Numerical and experimental investigation of the role of flexibility in flapping wing flight

Transcription

1 36th AIAA Fluid Dynamics Conference and Exhibit 5-8 June 26, San Francisco, California AIAA Numerical and experimental investigation ohe role of flexibility in flapping wing flight Jonathan Toomey and Jeff D. Eldredge Mechanical & Aerospace Engineering Department, University of California, Los Angeles, Los Angeles, CA The role of flexibility in flapping wing flight is explored using dynamically-scaled experiments and numerical simulations. To limit the dimension ohe parameter space, the target of study is a two-dimensional two-component wing connected by a hinge with a torsional spring. The motion ohe lead body is prescribed with flapping kinematics, while the trailing body motion is passive. Experiments are conducted in a water tank, enabling flow visualization with suspended particles. Numerical simulations rely on the viscous vortex particle method (VVPM) for coupled fluid-body dynamics. In both the experiments and the computations, the behavior ohe wing is characterized by monitoring the evolution ohe hinge angle, with favorable agreement between them. Flow structures are identified and compared for representative cases. Analysis of lift force and energy consumption from numerical simulations indicates that wing flexibility can improve wing performance when measured in terms of energy spent per unit lift. I. Introduction Recent interest in the development of micro-scale aerial vehicles (MAVs) has led to a reexamination of basic flight modes, particularly those that are inspired by biological observation. The size and mission requirements for MAVs (for example, hovering and precise maneuvering) are reminiscent of similarly-scaled airborne animals, so biologically-inspired mechanisms such as the flapping wing flight observed in insects provide natural candidates for MAV flight. However, the aerodynamics of such mechanisms are poorly understood, posing a great challenge to their adoption in artificial fliers. Specific issues, such as the role of vorticity shed into the wake, the sensitivity to kinematics and morphology ohe flight apparatus, and the importance ohree-dimensional flow features, all remain incompletely addressed. Conventional models for studying flapping wing aerodynamics generally rely on a rigid wing on which flapping kinematics are imposed. However, an insect wing is a highly sophisticated elastic structure, with variable anisotropic stiffness provided by longitudinal veins, calling into question the validity ohis basic assumption. Furthermore, flexibility may have aerodynamic benefits, such as damping of unsteady forces and elastic energy storage, that should be characterized in order to optimize the design of MAVs. Therefore, the goal ohis work is to explore the effects of wing flexibility on the aerodynamics of flapping wing flight. Though the wing structures of insects are diverse, natural observations have been distilled into a generic pattern that consists of a rigid anterior portion and a soft posterior portion, separated by a median flexion Graduate Research Assistant, toomey@seas.ucla.edu. Member, AIAA. Assistant Professor, eldredge@seas.ucla.edu. Member, AIAA. 1 of 15 Copyright 26 by the authors. Published by the American American Institute of Aeronautics Institute of and Aeronautics Astronautics, and Inc., Astronautics with permission.

2 Constrained component X 1 X h d 1 Hinge/Torsion spring d 2 θ X2 Passive component Figure 1. Model system consisting owo rigid elliptical sections connected by a hinge with torsion spring. line that runs the length ohe wing span at mid-chord. 1, 2 The wingbeat cycle of an insect involves an upstroke with constant positive angle of attack, followed by wing pronation in which the leading edge rotates downward for a positive angle of attack on the downstroke, and completed by wing supination to rotate the leading edge upward for the next stroke. The wing is continuously twisted and untwisted to achieve the proper angle during each stroke, and high-speed films ohis process reveal that the rear portion generally lags the stiff front during the rapid rotations at the ends of each half-stroke. 3 Furthermore the center of mass ohe wing is located aft ohe rotation axis in many insects, such as the dragonfly, 4 so inertia may serve an important role in aiding this rotation. Previous investigators in insect flight aerodynamics have speculated at a possible aerodynamic role served by this variable stiffness. Ellington 3 postulated a flex mechanism, wherein the vorticity production and shedding from the trailing edge was delayed by the flexion, thereby allowing a sufficient amount of bound circulation to be set up for the ensuing halfstroke. In flapping wing dynamics, vortex shedding and wake dynamics are inextricably connected with the requisite aerodynamic forces for lift and thrust. However, an important open question is the degree to which the dynamics ohe fluid and elastic wing are coupled. Combes and Daniel 5, 6 have presented some evidence to suggest that the aerodynamic forces can be ignored relative to the inertial and elastic forces within the wing, so that dynamic shape changes can be determined a priori. Unfortunately, the inherent challenge of obtaining flowfield measurements in the wake of a living insect makes such postulations difficult to test. The conclusions are often based on simple models for the aerodynamic forces that generally neglect much ohe important unsteady fluid dynamics. Therefore, much can be gained by using numerical simulations and experiments of simplified systems to evaluate the relationship between wing flexibility and unsteady aerodynamics. Recent experiments on a plunging two-dimensional wing section with a thick leading portion attached to a flexible trailing plate, conducted with both zero 7 and low Reynolds number 8 freestream velocity, have revealed that vortex shedding patterns are indeed influenced by flexibility. Furthermore, work on low aspect ratio fixed MAV wings has demonstrated that flexibility in a wing membrane can improve flight stability by reducing the fluctuations ohe generated forces in unsteady atmospheric conditions. 9 An important finding ohis previous work is that, because of self-initiated wing vibrations, time synchronization is needed between between the fluid and structural components for a coupled computational scheme. 1 In the present work, the relationship between wing stiffness and fluid dynamics in flapping wing flight will be explored, using a two-dimensional model wing owo rigid elliptical sections connected by a hinge fitted with a damped torsion spring, as illustrated in Figure 1. The motion ohe leading section is prescribed, using pitching and plunging kinematics that are representative of a section of a flapping insect wing with high aspect ratio. The aft section, which represents the soft posterior portion of an insect wing, moves passively under the influence ohe torsion spring and fluid dynamic forces. Parameters that will be explored in this 2 of 15

3 work are the kinematics ohe prescribed flapping and the stiffness and damping ohe spring. High-fidelity numerical simulations ohis problem will rely on the viscous vortex particle method (VVPM), which has 11, 12 recently been extended to the coupled dynamics of rigid bodies in a viscous fluid. The simulations are complemented by physical experiments using a dynamically-scaled model in a water tank. The results of both ohese methodologies will be presented and discussed. II. Methodology A. Numerical simulations The computational investigation ohis problem will rely on the viscous vortex particle method (VVPM), coupled with momentum equations for the rigid-body dynamics. The method relies on a fractional stepping procedure, in which the fluid convection, fluid diffusion, body motion, and vorticity creation are treated in separate substeps of each time increment. An advantage ohis method over conventional fixed-grid schemes is that the computational elements are convecting particles that automatically adapt to the flow, which obviates the need for expensive mesh regeneration. The treatment ohe solid-fluid interface shares many features with some recently-developed immersed boundary methods. In these other methods, a singular force distribution on the grid points is computed to enforce the no-slip condition on the generally nonconforming boundary. In the present method, a vorticity flux distribution is identified on the surface to annihilate spurious slip, which is then used to create new vorticity that is diffused to the nearby particles. The algorithm is briefly summarized here; further details can be found in Vortex particles The Navier Stokes equations are solved in vorticity form, Dω Dt = ν 2 ω. (1) The vorticity field in the fluid is regarded as the composition of a set of regularized vortex particles, ω(x, t) = p Γ p (t)ζ ε (x x p (t)), (2) where Γ p is the strength of a particle and x p is its position. The vorticity distribution of each particle is determined by ζ ε, which is generally radially symmetric, and ε, the radius ohe distribution. Fractional stepping can be used to split the solution of (1) into separate convection and diffusion substeps. In the first substep, vortex particle positions are updated with the local velocity field, which in turn is determined by the vorticity field by a discrete Biot-Savart integral, dx p dt = q Γ q (t)k ε (x p (t) x q (t)) + φ. (3) In this expression, K ε is the regularized version ohe velocity field associated with a single point vortex. The additional potential field, φ, is necessary to account for the effects of body rotation and possible uniform flow. The diffusion substep is treated with the technique of particle strength exchange, 13, 14 which approximates the Laplacian operator in (1) with a weighted summation of particle interactions, dγ p = ν dt ε 2 (Γ q Γ p )η ε (x p x q ), (4) q 3 of 15

4 where η ε is the radially symmetric interaction kernel. Diffusion is thereby simulated by the exchange of strength between particles, which can be made as accurate as desired by using a tailored kernel. 2. Vorticity creation The first substeps ohe algorithm update the fluid configuration according to the Navier-Stokes equations (1), keeping the body fixed. The spurious slip between the body and fluid that exists after this evolution is corrected by identifying the slip with an equivalent vortex sheet and diffusing it to the adjacent vortex particles. The vortex sheet strength, γ, is solved for with a classical boundary element technique, with the simultaneous enforcement of Kelvin s circulation theorem for each body, S j γ(s) ds = 2A bj (Ω bj (t + t) Ω bj (t)) + Γ lost,bj. (5) This constraint ensures that any change in circulation in interval t associated with the body rotation is accounted for by an equal and opposite change in circulation in the fluid. It also corrects for circulation leaks in other parts ohe algorithm by adding Γ lost,bj for each body. Once the surface vortex sheet on each body is computed, then a linear diffusion problem is solved, with Neumann boundary condition for the vorticity flux in terms of γ: ν ω n = γ t. (6) In the coupled algorithm, this slip removal by vorticity creation is carried out simultaneously with the body evolution, to ensure that at the end of a time step, the no-slip condition is enforced and the conservation of momentum ohe entire system has been obeyed. 3. Rigid body dynamics In the problems considered in this work, two rigid bodies are connected by a damped torsion spring, and the kinematics ohe leading body (body 1) are prescribed. The motion ohe trailing body (body 2) is determined by the forces and moments exerted by the fluid and damped torsion spring, as well as the holonomic constraints imposed by the hinge. These dynamics are described by the equations d 2 θ dθ I 2,h = R dt2 dt Kθ + M f 2 m 2 (X 2 X h ) du h dω 1 I 2,h h dt dt (7) X h = X 1 + d 1 (cos α 1, sin α 1 ) T (8) X 2 = X h + d 2 (cos α 2, sin α 2 ) T (9) U h = U 1 + Ω 1 (X h X 1 ) (1) U 2 = U h + Ω 2 (X 2 X h ) (11) θ = α 2 α 1 (12) In this coupled set of equations, the angular momentum equation is represented in terms ohe hinge angle θ. The configuration of body j is represented by its centroid position, X j, and angle, α j ; their rates of change are U j and Ω j, respectively. The position ohe hinge is X h ; the fixed distance between the rotational axis (lead body centroid) and the hinge, and between the hinge and trailing body centroid, are denoted by d 1 and d 2, respectively. The spring has stiffness K and damping coefficient R. The mass of body 2 is m 2, and its moment of inertia about the hinge is denoted by I 2,h. The dynamics ohe trailing body are influenced by the fluid dynamic moment, M f 2 h, exerted about the 4 of 15

5 hinge. An expression for this moment is M f h = µ (x X h ) S 2 [ 1 2 (x Xh ) ω ] n n ω ds 4µA 2 Ω 2 + ρ f ( 2B2 + A 2 X 2 X h 2) dω 2 dt + (X 2 X h ) ρ f A 2 du 2 dt where A 2 is the area and B 2 the second area moment of inertia of body 2 about its centroid, and n is a surface normal vector directed into the fluid. The density ohe the fluid is denoted by ρ f. The integral is taken over the surface contour of body 2. The vorticity flux in this expression is replaced with the vortex sheet strength using equation (6). (13) 4. Algorithm The coupled fluid/body dynamics are solved for using a fractional stepping procedure, with each time step divided into the following substeps: 1. Fluid evolution. The vortex particles positions and strengths are updated by integrating equations (3) and (4) with a 4th-order Runge-Kutta method. 2. Body evolution and slip removal. Implicit time integration ohe body dynamics equation (7) is carried out (using a backward Euler scheme). In this context, the hinge angle and velocity at the end ohe time-step are solved for iteratively with a globally convergent Newton-Raphson scheme that uses the following internal steps: (a) Using the current values for the hinge position and velocity (and the associated body configurations, via the constraints), the spurious slip is removed with the procedure of 2 and the resulting fluid dynamic moment computed with (13). (b) The remaining right-hand side terms of (7) are computed, and the next iteration values found. (c) Ihe implicit-time-discretized momentum equation is not sufficiently in balance, remove the diffused vortex sheet and repeat these steps with the new iteration values. In practice, only two or three iterations are required to ensure the simultaneous removal of slip and the balance ohe implicit discretized momentum equations. B. Experimental apparatus Experimental simulations are carried out in a 1m x 1m x.61m acrylic tank, using water as the working medium as seen in Figure 2. The wing models are dynamically scaled to coincide with the flow regime consistent with flapping insect wings. Motion inside ohe tank is generated by a two-axis motion stage capable of directly producing translational Reynolds numbers from 19 to 96 with a spatial resolution greater than.5 mm and a rotational Reynolds number from 23 to 3125 with an angular resolution of.2 rad. The flow visualization is carried out using a 4 mw HeNe laser and an optical sweep generator to create an illuminated plane, approximately 1 mm thick, through the tank at the mean depth ohe water. A Rilsan powder is used to seed the water as tracer particles, illuminated by the laser plane, due to its near neutral density, translucency, anti-clumping properties and its relative inertness in the environment. The illuminated particles are captured by means of a Canon 8 megapixel digital SLR camera moving in the translational reference frame ohe primary body, varying the frame rate and exposure to best suit the visualization. 5 of 15

6 Figure 2. Schematic drawings of experimental apparatus. Stiffness between the driven and following bodies is provided by a torsional spring with a torque of.42 Nm at a deflection of π from the rest state. Both the driven and following bodies are 5:1 aspect ratio elliptical extrusions capped with high precision hinges, giving a full body height of.61 m. The chord length of each body (the length ohe major axis) is 5.8 cm. The bodies were manufactured from molded fiberglass, stiffened with two 33 stainless steel rods and coated with gloss black paint. The trailing body was filled with lead shot, increasing its density to 58 kg/m 3, giving an approximate density ratio ρ b /ρ f = 5. The hinges that hold the driven and trailing bodies together reside at the upper and lower extents ohe fiberglass bodies and not along the height, thus preventing obstruction ohe flow. High precision steel shafting ride in ABEC-5 miniature high precision stainless steel bearings that rest in the hinges to reduce the resistance between the bodies as much as possible. Position recording ohe trailing body was measured relative to the leading, driven body by means of a HP HEDS-554 optical quadrature encoder. Encoder data acquisition is captured through a NI DAQmx PCI data acquisition board with a 1Mhz timebase. The number of data points written per translational cycle was kept effectively constant at 1*translational frequency. III. Results The primary goal ohis work is to examine the effects of wing flexibility on flapping flight fluid dynamics. The model chosen for this study has been designed to limit the dimension ohe parametric space, but retain crucial physical features of a continuously deformable wing. 6 of 15

7 The lead-body kinematics are described parametrically as follows: X 1 (t) = 1 2 A sin(2πft) (14) 1, t < t 1 cos(2πf r (t t 1 ) + Φf r /f), t 1 t < t 2 α 1 (t) = α + β 1, t 2 t < t 3 (15) cos(2πf r (t t 3 ) + Φf r /f), t 3 t < t 4 1, t 4 t < 1/f where t 1 = 1/(4f) 1/(4f r ) Φ/(2πf), t 2 = t 1 + 1/(2f r ), t 3 = t 1 + 1/(2f), and t 4 = t 2 + 1/(2f). The amplitude oranslation is A and the basic flapping frequency is f. The body is rotated between the extreme angles α ± β with a half-sinusoid of frequency f r (henceforth referred to as the rotational frequency). Thus, if f r = f, then the angle variation is purely sinusoidal, but f r > f allows intervals of constant angle. The phase Φ represents the lead ohe instant of peak rotation rate to the instant of change of direction; for Φ = these instants coincide. Kinematics ohis form, which have been used previously in flapping experiments by Dickinson, Lehmann and Sane, 15 provide sufficient richness for a wide variety of dynamic behavior. For all kinematics, the phase angle, Φ, and sweep angle, β were varied at the discrete intervals ( π/4,, π/4) and π/8, π/4, π/3, respectively. The translational amplitude (relative the chord length, c), A /c, was varied with the values (.7, 1.4, 2.8). The base flapping frequency, f, was generally kept at.15 Hz, and the rotational frequency ratio, f r /f, assigned the values , 5, and Note, for reference, that the Reynolds number based on peak rotation is 637. A. Effect oranslational amplitude For fixed flapping frequency, the scaled translational amplitude, A /c, controls the linear velocity and acceleration ohe wing. The results shown in this section demonstrate its effect, with other parameters fixed as follows: the rotational frequency ratio, f r /f = ; the rotation amplitude, β = π/4, and the rotation phase, Φ =. Figure 3 depicts the measured deflection angle, θ, over four flapping cycles, for three translational amplitudes: A /c = 2.8, 1.4 and.7; the prescribed kinematics are shown for reference. The lead body begins in the middle of its translation, traveling in the x direction at constant angle, 3π/4 (defined with respect to the +x axis). The wing deflects slightly rearward (positive deflection angle) in response to the impulsive acceleration. At around =.2, the wing begins its flip (characterized by a rapid rotation nose upward, or clockwise); the deflection angle reaches its maximum at the peak lead-body rotation rate at ft =.25, which coincides with the change of direction ohe lead body. The springfitted hinge produces a small recoil deflection immediately after the flip, and a constant, slightly negative, deflection angle is preserved for the translation in the opposite direction. The deflection signature does not vary significantly from case to case, though the recoil is slightly stronger as the translational acceleration increases. B. Effect of rotation rate In this section, the ratio of rotational to base frequency, f r /f, is explored, with other parameters held constant. This ratio determines the speed ohe wing flip in each half-cycle of flapping. Other parameters are set as follows: the translational amplitude, A /c = 2.8; the rotation amplitude, β = π/4, and the rotation phase, Φ =. Figure 4 displays the deflection angle over four cycles. The most notable effect can be observed in the peak deflection occurring in the middle ohe flip. The rotational acceleration increases with f r /f, 7 of 15

8 6 θ(t) -6 5 α 1 (t)-α -5 2 X 1 (t)/c Figure 3. Experimental measurement of deflection angle for varying translation amplitude. For all cases, f =.15 Hz, f r/f = , β = π/4, and Φ =. A /c = 2.8, ; A /c = 1.4, ; A /c =.7,. producing a larger forcing amplitude for the trailing body. However, the recoil is not significantly affected by the change of rotation rate. C. Effect of rotation amplitude The rotational amplitude, β, has a similar effect to the rotational rate, as can be seen in Figure 5, which correspond to amplitudes of β = π/8 and π/4. In these results, the other parameters are maintained at the values f r /f = 5, A /c = 2.8 and Φ =. However, in this case, both the peak deflection in the flip as well as the resulting recoil are affected. The peak deflection is approximately doubled, the same factor that the rotation amplitude is increased by. The recoil, on the other hand, is larger for the smaller rotation amplitude. This latter effect is likely due to a weaker fluid inertial reaction upon recoil after a weaker flip deflection. D. Effect of rotation phase Previous researchers (e.g. Dickinson, Lehmann and Sane 15 ) have identified the phase lead of rotation to translation, Φ, as important for a mean lift to be generated in hovering. Its effect on the deflection angle can be seen in Figure 6, in which the other parameters are kept at f r /f = 5, A /c = 2.8 and β = π/4. 8 of 15

9 6 θ(t) -6 5 α 1 (t)-α -5 2 X 1 (t)/c Figure 4. Experimental measurement of deflection angle for varying rotational frequency. For all cases, f =.15 Hz, A /c = 2.8, β = π/4, and Φ =. f r/f = , ; f r/f = 5, ; f r/f = ,. As the phase angle decreases from π/4, to, to π/4, the instant of peak lead-body rotation is delayed relative to the moment of direction change. In kind, the instant of peak hinge deflection is also delayed by the same interval oime. A small decay in this peak value can also be observed, as can a small increase in the recoil deflection. These phenomena can be attributed to the increasing coincidence ohe flip recoil with the steady translational velocity in the middle ohe cycle. For positive phase, the recoiling trailing body encounters an inertial force from the fluid composed of influence of both rotational and translational acceleration. As the phase decreases, the translational component ohis becomes smaller, allowing a larger recoil deflection. E. Numerical simulations The experimental measurements ohis paper are complemented with numerical simulation results, obtained from the viscous vortex particle method. In this section, we compare the results from one kinematic case, corresponding to f r /f = , A /c = 2.8, β = π/4 and Φ =. The Reynolds number ohe simulations, defined here based on the peak rotation rate, is 247, which is a factor of 25 smaller than the experiments (this choice was made to limit the computational cost ohe simulations). The comparison of wing deflection angle can be seen, for one cycle, in Figure 7. The agreement is reasonably good, in spite ohe linear models used for stiffness and damping (with coefficients determined by trial and error). All ohe qualitative features of the deflection angle coincide, and the magnitude agrees well for much ohe cycle. 9 of 15

10 6 θ(t) -6 5 α 1 (t)-α -5 2 X 1 (t)/c Figure 5. Experimental measurement of deflection angle for varying rotation amplitude. For all cases, f =.15 Hz, A /c = 2.8, f r/f = 5 and Φ =. β = π/8, ; β = π/4,. Based on the agreement of simulation results with experiments, we can draw conclusions about the fluid behavior during flexible wing flapping by examining the simulation data. Figure 8 displays several panels of vorticity taken at uniform intervals during one flapping cycle. Vorticity is produced at the leading and trailing edges of each body during the initial impulsive start. The leading edge vortex remains attached during the entire process, with each successive vortex recaptured during the flip and absorbed into the boundary layer. The trailing edge vortex produced during each flip is paired with one of opposite sign generated during the ensuing translation, forming a dipole that propagates in the opposite direction of wing motion. The pairing process and subsequent propagation is strongly affected by the phase between rotation and translation. Net lift on the wing is expected to be enhanced when these dipoles travel with a substantial downward trajectory (indicative of a net flux of momentum downward). The role of flexibility in this simplified system can be explored by comparing the forces and energy expenditure between this flexible wing and one in which the hinge is rigidly held. The lift force over three half-cycles is compared in Figure 9. In both cases, a substantial transient lift arises from the flip (contributed mostly by the trailing body), followed by a small peak from the jump in rotational acceleration at the end ohe flip, then a gradual growth and decay during the translational portion (mostly from the lead body). The lift developed by the rigid wing is substantially larger during all portions ohe cycle, and the mean lifts (computed over a single cycle) are F L = 1.5 for the flexible wing and 2.45 for the rigid wing. Note that the force has been scaled here by 1 2 ρ f U 2 c, where U is a reference velocity given by 6.667fc. The energy expenditure ohese two wings is depicted in Figure 1, which in each case is analyzed into all contributions. Most ohe total power goes directly into the fluid. However, some ohis power is taken by 1 of 15

11 6 θ(t) -6 5 α 1 (t)-α -5 2 X 1 (t)/c Figure 6. Experimental measurement of deflection angle for varying rotation phase. For all cases, f =.15 Hz, A /c = 2.8, β = π/4, and f r/f = 5. Φ = π/4, ; Φ =, ; Φ = π/4,. kinetic energy ohe wing (though the net change ohis is zero over a cycle). For the flexible wing, some energy is lost to dissipation by the hinge damping, and some is stored as elastic energy in the spring (though this is eventually returned to the system). A dramatic increase in the necessary power is clearly apparent in the rigid wing case, particularly during the flip, when the trailing body encounters the largest resistive and reactive forces. The mean power consumption, Ẇ, is 3.84 for the flexible case, and 6.87 for the rigid case; the power is normalized by ρ f U 3 c. We define here a performance parameter as the mean power required per unit mean lift generated, Ẇ /F L. For the two cases analyzed here, this performance parameter takes the value 2.56 and 2.81, for the flexible and rigid cases, respectively. This simple measure indicates that the flexible wing can develop lift more efficiently than a rigid one. However, more analysis is required. IV. Conclusions In this work, we have demonstrated that a complementary effort of experiments and numerical simulations can be used to investigate flapping wing flight with flexible wings. Our approach has been to use a simplified system that can be easily described and explored parametrically. The dynamical system, composed on both fluid and bodies, shows a rich complexity. The results of wing deflection depend crucially ohe magnitudes of rotation and translation, as well as their relative timing. 11 of 15

12 6 4 2 θ(t) Figure 7. Comparison of numerical simulation with experimentally measured deflection angle, with f =.15 Hz, A /c = 2.8, f r/f = , Φ =, and β = π/4. Experiment, ; VVPM simulation,. Results of wing deflection from numerical simulations agree quite well with those of experiments, despite a large discrepancy in Reynolds number. The simulation results have allowed us to more deeply investigate the dynamics of shed flow structures, as well as the generated forces and power requirements during the flapping cycle. We have shown that, for the system considered in this work, a flexible wing requires less power than a rigid one for the same kinematics, at the expense of some mean lift. However, a simple performance measure indicates that less power per unit lift is required by the flexible wing. The results shown in this paper are merely suggestive of more interesting phenomena, to be explored in greater detail in ongoing work. References 1 Dudley, R., The Biomechanics of Insect Flight: Form, Function and Evolution, Princeton University Press, Princeton, N. J., 2. 2 Weis-Fogh, T., Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production, J. Exp. Biol., Vol. 59, 1973, pp Ellington, C. P., The aerodynamics of hovering insect flight. III. Kinematics, Phil. Trans. R. Soc. Lond. B, Vol. 35, 1984, pp Norberg, R. A., Hovering flight ohe dragonfly Aeschna Juncea L., kinematics and aerodynamics, Swimming and Flying in Nature, edited by T. Y. Wu, C. J. Brokaw, and C. E. Brennen, Vol. 2, Plenum Press, New York, Daniel, T. L. and Combes, S. A., Flexible Wings and Fins: Bending by Inertial or Fluid-Dynamic Forces? Integ. Comp. Biol., Vol. 42, 22, pp Combes, S. A. and Daniel, T. L., Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta, J. Exp. Biol., Vol. 26, 23, pp Heathcote, S., Martin, D., and Gursul, I., Flexible Flapping Airfoil Propulsion at Zero Freestream Velocity, AIAA J., 24, pp Heathcote, S. and Gursul, I., Flexible flapping airfoil propulsion at low Reynolds numbers, AIAA Paper , Ifju, P. G., Jenkins, D. A., Ettinger, S., Lian, Y., Shyy, W., and Waszak, M. R., Flexible-wing-based micro air vehicles, AIAA Paper 22-75, Lian, Y., Shyy, W., Viieru, D., and Zhang, B., Membrane wing aerodynamics for micro air vehicles, Prog. Aerosp. Sci., Vol. 39, 23, pp Eldredge, J. D., Numerical simulation ohe fluid dynamics of 2D rigid body motion with the vortex particle method, with application to biological locomotion, Submitted to J. Comput. Phys., of 15

13 ft = Figure 8. Computed vorticity field from VVPM, with f =.15 Hz, A /c = 2.8, f r/f = , Φ =, and β = π/4. 12 Eldredge, J. D., Coupled dynamical simulations owo-dimensional rigid bodies in a viscous fluid, Submitted to J. Comput. Phys., Degond, P. and Mas-Gallic, S., The Weighted Particle Method for Convection-Diffusion Equations, Part 1: The Case of an Isotropic Viscosity, Math. Comp., Vol. 53, No. 188, 1989, pp Eldredge, J. D., Leonard, A., and Colonius, T., A general deterministic treatment of derivatives in particle methods, J. Comput. Phys., Vol. 18, 22, pp Dickinson, M. H., Lehmann, F.-O., and Sane, S. P., Wing Rotation and the Aerodynamic Basis of Insect Flight, Science, Vol. 284, 1999, pp of 15

14 F/ρ f U 2 c 2F/ρ f U 2 c Figure 9. Computed lift coefficient from VVPM, for flexible (left) and rigid (right) wings, with f =.15 Hz, A /c = 2.8, f r/f = , Φ =, and β = π/4. Contribution from trailing body, ; leading body, ; total, W/ρ f U 3 c W/ρ f U 3 c Figure 1. Computed power expenditure from VVPM, for flexible (left) and rigid (right) wings, with f =.15 Hz, A /c = 2.8, f r/f = , Φ =, and β = π/4. Contribution from work done on fluid, ; body kinetic energy, ; damper dissipation, ; spring potential energy, ; total,. 14 of 15