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A three-dimensional computational study of the aerodynamic mechanisms of insect flight

Ravi Ramamurti^* and William C. Sandberg

Laboratory for Computational Physics and Fluid Dynamics, Naval Research Laboratory, Washington, DC 20375, USA

View larger version (16K): [in a new window]   Fig. 1. (A) Schematic diagram of a hovering Drosophila showing the orientation of the x,y,z coordinate system. (B) Schematic diagram of the flapping Drosophila wing. The position of the wing is shown at three different times during the flapping cycle. The coordinate system (x',y',z') is fixed to the wing, and the wing rotates about the z' axis throughout the cycle. R, wing length; , wingbeat amplitude.

View larger version (24K): [in a new window]   Fig. 2 . Kinematics of the flapping wing. (A) Angle of rotation of the wing about the x axis (roll), and the z' axis (pitch) for three different phases between wing rotation and stroke reversal. (B) Translational velocity of the wing tip and rotational (angular) velocity of the wing for three different phases.

View larger version (23K): [in a new window]   Fig. 3 . Time history of thrust (A) and drag (B) forces during one wingbeat. The red lines are from the present study; the blue lines are from Dickinson et al. (1999). The numbered broken lines in A refer to times discussed in the text.

View larger version (14K): [in a new window]   Fig. 4 . (A) Position of the wing (red outline) at the beginning of the downstroke. The wing chord is aligned with the x axis of the x,y,z coordinate system at this instant. The orientation of the y=10 cm plane for which the velocity vectors are shown in Fig. 6 is indicated. (B) Position of the wing at t=12.5 s. The orientation of the two planes for which velocity vectors are shown in Fig. 7 is indicated.

View larger version (74K): [in a new window]   Fig. 5 . Instantaneous particle traces at the beginning of the downstroke. Particles were released from a rake of rectangular grid of points in a plane 0.8 mm away from the bottom surface of the wing. Using the instantaneous velocity field, the positions of these particles were obtained by integrating the velocity at these rake points until the length of the traces exceeded a specified length or the particles ended on a solid boundary or exited the computational domain. These particle traces are colored according to the magnitude of the velocity (in cm s-1) at that location. A leading edge vortex is seen rotating in the counterclockwise direction, and a stagnation line is shown near the z' axis of rotation (dark blue traces).

View larger version (37K): [in a new window]   Fig. 6. Velocity vectors near the leading edge at three instants at the beginning of the downstroke on a plane y=10 cm (see Fig. 4A for the orientation of the plane relative to wing). L.E., leading edge. The vectors are colored according to the magnitude of absolute velocity (cm s-1) and are of constant length. The flow separates at the leading edge (A) and the separation point moves down (B,C) as the downstroke is continued.

View larger version (58K): [in a new window]   Fig. 7. Velocity vectors (in cm s-1; see color scale) during the translation phase of the downstroke (between t3 and t4 in Fig. 3A). (A,B,C) Velocity vectors near the trailing edge on the xz plane at y=10 cm; (D,E,F) velocity vectors near the wing tip on the yz plane at x=0 cm (see Fig. 4B for the orientation of planes relative to the wing). T.E., trailing edge.

View larger version (38K): [in a new window]   Fig. 8. Translational and angular accelerations of the wing for three different phases between wing rotation and stroke reversal. The broken vertical lines refer to the sections of the wing stroke identified in Fig. 3A.

View larger version (32K): [in a new window]   Fig. 9. Particle streaklines produced during the downstroke at t=12.93s. The particles were release from a rake of rectangular grid on a plane 3.0 cm above (A) and 3.5 cm below (B) the wing parallel to the wing and near the leading edge. The absolute velocities of the particles (in cm s-1) are given in the color scale.

View larger version (39K): [in a new window]   Fig. 10. Flow patterns during the middle of the downstroke at t=13.79 s. (A) Particle traces (streaklines). The particles were released on a plane parallel to the wing and 3.9 cm below it, and velocity vectors are colored according to the magnitude of absolute velocity (in cm s-1). (B) Velocity vectors on the xy plane at z=20 cm. Velocity vectors are colored according to the magnitude of absolute velocity (in cm s-1) and are of constant length. (C) Pressure contours. Pressure is non-dimensionalized with respect to the dynamic head.

View larger version (38K): [in a new window]   Fig. 11. Particle traces prior to the end of the downstroke, t=14.65 s. The particles were release from a rake of rectangular grid on a plane 3.2 cm above the wing parallel to the wing and near the leading edge. Traces are colored according to the magnitude of absolute velocity (in cm s-1).

View larger version (26K): [in a new window]   Fig. 12. Effect of grid refinement on the computed thrust. The finer grid had twice the resolution and the time step was halved.

View larger version (22K): [in a new window]   Fig. 13. Effects of viscosity on the computed thrust (A) and drag (B) forces, Re=120.

View larger version (26K): [in a new window]   Fig. 14. Thrust (A) and drag (B) forces for the `advanced' case in which the wing rotation precedes stroke reversal. The results from the present computational model are compared with the experimental data of Dickinson et al. (1999).

View larger version (41K): [in a new window]   Fig. 15. Velocity vectors near the leading edge early in the downstroke for the `advanced' wing rotation in which rotation precedes stroke reversal. Velocity vectors are shown in the xz plane at y=10 cm. The vectors are colored according to the magnitude of velocity (in cm s-1) and are of constant length. L.E., leading edge.

View larger version (25K): [in a new window]   Fig. 16. Thrust (A) and drag (B) forces for the `delayed' case in which wing rotation is delayed with respect to stroke reversal. The results from the present computational model are compared with the experimental data of Dickinson et al. (1999).

View larger version (72K): [in a new window]   Fig. 17. Velocity vectors near the leading edge (A,B) and the trailing edge (C,D) early in the downstroke for the `delayed' case of wing motion in which rotation is delayed relative to stroke reversal. Velocity vectors are shown in the xz plane at y=10 cm. The vectors are colored according to the magnitude of velocity (cm s-1) and are of constant length. L.E., leading edge; T.E., trailing edge.

View larger version (66K): [in a new window]   Fig. 18. Magnitude of velocity in the wake of the wing at the beginning of the downstroke for the three cases of wing motion: rotation advanced (A), symmetrical (B) or delayed (C) relative to stroke reversal. Velocity contours are shown in the xz plane at y=10 cm. The contours are colored according to the magnitude of absolute velocity (in cm s-1).

View larger version (28K): [in a new window]   Fig. 19. Spanwise contribution to thrust for the `symmetrical' case of wing rotation relative to stroke reversal. The thrust generated at three spanwise locations (quarter, half and three-quarter span) was calculated and is shown together with the total thrust produced by the wing and half the total thrust.

View larger version (25K): [in a new window]   Fig. 20. Moments about the wing coordinate system (x',y',z') for the `symmetrical' case of wing rotation. Mx', My', Mz', moment about wing rotation axis x', y' and z', respectively.

View larger version (32K): [in a new window]   Fig. 21. Power requirement for one wing for three different phases between wing rotation and stroke reversal.