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First published online December 22, 2003
Journal of Experimental Biology 207, 449-460 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00739

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Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments

Z. Jane Wang^1,*, James M. Birch² and Michael H. Dickinson³

¹ Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
² Integrative Biology, University of California, Berkeley, CA 94720, USA
³ Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA

View larger version (97K): [in a new window]   Fig. 6. Vorticity plot in the case of A0/c=4.8, =0. Ten frames are shown in the fourth stroke. Red, counterclockwise rotating vortices; blue, clockwise rotating vortices. The wing is in black. (A,C) Computed vorticity; (B,D) digital particle image vorticity data in a 2D slice at 0.65R. See Materials and methods for details. Each pair in A,B and C,D corresponds to the same time during a stroke. The time sequence is indicated by the numbers on each plate. The color scale for vorticity of computation and experiments do not correspond to the exact same contour values, so the figure should be viewed as a qualitative comparison.

View larger version (22K): [in a new window]   Fig. 2. Computational and experimental lift and drag coefficients during advanced rotation (=/4; A0/c=2.8). (A) Lift (CL) and drag (CD) during the first four complete strokes. Red, experimental measurements; blue, computations. The time is normalized with the flapping period. The force is normalized by the maxima of the corresponding quasi-steady forces. (B) Experimental and (C) computational force vectors superimposed on wing positions, plotted at equal time intervals. The green line represents the wing chord; filled circles, the leading edge; arrows indicate force vectors on the wing. R, right; L, left.

View larger version (16K): [in a new window]   Fig. 5. Force traces for different amplitudes. (A) Experimental force coefficients for advanced rotation (=/4) and three stroke amplitudes, A0/c=2.8 (red), 3.6 (blue) and 4.2 (green). (B) Computational force coefficients for the same parameters. The time is normalized with the flapping period. The force is normalized by the maxima of the corresponding quasi-steady forces.

View larger version (13K): [in a new window]   Fig. 1. Quasi-steady lift CL (circles) and drag CD (crosses) coefficients measured from computation compared to the empirical formulae described by Equations 14,15 (solid and broken lines, respectively).

View larger version (25K): [in a new window]   Fig. 3. Computational and experimental lift and drag coefficients during symmetrical rotation (=0; A0/c=2.8). Details as in Fig. 2.

View larger version (23K): [in a new window]   Fig. 4. Computational and experimental lift and drag coefficients during delayed rotation (=–/4; A0/c=2.8). Details as in Fig. 2.

View larger version (46K): [in a new window]   Fig. 7. Force comparison between experiment, computation and quasi-steady predictions for (A,D) advanced (=/4), (B,E) symmetric (=0) and (C,F) delayed (=–/4) rotations. (A–C) Force traces of four strokes starting from rest. Red, experimental force; blue, computational force; green, quasi-steady estimates using Equations 14, 15, 16, 17. Forces are normalized as in Figs 2, 3, 4. (D–F) Difference trace (red) between the experimental forces and quasi-steady forces. Linear acceleration du(t)/dt (blue), and an estimate of rotational force u(t)(t) (green) are shown in arbitrary scale since we are only interested in their basic time course. Broken vertical lines mark the wing reversal during t=[1,2]. Features associated with rotation (r) and unsteady circulation due to wing acceleration (u), are labelled accordingly.

View larger version (10K): [in a new window]   Fig. 8. Contributions to drag coefficients. Solid line, total drag; broken line, contribution due to vorticity flux alone. The parameters in the wing kinematics are: A0/c=2.8, =/4.