This manuscript has been submitted to Journal of Experimental Biology in 2004.

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1 This manuscript has been submitted t Jurnal f Experimental Bilgy in A cmputatinal study f the aerdynamic frces and pwer requirements f dragnfly Aeschna juncea hvering Ma Sun* and Shi ng an Institute f Fluid Mechanics, Beijing University f Aernautics & Astrnautics, Beijing , Peple s Republic f hina rrespnding authr: Ma Sun Address: Institute f Fluid Mechanics Beijing University f Aernautics & Astrnautics Beijing , P.R. hina * sunma@public.fhnet.cn.net Summary Aerdynamic frce generatin and mechanical pwer requirements f a dragnfly (Aeschna juncea) in hvering flight are studied. The methd f numerically slving the Navier-Stkes equatins in mving verset grids is used. When the midstrke angles f attack in the dwnstrke and the upstrke are set t 52 and 8, respectively (these values are clse t thse bserved), the mean vertical frce equals the insect weight and the mean thrust is apprximately zer. There are tw large vertical frce peaks in ne flapping cycle. One is in the first half f the cycle, which is mainly due t the hindwings in their dwnstrke; the ther is in the secnd half f the cycle, which is mainly due t the frewings in their dwnstrke. Hvering with a large strke plane angle ( 52 ), the dragnfly uses drag as a majr surce fr its weight supprting frce (apprximately 65% f the ttal vertical frce is cntributed by the drag and 35% by the lift f the wings). The vertical frce cefficient f a wing is twice as large as the quasi-steady value. The interactin between the fre- and hindwings is nt very strng and is detrimental t the vertical frce generatin. mpared with the case f a single wing in the same mtin, the interactin effect reduces the vertical frces n the fre- and hindwings by 14% and 16% f that f the crrespnding single wing, respectively. The large vertical frce is due t the unsteady flw effects. The mechanism f the unsteady frce is that in each dwnstrke f the hindwing r the frewing, a new vrtex ring 1

2 cntaining dwnward mmentum is generated, giving an upward frce. The bdy-mass-specific pwer is 37 W kg -1, which is mainly cntributed by the aerdynamic pwer. Key wrds: dragnfly, hvering flight, unsteady aerdynamics, pwer requirements, Navier-Stkes simulatin Intrductin Dragnflies are capable f lng-time hvering, fast frward flight and quick maneuver. Scientists have always been fascinated by their flight. Kinematic data such as strke amplitude, inclinatin f the strke-planes, wing beat frequency and phase-relatin between the fre- and hindwings, etc., were measured by taking high-speed pictures f dragnflies in free-flight (e.g. Nrberg, 1975; Wakeling and Ellingtn, 1997b) and tethered dragnflies (e.g. Alexander, 1984). Using these data in quasi-steady analyses (nt including the interactin effects between frewing and hindwing), it was shwn that the lift cefficient required fr flight was much greater than the steady-state values that measured frm dragnfly wings (Wakeling and Ellingtn, 1997a). This suggested that unsteady wing mtin r/and flw interactin between the fre- and hindwings must play imprtant rles in the flight f dragnflies (Nrberg, 1975; Wakeling and Ellingtn, 1997c). Frce measurement n a tethered dragnfly was cnducted by Smps and uttges (1985). It was shwn that ver sme part f a strke cycle, lift frce was many times larger than that measured frm dragnfly wings under steady-state cnditins. This clearly shwed that the effect f unsteady flw and/r wing interactin were imprtant. Flw visualizatin studies n flapping mdel dragnfly wings were cnducted by Saharn and uttges (1988,1989), and it was shwn that cnstructive r destructive wing/flw interactins might ccur, depending n the kinematic parameters f the flapping mtin. In these studies, nly the ttal frce f the fre- and hindwings was measured, and mrever, frce measurements and flw visualizatins were cnducted in separated wrks. In rder t further understand the dragnfly aerdynamics, it was desirable t have the aerdynamic frce and flw structure simultaneusly and als t knw the frce n the individual frewing and hindwing during their flapping mtins. Freymuth (1990) cnducted frce measurement and flw visualizatin n an airfil in hver mdes. One f the hver mdes was fr hvering dragnflies. Only mean vertical frce was measured. It was shwn that large mean vertical frce cefficient culd be btained and the frce was related t a wake f vrtex pairs which prduced a dwnward jet f stream. Wang (2000) used cmputatinal fluid dynamics (FD) methd t study the aerdynamic frce and vrtex wake structure f an airfil in dragnfly hvering mde. Time variatin f the aerdynamic frce in each flapping cycle and the vrtex shedding prcess were btained. It was shwn that large vertical frce was prduce during each dwnstrke and the mean vertical frce was enugh t supprt the weight f a typical dragnfly. During each dwnstrke, a vrtex pair was 2

3 created. The large vertical frce was explained by the dwnward tw-dimensinal jet induced by the vrtex pair. In the wrks f Freymuth (1990) and Wang (2000), nly a single airfil was cnsidered. an and Sun (2001c) studied tw airfils in dragnfly hvering mde using FD methd. Fr cmparisn, they als cmputed the flw f a single airfil. Fr the case f single airfil, their results f aerdynamic frce and flw structure were similar t that f Freymuth s (1990) experiment and Wang s (2000) cmputatin. Fr the fre and aft airfils flapping with 180º phase difference (cunter strking), the time variatin f the aerdynamic frce n each airfil was bradly similar t that f the single airfil; the majr effect f interactin between the fre and aft airfils was that the vertical frces n bth the airfils were decreased by apprximately 20% in cmparisn with that f the single airfil. The abve wrks (Freymuth, 1990; Wang, 2000; an and Sun, 2001c) which btained aerdynamic frce and flw structure simultaneusly were dne fr airfils. It is well knwn that the lift n an airplane wing f large aspect rati can be explained by a tw dimensinal wing thery. But fr a dragnfly wing, althugh its aspect rati is relatively large, its mtin is much mre cmplex than that f an airplane wing. Three dimensinal effect shuld be investigated. Mrever, the effect f aerdynamic interactin between the fre- and hindwings in three-dimensinal case are unknwn. The wrk f an and Sun (2001c) n tw airfils flapping with 180º phase difference shwed that interactin between the tw airfils was detrimental t their aerdynamic perfrmance. This result is ppsite t the cmmn expectatin that wing interactin f a dragnfly wuld enhance its aerdynamic perfrmance. It is f interest t investigate hw interactin effect will be in the three dimensinal case. In the present study, we extend ur previus tw-dimensinal study (an and Sun, 2001c) t three dimensinal case. As a first step, we study the case f hvering flight. Fr dragnfly Aeschna juncea in free hvering flight, detailed kinematic data were btained by Nrberg (1975). Mrphlgical data f the dragnfly (wing shape, wing size, wing mass distributin, weight f the insect, etc.) are als available (Nrberg, 1972). On the basis f these data, the flws and aerdynamic frces and the pwer required fr prducing the frces are cmputed and analyzed. Because f the unique feature f the mtin, i.e. the frewing and the hindwing mve relative t each ther, the apprach f slving the flw equatins ver mving verset grids is emplyed. Materials and methds The mdel wings and their kinematics The fre- and hindwings f the dragnfly are apprximated by tw flat plates. The thickness f the mdel wings is 1%c (where c is the mean chrd length f the frewing). The planfrms f the mdel wings (see Fig. 1A) are similar t thse f the real wings (Nrberg, 1972). The tw wings have the same length but the chrd length f the hindwing is larger than that f the frewing. The radius f the secnd mment 2 f the frewing area is dented by r 2 ( r 2 = r dsf Sf, where r is radial distance S f 3

4 and S f is the area f frewing); r = 0. 61R (R is the wing length). The flapping 2 mtins f the wings in hvering flight are sketched in Fig. 1B. The hindwing leads the frewing in phase by 180 (Nrberg, 1975). The azimuthal rtatin f the wing abut the z axis (see Fig. 1) is called translatin and the pitching (r flip) rtatin f the wing near the end f a half-strke and at the beginning f the fllwing half-strke is called rtatin. The speed at r 2 is called the translatinal speed. The flapping mtin f a wing is simplified as fllws. The wing translates dwnward and upward alng the strke plane and rtates during strke reversal (Fig. 1B). The translatinal velcity is dented by u t and is given by + u =.5πsin(2πτ/ τ + γ), (1) t 0 c where the nn-dimensinal translatinal velcity velcity); the nn-dimensinal time + u u / U t = t (U is the reference τ = tu / c (t is the time; c is the mean chrd length f the frewing, used as reference length in the present study); τ c is the nn-dimensinal perid f the flapping cycle; and γ is the phase angle f the translatin f the wing. The reference velcity is U = 2Φnr2, where Φ and n are the strke amplitude and strke frequency f the frewing, respectively. Denting the azimuth-rtatinal velcity as φ &, we have φ & = u t r2. The angle f attack f the wing is dented by α. It assumes a cnstant value in the middle prtin f a half-strke. The cnstant value is dented by α d fr the dwnstrke and by α u fr the upstrke. Arund the strke reversal, α changes with time and the angular velcity ( α& ) is given by: + + α& =.5α& {1 cs[2π( τ τ ) / ]}, τ r τ τr + τ r, (2) 0 0 r τr where the nn-dimensinal frm α & + = αc & / U ; + α& 0 is a cnstant; τ r is the time at which the rtatin starts; and τr is the time interval ver which the rtatin lasts. In the time interval f τr, the wing rtates frm = α d α t α = 180 α u. Therefre, when α d, α u and are specified, & + can be determined (arund τr α 0 the next strke reversal, the wing wuld rtate frm α = 180 α u t α = α d, the 4

5 sign f the right-hand side f equatin 2 shuld be reversed). The axis f the flip rtatin is lcated at a distance f 1/4 chrd length frm the leading edge f the wing. The Navier-Stkes equatins and slutin methd The Navier-Stkes equatins are numerically slved using mving verset grids. Fr flw past a bdy in arbitrary mtin, the gverning equatins can be cast in an inertial frame f reference using a general time-dependent crdinate transfrmatin t accunt fr the mtin f the bdy. The nn-dimensinalized three-dimensinal incmpressible unsteady Navier-Stkes equatins, written in the inertial crdinate system xyz (Fig. 1), are as fllws: u v w + + = 0, (3) x y z u u u u + u + v + w τ x y z v v v v + u + v + w τ x y z p 1 u u u = + ( + + ), (4) x Re x y y p 1 v v v = + ( + + ), (5) y Re x y y w w w w p 1 w w w + u + v + w = + ( + + ), (6) τ x y z z Re x y y where u, v and w are three cmpnents f the nn-dimensinal velcity and p is the nn-dimensinal pressure. In the nn-dimensinalizatin, U, c and c/u are taken as the reference velcity, length and time, respectively. Re dentes the Reynlds number and is defined as Re = cu υ, where υ is kinematic viscsity f the air. Equatins 3 t 6 are slved using an algrithm based n the methd f artificial cmpressibility. The algrithm was first develped by Rgers and Kwak (1990) and Rgers et al. (1991) fr single-zne grid, and it was extended by Rgers and Pulliam (1994) t verset grids. The algrithm is utlined belw. The equatins are first transfrmed frm the artesian crdinate system ( x,y,z, τ ) t the curvilinear crdinate system ( ξ, η, ζ, τ ) using a general time-dependent crdinate transfrmatin. Fr a flapping wing, in rder t make the transfrmatin simple, a bdy-fixed crdinate system ( 'x'y'z ' ) is als emplyed (Fig.1). In terms f the Euler angles α and φ (defined in Fig.1), the inertial crdinates (,x,y,z) are related t the bdy-fixed crdinates ( 'x'y'z ' ) thrugh the fllwing relatinship: 5

6 x csα csφ = y sin α z sin φcsα sin φ 0 csφ csφsin α x csα y. (7) sin φsin α z Using equatin 7, the transfrmatin metrics in the inertial crdinate system, ξ, ξ, ξ, ξ ) η, η, η, η ) and ζ, ζ, ζ, ζ ), which are needed in the ( x y z τ ( x y z τ ( x y z transfrmed Navier-Stkes equatins, can be calculated frm thse in the bdy-fixed, nn-inertial crdinate system, ξ, ξ, ξ ), η, η, η ) and ζ, ζ, ζ ), ( x' y' z' τ ( x' y' z' ( x' y' z' which need t be calculated nly nce. As a wing mves, the crdinate transfrmatin functins vary with ( x,y,z, τ ) such that the grid system mves and always fits the wing. The bdy-fixed nn-inertial frame f reference ( x y z ) is used in the initial grid generatin. The time derivatives f the mmentum equatins are differenced using a secnd-rder, three-pint backward difference frmula. T slve the time discretized mmentum equatins fr a divergence free velcity at a new time level, a pseud-time level is intrduced int the equatins and a pseud-time derivative f pressure divided by an artificial cmpressibility cnstant is intrduced int the cntinuity equatin. The resulting system f equatins is iterated in pseud-time until the pseud-time derivative f pressure appraches zer, thus, the divergence f the velcity at the new time level appraches zer. The derivatives f the viscus fluxes in the mmentum equatin are apprximated using secnd-rder central differences. Fr the derivatives f cnvective fluxes, upwind differencing based n the flux-difference splitting technique is used. A third-rder upwind differencing is used at the interir pints and a secnd-rder upwind differencing is used at pints next t bundaries. Details f this algrithm can be fund in Rgers and Kwak (1990) and Rgers et al.(1991). Fr the cmputatin in the present wrk, the artificial cmpressibility cnstant is set t 100 (it has been shwn that when the artificial cmpressibility cnstant varied between 10 and 300, the number f sub-iteratin changes a little, but the final result des nt change). With verset grids, as shwn in Fig. 2, fr each wing there is a bdy-fitted curvilinear grid, which extends a relatively shrt distance frm the bdy surface, and in additin, there is a backgrund artesian grid, which extends t the far-field bundary f the dmain (i.e. there are three grids). The slutin methd fr single-grid is applied t each f the three grids. The wing grids capture features such as bundary layers, separated vrtices and vrtex/wing interactins, etc. The backgrund grid carries the slutin t the far field. The tw wing grids are verset nt the backgrund artesian grid and parts f the tw wing-grids verlap when the tw wings mve clse t each ther. As a result f the versetting f the grids, there are hle regins in the wing grids and in the backgrund grid. As the wing grids mve, the hles and hle bundaries change with time. T determine the hle-fringe pints, 6

7 the methd knwn as dmain cnnectivity functins by Meakin (1993) is emplyed. Intergrid bundary pints are the uter-bundary pints f the wing grids and the hle-fringe pints. Data are interplated frm ne grid t anther at the hle-fringe pints and similarly, at the uter-bundary pints f the wing grids. In the present study, the backgrund grid des nt mve and the tw wing-grids mve in the backgrund grid. The wing grids are generated by using a Pissn slver which is based n the wrk f Hilgenstck (1988). They are f O-H type grids. The backgrund artesian grid is generated algebraically. Sme prtins f the grids are shwn in Fig. 2. Fr far field bundary cnditins, at the inflw bundary, the velcity cmpnents are specified as freestream cnditins while pressure is extraplated frm the interir; at the utflw bundary, pressure is set equal t the free-stream static pressure and the velcity is extraplated frm the interir. On the wing surfaces, impermeable wall and n-slip bundary cnditins are applied, and the pressure n the bundary is btained thrugh the nrmal cmpnent f the mmentum equatin written in the mving crdinate system. On the plane f symmetry f the dragnfly (the XZ plane; see Fig.1A), flw-symmetry cnditins are applied (i.e. w and the derivatives f u, v, and p with respect t y are set t zer). Evaluatin f the aerdynamic frces The lift f a wing is the cmpnent f the aerdynamic frce n the wing that is perpendicular t the translatinal velcity f the wing (i.e. perpendicular t the strke plane); the drag f a wing is the cmpnent that is parallel t the translatinal velcity. l f and d f dente the lift and drag f the frewing, respectively; l h and dente the lift and drag f the hindwing, respectively. Reslving the lift and drag int the Z and X directins gives the vertical frce and thrust f a wing. dente the vertical frce and thrust f the frewing, respectively; the vertical frce and thrust f the hindwing, respectively. Fr the frewing, d h f and T f h and T h dente = l csβ + d sin φsin β, (8) f f f T = l sin β d sin φcsβ. (9) f f f These tw frmulae als apply t the case f hindwing. The cefficients f l f, d f, l h, d h, f, T f, h and T h are dented as l, f, d, f, l, h, d, h,, f, T, f,,h and T, h, respectively. They are defined as l,f = lf 2.5ρU ( S + S ), etc. (10) 0 f h 7

8 where ρ is the fluid density, S f and S h are the areas f the fre- and hindwings, respectively. The ttal vertical frce cefficient ( ) and ttal thrust cefficient ( ) f the fre- and hindwings are as fllws:,f,h = +, (11) T = +. (12) T T,f T,h Data f hvering flight in Aeshna juncea High-speed pictures f dragnfly Aeshna juncea in hvering flight was taken by Nrberg (1975) and the fllwing kinematic data were btained. Fr bth the fre- and hingwings, the chrd is almst hrizntal during the dwnstrke (i.e. α d β ) and is clse t the vertical during the upstrke; the strke plane angle ( β ) is apprximately 60 ; the strke frequency ( n ) is 36 Hz, the strke amplitude ( Φ ) is 69 ; the hindwing leads the frewing in phase by 180. The mass f the insect ( m ) is 754 mg; frewing length is 4.74 cm; hindwing length is 4.60 cm; the mean chrd lengths f the fre- and hindwings are 0.81 cm and 1.12 cm, respectively; the mment f inertial f wing-mass with respect t the fulcrum ( I ) is 4.54 g cm 2 fr the frewing and 3.77 g cm 2 fr the hindwing (Nrberg, 1972). On the basis f the abve data, the parameters f the mdel wings and the wing kinematics are determined as fllwing. The lengths f bth wings ( R ) are assumed as 4.7 cm; the reference length (the mean chrd length f the frewing) c = cm; the reference velcity U = 2 Φnr = ms-1 ; the Reynlds number Re = Uc υ 1350 ; the strke perid τc = U nc = γ is set as 180 and zer fr the fre- and hindwings, respectively. Nrberg (1975) did nt prvide the rate f wing rtatin during strke reversal. Reavis and uttges (1988) made measurements n similar dragnflies and it was fund that maximum α& was 10000~30000 deg./sec. Here, α& is set as deg./sec., giving α & and τ r = Results and analysis Test f the slver A single-grid slver based n the cmputatinal methd described abve was develped by an and Sun (2001a). It was tested by the analytical slutins f the bundary layer flw n a flat plate (an and Sun, 2001a) and by the measured unsteady frces n a flapping mdel fruit fly wing (Sun and Wu, 2003). A mving 8

9 verset-grid slver fr tw dimensinal flw based n the abve methd was develped by the same authrs and it was tested by cmparisn with the analytical slutin f the starting flw arund an elliptical airfil (an and Sun, 2001b,c). The tw-dimensinal mving verset-grids slver is extended t three-dimensin in the present study. The three-dimensinal mving verset-grids slver is tested here in three ways. First the flw past a starting sphere is cnsidered, fr which the apprximate slutin f the Navier-Stkes equatins is knwn. Secndly, the cde is tested by cmparing with the results f the single-grid. Finally, the cde is tested against experimental data f a flapping mdel fruit fly wing by Sane and Dickinsn (2001). As a first test, it is nted that in the initial stage f the starting mtin f a sphere, because the bundary layer is still very thin, the flw arund the sphere can be adequately treated by ptential flw thery, and the flw velcity arund the sphere and the drag (added-mass frce) n the sphere can be btained analytically. The acceleratin f the sphere during the initial start is a csine functin f time; after the initial start, the sphere mves at cnstant speed ( U ). In the numerical calculatin, the s Reynlds number [based n U s and the radius ( a ) f the sphere] is set as Fig. 3A shws the numerical and analytical drag cefficients ( ) vs. nn-dimensinal time ( τ s ) ( d = drag 2 2 ρ U s π ; s = tu s 2a 0.5 a τ ). Between τ s = 0 and τ s 0. 2, the numerical result is in very gd agreement with the analytical slutin. Fig. 3B shws the azimuthal velcity ( u θ ) at τ s = 0. 1 as a functin f r 2 a (r is radial distance) with fixed azimuthal angle π 2. The numerical result agrees well with the analytical slutin utside the bundary layer. In the secnd test, the flw arund the starting sphere is cmputed by the single-grid cde, and the results cmputed using the single-grid and mving verset-grid are cmpared (als in Fig. 3). They are in gd agreement. Fr the case f single-grid, the grid is f O-O type, where the numerical crdinates ( ξ, η, ζ ) lie alng the standard spherical crdinates. It has dimensins The uter bundary is set at 30 a frm the sphere. The nn-dimensinal time step is Grid sizes f and were als used. By cmparing the results frm these three grids, it was shwn that the grid size f was apprpriate fr the cmputatin. Fr the case f mving verset-grids, the grid system cnsists tw grids, ne is the curvilinear grid f the sphere; anther is the backgrund artesian grid. The uter bundary f the sphere grid is at 1.4a frm the sphere surface and the ut bundary f the backgrund grid is 30 a frm the sphere. The grid density is made similar t that f the single-grid. In the third test, the set up f Sane and Dickinsn (2001) is fllwed and the d 9

10 aerdynamic frces are cmputed fr the flapping mdel fruit fly wing. The cmputed lift and drag are cmpared with the measured in Fig.4. In the cmputatin, the wing grid has dimensins arund the wing sectin, in nrmal directin and in spanwise directin, respectively; the uter bundary f the wing grid is apprximately 2.0 c frm the wing. The backgrund artesian grid has dimensins and the uter bundary is 20 c frm the wing. The nn-dimensinal time step is Grid density test was cnducted and it was shwn that abve verset grids were apprpriate fr the cmputatin. In Figs 4A,B, the flapping amplitude is 60 and the midstrke angle f attack is 180 and 50 ; in Figs 4,D, these quantities are 50, respectively. The magnitude and trends with variatin ver time f the cmputed lift and drag frces are in reasnably gd agreement with the measured results. The ttal vertical frce and thrust; cmparisn with insect weight In the calculatin, the wings start the flapping mtin in still air and the calculatin is ended when peridicity in aerdynamic frces and flw structure is apprximately reached (peridicity is apprximately reached 2-3 perids after the calculatin is started). Figure 5 shws the ttal vertical frce and thrust cefficients in ne cycle, cmputed by tw grid systems, grid-system 1 and grid-system 2. In bth grid-systems, the uter bundary f the wing-grid was set at abut 2 c frm the wing surface and that f the backgrund-grid at abut 40 c frm the wings. Fr grid-system 1, the wing grid had dimensins in the nrmal directin, arund the wing and in the spanwise directin, respectively, and the backgrund grid had dimensins in the X (hrizntal), Z (vertical) and Y directins, respectively (Figure 3 shws a prtin f grid-system 1). Fr grid-system 2, the crrespnding grid dimensins were and Fr bth grid systems, grid pints f the backgrund grid cncentrated in the near field f the wings where its grid density was apprximately the same as that f the uter part f the wing-grid. As seen in Fig. 5, there is almst n difference between the frce cefficients calculated by the tw grid-systems. alculatins were als cnducted using a larger cmputatinal dmain. The dmain was enlarged by adding mre grid pints t the utside f the backgrund grid f grid-system 2. The calculated results shwed that there was n need t put the uter bundary further than that f grid-system 2. It was cncluded that grid-system 1 was apprpriate fr the present study. The effect f time step value was cnsidered and it was fund that a numerical slutin effectively independent f the time step was achieved if τ Therefre, τ = 0. 02, was used in the present calculatins. Frm Fig. 5, it is seen that there are tw large peaks in ne cycle, ne in the first half f the cycle (while the hindwing is in its dwnstrke) and the ther is in the secnd half f the cycle (while the frewing is in its dwnstrke). It shuld be nted 10

11 that by having tw large peaks alternatively in the first and secnd halves f a cycle, the flight wuld be smther. Averaging (and T ) ver ne cycle gives the mean vertical frce cefficient ( ) [and the mean thrust cefficient ( )]; = 1.35 and T = The value f 1.35 gives a vertical frce f 756 mg, apprximately equal t the insect weight (754 mg). The cmputed mean thrust (11 mg) is clse t zer. That is, the frce balance cnditin is apprximately satisfied. In the calculatin, the strke plane angle, the midstrke angles f attack fr the dwnstrke and the upstrke have been set as β = 52, α d = 52 and α u = 8, respectively. T These values f β, α d and α u give an apprximately balanced flight and they are clse t the bserved values [ β 60, during the dwnstrke the chrd is almst hrizntal (i.e. α d β ), and during the upstrke the chrd is clse t vertical]. The frces f the frewing and the hindwing The ttal vertical frce (r thrust) cefficient is the sum f vertical frce (r thrust) cefficient f the fre- and hindwings. Figure 6 gives the vertical frce and thrust cefficients f the fre- and hindwings. The hindwing prduces a large, h peak during its dwnstrke (the first half f the cycle) and very small, h in its upstrke (the secnd half f the cycle); this is true fr the frewing, but its dwnstrke is in the secnd half f the cycle. mparing Fig. 6 with Fig. 5 shws that the hindwing in its dwnstrke is respnsible fr the large peak in the first half f the cycle and the frewing in its dwnstrke is respnsible fr the large peak in the secnd half f the cycle. The cntributins t the mean ttal vertical frce by the frewing and hindwing are 42% and 58%, respectively. The vertical frce n the hindwing is 1.38 times f that n the frewing. Nte that the area f the hindwing is 1.32 times f that f the frewing. That is, the relatively large vertical frce n the hindwing is mainly due t its relatively large size. The vertical frce and thrust cefficients f a wing are the results f the lift and drag cefficients f the wing. The crrespnding lift and drag cefficients l, f, d, f, l,h, and d, h are shwn in Fig. 7. Fr the hindwing, d, h is larger than l, h during the dwnstrke f the wing; and β is large ( 52 ). As a result, a large part f 11

12 ,h is frm d, h (apprximately 65% f, h is frm d, h and 35% is frm l,h ). This is als true fr the frewing. That is, the dragnfly uses drag as a majr surce fr its weight supprting frce when hvering with a large strke plane angle. The mechanism f the large vertical frce As shwn in Fig. 6, the peak value f, h is apprximately 3.0 (that f, f is apprximately 2.6). Nte that in the definitin f the frce cefficient, that ttal area f the fre- and hindwings ( S + S f h ) and the mean flapping velcity U are used as reference area and reference velcity, respectively. Fr the hindwing, if its wn area and the instantaneus velcity are used as reference area and reference velcity, respectively, the peak value f the vertical frce cefficient wuld be [( S + S ) S ] U ( πu 2) 2 f h h = 2.1. Similarly, fr the frewing, the peak value 2 2 wuld be 2.6 [( S + S ) S ] U ( πu 2) Since the thrust cefficients T,f and T, h f h f = are small,, f and, h can be taken as the cefficients f the resultant aerdynamic frce n the fre- and hindwings, respectively. The abve shws that the peak value f resultant aerdynamic frce cefficient fr the frewing r hindwing is (when using the area f the crrespnding wing and the instantaneus velcity as reference area and reference velcity, respectively). This value is apprximately twice as large as the steady-state value measured n a dragnfly wing at Re = (steady-state aerdynamic frces n the freand hindwings f dragnfly Sympetrum sanguineum were measured in wind-tunnel by Wakeling and Ellingtn, 1997a; the maximum resultant frce cefficient, btained at angle f attack f arund 60, was apprximately 1.3). There are tw pssible reasns fr the large vertical frce cefficients f the flapping wings: ne is the unsteady flw effect; the ther is the effect f interactin between the fre- and hindwings (in steady-state wind-tunnel test, interactin between fre- and hindwings was nt cnsidered). The effect f interactin between the fre- and hindwings In rder t investigate the interference effect between the fre- and hindwings, we cmputed the flw arund a single frewing (and als a single hindwing) perfrming the same flapping mtin as abve. Figure 8A,B gives vertical frce (, sf ) and thrust ( T, sf ) cefficients f the single frewing, cmpared with, f and T,f, respectively. The differences between, sf and, f and between T, sf and T,f shw the interactin effect. Similar cmparisn fr the hindwing is given in Fig. 12

13 8,D. Fr bth the fre- and hindwings, the vertical frce cefficient n single wing (i.e. withut interactin) is a little larger than that with interactin. Fr the frewing, the interactin effect reduces the mean vertical frce cefficient by 14% f that f the single wing; fr the hindwing, the reductin is 16% f that f the single wing. The interactin effect is nt very large and is detrimental t the vertical frce generatin. The unsteady flw effect The abve results shw that the interactin effect between the fre- and hindwings is small and, mrever, is detrimental t the vertical frce generatin. Therefre, the large vertical frce cefficients prduced by the wings must be due t the unsteady flw effect. Here the flw infrmatin is used t explain the unsteady aerdynamic frce. First, the case f single wing is cnsidered. Figure 9 gives the is-vrticity surface plts at varius times during ne cycle. In rder t crrelate frce and flw infrmatin, we express time during a strke cycle as a nn-dimensinal parameter, tˆ, such that t ˆ = 0 at the start f the cycle and t ˆ = 1 at the end f the cycle. After the dwnstrke f the hindwing has just started ( t ˆ = ; Fig. 9A), a starting vrtex is generated near the trailing edge f the wing and a leading edge vrtex (EV) is generated at the leading edge f the wing; the EV and the starting vrtex are cnnected by the tip vrtices, frming a vrtex ring. Thrugh the dwnstrke (Fig. 9B,), the vrtex ring grws in size and mves dwnward. At strke reversal (between t ˆ and t ˆ ), the wing rtates and the EV is shed. During the upstrke, the wing almst des nt prduce any vrticity. The vrtex ring prduced during the dwnstrke is left belw the strke plane (Fig. 9D,E,F) and will cnvect dwnwards due t its self-induced velcity. The vrtex ring cntains a dwnward jet (see belw). We thus see that in each cycle, a new vrtex ring carrying dwnward mmentum is prduced, resulting in an upward frce. This qualitatively explains the unsteady vertical frce prductin. Figure 10 gives the velcity vectrs prjected in a vertical plane that is parallel t and 0.6R frm the plane f symmetry f the insect. The dwnward jet is clearly seen. Figure 11 gives the is-vrticity surface plts fr the fre- and hindwings (in the first half f the cycle the hindwing is in its dwnstrke; in the secnd half f the cycle the frewing in its dwnstrke). Similar t the case f single wing, just after the start f the first half f the cycle, a new vrtex ring is prduced by the hindwing (Fig. 11A); this vrtex ring grws in size and cnvects dwnwards (Fig. 11A,B,). Similarly, just after the start f the secnd half f the cycle, a new vrtex ring is prduced by the frewing (Fig. 11D), which als grws in size and cnvects dwnwards as time increasing. Figure12 gives the crrespnding velcity vectr plts. The qualitative explanatin f the large unsteady frces n the fre- and hindwings is similar t that fr the single wing. On the basis f the abve analysis f the aerdynamic frce mechanism, we give a preliminary explanatin fr why the frewing-hindwing interactin is nt strng and 13

14 is detrimental. The new vrtex ring, which is respnsible fr the large aerdynamic frce n a wing, is generated by the rapid unsteady mtin f the wing at a large angle f attack. As a result, the effect f the wake f the ther wing is relatively small. Mrever, the wake f the ther wing prduces dwnwash velcity, resulting in the detrimental effects. Pwer requirements As shwn abve, the cmputed lift is enugh t supprt the insect weight and the hrizntal frce is apprximately zer; i.e. the frce balance cnditins f hvering are satisfied. Here we calculate the mechanical pwer utput f the dragnfly. The mechanical pwer includes the aerdynamic pwer (wrk dne against the aerdynamic trques) and the inertial pwer (wrk dne against the trques due t accelerating the wing-mass). As expressed in equatin 20 f Sun and Tang (2002), the aerdynamic pwer cnsists f tw parts, ne due t the aerdynamic trque fr translatin and the ther t the aerdynamic trque fr rtatin. The cefficients f these tw trques (dented by Q,a, t and Q,a, r, respectively) are defined as Q,a,t Q,a,r = Qa,t 2 0.5ρU ( Sf + S h ) c (13) = Qa,r 2 0.5ρU ( Sf + S h ) c (14) where Q a, t and a, r Q are the aerdynamic trques arund the axis f azimuthal rtatin ( z axis) and the axis f pitching rtatin, respectively. Q,a, t and Q,a, r are shwn in Fig.13A,B. It is seen that Q,a, t is much larger than Q,a, r. The inertial pwer als cnsists f tw parts (see equatin 35 f Sun and Tang, 2002), ne due t the inertial trque fr translatin and the ther t the inertial trque fr rtatin. The cefficient f inertial trque fr translatin ( = I & φ Q,i, t ) is defined as + Q,i,t, (15) 3 0.5ρ( Sf + S h ) c where & φ &+ is the nn-dimensinal angular acceleratin f wing translatin. Q,i, t is shwn in Fig.13. The inertial trque fr rtatin can nt be calculated since the mment f inertial f wing-mass with respect t the axis f flip rtatin is nt available. Because mst f the wing-mass is lcated near the axis f flip rtatin, it is expected that the inertial trque fr rtatin is much smaller than that fr translatin. That is, bth the aerdynamic and inertial trques fr rtatin might be much smaller 14

15 than thse fr translatin. In the present study, the aerdynamic and inertial trques fr rtatin are neglected in the pwer calculatin. The pwer cefficient ( p ), i.e. pwer nn-dimensinalized by 3 0.5ρ U ( Sf + S h ), is = +, (16) p p,a p,i where + p, a Q,a,tφ, (17) = & + p, i Q,i,tφ. (18) = & p f the fre- and hindwings are shwn in Fig.14. In the figure, cntributins t p by the aerdynamic and inertial trques (represented by p, a and p, i, respectively) are als shwn. Fr the frewing (Fig.14A), the time curse f p is similar t that f p, a in the dwnstrke and t that f p, i in the upstrke; i.e. the aerdynamic pwer dminates ver the dwntrke and the inertial pwer dminates ver the upstrke. This is als true fr the hindwing (Fig. 14B). Integrating p ver the part f a wingbeat cycle where it is psitive gives the + cefficient f psitive wrk ( W ) fr translatin. Integrating p ver the part f the cycle where it is negative gives the cefficient f negative wrk ( the wing in this part f the cycle. W ) fr braking + W and W fr the frewing are 8.33 and -2.16, respectively. Fr the hindwing, they are 8.93 and -1.14, respectively. The mass specific pwer ( P ) is defined as the mean mechanical pwer ver a flapping cycle divided by the mass f the insect, and it can be written as fllws (Sun and Tang, 2002): 3 P =.5ρU (2S + 2S )( τ + τ ) 0 f h W,f c W,h c m, (19) where W, f and W, h are the cefficients f wrk per cycle fr the fre- and hindwings, respectively. When calculating W, f r W, h, ne needs t cnsider hw the negative wrk fits int the pwer budget. There are three pssibilities (Weis-Fgh, 1972; Ellingtn, 1984). One is that the negative pwer is simply 15

16 dissipated as heat and sund by sme frm f an end stp, then it can be ignred in the pwer budget. The secnd is that in the perid f negative wrk, the excess energy can be stred by an elastic element, and this energy can then be released when the wing des psitive wrk. The third is that the flight muscles d negative wrk (i.e. they are stretched while develping tensin, instead f cntracting as in psitive wrk) but the negative wrk uses much less metablic energy than an equivalent amunt f psitive wrk, and again, the negative pwer can be ignred in the pwer budget. That is, ut f these three pssibilities, tw ways f cmputing W,h can be taken. One is neglecting the negative wrk, i.e.: W, f r + W, f = ( W ) frewing, (20) + W, h = ( W ) hindwing. (21) The ther is assuming the negative wrk can be stred and released when the wing des psitive wrk, i.e.: W, f = ( + W + W ) frewing, (22) W, h = ( + W + W ) hindwing. (23) Here equatins 20 and 21 are used, the cmputed P is 37 W kg -1 (when equatins 22 and 23 are used, P is 30 W kg -1 ). Discussin mparisn with previus tw-dimensinal results Wang (2000) and an and Sun (2001c) have presented tw-dimensinal (2D) cmputatins based n wing kinematics similar t thse used in this study. Wang (2000) investigated the case f a single airfil; an and Sun (2001c) investigated bth the cases f a single airfil and fre and aft airfils. It is f interest t make cmparisn between the present three-dimensinal (3D) and the previus 2D results. The value (single airfil) cmputed by Wang (2000) is apprximately is 1.97 [in figure 4 f Wang (2000), maximum f u t is used as reference velcity and the the value is apprximately 0.8; if the mean f u t is used as reference velcity, value becmes 0.8 (0.5π) 2 = ]; apprximately the same value (single airfil) was btained by an and Sun (2001c). In the present study, the values fr the single frewing and single hindwing are 1.51 and 1.64, respectively, 16

17 apprximately 20% less than the 2D value. This shws that the 3D effect n is significant. The wing length-t-chrd rati is nt small (apprximately 5); ne might expect a small 3D effect. But fr a flapping wing (especially in hver mde), the relative velcity varies alng the wing span, frm zer at the wing base t its maximum at the wing tip, which can increase the 3D effect. Nte that althugh is reduced by 3D effect significantly, the time curse f f the frewing r the hindwing is nearly identical t that f the airfil (cmpare Fig. 6A with figure 3 f Wang, 2000). an and Sun s (2001c) results fr the fre and aft airfils shwed that the interactin effect decreased the vertical frces n the airfils by apprximately 22% cmpared t that f the single airfil. Fr the fre- and the hindwings in present study, the reductin is apprximately 15%, shwing that 3D frewing-hindwing interactin is weaker than the 2D case. Aerdynamic frce mechanism and frewing-hindwing interactin Recent studies (e.g. Ellingtn et al, 1996; Dickinsn et al, 1997; Wu and Sun, 2004) have shwn that the large unsteady aerdynamic frces n flapping mdel insect wings are mainly due t the attachment f a EV r the delayed stall mechanism. This als true fr the fre- and hindwings in the present study. The EV dse nt shed befre the end f the dwnstrke f the fre- r hindwing (Fig. 11). If the EV sheds shrtly after the start f the dwnstrke, the EV wuld be very clse t the starting vrtex and a vrtex ring that carries a large dwnward mmentum (i.e. the large aerdynamic frces) culd nt be prduced. Generatin f a vrtex ring carrying large dwnward mmentum is equivalent t the delayed stall mechanism. Data presented in Fig. 8 shw that the frewing-hindwing interactin is nt very strng and is detrimental. In btaining these data, the wing kinematics bserved fr a dragnfly in hvering flight (e.g. 180 phase difference between the frewing and the hindwing; n incming free-stream) has been used. Althugh sme preliminary explanatin have been given fr this result, at the present, we cannt distinguish whether r nt this result will exist when the phasing, the incming flw cnditin, etc., are varied. Analysis based n flw simulatins in which the wing kinematics and the flight velcity are systematically varied is needed. Pwer requirements cmpared with quasi-steady results and with Drsphila results Wakeling and Ellingtn (1997b,c) cmputed the pwer requirements fr the dragnfly Sympetrum sanguineum. In mst cases they investigated, the dragnfly was in accelerating and/r climbing flight. Only ne case is clse t hvering (flight SSan 5.2); in this case, the flight speed is rather lw (advance ratin is apprximately 0.1) and the resultant aerdynamic frce is clse t the insect weight (see figure 7D f Wakeling and Ellingtn, 1997b and figure 5 f Wakeling and Ellingtn, 1997c). Their 17

18 cmputed bdy-mass specific aerdynamic pwer is 17.1 W kg -1 (see table 3 f Wakeling and Ellingtn, 1997c; nte that we have cnverted the muscle specific pwer given in the table t the bdy-mass specific pwer), nly apprximately half the value calculated in the present study. ehman and Dickinsn (1997) and Sun and Tang (2002), based n experimental and FD studies, respectively, shwed that fr fruit flies, calculatin by quasi-steady analysis might under-estimate the aerdynamic pwer by 50%. Similar result is seen fr the hvering dragnflies. It is f interest t nte that the value f P fr the dragnfly in the present study (37 W kg -1 ) is nt very different frm that cmputed fr a fruit fly (30 W kg -1 ; Sun and Tang, 2002), even their sizes are greatly different (the wing length f the fruit fly is 0.3 cm and that f the dragnfly is 4.7 cm). Fr the fruit fly, the mechanical pwer is mainly cntributed by aerdynamic pwer (Sun and Tang, 2002). It is apprximately the case with the dragnfly in the present study (see Fig.14). Frm equatin 15 f Sun and Tang (2002), the aerdynamic trque f a wing can be written as Q ~ rˆ d dr, (24) a,t where d is the mean drag f the wing; ˆr d is the radius f the first mment f the drag nrmalized by R. When the majrity f the pwer is due t aerdynamic trque, P can be apprximated as P ~ d rˆ nφr d, (25) d is the rati f the mean drag t the mean vertical frce f the wing. Fr the fruit fly, this rati is arund 1 (Sun and Tang, 2002). Fr the dragnfly in this study, since a large part f the vertical frce is cntributed by the drag, this rati is nt very different frm 1. We assume ˆr d fr the tw insects is nt very different. Then, P depends mainly n nφ R (half the mean tip speed). The dragnfly s R is apprximately 16 times f that f the fruit fly; but its n Φ ( 36 Hz 69 ) is apprximately 1/14 f that f the fruit fly ( 240Hz 150 ). This explains why different frm that f the fruit fly. P f the dragnfly is nt very We thank the tw referees whse thughtful questins and valuable suggestins greatly imprved the quality f the paper. This research was supprted by the Natinal Natural Science Fundatin f hina ( ). 18

19 References Alexander, D. E. (1984). Unusual phase relatinships between the frewings and hindwings in flying dragnflies. J. Exp. Bil. 109, Dickinsn, M. H., ehman, F. O. and Sane, S. P. (1999). Wing rtatin and the aerdynamic basis f insect flight. Science 284, Ellingtn,. P. (1984). The aerdynamics f hvering insect flight. (6). ift and pwer requirements. Phil. Trans. R. Sc. nd. B 305, Ellingtn,. P., van den Berg,. and Willmtt, A.P. (1996). eading edge vrtices in insect flight. Nature 384, Freymuth, P. (1990). Thrust generatin by an airfil in hver mdes. Experiments in Fluids. 9, Hilgenstck, A. (1988). A fast methd fr the elliptic generatin f three dimensinal grid with full bundary cntrl. In Numerical Grid Generatin in FM 88 (ed. S. Sengupta, J. Hauser, P. R. Eiseman, and J. F. Thmpsn), pp Swansea UK: Pineridge Press td. an, S.. and Sun, M. (2001a). Aerdynamic prperties f a wing perfrming unsteady rtatinal mtins at lw Reynlds number. Acta Mech. 149, an, S. and Sun, M. (2001b). Aerdynamic interactins f tw fils in unsteady mtins. Acta Mech. 150, an, S. and Sun, M. (2001c). Aerdynamic frce and flw structures f tw airfils in flapping mtins. Acta Mech. Sinica. 17, ehmann, F.-O. and Dickinsn, H. D. (1997). The changes in pwer requirements and muscle efficiency during elevated frce prductin in the fruit fly Drsphila melangaster. J. Exp. Bil. 200, Meakin, R. (1993). Mving bdy verset grid methds fr cmplete aircraft tiltrtr simulatins. AIAA Paper Nrberg, R. A. (1972). The pterstigma f insect wings and inertial regulatr f wing pitch. J. mp. Physil. 81, Nrberg, R. A. (1975). Hvering flight f the dragnfly Aeschna juncea., kinematics and aerdynamics. In Swimming and Flying in Nature (ed. T. Y. Wu,. J. Brkaw and. Brennen), pp NewYrk: Plenum Press. Reavis, M. A. and uttges, M. W. (1988). Aerdynamic frces prduced by a dragnfly. AIAA Paper Rgers, S.E. and Kwak, D. (1990). Upwind Differencing scheme fr the time-accurate incmpressible Navier-Stkes equatins. AIAA J. 28, Rgers, S.E., Kwak, D. and Kiris,. (1991). Steady and unsteady slutins f the incmpressible Navier-Stkes equatins. AIAA J. 29, Rgers, S.E. and Pulliam, T.H. (1994). Accuracy enhancements fr verset grids using a defect crrectin apprach. AIAA Paper Saharn, D. and uttges, M. (1988). Visualizatin f unsteady separated flw prduced by mechanically driven dragnfly wing kinematics mdel. AIAA Paper Saharn, D. and uttges, M. (1989). Dragnfly unsteady aerdynamics: the rle f the wing phase relatins in cntrlling the prduced flws. AIAA Paper

20 Sane, S.P. and Dickinsn, M.H. (2001). The cntrl f flight frce by a flapping wing: lift and drag prductin. J. Exp. Bil. 204, Smps,. and uttges, M. (1985). Dragnfly flight: nvel uses f unsteady separatin flws. Sicence 28, Sun, M. and Tang, J. (2002). ift and pwer requirements f hvering flight in Drsphila virilis. J. Exp. Bil. 205, Sun, M. and Wu, J. H. (2003). Aerdynamic frce generatin and pwer requirements in frward flight in a fruit fly with mdeled wing mtin. J. Exp. Bil. 206, Wakeling, J.M. and Ellingtn,.P. (1997a). Dragnfly flight. (1). Gliding flight and steady-state aerdynamic frces. J. Exp. Bil. 200, Wakeling, J.M. and Ellingtn,.P. (1997b). Dragnfly flight, (2). velcities, acceleratins and kinematics f flapping flight. J. Exp. Bil. 200, Wakeling, J.M. and Ellingtn,.P. (1997c). Dragnfly flight, (3). Quasi-steady lift and pwer requirements. J. Exp. Bil. 200, Wang, Z. J. (2000). Tw dimensinal mechanism fr insect hvering. Phys. Rev. ett. 85, Weis-Fgh, T. (1972). Energetics f hvering flight in hummingbirds and in Drsphila. J. Exp. Bil. 56, Wu, J. H. and Sun, M. (2004). Unsteady aerdynamic frces f a flapping wing. J. Exp. Bil. 207, Figure egends Fig. 1 Sketches f the mdel wings, the flapping mtin and the reference frames. FW and HW dente fre- and hindwings, respectively. OXYZ is an inertial frame, with the X and Y axes in the hrizntal plane; xyz is anther inertial frame, with the x and y axes in the strke plane; x y z is a frame fixed n the wing, with the x axis alng the wing chrd and y axis alng the wing span. β, strke plane angle; φ, psitinal angle; α, angle f attack; R, wing length. Fig. 2 Sme prtins f the mving verset grids. Fig. 3 mparisn between numerical and analytical slutins f a starting sphere. (A) Drag cefficient ( d ) vs. nn-dimensinal time ( τ s ). (B) Azimuthal velcity ( u θ ) vs. nn-dimensinal radial distance ( r 2 a ). Fig. 4 mparisn f the calculated and measured lift and drag frces. The experimental data are reprduced frm figs 3, D f Sane and Dickinsn (2001). (A, B), the midstrke angle f attack is 50 and strke amplitude is 60 ; (, D), the 20

21 midstrke angle f attack is 50 and strke amplitude is 180. Fig. 5 (A) Nn-dimensinal angular velcity f flip rtatin ( α& + ) and azimuthal + rtatin ( φ & ) f hindwing and (B) frewing; () time curses f ttal vertical frce cefficient ( ) and (D) ttal thrust cefficient ( ) in ne cycle. Fig. 6 (A) Time curses f vertical frce cefficients f frewing (, f T ) and hindwing (, h ) and (B) thrust cefficients f the frewing ( T, f ) and the hindwing ( T, h ) in ne cycle. Fig. 7 (A) Time curses f lift cefficients f frewing ( l, f ) and hindwing ( l, h ) and (B) drag cefficients f the frewing ( d, f ) and the hindwing ( d, h ) in ne cycle. Fig. 8 (A) Time curses f vertical frce cefficients f frewing (, f ) and single frewing (, sf ); (B) thrust cefficients f the frewing ( T, f ) and single frewing ( T, sf ); () vertical frce cefficients f hindwing (, h ) and single hindwing (, sh ) and (D) thrust cefficients f the hindwing ( T, h ) and single hindwing ( T, sh ) in ne cycle. Fig. 9 Is-vrticity surface plts at varius times in ne cycle (single hindwing). Nte that the X axis is alng the bdy f the dragnfly and XZ plane is the plane f symmetry f the insect. tˆ, nn-dimensinal time. The magnitude f the nn-dimensinal vrticity is 1. Fig. 10 Velcity vectrs in a vertical plane parallel t and 0.6R frm the plane f symmetry at varius times in ne cycle (single hindwing). The hrizntal arrw at the tp left represents the reference velcity (U ). tˆ, nn-dimensinal time. Fig. 11 Is-vrticity surface plts at varius times in ne cycle (fre- and hindwings). Nte that the X axis is alng the bdy f the dragnfly and XZ plane is the plane f symmetry f the insect. tˆ, nn-dimensinal time. The magnitude f the nn-dimensinal vrticity is 1. Fig. 12 Velcity vectrs in a vertical plane parallel t and 0.6R frm the plane f symmetry at varius times in ne cycle (fre- and hindwings). The hrizntal arrw at the tp left represents the reference velcity (U ). tˆ, nn-dimensinal time. 21

22 Fig. 13 (A) Time curses f aerdynamic trque cefficients fr translatin ( Q,a, t ) and rtatin ( Q,a, r ) f frewing and (B) hindwing in ne cycle; () time curses f inertial trque cefficient fr translatin ( Fig. 14 ne cycle. Q,i, t ) in ne cycle. Time curses f pwer cefficients f frewing (A) and hindwing (B) in p, pwer cefficient; p, a, cefficient f pwer due t aerdynamic frce; p, i, cefficient f pwer due t inertial frce. Fig.1A Fig.1B Fig.1 22

23 Fig.2 Fig.3A Fig.3B 23

24 Fig.4 Fig.5 24

25 Fig.6 Fig.7 25

26 Fig.8 26

27 Fig.9(1) 27

28 Fig.9(2) Fig.10A 28

29 Fig.10B Fig.10 Fig.11(1) 29

30 Fig.11(2) Fig.12A Fig.12B 30

31 Fig.12 Fig.12D Fig.12E Fig.12F 31