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Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion

Mao Sun^* and Jian Tang

Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China

View larger version (13K): [in a new window]   Fig. 1. Sketches of the reference frames and wing motion. (A) oxyz is an inertial frame, with the xz plane in the stroke plane. o'x'y'z' is a frame fixed on the wing, with the x' axis along the wing chord and the z' axis along the wing span. is the azimuthal angle of the wing, is the angle of attack and l is the distance between the y axis and the wing tip or the wing length. (B) The motion of a section of the wing.

View larger version (120K): [in a new window]   Fig. 2. Portions of the body-conforming grid near the wing surface. (A) In a sectional plane; (B) in the y'=0 plane (see Fig. 1A for a definition of this plane).

View larger version (20K): [in a new window]   Fig. 3. Wing translational (ut+) and rotational (+) non-dimensional velocities (A), lift coefficient CL (B) and drag coefficient CD (C) plotted against non-dimensional time . In B, the quasi-steady lift coefficient is also plotted for comparison.

View larger version (81K): [in a new window]   Fig. 4. (A–H) Vorticity plots at three spanwise locations at various times during one stroke. r1, r2 and r3 denote locations 0.75, 0.5 and 0.25 wing lengths from the wing root, respectively. , non-dimensional time; , angle of attack. Solid and broken vorticity lines indicate positive and negative vorticity, respectively. The magnitude of the non-dimensional vorticity at an outer contour is 1. Starting from the outer contour, for the first 21 contours, the contour interval is 0.2, for the next 30 contours, the contour interval is 0.5, and for the remainder of the contours, the contour interval is 5.

View larger version (120K): [in a new window]   Fig. 5. Sectional streamline plots at three spanwise locations at various times during one stroke. r1, r2 and r3 denote locations 0.75, 0.5 and 0.25 wing lengths from the wing root, respectively. , non-dimensional time; , angle of attack (the spatial interval of the incoming streamlines can be seen from the right of each plot).

View larger version (20K): [in a new window]   Fig. 6. Effects of the period of acceleration (t). The wing translational (ut+) and rotational (+) non-dimensional velocities (A), lift coefficient CL (B) and drag coefficient CD (C) plotted against non-dimensional time .

View larger version (16K): [in a new window]   Fig. 7. Effects of the wake from previous strokes. Lift CL and drag CD coefficients plotted against non-dimensional time during one stroke. The solid lines represent results replotted from Fig. 6; the dashed lines represent results from a wing started in still air. t, period of acceleration.

View larger version (113K): [in a new window]   Fig. 8. Velocity vectors in the middle section of the wing near the start of a stroke, corresponding to the vorticity field in Fig. 4A. The horizontal arrow at the top right represents the velocity of the wing in the phase of constant-speed translation and it serves as a scale for the velocity vectors in the figure.

View larger version (15K): [in a new window]   Fig. 9. Effects of pitching-axis location (as a percentage of mean chord length c from the leading edge of the wing). Lift CL and drag CD coefficients plotted against non-dimensional time during one cycle.

View larger version (21K): [in a new window]   Fig. 10. Effects of the pitching-up rotation rate r. The wing translational (ut+) and rotational (+) non-dimensional velocities (A), lift coefficient CL (B) and drag coefficient CD (C) plotted against non-dimensional time .

View larger version (21K): [in a new window]   Fig. 11. Effects of the timing of wing rotation. The wing translational (ut+) and rotational (+) non-dimensional velocities (A), lift coefficient CL (B) and drag coefficient CD (C) plotted against non-dimensional time . Rotation is defined as advanced, symmetrical or delayed with respect to stroke reversal.

View larger version (52K): [in a new window]   Fig. 12. (A–E) Vorticity plots at three spanwise locations at various times during one stroke (symmetrical rotation). r1, r2 and r3 denote locations at 0.75, 0.5 and 0.25 wing lengths from the wing root, respectively. , non-dimensional time; , angle of attack. The magnitude of the non-dimensional vorticity at an outer contour is 1. Starting from the outer contour, for the first 21 contours, the contour interval is 0.2, for the next 30 contours, the contour interval is 0.5, and for the reminder of the contours, the contour interval is 5.

View larger version (56K): [in a new window]   Fig. 13. (A–F) Vorticity plots at three spanwise locations at various times during one stroke (rotation delayed). r1, r2 and r3 denote locations at 0.75, 0.5 and 0.25 wing lengths from the wing root, respectively. , non-dimensional time; , angle of attack. The magnitude of the non-dimensional vorticity at an outer contour is 1. Starting from the outer contour, for the first 21 contours, the contour interval is 0.2, for the next 30 contours, the contour interval is 0.5, and for the reminder of contours, the contour interval is 5.

View larger version (15K): [in a new window]   Fig. 14. Comparison of the lift coefficient CL, calculated here for the rotation-advanced case, with that taken from the experimental data of fig. 3A of Dickinson et al. (1999).