The wings and the body shape of Manduca sexta and Agrius convolvuli are compared in

Transcription

1 1 Wing and body shape of Manduca sexta and Agrius convolvuli The wings and the body shape of Manduca sexta and Agrius convolvuli are compared in terms of the aspect ratio of forewing AR fw (wing length /wing area), the ratio of body length and wing length R bw, and the ratio of thorax radius and wing length R tw. The results are summarized in Table S1. AR fw of Manduca sexta is calculated from an image of Manduca sexta forewing in Combes & Daniel (3). R bw and R tw of Manduca sexta is from the measurement by Hedrick & Daniel (6). References Combes, S. A. & Daniel, T. L. 3 Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta. J. Exp. Biol. 6, Hedrick, T. L. & Daniel, T. L. 6 Flight control in the hawkmoth Manduca sexta: the inverse problem of hovering. J. Exp. Biol. 9,

2 Table S1. Morphological parameters of Manduca sexta and Agrius convolucli aspect ratio of forewing, AR fw ratio of body and wing length, R bw ratio of thorax radius and wing length, R tw Manduca sexta Agrius convolvuli

3 3 Kinematic modeling of hawkmoth hovering for fluid-structure interaction analysis The wing-body kinematics of a hovering hawkmoth, Manduca sexta, is modeled by the flapping wing kinematics, the stroke plane angle, and the body angle as illustrated in figure S1 based on the measurements (Willmott & Ellington 1997). With consideration of the negligible small oscillation of the insect body during hovering (Willmott & Ellington 1997), we employ a tethered model in this study, in which 6 degrees of freedom of the hawkmoth body are locked up. The wing kinematics is defined by three angles within stroke plane as shown in figure S1b, in which each angle is represented by a third order Fourier series, such that: outer inner 3 cn cosnt sn sinnt n 3 outer, cn cosnt outer, sn sinnt n 3 inner, cn cosnt inner, sn sinnt n 3 cn cosnt sn sinnt n (S1) where is the flapping angular velocity and the coefficients of the Fourier series are determined from the measured data (Willmott & Ellington 1997). Terms inner and outer denote the feathering angles at wing tip of fore- and hind-wing, respectively. Note that the feathering angle outer has been utilized in our previous studies (Liu et al. 1998; Liu 9). The wing deformation due to inertial force, aerodynamic force and physical wing-body interaction is in general inherently included in the measured wing kinematics. Hence, subtraction of the wing deformation is essentially difficult, and a simplified harmonic wing kinematics is hereby used as an input at wing base. The realistic three angles at wing base are simplified into the first order and asymmetric movements of the flapping wing as shown in figure S1b; the elevation angle is ignored here because of the

4 4 small magnitude and its marginal effect on mean lift and drag forces as reported by Sane & Dickinson (1) and Bos et al. (8). Therefore, the wing kinematics for FSI analysis is written as 1 c cos nt 1 s sin nt (S) where s1 outer, c1 c1 c1 s1 outer, s1 inner, c1 inner, s1 (S3) Note that, for simplicity, this virtual kinematics is designed to be symmetric with respect to stroke plane without high order harmonics. The amplitude of the positional angle c1 is set to be the same as used in Liu (9) but the amplitude of feathering angle si is modified so as to remove a nose-down twist (Wootton 199). Since the feathering angles measured in hawkmoth hovering in general include two angles for fore- and hind-wing, which together forms a flex in-between, we hereby define the amplitude of the feathering angle by averaging these two feathering angles of fore-and hind-wing. References Bos, F. M., Lentink, D., Van oudheusden, B. W. & Bijl, H. 8 Influence of wing kinematics on aerodynamic performance in hovering insect flight. J. Fluid Mech. 594, Liu, H. 9 Integrated modeling of insect flight: from morphology, kinematics to aerodynamics. J. Comput. Phys. 8,

5 5 Liu, H., Ellington, C. P., Kawachi, K., van den Berg, C. & Willmott, A. P A computational fluid dynamic study of hawkmoth hovering. J. Exp. Biol. 1, Sane, S. P. & Dickinson, M. H. 1 The control of flight force by a flapping wing: lift and drag production. J. Exp. Biol. 4, Willmott, A. P. & Ellington, C. P The mechanics of flight in the hawkmoth Manduca sexta. I. kinematics of hovering and forward flight. J. Exp. Biol., Wootton, R. J. 199 Functional morphology of insect wings. Annu. Rev. Entomol. 37,

6 6 (a) z x Stroke plane angle Body angle (c) Feathering angle (inner) Feathering angle (outer) (b) x z y Stroke plane : Positional angle : Elevation angle : Feathering angle Angles,,, /deg (d) Elevation angle Positional angle Downstroke Upstroke Feathering angle Positional angle Stroke plane y -3-6 Downstroke Stroke cycle, t/t Elevation angle Upstroke Figure S1. Schematic diagram of wings and body kinematics. (a) Definition of a global coordinate system (x, y, z), body angle and stroke plane angle. (b) Wing position parameters within stroke plane: positional angle, elevation angle and feathering angle. (c) Realistic wing kinematics of hovering hawkmoth and (d) simplified kinematics at wing base for FSI analysis.

7 7 Structural modeling of hawkmoth wing The hawkmoth wing is treated to be composed of three regions involving the leading edge, the forewing and the hind wing; and in each region the wing venation is modeled to have a structure of the distributed fractions and running directions of fibers. Material properties of the wing model in terms of Young s modulus, Poisson s ratio, and density of veins and membranes are given in table S, in which a thickness distribution of both veins and membranes is assumed to vary exponentially from wing base to wing tip. Colored contours in figure 1b represent the thickness distribution of veins. The anisotropic and distributed elastic modulus is calculated based on the rule of mixture, which is commonly used in the modeling of composite materials (Jones 1999). Validation of the present FSI-based hovering simulation is first implemented for a rotating hawkmoth forewing (figure Sa) by comparing the bending angles of wing deformation with the measurements by Combes and Daniel (3b). Note that some high-order oscillations in the measured bending angles are observed at trailing edge, which is obviously induced in the very flexible edge zone areas (Combes & Daniel 3b). Such wing deformation at trailing edge, however, does not often occur during flapping flight of hawkmoth because of the physical link between fore- and hind-wing. A second-order Fourier series is hereby introduced to smoothen the experimental results. The simulated and measured smooth bending angles are in reasonable agreement, both showing large displacements at wing tip and trailing edge (figure Sb). The wing model has a mass of approximately 4 mg, very close to the average wing mass of hawkmoth, i.e., % of the body mass (Ellington 1984; Hedrick & Daniel 6). Moreover, the static span-and chord-wise, bending stiffness of forewing is simulated by fixing the wing base or leading edge and applying a static force on the forewing model (Combes & Danliel 3a). The spanwise stiffness is calculated to be 1.5 x 1-4, which agrees well with the measurements ( x 1-4 Nm, Combes &

8 8 Daniel 3a). The chordwise stiffness is calculated to be 1.6 x 1-6 Nm, apparently under-estimated compared with the measurements (.5 x x 1-5 Nm, Combes & Daniel 3a). This implies that the current model may need to be further improved probably by introducing the non-linear structural components. References Combes, S. A. & Daniel, T. L. 3a Flexural stiffness in insect wings I. scaling and the influence of wing venation. J. Exp. Biol. 6, Combes, S. A. & Daniel, T. L. 3b Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta. J. Exp. Biol. 6, Ellington, C. P The aerodynamics of hovering insect flight. II. morphological parameters. Phil. Trans. R. Soc. B 35, Hedrick, T. L. & Daniel, T. L. 6 Flight control in the hawkmoth Manduca sexta: the inverse problem of hovering. J. Exp. Biol. 9, Jones, R. M Mechanics of composite materials, nd edn. Philadelphia, PA: Taylor & Francis.

9 9 Table S. Material properties of insect wings for CSD analysis. volume fraction of vein (%) Young s modulus of vein (GPa) density of Young s modulus Poisson s vein of membrane (GPa) ratio (kg/m 3 ) density of membrane (kg/m 3 ) Flexible wing Leading edge Forewing Hind wing Rigid wing

10 1 (a) Camera wt te Wing Camera (b) Bending angle at wing tip/rad : Computation : Experiment : nd order Fourier Motor Motor wt Wing te wt: wing tip te: trailing edge Bending angle at trailing edge/rad t/t Figure S. (a) Experimental apparatus for the measurement of hawkmoth wing bending in air and helium, and angular position converted from the coordinates of the marked points (Combes and Daniel 3b). (b) Difference of bending angles with a rigid rotating wing at wing tip and at trailing edge of a hawkmoth forewing in air. Experimental data are traced from the literature (Combes & Daniel 3b).

11 11 An evaluation method for hovering efficiency A method to evaluate the aerodynamic efficiency of hovering flight is proposed here. In the case of hovering flight, the induced power that is received by the surrounding fluid must be used in evaluating its aerodynamic efficiency, which may be expressed by the following formula: P ind wf v (S4) where w denotes the vertical fluid velocity in the vicinity of the wing. In the case of the Rankin-Froude momentum (RFm) efficiency, the velocity w is calculated based on the Bernoulli s theorem and the law of momentum under the assumption that F v equals to the insect s weight (Ellington 1984). Since the NS solution-based computation provides detailed information on the unsteady and three-dimensional flow fields around wings and body, the velocity w can be computed directly. Considering that the induced downward flow is created by wing flapping as illustrated in figure S3, we hereby introduce a closed-loop virtual surface to wrap up each single wing, which moves with the flapping wing, and define the downward velocity w as: w i where i denotes the i-th cell, or, control volume on the virtual surface at which the downward flow is created; w i expresses the downward velocity at cell i; and A i is the surface area of the cell i. Note that we don t take into account any contributions from the cells where no downward flow is created. A distance between the wing surface and the virtual surface must be selected appropriately not only far sufficient to minimum the divergence of fluid velocities into the velocity of wing surface but also near enough to minimum the viscosity-based damping. After a systematically parametric study, we find that a distance of.1 c m is reasonable. Then the hovering aerodynamic efficiency is calculated by dividing the mean induced power P ind with the mean aerodynamic i A w i A i i (S5)

12 1 power P aero. References Ellington, C. P The aerodynamics of hovering insect flight. VI. lift and power requirements. Phil. Trans. R. Soc. B 35,

13 13 virtual surface virtual surface.1 c m wing cross section wing cross section Figure S3. Closed-loop virtual surface for computation of downward flow in the vicinity of a flapping wing.

14 14 Decomposition and interpolation of FSI-based wing deformations In this study, the wing deformation at each cross section is evaluated by using the three-dimensional wing configuration based on FSI analysis. As shown in figure S4, a spanwise bending is defined as the slope of spanwise line through a pivot; a twist is defined by a chordwise wing rotation; and a camber is defined by a ratio between chordwise deflection and chord length. The FSI analysis-based wing deformation is decomposed into a Fourier series, such as: N x Nt nx nx c nx, nt l cosntt s nx, nt l sinntt n n x t (S6) where is the deformation as defined in figure S4(i); c and s are the coefficients for Fourier series; l denotes the spanwise position on wing surface relative to wing length. On the basis of the Fourier series s decomposition, an interpolated wing shape is illustrated in figure S4.

15 15 (a) Spanwise bending (i) z w Flexible wing 1 (rad).3 x w yw/r Rigid wing sb y w (iii) 1 1 Stroke cycle, t/t -.3 (b) Twist (i) z w Deformed wing with spanwise bending 1 (rad). x w yw/r tw y w (iii) 1 1 Stroke cycle, t/t -. (c) Camber (i) x w z w d c cb =d/c cross section yw/r 1 (%) 3 Deformed wing with spanwise bending and twist y w (iii) 1 1 Stroke cycle, t/t -3 Figure S4. (i) Definition of the simulated (a) spanwise bending anlge sb, (b) twist angle tw and (c) camber cb in the flexible wing. Time-courses and spanwise distribution of the three kinds of deformations of a flexible wing and (iii) their interpolated deformations.

16 16 (a) Spanwise bending : Flexible wing (b) Twist (c) Camber.3 :Rigid wing. 3. : Interpolated (i).1.1 (i) 1 ABC D ABC D Vertical force coefficient, Cv Bending angle/rad Twist angle/rad Stroke cycle, t/t Stroke cycle, t/t Stroke cycle, t/t (iii) (iii) (iii) Camber/% 1.5 (i) Downstroke Upstroke Figure S5. Effects of interpolated (a) spanwise bending, (b) twist and (c) camber on the aerodynamic force production of flapping wings. (i) Time-courses of three kinds of interpolated deformations. Time-courses of vertical force coefficients created by rigid wing models flapping with the prescribed wing deformations (a), (b), and (c). (iii) Time-averaged aerodynamic force vectors generated during down-and up-stroke. Note that for comparison the gray and red lines in (i), and (iii) represent results of the rigid and flexible wings, respectively.

17 17 Table S3. Time-averaged vertical and horizontal forces acting on wings and body at down- (F v, down, F h, down ) and up-stroke (F v, up, F h, up ), in a wing-beat cycle (F v, F h ), muscle-mass-specific aerodynamic and induced powers of a single flapping wing (P a, P ind ) and aerodynamic efficiencies (). Vertical force (mn) Horizontal force (mn) Aerodynamic power, F v, down F v, up F v F h, down F h, up F h P a (W/kg) Induced power, P ind (W/kg) Aerodynamic efficiency, (%) flexible rigid (base) rigid (tip) spanwise bending twist camber

18 18 Bending angle/rad (a) Effect of Young s modulus (i) (b) Effect of thickness (i) : EI/EI moth = 3 : EI/EI moth = : EI/EI moth = 1 : EI/EI moth = 1/ : EI/EI moth = 1/ Twist angle/rad (iii) (iii) Camber/% Stroke cycle, t/t Stroke cycle, t/t Figure S6. Effects of (a) Young s modulus and (b) wing thickness on wing deformation including (i) spanwise bending, twist and (iii) camber.

19 19 Angles,, /deg (a) Effect of Young s modulus Positional angle Feathering angle (i) (b) Effect of thickness Positional angle Feathering angle (i) : EI/EI moth = 3 : EI/EI moth = : EI/EI moth = 1 : EI/EI moth = 1/ : EI/EI moth = 1/3 : Rigid wing Vertical force coefficient, Cv (iii) (iii) Downstroke Upstroke Figure S7. Effect of (a) Young s modulus and (b) wing thickness on the (i) wing kinematics at.8 R from wing base. Time-courses of vertical force coefficients. (iii) Time-averaged aerodynamic force vectors generated during down-and up-stroke.