Berg and Ellington, 1997), this leading-edge vortex athe axial (spanwise) flow at the core of the vortex were cleaobserved and confirmed to be essential for enhancing liftinsect flight.

The second approach, obtaining the time course of fogeneration by flapping wings, has also been followed boexperimentally and numerically. There are relatively femeasurements of the instantaneous forces produced bythree-dimensional flapping wings of animals. Jensen (19measured the mean lift of isolated locust wings. Cloupe(1979) measured instantaneous lift in tethered locusts, concluded that unsteady mechanisms must generate extraforces. Wilkin (1990) also succeeded in measuring tfluctuating lift forces on tethered locusts flying in a wintunnel, finding that the peak lift was approximately twice thquasi-steady value. Wilkin (1993) and Wilkin and William(1993) produced further estimates of the flight forces of bolocusts and moths, determining aerodynamic forces subtracting estimates of the inertial forces from the measuforces. Dickinson (1996) used flow visualisation aninstantaneous force measurements of tethered fruit fl(Drosophila melanogaster) to study the dynamics of forcegeneration during flight, but could not obtain a cohesive pictuof their relationship mainly because of the time delay involvin the force measurements.

Numerical studies of the unsteady flow around thredimensional flapping wings are even fewer, largely becausethe difficulties in modelling the complicated geometry and simulating the three-dimensional movements of wings. Opopular method that is available for computing the unsteaflow about arbitrary bodies is the potential flow panel methoa review of the various panel methods is given by Katz (199More recently, an extension of such a three-dimensional pamethods was developed by Vest (1996), in which the motof the body is prescribed and each wing is assumed to be rand to rotate about a common axis. By using a time-steppsolution procedure, the shape of the trailing wake determined; all vorticity is assumed to be confined to tboundary layer on the surface of the body and in a thin vorwake shed behind the body. Smith (1996) also developedunsteady aerodynamic three-dimensional panel method wh

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can describe the kinematics of a flapping wing relativerealistically and can also account for wing flexibility using finite element method in analysing the dynamic structure.

These panel methods are generally coupled with the discvortex method, and thus can predict the wake as well asunsteady force generation. However, by ignoring the viscosof fluid, they fail to predict reasonably the near-field flow, e.a large attached leading-edge vortex that has been recognrecently (Ellington et al. 1996) to play a key role in liftgeneration for insect flight at Reynolds numbers (Re) around103. At such low Re values, flows around flapping wingsbehave unsteadily, usually demonstrating occurrence/bredown of vortices, separation and reattachment quite differfrom the behaviour of a fluid at high Reynolds numbers.

As a long-standing goal of studying the aerodynamics flapping flight, the best approach is to solve the complviscous and unsteady near-field flows around the flappwings of animals. Solutions of the full Navier–Stokeequations for the three-dimensional, unsteady flow fields challenging, and for the flow around variable-geometry bodthis problem is complicated even further. To understand unsteady aerodynamics induced by a flapping wing, we hdeveloped a computational fluid dynamic (CFD) model (Liu etal. 1995, 1996; Liu, 1997) designed to study both forwaflight and hovering of a three-dimensional virtual wing whiccan have arbitrary variable geometry and can undergo realthree-dimensional motions, accurately mimicking thmovements of insect/bird wings. This CFD model is baseda robust and effective solver of the three-dimensionincompressible, unsteady Navier–Stokes equations (Liu et al.1995, 1996), which incorporates recent techniques. In present study, by defining the three-dimensional geometrya Manduca-based robotic wing, and by hovering the objewith the three-dimensional movements of the wing of hovering hawkmoth, we address an overall understandingviscous and unsteady flows around a three-dimensioflapping wing during the whole process of translational arotational motions, as well as the corresponding time courseinstantaneous force generation.

Materials and methodsResolution of the Navier–Stokes equations

The three-dimensional incompressible unsteady NavieStokes equations in the conservative form of momentum amass are nondimensionalized in a local wingbase-fixcoordinate system (see Fig. 1A), written in an integral form

where vector q=[u, v, w, p]T in which u, v and w are threecomponents of velocity, and p is pressure. The term Fi

expresses flow flux coming entering and leaving the cboundary surfaces. The term V(t) denotes the volume of the cel

(1)∂∂τ $⌠

⌡V(t) S(t)

qdV + (Fi −qug) ×ndS= 0 ,

463CFD study of hawkmoth hovering

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a

Z (vertical)

X (horizontal) Y (horizontal)

y

Downwash

Stroke plane

0

r00′

(x, y, z): local coordinate system(X, Y, Z): global coordinate system

A

Coordinate systems

φ

φ, positional angle θ, elevation angleαF, angle of attack of the leading edge (forewing)αH, angle of attack of the trailing edge (hindwing) β, inclination of the stroke plane

θ

αH

B

Definition of three-dimensional motions

y

z

z

x

y

Stroke plane

Stroke plane

x

z

X

Z

βαF

0′Fig. 1. Schematic diagram of thevirtual wing. (A) The local stroke-plane-fixed and the global space-fixed coordinate systems. Note thatr00′ denotes the positional vector ofthe origin of the local stroke-plane-fixed system and 0′ represents thecentre of the stroke plane (the wingbase). The local wingbase-fixedcoordinate system (x, y, z) is fixed onthe centre of the stroke plane (thewing base) with the x-directionnormal to the stroke plane, the y-direction vertical to the body axis,and the z-direction parallel to thestroke plane. (B) Three basiccomponents of flapping motion(positional angle φ), elevationmotion (elevation angle θ), andfeathering motion of the fore- andhindwing [angle of attack of theleading edge (forewing) αF and angleof attack of the trailing edge(hindwing) αH]. These angles ofattack are defined relative to the x–yplane. Rotating the local coordinatesystem by the stroke plane angle (β)gives the global coordinate system(X, Y, Z); X and Y form the horizontalplane, and the Z-direction is vertical.Note that the axes y and Y areparallel.

(i,j,k), a hexahedron constructed from eight grid points, and S(t)are its six boundary surfaces with unit outward normal vectof n. The influence of the moving grid is contributed by thvelocity of the moving cell ug. The term τ denotes thecomputational time. The governing equations are built asolved in the local wingbase-fixed coordinate system illustrain Fig. 1A. A moving grid system is introduced to make grifit the moving and deforming wing at each physical time stand therefore to ensure sufficient grid density in resolving viscous and unsteady flows around the wing surface.

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The governing equations are discretized using the finivolume method (FVM) and are solved in a time-marchinmanner using the pseudo-compressibility technique. The timdependent solution to the governing equations is realised introducing an inner iteration to allow the divergence ovelocity to vanish at each physical time step, and the timderivatives in the momentum equations are differenced usia first-order, two-point, backward-difference implicit formula(Liu et al.1995).

In developing boundary conditions on the body surface, th

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20

40

60

80

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Positional angle

Elevation angleAngle of attack of forewingAngle of attack of hindwing

Ang

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Time in flapping cycle

Fig. 2. The positional angle, elevation angle and the angles of attackof the fore-and hindwing versustime during one complete flappingcycle. The curves are based on the first three Fourier series terms(equations 2–5), which were fitted to measurements of a hoveringhawkmoth by Willmott and Ellington (1997).

no-slip condition is used for the velocity components: i.e. tvelocity of a fluid particle is equal to the velocity of the winat that point. This means that the wing has a smooth surfalthough the roughness of the real hawkmoth wing may rein some discrepancy. For the dynamic effect due to raacceleration/deceleration of the wing, pressure divergencthe surface stencils is used for the pressure condition derfrom the local momentum equation. At open boundary stencto satisfy the conservation of momentum, zero-gradieconditions are utilised for velocity components, which meathat the local fluid can enter and leave the domain freePressure is unchangeable at the open boundary. Details relto the algorithm will be published (H. Liu, in preparation).

Kinematics of the flapping wing

The kinematics of the hovering wing for CFD analyses abased on the kinematic analysis of a hovering hawkmothWillmott and Ellington (1997), which was also used by Vaden Berg and Ellington (1997) as the motions for a roboinsect wing model (the flapper). Three-dimensionmovements of the wing as illustrated in Fig. 1B consist of thrbasic motions: (1) flapping in the stroke plane by the positioangle (φ); (2) rotation of the wing on either side, out of thstroke plane, by the elevation angle (θ); (3) rotation(feathering) of the forewing (leading edge) and hindwin(trailing edge) with respect to the stroke plane by the angleattack of the forewing (αF) and the angle of attack of thehindwing (αH). The positional angle, elevation angle and thangles of attack of the fore- and hindwing are defined (in rusing the first three Fourier terms in Willmott and Ellingto(1997) as illustrated in Fig. 2:

Note that time t is dimensionless and parameter K is thereduced frequency defined by 2πfcm/2Uref, where fis flappingfrequency, cm is the mean wing chord length (also used reference length) and Uref is a reference velocity at the radiuof the second moment of wing area [r2(s): as a fraction of winglength R] (Willmott and Ellington, 1997). The coefficients φcn,φsn, θcn, θsn, αHcn, αFsn, αHcn andαHsnare determined from theempirical kinematic data as in Willmott and Ellington (1997where n is an integer varying from 0 to 3. The referenc

^3

n=0

(2)[φcncos(nKt) + φsnsin(nKt)]φ(t) =

^3

n=0

(3)[θcncos(nKt) + θsnsin(nKt)]θ(t) =

^3

n=0

(4)[αFcncos(nKt) + αFsnsin(nKt)]αF(t) =

^3

n=0

(5)[αHcncos(nKt) + αHsnsin(nKt)] .αH(t) =

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velocity equals ω–r2(s)R, where ω– is the mean angular velocityof the wing. ω– is equal to twice the wingbeat amplitude (Φ, inrad) multiplied by the wingbeat frequency (f, in s−1). TheReynolds number Re is given by:

where ν is the kinematic viscosity of air, 1.5×10−5m2s−1. Forthe hawkmoth, cm=1.83 cm, r2(s)=0.52, R=4.83 cm, Φ=2.0 radand f=26.1 s−1, so Reis between approximately 3000 and 4000The reduced frequency parameter K was calculated as 0.37,resulting in a dimensionless computational period of 8.5. Tdimensionless parameters Reand Kdescribing the flow werethe same for the real moth, the flapper and the computatiocase.

Geometry of the virtual wing and domain discretization

We base our geometry of the object on a tracing of toutline of a fore-and hindwing of a male M. sexta. The flapperwing shape was based on the same tracing (Van den Berg Ellington, 1997), whose aspect ratio is 5.3. A uniformthickness was used, but with elliptic smoothing at the leadiand trailing edges as well as at the tip. On the basis of kinematics of the hawkmoth, in which the two wings do noclap but maintain a certain distance between them, we moa single virtual wing to mimic the hovering mode employed binsects that do not take advantage of the clap-and-flimechanism.

Since the main spars of insect wings radiate from the baof the wings, we constructed our virtual wing in a similamanner. Also, since a hawkmoth wing couple consists of

(6)Re= ,cmUref

ν

465CFD study of hawkmoth hovering

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Fig. 3. (A) Geometry of the virtual hawkmoth wing. The planfor(two-dimensional shape) of the virtual wing is based on a tracinthe outline of a fore- and hindwing of a male Manduca sexta, with anaspect ratio of 5.3. The wing has a constant thickness but smoothed elliptical curves at the leading and trailing edges as weat the wing tip. (B) A O-O type grid system that discretizes physical space in the form of a semi-ellipsoid around the wing. leading-edge vortex was covered by a sufficiently large numbegrid points (15–20) in the streamwise direction such that the vostructure was modelled with high resolution.

A Geometry of hawkmoth wing

B Grid system

Forewing

Open boundary

Symmetrical plane

Co-axis

Hindwing

connected forewing and hindwing, a two-block method wdeveloped in which each block of grids corresponded to of the wings. The two parts were joined by a co-grid line at co-axis between them (Fig. 3A).

To facilitate resolution of the viscous flow at the leadiedge, trailing edge and wing tip as well as to save computime, an O-O type grid topology was introduced (Fig. 3B). this scheme, the whole physical domain is a semi-ellipsoidgrid system with 72 000 (79×29×29) nodes was generatearound the wing. Grids were clustered to the wing surfaceparticular to the leading edge, the trailing edge and the wingTo avoid the unstable reflection of the solution at the opboundary, the distance far from the wing was six times the mchord length.

To model the moving/deforming boundary during movemeof the wing, a special method was developed to regenerate explicitly at each time step, which minimizes the computing tiin regridding. To deal with large deformations of the win

asonethe

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ntsgridsmeg,

regridding is implemented in two steps: (1) rotation of the gridin the whole domain according to the ‘rigid’ rotational motion othe wing, i.e. flapping, elevation and feathering, which keeps tqualities of the original grids; followed by (2) nonlinear twistingof the grids to incorporate flexing and camber, and stretching grid lines to fit the deformed wing completely. In the movemenof the flapper (Van den Berg and Ellington, 1997), camber formed as a result of the difference in angle of attack betwethe forewing and hindwing. This camber causes shear in the wmembrane, which is particularly pronounced during supinatioand pronation, and which is relieved by a small spanwise slidimotion of the hindwing. We realise this deformation numericallby first rotating the grids according to the angle of attack of thflexing forewing, and then flexing the hindwing to account fothe difference between the angles of attack. To mimic the realiswing, this flexing of the hindwing is achieved by moving gridpoints vertically (H. Liu, in preparation).

Evaluation of forces

The Navier–Stokes equations, once solved, describe tnear-field flows and the resultant wake in terms of the veloccomponents and pressure at discretized grid points or ccentres. The lift and drag forces acting on the wing surface cbe evaluated from the pressure and stresses along its surfThe resultant lift and thrust forces are calculated first in thlocal wingbase-fixed coordinate system (x, y, z), as illustratedin Fig. 1A, and then transformed into the global space-fixecoordinate system (X, Y, Z) by accounting for the stroke planeangle.

The local forces are evaluated by a summation of viscoand inviscid fluxes on the wing surface, which are built ansaved during computation, leading to three force componein the x- ,y- and z-directions (Fig. 1A); this can minimise thenumerical error arising from integrating the projection opressure and stresses. All the force-related quantities nondimensionalized with respect to the reference velocity Uref,the reference length cm and the platform area of the wing Swing

such that:

where L is lift, T is thrust, CL and CT are the lift coefficientand the thrust coefficient, respectively, and ρ is the air density.

To assess the vertical force component that supports insect’s weight, we need to transform the lift and thrust intthe global coordinate system. Considering the inclination ang(β) of the stroke plane with respect to the horizontal plane, wcan obtain the vertical (Cv) and horizontal (Ch) forcecoefficients respectively as

Cv =CLsin(β) + CTcos(β) , (9)and

Ch=CLcos(β) −CTsin(β) . (10)

(8)CT = ,T

GρUref2Swing

(7)CL = ,L

GρUref2Swing

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H. LIU AND OTHERS

The force balance of hovering requires the mean vertical foto equal the weight, and the mean horizontal force to be zeIn addition to inserting the observed stroke plane angle β intoequations 9 and 10, therefore, we can also calculate β suchthat:

where CL–

and CT–

are the mean values of CL and CT,respectively.

Results and discussionValidation

A series of validation studies have been undertaken and hshown that this computational model is robust and capablereasonably estimating unsteady fluid–structure interactionsbiomechanical problems (Liu et al. 1995, 1996; H. Liu, inpreparation). To establish further the validity of the model simulating the aerodynamics of hawkmoth hovering, wpresent below a comparison between our computed flows the experimental results of Van den Berg and Ellington (199Details of this comparison will be published (H. Liu, inpreparation).

Van den Berg and Ellington (1997) used video and 35 mphotographs to record the flow around the flapper using smreleased from the leading edge. The visualised flows wgiven at five spanwise cross sections at distances of 0.2R,0.5R, 0.63R, 0.75Rand 0.87Rfrom the base, respectively. Thecore positions of the leading-edge vortices and treattachment points are shown in Fig. 4. The flow at thrpositional angles φ during the downstroke with values of 30 °0 ° and −36 ° is also illustrated in Fig. 4. For comparison, thcomputed flows were visualised at the same three positioand the instantaneous streamlines are plotted in top and views relative to the stroke plane (Fig. 5). Pressure contoover the wing are coloured: red indicates high pressure, whblue represents low pressure. There is good qualitatagreement between our CFD analysis (Fig. 5) and experimental results (Fig. 4) in terms of three-dimensionleading-edge vortex flow. The CFD results show clear axflow at the core of the leading-edge vortex during the middand late downstroke, giving the vortex a spiral conical shaAlso, the negative pressure region over the wing surface nthe leading edge matches quite well the location of tmeasured cores of the leading-edge vortices. Additionallyspanwise velocity gradient can be seen in Fig. 6. The svortex from the early half of the downstroke can still be seabove the wing, which is in good agreement with the smovisualised flow in Fig. 4B (the red-coloured vortex). A strontip vortex with large velocity magnitudes (Fig. 6) is detecteat the position where the leading-edge vortex breaks down is shed from the wing tip.

In contrast, the computed leading-edge vortex is slighsmaller than the experimental one (Figs 4, 5). In a study of visualised airflow around a tethered hawkmoth in a wind tun

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CT

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(Willmott and Ellington, 1997), the leading-edge vortex waalso observed to be small at low near-hovering airspeeds. Nthat streaklines, not streamlines, were observed in the flapand hawkmoth experiments.

We had no measured time course of instantaneous forcevalidate the computed force-related quantities. With the goqualitative agreement of the flows between the simulation athe measurements, however, we think that the present Cmodel can give a reasonable understanding of the complicaunsteady fluid phenomena in insect flight. In the next sectiowe address an overall understanding of how the viscous aunsteady flows near the wing behave and how tinstantaneous forces are produced during the course ocomplete flapping cycle.

CFD-visualised flows during a flapping cycle

During downstroke

As illustrated in Fig. 5, a large leading-edge vortex is readproduced during most of the downstroke. During the eardownstroke (Fig. 5A), a coherent leading-edge vortex (LEV1)stretches from the base to approximately 60–75 % of the wlength. LEV1 is quite two-dimensional in structure, althoughsome axial flow is visible, with the swirls closely attached the leading edge of the upper surface and with the coenlarging towards the wing tip.

As the wing moves to a horizontal position (Fig. 5B), LEV1becomes a large, spiralling vortex with strong axial flow at thcore, towards the wing tip. The average angle of attack relatto the local flow at this instant is approximately 50 °, ignorinthe mean induced velocity. This is much higher than the angapproximately 20–30 °, at which stall is initiated for astationary hawkmoth wing (Willmott and Ellington, 1997)LEV1 is very probably created by dynamic stall. Note thaLEV1 is very three-dimensional in structure, consisting of conical spiral, enlarging as it is swept along the wing. UndLEV1, there is a region of negative pressure (coloured blue)the wing surface, which corresponds to the diameter alocation of the vortex and which will augment the lift force(Fig. 5B).

The axial (spanwise) flow is probably closely related to thspanwise pressure gradient resulting from the spanwvelocity gradient along the flapping wing. The axial flow iaccelerated by the spanwise pressure gradient to approxima50–75 % of the wing length, where a negative pressure islais observed (Fig. 5B). The low-pressure region is quite thredimensional in structure where the spanwise pressure gradreverses. This corresponds to a large iso-speed area, wheraxial flow may reach its peak (Fig. 6) and then starts decelerate due to a reversepressure gradient (Fig. 5B). Atapproximately 75 % of the wing length, the axial flow ireduced to zero (Fig. 5B). These results strongly support thoof Van den Berg and Ellington (1997), who found that thlargest axial flow velocities were comparable with the meawing tip velocity and were reached at approximately 50–60of the wing length.

467CFD study of hawkmoth hovering

Fig. 4. Smoke-visualised flows from Van den Berg and Ellington (1997) during the downstroke at positional angles of (A) φ=30 °, (B) φ=0 °and (C) φ=−36 °. The smoke is released from a stiff brass smoke rake extending along the leading edge. A large spiral leading-edge vortex isclearly observable in B and C (coloured blue). Top views (left) from Van den Berg and Ellington (1997) illustrate the location and diameter ofthe leading-edge vortices at the three positional angles; the dashed lines show reattachment points. Note that the area coloured red is theleading-edge vortex observed in the early downstroke and the blue area is the leading-edge vortex in the middle of the downstroke. R, winglength.

468

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H. LIU AND OTHERS

Top view

During downstroke

A

B

C

Side view1.02e+005.23e−012.203−02−4.79e−01−9.80e−01−1.48e+00−1.98e+00

Immediately after the middle of the downstroke, the coreLEV1 breaks down at 75 % of the wing length, and the region separates from the wing early in the second half of

oftipthe

downstroke. We think that this breakdown and separation LEV1into the tip vortex is because of the instability of the flowwith the reverse pressure gradient combined with the onset

469CFD study of hawkmoth hovering

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Fig. 5. Top and side views relative to the stroke plane of CFvisualised flows at three positional angles during the downstrok(A) φ=30 °, (B) φ=0 ° and (C) φ=−36 °. Pressure contours on the winsurface are plotted; red denotes high pressure while blue reprelow pressure. Instantaneous streamlines are released at the leedge from both the upper and lower surfaces of the wing, andfollowed until the longest ones leave the computational domain. Tlarge, coherent leading-edge vortices (LEV1 and LEV2) with strongaxial flow are present in B and C, which show very good agreemwith the experimental results (see Fig. 4). In A and B, a spanwpressure gradient is observed, which reverses direction near thetip. The opposing pressure gradients stop the axial flow at 80 %50 % of the wing length (B and C), respectively, at which points leading-edge vortices break down. −press. island, negative pressuisland.

wing deceleration after the middle of the downstroke. LEV1does not show a significant change from the base to the 7position.

Towards the end of the downstroke, a new leading-ed

Iso-speed contours an

Top

2.39e+002.15e+001.91e+001.68e+001.44e+001.20e+009.57e−017.18e−014.79e−012.39e−010.00e+00

Fig. 6. Top view of iso-velocity contours in a plane cutting the leais a spanwise velocity gradient from the base to the tip of the winleading-edge vortex. The shed vortex of the early downstroke is where the leading-edge vortex (LEV1) breaks down and is shed.

5 %

ge

vortex (LEV2) is created between the wing tip and the brokendown position (Fig. 5C). LEV2also shows a spiral axial flow,but directed towards the base, and it connects to the originLEV1. This reverseaxial flow is apparently created by thereversepressure gradient originally generated by the tip vorteLEV2 is strongly affected by the tip vortex and is relativelyunstable; the reverse pressure gradient meeting the spanwgradient leads to a weak axial flow.

As the wing approaches the end of the downstroke and stasupinatory rotation, LEV2begins to be pushed off the leadingedge because of the deceleration of the wing, and tbreakdown point of LEV1moves in to 50 % of the wing length(Fig. 5C). Note that LEV2looks like a free vortex but is notshed, maintaining a constant vorticity and staying connectedthe tip vortex because of the axial flow.

During supination

When the positional angle of the wing reaches its minimum(−45 °), i.e. at the bottom of the stroke (Fig. 7), the angles

D-e atgsentsading arewo

entise

wing andthere

d instantaneous streamlines

view

ding edge of the wing together with instantaneous streamlines at φ=0 °. Thereg, but the flow becomes more complicated near the point of breakdown of thestill visible above the wing. A strong tip vortex with high velocity is detected

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H. LIU AND OTHERS

attack of the fore- and hindwing have risen to 25 ° and 1(Fig. 2), respectively, resulting in a cambered profile. TLEVs produced during the downstroke and the tip vorappearing during early supination are not shed yet but, witslight deformation, turn into a ‘hook-shaped vortex’ (Fig. 7BThe hook-shaped vortex stays attached to the leading edgewraps around the wing tip, producing an island at the centrethe ‘hook’ where all the reattachment (and/or attachmeflows meet. Hence, there is a stagnation point at the cenresulting in a positive pressure region. Additionally, a shelayer vortex (SLV, a vortex created by the steep gradient of tshear layer close to the wing surface) is observed on the losurface of the wing, running from the base to the wing tcausing a region of negative pressure (Fig. 7C).

During the latter half of supination, when the wing translatupwards, rotates, flexes and increases its angle of attackhook-shaped vortex is strongly deformed and very quiccollapses beyond the mid-span joint of the two LEVs. LEV1 ispushed onto the upper surface of the wing with a straight of axial flow, but is virtually separated at the mid-span aconnects to a large flux of flow going over the upper surfaand shedding from the trailing edge. This shedding over trailing edge during late supination is also observed in the flovisualisation experiments of a tethered peacock butteInachis io(Brodsky, 1991). LEV2 rolls over the leading edgeonto the lower surface, connects to the SLVand is eventuallyshed from the trailing edge.

During upstroke

Immediately after shedding of the hook-shaped vortex at end of supination, a very small leading-edge vortex appearthe wing tip at a positional angle of −36 ° (Fig. 8A). It is notseparated, like a tip vortex. The vortex is quite twdimensional in structure, although there is some spiral axflow inside. We did not observe a separate starting vortexthe early upstroke. The leading-edge vortex is limited tonarrow region close to the wing tip and barely affects forgeneration at this instant (see below).

By the time that the wing reaches the middle positi(Fig. 8B), the vortex extends from the wing tip to the base. Tvortex still has a two-dimensional structure and is very smprobably because of the small angle of attack with respecthe local flow. The vortex is tightly attached to the leading edbut merely leads to a very small negative pressure region awing tip. The flow follows both the upper and lower surfacsmoothly. The axial flow at the core of the vortex initialtravels from the wing tip to the base, but reverses directduring the latter half of the upstroke.

After the middle of the upstroke, the leading-edge vortgrows rapidly and hence enlarges the negative pressure reWhen the wing reaches a positional angle of 51 ° (Fig. 8C), vortex dimensions are comparable with those observed at edownstroke. Just as during the later part of the downstrokreverse pressure gradient and a reverse axial flow run frombreak-down point of the upstroke leading-edge vortex to wing tip. The vortex breaks down at approximately 60–70

0 °hetexh a). and of

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of the wing length, but there is no significant shedding of ttip vortex, which is quite different from the downstroke. Wbelieve that, as in the downstroke, the position of breakdoresults from the balance between the spanwise pressgradient and the reverse pressure gradient at the wing tip.

During pronation

During pronation, the wing has a small velocity as thpositional angle reaches its peak (65 °) and the angles of atof the fore-and hindwing approach zero (Fig. 2). The near-fieflow around the wing is visualised in top and side views Fig. 9. The upstroke LEV remains attached to the leading edgof the lower wing surface (Fig. 9A).

A trailing-edge vortex (TEV) is also detected (Fig. 9B,C)below the lower surface, which is probably due to winrotation during the first half of pronation. This vortex runs frothe base to the tip, is larger than the LEV and lies below it.Note that the LEVspirals to the wing tip, crosses the lowesurface of the wing and eventually connects to the TEV. Thisleads to a large area of flow attachment and hence a pospressure region (Fig. 9B). Furthermore, a shear-layer vor(SLV) (Fig. 9C) considerably larger than that at supinaticovers the upper surface and leads to a large region of negapressure (Fig. 9C).

As the wing continues to translate and rotate, changingangle of attack, the separated region of the SLV graduallygrows and moves towards the trailing edge, where it eventually shed into the wake. As the TEV is shed into thewake, the LEV breaks down below the wing with a large fluxof flow at the midwing. At the end of pronation, when the winis about to start the downstroke, air flows smoothly over tlower surface but leaves a large messy flow from the shed Sof the upper surface. This implies that all the vortices observduring pronation are shed before the downstroke and thus tin order to enhance lift generation, the wing must producenew leading-edge vortex on the downstroke.

Estimation of force generation

To link the CFD-visualized flows to the time course oinstantaneous forces, the vortex system is illustrated in Fig.The time variation curves (Fig. 11) of the vertical anhorizontal forces are given in the global space-fixed coordinsystem (see Fig. 1A). The mean horizontal force must be zin hovering, and substitution of the computed forces inequation 11 suggests that a stroke plane angle (β) of 23.6 ° isrequired to satisfy the force balance. This value is in excellagreement with the range of approximately 20–25 ° reportby Willmott and Ellington (1997) for hovering Manduca sextand it has been used in equations 9 and 10.

Comparing Fig. 10 with Fig. 11, we see that the sheddingall the vortices at the end of pronation apparently leads tosmall decrease in vertical force. The first half of thdownstroke shows a steep increase in lift, and the vertical focoefficient rises almost linearly to a peak of approximately 0This is probably due to delayed stall, and the increasing sand hence strength, of the leading-edge vortex (LEV1 in

471CFD study of hawkmoth hovering

During supination

ATop view

BSide view

(upper surface)

CSide view

(lower surface)Fig. 7. Top and side views of CFD-visualised flows with instantaneousstreamlines and surface pressurecontours during supination. Red denoteshigh pressure, and blue denotes lowpressure. (A) Top view of the flowshowing that the downstroke leading-edge vortices (LEV1 and LEV2) are stillattached to the leading edge andconnected to each other, breaking downat the mid-span. (B) Side view of theupper surface of the wing showing ahigh-pressure island near the wing tipand a ‘hook-shaped vortex’. (C) Sideview of the lower surface of the wingshowing a shear-layer vortex (SLV)covering most of the front of the wing.

472 H. LIU AND OTHERS

Fig. 8. Top and side views of CFD-visualised flows with instantaneous streamlines and surface pressure contours during the upstroke atpositional angles of (A) φ=−36 °, (B) φ=30 ° and (C) φ=51 °. Red denotes high pressure; blue denotes low pressure. A very small leading-edgevortex (LEV) is observed, quite two-dimensional in structure during the first half of the upstroke, but it gradually grows into a larger three-dimensional spiral vortex during the second half of the upstroke. Note that the upstroke LEV is considerably smaller and weaker than thedownstroke LEV.

Top view

During downstroke

Side view

A

B

C

473CFD study of hawkmoth hovering

h its

During pronation

ATop view

BSide view

(upper surface)

CSide view

(lower surface)

Fig. 9. CFD-visualised flows with instantaneousstreamlines and surface pressure contours duringpronation. Red denotes high pressure; blue denoteslow pressure. (A) Top view of the flow showingthe upstroke leading-edge vortex (LEV) stillattached to the leading edge but breaking down at60–75 % of the wing length. (B) Side view of thelower surface of the wing with a high-pressureregion near the wing tip. A trailing-edge vortex(TEV) connects to the broken-down leading-edgevortex. (C) Side view of the upper surface showinga large shear-layer vortex (SLV) covering most ofthe fore- and hindwing and leading to a lower-pressure region that contributes to lift generation.

Fig. 5A,B) produces an increasing negative pressure regioncontrast, the lift shows relatively non-linear features during tlatter half of the downstroke (Fig. 11) but maintains a vehigh value. We believe that these forces are predomina

. Inheryntly

produced by the two leading-edge vortices LEV1 and LEV2(Fig. 5C), which are also created by delayed stall. LEV2,withreverse axial flow originating from the wingtip vortex witstrong non-linearity, rapidly increases its strength and hence

474

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H. LIU AND OTHERS

Downstroke

Tran

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ion

Pronation

LEV

TEV

SLV

Axial flow

Vortex

Supination

LEV1/2

SLV

Upstroke

LEV

LEV1/2 and SLV she

LEV, TEV and SLV shed

Reverse point

Hook-shaped vortex

LEV

Rotation

Rotation

Fig. 10. Diagram of thevortex system during thecomplete wingbeat cycle.The shaded area atpronation denotes themorphological lower wingsurface on the insectdiagram (insets). A largeleading-edge vortex (LEV)with strong axial flow isobserved during thedownstroke. This LEVisstill present duringsupination, but turns into ahook-shaped vortex. Asmall LEV is also detectedduring the early upstroke,and gradually grows into alarge vortex in the latterhalf of the upstroke. ThisLEV is still observedclosely attached to the wingduring the subsequentpronation, where a trailing-edge vortex (TEV) and ashear-layer vortex (SLV) arealso formed, togetherforming a complicatedvortex system.

fractional contribution to lift production. Therefore, thaerodynamics of the flapping wing, particularly during thlatter half of the downstroke, is strongly dependent upon unsteady three-dimensional movements of the wing.

During the upstroke, the lift coefficient shows a moderaincrease, from a slightly negative value immediately afsupination to a peak of approximately 0.4 just prior pronation (Fig. 11). A negative horizontal force, or thrust, produced throughout the upstroke, which balances backwardly directed horizontal force of the downstrokDuring the first half of the upstroke, therefore, the dominaforces are very likely to be drag-based (both pressure friction drag). In the late upstroke (Fig. 8C), we see a LEVwithaxial flow developing quite quickly, and this non-linearises tforce variation (Fig. 11) as pronation is approached.

To investigate the extent to which the translational motiis responsible for force production during one compleflapping cycle, we calculated the mean vertical and horizonforces during the downstroke and upstroke by integrating instantaneous forces. Mean vertical and horizontal focoefficients are respectively 0.46 and −0.31 during thedownstroke, and 0.14 and 0.31 during the upstroke. Note most of the vertical force, nearly 77 % of the total in one cyc

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is produced during the downstroke, but 23 % is still produceduring the upstroke. The backward thrust force during thdownstroke is completely cancelled out by a forward thruforce during the upstroke because of the stroke plane ancalculated using equation 11.

The complex vortex structure during the upstroke, with thLEV, the TEVand the SLV(Fig. 9), complicates force productionduring pronation. On the lower wing surface, the interactiobetween the LEVand the TEVresults in a large region of positivepressure (Fig. 9C); on the upper surface, we see a large regof negative pressure at the SLV core. Note that the lift forceremains positive during pronation (Fig. 11). With the calculatestroke plane angle of 23.6 °, we see a large instantaneous verforce coefficient of approximately 0.35 immediately after thhighest wing position, which is even larger than the mean valof 0.3 during one complete cycle (Fig. 11). Because pronatiis so brief, however, it contributes only slightly to lift production

Analysis of the aerodynamics during pronation suggests ththe different three-dimensional movements (Fig. 2) of the winduring supination are of great importance in avoiding negatilift production. The asymmetric kinematics of supinationcreates a complex ‘hook-shaped’ vortex on the upper surfa(Fig. 7B) with a small SLV on the lower surface (Fig. 7C),

475CFD study of hawkmoth hovering

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Fig. 11. Coefficients of vertical and horizontal forces versuspositional angle. Arrows for each phase of the wingbeat cycleindicate the direction of time. The mean vertical force coefficient is0.3, equivalent to a vertical force of approximately 11.05 mN andsufficient to support half the weight of a hawkmoth (7.8 mN).

Fig. 12. The angles of attack relative to the local flow during onecomplete flapping cycle. The angles are calculated for the spanposition r2(s)R. Note that the effect of the downwash is not takeninto account. The relative angle of attack in most of the downstrokeis roughly flat with a value of approximately 50 °, which would bereduced by approximately 15 ° by the downwash.

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−40 −20 0 20 40 60

Forc

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Positional angle (degrees)

Upstroke

Downstroke

Supination

Pronation

Vertical force

Horizontal force

−50

−100

−150

150

50

0

100

Ang

le (

degr

ees)

−1.6 −1.2 −0.8 −0.4 0.40 0.8 1.2 1.6

Time in wingbeat cycle

Downstroke Upstroke

Positional angleAngle of attack of the leading edge (forewing)Relative angle of attack

which is quite different from that seen during pronatio(Fig. 9). Therefore, little negative lift and thrust forces aproduced during supination (Fig. 11).

To assess whether the vertical force from the prescomputation is capable of supporting the hovering moth, further integrated the vertical coefficients during one compleflapping cycle and obtained a mean value of 0.3. Note thatforce coefficients are nondimensionalised by normalizationthe maximum velocity during one flapping cycle. Given threference velocity Uref=2.6 m s−1 and the surface area Swing=8.8cm2 of a real Manduca sextawing (Willmott and Ellington,1997), the vertical force Fv is equal to:

Fv = 0.5rUref2SwingCv , (12)

where the density of air ρ is taken as 1.225×10−6kg m−3. Thetotal vertical force is therefore equal to 11.05 mN for one winThe mean force required per wing is half the weight of thawkmoth (mass 1.58 g) or approximately 7.8 mN. Hence, calculated mean force in a complete flapping cycle is 1times that required to support the weight of the hawkmoth. Tinsect body may interact with the flow, but it is very unlikethat this would result in an almost 41 % reduction of thvertical force.

Leading-edge vortices during up- and downstroke

Large leading-edge vortices that enhance the circulatoryand that are likely to feature prominently in the delayed stof insect wings are commonly observed in two-dimensionmodel experiments at appropriate Reynolds numbers (Newman 1977; Saharon and Luttges, 1989; Dickinson aGötz, 1993). However, these large two-dimensional leadinedge vortex mechanisms can only give approximaexplanations of insect flight because they lack the axial flgenerated by the more sophisticated three-dimensio

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mechanism that couples both three-dimensional motions the geometry of the wing.

A leading-edge vortex is observed (Figs 4B, 5B) and a lalift is detected (Fig. 11) during the downstroke, but how is tleading-edge vortex created? Using the instantaneous velobased on the kinematic data (Fig. 2), we first calculated angles of attack with respect to the local flow at the spposition r2(s)R and plotted them in Fig. 12. These values dnot include the effect of the downwash from the inducvelocity field. The angle of attack is roughly flat during moof the downstroke with a value of approximately 50 °, whicwould be reduced by approximately 15 ° by the downwa(Willmott and Ellington, 1997). It is during this period that thlarge vortices (LEV1 and LEV2) are closely attached to thleading edge on the upper wing surface. Compared with stall angle of attack (approximately 20–30 °) of a hawkmowing in steady flow (Willmott and Ellington, 1997), thisindicates that the leading-edge vortices are created by delastall. The constancy of the angle of attack further implies tit might be optimal for creating the leading-edge vorte(Fig. 5) and hence the large lift production (Fig. 11).

Why is the leading-edge vortex spiral? The pressure graddue to the velocity gradient and/or spanwise centrifugacceleration may well have been responsible for the obseraxial flow. However, these conditions also apply to helicoprotors and wind turbine blades, but large-scale spanwise flohave not been observed for them (De Vries, 1983). So whthe spiral leading-edge vortex not formed for rotary winwhereas it is clearly present for flapping wings?

On the basis of our analysis given above, we think that thmay be two essential conditions for the existence of a thr

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H. LIU AND OTHERS

dimensional vortex with axial flow: vorticity and a pressugradient at the core of the vortex. Flapping insect wings alarge angle of attack can produce a strong leading-edge vovery probably by delayed stall. At the low Reynolds numbeat which insects fly, this LEV, when it occurs, would usuamaintain chordwise flow (flow from the leading-edge to thtrailing-edge) without shedding, which would further increaits strength. The steep spanwise pressure gradient, whicprobably due to the velocity gradient, can produce the stroaxial flow that concentrates the LEV, preventing it fromaccumulating into a large vortex that would be unstable,observed in previous two-dimensional experiments relevaninsect flight (Edwards, 1982; Dickinson, 1996).

In helicopter rotors and wind turbine blades, the fircondition is probably not satisfied. The high Reynolds numb(up to approximately 106) together with a constant small anglof attack will not result in a strong and stable vortex; evena vortex were generated during dynamic stall at high angleattack, it would not be able to resist the very strong chordwflow and would immediately collapse. Thus, even though second condition may be fulfilled, we observe only a smradial velocity component (Chee, 1990).

How is the reverse pressure gradient at the wing (Fig. 5B) produced? The large angle of attack of the wiusually leads to a steep velocity gradient at the wing tip athus produces a pressure gradient between the lower and usurfaces (Fig. 5B). As a result, as is often seen in flows aroa delta wing, a wing tip vortex is observed. Differences velocities at the wing tip during the downstroke therefore leto a reverse pressure gradient against the spanwise graproduced by the velocity gradient (Figs 5B, 6). Note that boflow and pressure distribution at the wing tip are very muthree-dimensional in structure and therefore are strondependent upon both the local geometry and the wing motioCombining with the spanwise pressure gradient, the revepressure gradient decelerates the axial flow to zeroapproximately the 75 % wing length (Fig. 5A,B). During thlatter half of the downstroke, the wing decelerates and hethe velocity gradient decreases, leading to a reduced spanpressure gradient. In addition, with increasing positional angthe velocity gradient over the wing tip between the upper alower surfaces may become steeper, which could result inincrease in the reverse pressure gradient. As a result,reverse pressure gradient becomes increasingly dominantthe point of breakdown is continuously pushed towards base of the wing. At the end of the downstroke, it is locanear the middle of the wing (Fig. 5C). This gives a goexplanation of what happens in the dark region from the mspan to the wing tip as seen in the smoke-visualised flowFig. 4C.

Aerodynamics during pronation and supination

The presence of the LEVduring pronation as well assupination poses a question: how can a LEVcreated duringprevious translational motion remain attached to the wing a

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not be shed? This is a general question that can be appliethe flapping flight of insects and other animals that do not taadvantage of the clap-and-fling mechanism (Weis-Fog1972). As described in previous sections, we believe that tis because of the appropriate coupling of unsteady motioinvolving the deceleration, the elevation and the rotationmotion, which in totodelays shedding of the vortex duringpronation and supination (Fig. 10).

Additionally, the appearance of the shear-layer vortex (SLV)can be explained by the fact that the rotational motion adeceleration of the wing lead to the separation point of the LErolling over the leading edge and attaching onto the uppsurface, which means that the leading edge cuts into the vorAdditionally, with the help of the elevation motion, flow withinthe separated region on the upper surface of the wing wobe accelerated, which could also enlarge this region of suctand lead to the production of the large shear-layer vortex. Nothat the vortex shows a three-dimensional conical structuTherefore, the region of suction leads to the production oflarge negative pressure region covering most of the uppsurface of the wing which, together with the positive pressuregion on the lower surface, may contribute to some extentlift production during pronation, depending upon theorientation of the wing.

Through our CFD analysis, we believe that the rotationflow structures are apparently dependent on the history of theaerodynamics, i.e. the leading-edge vortex created duriprevious translational motion, and also on the complex thredimensional motions of the wing. The complicated vortesystem observed during pronation and supination could noccur in the absence of the upstroke and the downstroke. Trotational motion seems to capture and deform the existivortex, but not create a new leading-edge vortex. For insetaking advantage of the clap-and-fling mechanism, the clapthe top of the stroke would probably shed the leading-edvortex generated during the upstroke, and the quick fling of twings would produce a large new leading-edge vortex on tupper surface that would enhance the circulation and lift on tdownstroke. Although many insects could use the circulatiocreated by a fling/peel motion, most seem to shun tmechanism in spite of the aerodynamic benefits. If nothinelse, there must be significant mechanical wear and damagthe wings caused by repeated clapping (Ellington, 1995). Fmost insects that use a near fling or near peel, where the wiremain separated by a fraction of the chord length and do actually touch, however, it is obvious that they receivaerodynamic benefit from the previous translational motion.

This research was supported by the Japan Science Technology Corporation and by the British Council. Thanks gto Dr Shigeru Sunada for valuable discussions and suggestio

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