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First published online December 28, 2007
Journal of Experimental Biology 211, 239-257 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.008649

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Near- and far-field aerodynamics in insect hovering flight: an integrated computational study

Hikaru Aono¹, Fuyou Liang² and Hao Liu^3,*

¹ Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
² Next-generation computation research group, RIKEN, 2-1 Hirosawa, Wako-Shi, Saitama 351-0198, Japan
³ Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

View larger version (59K): [in this window] [in a new window]   Fig. 1. A morphological model of a fruit fly Drosophila melanogaster. (A) A fruit fly with a computational model superimposed on the right half (http://www.tmd.ac.jp/artsci/biol/textlife/fruitfly.jpg). The fruit fly has a body length of 2.78 mm, a wing length of 2.39 mm (mean wing chord length =0.78 mm), and an aspect ratio of 3.06. (B) A multi-block grid system of the two wings and body of the fruit fly (wing, 45x45x31; body, 45x47x95) with a distance between the body surface and the outer boundary of 20.

View larger version (49K): [in this window] [in a new window]   Fig. 2. Schematic diagram of the computational system of a fruit fly Drosophila melanogaster. (A) The local wingbase-fixed (x, y, z) and the global earth-fixed (X, Y, Z) coordinate systems. The origin O' of the wingbase-fixed coordinate system lies at the wing base, with the x-axis normal to the stroke plane [the yz plane as defined by Ellington (Ellington, 1984b)], the y-axis vertical to the body axis and z-direction parallel to the stroke plane. (B) The wing kinematics are described by the positional angle , the feathering angle (angle of attack of the wing) , the elevation angle , and the stroke plane angle β; the link to the earth-fixed frame of reference comes through the body angle . We assume a body angle of 45° and a stroke plane angle β of 0° (Fry et al., 2005). (C) Instantaneous positional angle , feathering angle , and elevation angle of the fruit fly wing over one complete flapping cycle. Green solid, orange broken and blue dash-dot lines represent the positional angle , the feathering angle and the elevation angle , respectively. Red points a–g: (a) mid pronation, (b) early downstroke, (c) mid downstroke, (d) late downstroke, (e) early upstroke, (f) mid upstroke and (g) late upstroke. T, dimensionless period of one flapping cycle.

View larger version (283K): [in this window] [in a new window]   Fig. 3. Far-field flow structures around a hovering fruit fly. Absolute iso-vorticity surfaces and velocity vectors with a body (Ai,ii,Bi,ii,Ci,ii,Di) and without a body (Aiv,v,Biv,v,Civ,v,Diii) at four instances of (a), (b), (e) and (g), respectively, as illustrated in Fig. 2C. (Aiii,Biii,Ciii,Dii) The corresponding velocity vectors and contours of vertical velocity components in the stroke plane (red broken circle).

View larger version (65K): [in this window] [in a new window]   Fig. 4. Downward velocity maps at two cross sections around a hovering fruit fly at mid down- and upstroke. (A) Schematic diagram of the two cross sections around the hovering fruit fly, showing the position of velocity maps at the two cross sections at mid down- and upstroke (B–E). (B,C) Velocity maps at the plane parallel to the X-axis (a–a'); (D,E) Velocity maps at the plane parallel to the Y-axis (b–b'). Color of vectors indicates magnitude of downward velocity.

View larger version (72K): [in this window] [in a new window]   Fig. 5. Streamlines around the wing at mid down- and upstroke. Streamlines are color-coded to indicate the absolute flow speed. (A) The LEV and the TV at mid downstroke (see Fig. 2Cc). (B) The LEV and the TV at mid upstroke (see Fig. 2Cf). LEV, leading-edge vortex; TV, wing tip vortex (for further explanation, see text).

View larger version (67K): [in this window] [in a new window]   Fig. 6. Streamlines around a fruit fly at early supination and pronation. Streamlines around a hovering fruit fly at early supination (A) and at early pronation (B). A color map represents pressure contours on the wing and body surfaces: red is high and blue is low; streamlines are released at the leading edge from the wing surfaces, color of streamlines indicates the absolute flow speed. LEV, leading-edge vortex; TV, wing tip vortex; SV, stopping vortex (for further explanation, see text).

View larger version (57K): [in this window] [in a new window]   Fig. 7. Velocity vectors and contours of spanwise flow velocity at mid down- and upstroke. A wing–body computational model of a fruit fly (Ai) at mid downstroke, and (Bi) at mid upstroke. Velocity vectors at two cross sections at 50% and 60% of the wing length, (Aii) and (Aiii) at mid downstroke, and (Bii) and (Biii) at mid upstroke. A color map denotes the spanwise flow velocity; a negative magnitude of the spanwise velocity points to a flow from the wing base to the wing tip, and vice versa. L.E., leading edge; T.E., trailing edge.

View larger version (74K): [in this window] [in a new window]   Fig. 8. Comparison of near-field flow between a fruit fly and a hawkmoth at mid downstroke. A wing-body computational model of the hawkmoth (Re=6300, Uref=5.05 m s–1, cm=1.83 cm) (Ai) and the fruit fly (Re=134, Uref=2.54 m s–1, cm=0.78 mm) (Bi), with the LEVs visualized by streamlines; (Aii, Bii) the corresponding velocity vectors at the cross sections at 60% of the wing length. A color map denotes the spanwise flow velocity; a negative magnitude of the spanwise velocity points to a flow from the wing base to the wing tip, and vice versa. L.E., leading edge; T.E., trailing edge.

View larger version (21K): [in this window] [in a new window]   Fig. 9. Time courses of the vertical (lift; A) and the horizontal (drag and thrust; B) forces over a flapping cycle. Blue, red and yellow lines represent the measurements of the upper (Exp_u), average (Exp_a) and lower (Exp_l) values obtained by Fry et al. (Fry et al., 2005), respectively; broken line is the computed result (Com_a). T, dimensionless period of one flapping cycle.

View larger version (32K): [in this window] [in a new window]   Fig. 10. Time courses of aerodynamic and inertial torques of the wing over a flapping cycle. Aerodynamic and inertial torques of the wing as shown in Fig. 2B. (A) Rolling torques: aerodynamic rolling torque (blue; Tr_aero_t), inertial rolling torque (red; Tr_iner_t) and total rolling torque (green; Tr_total). (B) Yawing torques: aerodynamic yawing torque (blue; Ty_aero_t), inertial yawing torque (red; Ty_iner_t) and total yawing torque (green; Ty_total). (C) Pitching torques: aerodynamic pitching torque (blue; Tp_aero_t), inertial pitching torque (red; Tp_iner_t) and total pitching torque (green; Tp_total). T, dimensionless period of one flapping cycle.

View larger version (26K): [in this window] [in a new window]   Fig. 11. Time courses of muscle-mass-specific powers over a flapping cycle. (A) Inertial powers: solid line (Exp) and broken line (Com) express the experimental data (Fry et al., 2005) and the computational data, respectively. (B) Aerodynamic powers: upper (blue; Exp_u), average (red; Exp_a) and lower (yellow; Exp_l) data from Fry et al. (Fry et al., 2005; green broken line is the computed data (Com_a). (C) Total mechanical powers: blue solid line (Exp) and orange broken line (Com) express the experimental data (Fry et al., 2005) and the computational data. T, dimensionless period of one flapping cycle.

View larger version (24K): [in this window] [in a new window]   Fig. 12. Pressure gradient contours on the wings of a fruit fly (A) and a hawkmoth (B) at mid downstroke. The white arrows represent the core and direction of the spanwise pressure gradient on the wing surfaces.

View larger version (10K): [in this window] [in a new window]   Fig. 13. Time course of vertical (lift) force during the first three flapping cycles. The mean lift force generated during the second cycle is reduced significantly by 22.5% compared with the first cycle, but almost no difference is observed between the second and third cycles.

View larger version (15K): [in this window] [in a new window]   Fig. 14. Comparison of aerodynamic force coefficients between a fruit fly with and without the body over a flapping cycle. While the mean aerodynamic force coefficients over one flapping cycle are similar, time courses of aerodynamic force coefficients differ between the simulation `with body' and `without body'. Solid lines, lift coefficients with the body (orange) and without the body (brown); broken lines, drag coefficients with the body (dark blue) and without the body (sky blue); dotted lines, sideslip force coefficients with the body (pink) and without the body (purple). T, dimensionless period of one flapping cycle.

View larger version (25K): [in this window] [in a new window]   Fig. A1. Effects of grid and time step on aerodynamic force production. (A) Grid sensitive analysis for aerodynamic force coefficients over a flapping cycle. Three grid systems are used: c, a coarse grid system (wing: 33x35x19, body: 33x35x35); fn, a fine grid system (wing: 45x45x31, body: 45x47x95); and ft, a finest grid system (wing: 45x45x31, body: 57x55x121). (B) Time step sensitive analysis for aerodynamic force coefficients over a flapping cycle. Two time steps are used: t1, time step dt=0.005; t2, time step dt=0.0025.