Computation of dragonfly aerodynamics - Science Direct

Computer Physics Communications 65 (1991) 121-132 North-Holland

121

Computation of dragonfly aerodynamics Karl G u s t a f s o n Department of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA

and Robert Leben Colorado Center for Astrodynamies Research, University of Colorado, Boulder, CO 80309-0431, USA

Dragonflies are seen to hover and dart, seemingly at will and in remarkably nimble fashion, with great bursts of speed and effectively discontinuous changes of direction. In their short lives, their gossamer flight provides us with glimpses of an aerodynamics of almost extraterrestrial quality. Here we present the first computer simulations of such aerodynamics.

1. Introduction Conventional airplane flight is based upon postulated smooth flow of air over wing surfaces. Most conventional flight, e.g. that of airliners, therefore depends upon steady aerodynamics: the uniform motion of an airfoil into a uniform freestream flow. The fundamental physics of such flight then comes from the fact that a steady motion into an incompressible irrotational fluid generates a side force (lift) and no drag. In reality, one needs viscosity in order to generate and to maintain the circulation which sustains lift. Conventional aerodynamic design thus becomes a set of compromises aimed at producing efficient propulsive motion with reasonable lift (e.g. lift coefficient C L -- 2) and small drag. As is well known, attaining this has been a principal scientific and commercial enterprise of this century. Recently a number of researchers have ventured into unsteady aerodynamics: aerodynamics permitting maneuvers with rapid accelerations, quick plunging and pitching motions - in short, those motions your conventional airliner wishes to avoid. Roughly one can characterize this heightened activity as one of the last decade. Improved

wind tunnel techniques and better stroboscopic photography have been instrumental in enabling this research. Of most importance, from our point of view, has been the advent of very large, very high speed, computers, on which one can try to simulate the physics of such flows via the mathematical equations of motion.

2. Dragonfly aerodynamics Unsteady aerodynamics, although a recent endeavor for design engineers, appears to have been exploited for over 100 million years in nature. Hummingbirds, moths, wasps, and dragonflies appear to generate their own flows, thereby enabling them to dart and hover in a still air environment. Moreover, some hovering insects, such as the dragonfly, attain lift coefficients far too high (e.g. C L = 6) to be explainable by steady flow principles. One of the first to examine dragonfly wing motion was Chadwick [1]. An early study of flapping and oscillating airfoils may be found in Garrick [2], although such earlier studies were framed in a forward-flight (rather than still air) environment, and (roughly) in kinematic rather

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than full-fluid equation terms. Another early paper examining aspects of the flight mechanics of dragonflies was that of Neville [3]. One of the first explanations of how hovering flight is actually achieved was given by Weis-Fogh [4-6]. The chalcid wasp, which is simpler than the dragonfly in the sense of having only one set of wings, was studied by high-speed photographic techniques. From these experiments, Weis-Fogh hypothesized that the wasp's lift was generated by the periodic generation of equal but oppositely signed vortices controlled by "clap" and "fling" movements of the wings. For further investigations of the aerodynamics of insect flight in this period let us refer to the volumes by Wu, Brokaw and Brennan [7], Lighthill [8], the chapter by Maxworthy [9], and the papers included or cited therein. See also, of course, the book by Lugt [10]. Interest in dragonfly aerodynamics intensified with the work of Luttges et al. [11-15]. Stroboscopic photography of tethered dragonflies revealed in many test specimens a laminar vortexdominated flow downstream from the tethered insect. Peak lift forces of up to 20 times body weight could be generated, with the average being approximately twice body weight. Exact biological flow-field details were, however and in our opinion, rather hard to discern, due to the near-field interference of patterns from individual wings, and the far-field convective dissipation. The fact that 4 wings (2 pairwise) are involved in dragonfly aerodynamics, rather than just 2 (1 pairwise) in the chalcid wasp studies of Weis-Fogh [4], opens up possibilities of further lift enhancement by back-capture of fore-generated vortices, an idea that has of course occurred at various times elsewhere in aerodesign literature. Also intriguing regarding dragonfly flight itself is the question of how the dragonfly (neurally) controls such rapid self-generated flight. In the intensive studies of Saharon and Luttges [13-15], a mechanical model of a single dragonfly wing, and then tandem wings, was employed. This model corresponded well to the biological model and permitted more extensive visualizations of the flow kinematics. Toward a better understanding of observed complex oscillating cycles involving mix-

ed plunges, and pitch, eight elemental vortex structures were identified for further study: fore down pitch (1), fore up pitch (2), fore down plunge (3), fore up plunge (4), aft down pitch (5), aft down plunge (6), aft pitch up (7), aft plunge up (8). Based upon flow (smoke) visualizations at certain flow parameters (Re = Reynolds number, T = period of wing cycle, FR = frequency ratio between pitching and plunging), caricatures of these elemental vortex structures were presented. Principal conclusions of these wing-flow interaction studies were: an almost continuous production of transitional flow-field structures (mostly vortices), temporal-spatial developments into highly structured, reproducible flows, and separations that are sufficiently short-lived to prevent the development of turbulent stall conditions about the wings. Because such important vortex structures occur very adjacent to the wing or in the near wake, it would appear (our opinion) that such smoke visualizations, which in the current state of the art have at best limited precision in such near fields, can provide at best a qualitative description of such events. In the far field, natural intensity degradation makes it difficult to visualize the longer-term flow dynamics. And the all important mid-field vortex dynamics is still too complicated for us to understand adequately.

3. A simplified airfoil model

In an attempt to discover the fundamentals of the aerodynamics of such flight, Freymuth [16] initiated an experimental study of a single airfoil executing combined plunging and pitching sinusoidal motions. In the presence of a steady wind, he found [16] the downstream development of a reverse Karman vortex street, i.e. a two-dimensional jet. The dominant feature of this flow was trailing-edge vortex shedding during wing flapping. This simplified model would appear to explain the mean thrust generation similar to the propulsive force generated by the flapping of a bird wing or the fanning of a fish tail. Recall our discussion at the beginning of this paper distinguishing steady from unsteady mo-

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tions: the just described experimental thrust model is one of unsteady motion, but into a uniform free stream flow. The results are only flow-averaged quantities. Nonetheless, the unsteady generation of such a vortical thrust signature provides another mechanism for the explanation of the flight of moths, wasps, hummingbirds, and of course dragonflies. Secondly, such a mechanism clearly depends only upon the unsteady motion of a single wing. This is a conceptual simplification from the 4 wing dragonfly models and the two wing wasp models. The experimental apparatus utilizes a very simple airfoil, a high aspect-ratio rectangular cross section, thus obviating the complications (probably interesting ones, but not essential) of the changed wing shapes used by a hummingbird and, to a much lesser extent, by the insects. To understand dragonfly aerodynamics, and more generally that of all hovering birds, insect, and possibly future flight vehicles, in 1988 we launched, in concert with Professor Freymuth, a combined study of the hovering dynamics of a single airfoil in a still-air environment. Freymuth's experimental set up is very similar to that of ref. [16] and he has reported some of his early results in ref. [17], whereas we have reported early computational results in ref. [18]. A combined preliminary presentation of our results appeared in ref. [19], based upon initial studies. In the experimental studies of Freymuth, large time-averaged thrust coefficients C x (e.g. up to 6) were found in well-tuned hovering modes. These coefficients are measured experimentally from the formula (3.1) where V 2 is the mean-square velocity at a sufficient distance downstream where the vortical jet signature has acquired ambient pressure, f is the frequency of sinusoidal plunge (translation) oscillation, h a is the (maximum) amplitude of the plunging velocity, and c is the airfoil chord length. Resulting experimental vortical signatures of the " w a t e r treading" (mode 1) or " n o r m a l hovering model" (mode 2, degenerate figure 8) may be seen

Jet

/I

I \\

m j (a)

J

j

i

Jet

//

I \\ i

I I 1/11 I

(b)

(c) Fig. 1. Three hovering modes: (a) Sketch of combined translating-pitching motions of the airfoil for one cycle of mode 1 hovering (a 0 = 0 °, q~= 90 ° ). (b) Sketch of combined translating-pitching motions of the airfoil for one cycle of mode 2 hovering (a 0 = 90 °, ¢ = - 90 ° ). (c) Sketch of combined translating-pitching motions of the airfoil for one cycle of mode 3 hovering (a 0 is oblique, ~ = - 90 ° ).

in refs. [17-19]. We show mode 1 and mode 2, as well as an "oblique" (mode 3, darting) mode in fig. 1. Further specifics of the modes to be computed here will be given later.

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K. Gustafson, R. Leben / Computation of dragonfly aerodynamics

Fig. 2. A well-tuned mode 1 vortical signature.

Badly tuned hovering airfoils produce destructive, chaotic vortical signatures. Well-tuned hovering airfoils produce ordered vortical signatures equivalent to high-thrust jets amenable to quick high-lift changes of direction. An example of a well tuned model 1 hovering airfoil, and the resulting vortex street (going up rather than down, to avoid wind-tunnel floor effects) is shown in fig. 2.

4. Computation of hovering-mode dynamics As mentioned above, in 1988 we turned our attention to a mathematical study of dragonfly aerodynamics, by computer simulation. Our first

goal was to see if we could reproduce qualitatively the vortical jet signatures obtained by Freymuth in the wind tunnel. As will be seen in the following figures, we have been able to do this, to remarkably good agreement, even on relatively coarse computational meshes to date. With finer meshes we should be able to capture interesting details of the near-field vortex generation dynamics and the fluid-structure interactions. We show one of the latter here. In accord with Shen and Wu [20], we believe such study of fluid-wall dynamics to be a subject in itself, of great importance to understanding the effects of separated flows on helicopter blades, propellers, cascades, turbines and rapidly turning aircraft. Experimental flow visualizations, biological measurements, photography and wind tunnel studies cannot provide the instantaneous wing-lift pressure dynamics which are of great importance. We will give these elsewhere [21]. Our approach to the computation of hoveringmode dynamics was makeshift and practical due to working time limitations. Previous work [22,23] had provided us with a robust two-dimensional full Navier-Stokes solver for the study of unsteady flows past airfoils. This solver, which will be briefly discussed in the next section, possesses some unique advantages which are fortuitously transferable to the computation of hovering-mode aerodynamics. These hovering-mode computations are the first of their kind. Roughly speaking, studies of unsteady airfoil aerodynamics are of recent advent and have been predicated upon the development of large-scale high-speed computers, notably the C R A Y series. Excellent early studies were those of Mehta and Lavan [24], for impulsively started airfoils, and Lugt and Haussling [25], for constantly accelerated air foils. The reader can come pretty much up to date for recent computational work on accelerative modes for airfoils by Visbal, Ghia, Shen, McCune and others, by consulting the volumes by McMichael and Helin [26] and Ghia and Dalton [27]. The physical simulation of the hovering modes by Freymuth [17,19] was achieved by the rapid airfoil movement in still air. In our numerical simulations, the reference frame is fixed with ~he

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K. Gustafson, R. Leben /Computation of dragonfly aerodynamics

airfoil. T o allow simulation of the hovering mode, the freestream flow relative to the airfoil is rapidly varied about the airfoil. This simulation introduces a n u m b e r of dynamical considerations beyond those of our previous work [22,23].

sonable outflow condition at a d o w n s t r e a m b o u n d a r y were also removed. We have stressed this point in ref. [23] and do not further elaborate on it here. To apply this method to the hovering p r o b l e m we elected to model an elliptic airfoil. This simplifies the grid-generation procedure since a conformal m a p p i n g exists which m a p s the unit circle in the z plane onto the infinite physical d o m a i n exterior to an ellipse in the Z plane. This transformation is given by:

5. A Navier-Stokes solver for pitching and plunging In our earlier work on airfoils, see fig. 3, we conformally m a p p e d the exterior domain into a auxiliary interior, near-circular, domain. The latter was then provided with an orthogonal grid. The grid, fig. 4, was c o m p u t e d on a rectangular computational domain, employing the weak constraint m e t h o d of Ryskin and Leal [28] and a multigrid elliptic solver. In this way we were able to obviate the need to specify outflow b o u n d a r y conditions at truncated far-field boundaries. In addition, difficulties posed by the need to specify some rea-

yi=l(ze-X+z

F(Z)=X+

-1 eX),

where a = cosh ~,

(5.2a)

b = sinh ~,

(5.2b)

c2 = a z- b2

(5.2c)

are the semimajor axis, semiminor axis and the distance to the foci of the ellipse from the origin,

Physical Domain

Auxiliary Domain

Computational Domain

Exterior of Airfoil

Near Circular

Cavity, A ffi2

z-x+iy

ZffiX+iY Y

~=0

Multigrid Solver

Y

r ~

=1

~?=2

r/=l X

X

r/=0 ~:o

Analytic Map

Numerical Mapping

Inverse Joukowaki

Elliptic Orthogonal

112

G(z) = Z - (Z2- 11

(5.1)

Grid

Generation

Fig. 3. Conformally preconditioned domain mappings.

~:i

K. Gustafson, R. Leben / Computation of dragonfly aerodynamics

126

Physical Domain

Auxiliary Domain

Computational Domain

r

Fig. 4. Sample orthogonal numerical grids.

respectively. The thickness of the ellipse, t, is given by t = tanh ?~,

(5.3)

?t=½ l n ( l + t )

(5.4)

1 -tJ

We can now algebraically generate an orthogonal grid on the unit circle and use the conformal transformation above to determine an orthogonal ~, ~ grid on the infinite physical domain as we have done previously with a conventional airfoil. The equations of motion are formulated in terms of the stream function 4' and w, the vorticity component normal to the ~, 77 plane. All physical quantities are made dimensionless by a characteristic velocity U, a Reynolds number R, and a characteristic length L. In terms of these quantities, the dimensionless governing Navier Stokes equations on an orthogonal curvilinear coordinate system are

1 ( 3h2uw +

3h2v~o] +

L =

V'2~ = - w, where

~72

I

{ 0 {h 2

1 O~ u-

so ?~ may be calculated from

3~o

The velocities are defined by

Z(hlZ

(5.5)

h2 3 ~ '

1 3~b v = hi 3~"

(5.7)

The Poisson equation for the stream function + in (5.5), appropriately modified to a disturbance streamfunction relative to the background flow, see ref. [22,23] and below, is solved by a covariant multigrid solver, and the vorticity transport equation of (5.5) is solved by an A D I time-marching scheme. The coefficient matrices in the ( direction are tridiagonal; in the ~ direction the method for periodic line relaxation must be used to calculate the solution. As seen from refs. [22,23,29], our method provides excellent correspondence between the mathematical simulation and wind tunnel experiments, for both impulsively and constantly accelerated airfoils. The boundary-fitted grids are excellent for boundary-layer resolution and clearly resolved interesting flow phenomena such as vortex splitting, shredding, and higher-order vortical detail, even on a relatively coarse (33 × 65) grid using 5 multigrid levels. To undertake the numerical study of hovering airfoils, the Navier Stokes equations must be recast in terms of a frame of reference which is fixed on the rotating/translating airfoil. We nondimen-

K. Gustafson, R. Leben / Computation of dragonfly aerodynamics

sionalize the angular rotation rate of the airfoil as a& = ~,

(5.8)

where a -= characteristic length (approximately one-half the chord length), ti = angular rotation rate of the airfoil, U = translational velocity at midcycle. The stream function due to the flow at infinity is modified to include the effect of the angular rotation and translation:

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Modeling of the hovering of the airfoil is achieved by specifying the angle of attack and translational velocity of the airfoil as a function of time. This is determined by the hovering mode being modeled, as will be discussed in the next section. Boundary conditions are no slip velocity on the airfoil surface, irrotational flow at infinity, a firstorder differencing of the Laplacian of the streamfunction for the vorticity at the airfoil surface, and an impermeability condition for the streamfunction at the airfoil boundary.

6. Hovering-mode computations lPoo = lPfreest . . . .

(5.9)

"{- ~rotation"

Consider a thin airfoil with chord length c exposed to still air and executing a translating (plunging) motion h in horizontal direction,

Thus, ~oo = ft .... lational(COS(a) Y--sin(o~) X ) + l~-~(X2 + r2).

(5.10)

h = h a sin(2~rft),

(6.1)

Since the vorticity field observed in the reference frame fixed with the hovering airfoil differs only by a constant from one at rest, we introduce the disturbance stream function:

where h a is the amplitude of translation, f is the frequency of oscillation and t is time. Consider the airfoil to simultaneously execute a pitching motion around the half-chord axis,

~k* = ~k - ~k~.

O/ = a 0 -}- O/a sin(2~rft + 0 ) ,

(5.11)

Then, the governing equations to be solved for comparison to the flow-visualization experiments are

3w a--? +

1 { Oh2fiw

3htvw t] = XL xT'2 , (5.12)

~72~ * ~ 0~,

where

a~k* ~l = ~

+ ~loo = h 2 u ,

~= aa-~ + G =hlv, (5.13)

and:

?~ = h2u~

ago

aG~

art ' (5.14)

(6.2)

where a is the angle of attack with respect to the horizontal line, a 0 is the average angle of attack, ota is the pitch amplitude and 0 is the phase difference between pitching and plunging. The dimensionless parameters of the system are: ao, aa, q,, the dimensionless plunge amplitude h a / c and a Reynolds number Rf = 2~rfhac/V based on maximum plunge speed and on c, where p, the kinematic viscosity which for air, is taken as 0.15 c m / s 2. Three hovering modes were illustrated in fig. 1. There the jets are directed 180 ° different from that of actual insect flight, to reflect wind tunnel realities. Mode 1, or the "water-treading mode", is characterized by a 0 = 0 and ~ = 90 o. Note that the leading and trailing edges switch their role during each cycle of oscillation. Mode 2, or "normal hovering mode", is characterized by a 0 = 90 o and q ~ = - 9 0 ° . In this mode the leading and trailing edges do not switch their role. Mode 3, or

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K. Gustafson, R. Leben / Computation of dragonfly aerodynamics

"oblique mode", is a variation on mode 2, and corresponds to an angled direction of flight. Other hovering modes are under investigation and will be reported on elsewhere [21]. Here we confine attention only to the generic modes 1 and 2. The airfoil used in the wind-tunnel experiments with which we will compare was of cross section 25.4 m m × 1.6 mm, with slightly rounded edges, and long span to provide two-dimensional midwing dynamics for photographic visualization. Flow history is recorded by placing on the upper surface of the airfoil a liquid (e.g. titanium tetrachloride TIC14) which reacts with air to give off a dense white smoke, thereby tagging and making visible (in white on black) the vortex patterns of the flow. As can be seen from the figures, this technique gives excellent visualization in the middle flow field, but is overly bright in the very near field, and probably departs somewhat from the actual fluid vorticity in the far field. For our hovering-mode computational comparisons, we used an elliptical airfoil of 10% thickness ratio to approximate the laboratory airfoil. As the results will show, the slight differences, e.g. in roundedness of airfoil ends, between the laboratory and computational airfoils, did not enter significantly into the comparisons, at least at the (low) Reynolds numbers tested. The hovering mode computations to be reported here used a 65 × 65 mesh. Flow parameters were set to correspond to those experimental simulations made available to us by Professor Freymuth. Numerical flows were visualized in the 4th hovering-mode cycle. Computational flow visualization was performed using color intensity plots of the vorticity field. A multigrid interpolation scheme was developed to produce byte files for display, using available satellite-image display software. Because of publication expense, here we present only grey-image pictures, in which dark and light colors correspond to opposite rotation directions.

7. Comparison with experiment We present two simulations:

Simulation 1 (mode 1): c % = 0 , ~ = 9 0 °. The

tuneable parameters were set to: ~a=66°,

hJc=l.5,

f=l.0Hz,

Rf=340.

Simulation 2 (mode 2): a(> = 90 °, 0 = - 9 0 ° . The tuneable parameters were set to:

~,,=33 ° ,

hJc=l.O,

f = 1 . 3 Hz,

Rf=300.

Generally we set At = 1 / 1 6 s for frame images to compare to experiment (this is not the At of computation!), and the images shown here for mode 2 are at this At. More frame images per second would provide a better phase match between experiment and simulation, and we plan to do that in future runs. For the close-ups provided by the mode 1 simulation, pictures of the dynamics were produced at At -- 1 / 6 4 s to provide comparison to available corresponding close-ups from the laboratory tests. Turning now to our results, in fig. 5 the mode 1 computational results are on the right, the physical laboratory visualizations are on the left, thereby providing a side by side direct comparison. The two flows are still not exactly calibrated but the correspondence is excellent. In the laboratory images, the camera is slightly askew to view the airfoil, which sometimes makes the results a bit hard to make out. However~ in fig. 5, especially at the beginning of the sequence, the laboratory airfoil is visible right on top of the vortex on the lower right. Of considerable interest is the " v o r t e x severing" seen in this experiment, in both the laboratory test and the computer simulation. Note that far more detail of this fluid-structure dynamical event is available from computer simulation than from laboratory experiment. Following this vortex severing event, amalgamation of its residue with the "leading-edge" vortex, formed during the same sequence above the descending trailing edge, takes place. This combined reinforced vortex is then shed into the jet. A column of such vortices forms one side of the thrust jet, with a column, staggered, of oppositely signed vortices forming the other side of the thrust jet. Figure 6 shows a side by side comparison of the mode 2 experiment. Here the airfoil of the labora-

K. Gusta[son, R. Leben / Computation of dragonfly aerodynamics

Fig. 5. Mode 1 hovering dynamics: a a = 66 °, ha/c = 1.5, f = 1.0 Hz, R r = 340, At = 1/64 seconds. (a) Experimental sequence on left is slightly askew to view airfoil. Note the very good correlation of the computer simulation on the right. tory e x p e r i m e n t is h i d d e n by the hovering mechanism: the reader should n o t e the scale difference between corresponding laboratory-computational vortical events. Also, as n o t e d above, the At = 1.16 s interval for f r am e images limited the phase m a t c h b e t w e e n l a b o r a t o r y e x p e r i m e n t a n d simulation.

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Fig. 5. (b) Severing begins as trailing edge descends.

N o t e that at the b e g i n n i n g of the cycle, a small trailing-edge vortex forms at the top of the airfoil. This then a m a l g a m a t e s near the top of the airfoil with the leftover stall v o r t e x to the left of the airfoil. A strengthened, very strong trailing-edge vortex forms ab o v e- r i g h t of the airfoil. As the

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K. Gustafson, R. Leben / C o m p u t a t i o n o f dragonfly aeroclvnamics

Fig. 5. (c) Amalgamation will finally produce a single clockwise vortex to be shed into the jet. Following this, a single counterclockwise vortex will be produced in the same way and shed into the jet.

Fig. 6. Mode 2 hovering dynamics: c~ = 33, h,w ~ = 1.0, jr = 1.3 Hz, Rf - 300, At = 1/16 seconds. (a) The forming trailing-edge vortex at the top of the airfoil, and the previously formed stall vortex (both dark colored) to the left of the airfoil begin to amalgamate. Large dark region above left of airfoil is the result of the previous trailing-edge and stall-vortex amalgamation process.

airfoil straightens into the vertical again, this s t r o n g v o r t e x a w a i t s to t h e right. T h e t o p o f t h e a i r f o i l t h e n p i t c h e s t o t h e left, g e n e r a t i n g a s i m i l a r ,

but oppositely signed, amalgamated trailingedge-stall-vortex combination. T h e last f o u r f r a m e s , as t h e airfoil r e t u r n s to t h e r i g h t m o s t

K. Gustafson, R. Leben /Computation of dragonfly aerodynamics

Fig. 6. (b) Opposite rotation trailing-edge vortex begins to form at top of the airfoil. Below right of airfoil is the previously formed stall vortex. (Both are light colored.) Amalgamation of these two vortices takes place. Large light region above left of airfoil is the result of the previous amalgamation of same rotation.

p o s i t i o n , s h o w v e r y clearly the r e s u l t i n g s t r o n g j e t structure, with excellent near- and mid-field corres p o n d e n c e b e t w e e n the l a b o r a t o r y a n d c o m p u t a tional experiments. For well-tuned hovering

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Fig. 6. (c) The strong jet structure is apparent in the first frame, with two light-colored vortices coupled with two darkcolored vortices. The amalgamation dynamics of mode 2 plus the avoidance of the fluid/structure severing action produces higher thrust than in mode 1.

m o d e s , this s t r o n g a m a l g a m a t i o n w i t h n o s e v e r i n g w o u l d a p p e a r to e x p l a i n the h i g h e r t h r u s t s for m o d e 2 m e a s u r e d in the l a b o r a t o r y .

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8. Conclusions and observations Excellent spatial and temporal correlation with l a b o r a t o r y s t u d i e s serves to v a l i d a t e o u r c o m p u t a tional hovering model. The independent computational confirmation of near-field high-lift coeffic i e n t s C L, to c o n f i r m the h i g h - t h r u s t c o e f f i c i e n t s C T f o u n d d o w n - f i e l d in the l a b o r a t o r y , is a n imp o r t a n t a n d p r e s s i n g n e x t step. High-thrust and high-lift vortical signatures can a p p a r e n t l y b e a t t a i n e d b y a single h o v e r i n g airfoil, without necessarily involving more complicated p h a s i n g of m u l t i p l e w i n g b e a t s in d r a g o n f l i e s , o r the c l a p a n d fling m e c h a n i s m s o f wasps. It is a p p a r e n t l y s u f f i c i e n t to h a v e a single p r o p e r l y t u n e d airfoil, e x e c u t i n g s h o r t p l u n g e c o m b i n e d w i t h r e l a t i v e l y large pitch, a n d the t u n i n g is relatively simple. F u r t h e r c o m p u t a t i o n a l studies s h o u l d p r o v i d e a better understanding of such unsteady separated flows, u n c o n v e n t i o n a l in c u s t o m a r y a i r c r a f t design, b u t q u i t e c o n v e n t i o n a l in n a t u r e .

Acknowledgements The authors appreciate grants of NAS comput a t i o n a l r e s o u r c e s at N A S A Ames Research Laboratories, on which higher resolution studies are c u r r e n t l y b e i n g p e r f o r m e d .

References [1] L.E. Chadwick, Bull. Brooklyn Entomol. Soc. 35 (1940) 109. [2] I.E. Garrick, Propulsion of a flapping and oscillating airfoil, NACA Report 567 (1936). [3] A.C. Neville, J. Exp. Biol. 37 (1960) 631.

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