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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
Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments
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
* Author for correspondence (e-mail: jane.wang{at}cornell.edu)
Accepted 3 October 2003
We compare computational, experimental and quasi-steady forces in a generic hovering wing undergoing sinusoidal motion along a horizontal stroke plane. In particular, we investigate unsteady effects and compare two-dimensional (2D) computations and three-dimensional (3D) experiments in several qualitatively different kinematic patterns. In all cases, the computed drag compares well with the experiments. The computed lift agrees in the cases in which the sinusoidal changes in angle of attack are symmetrical or advanced with respect to stroke positions, but lags behind the measured 3D lift in the delayed case.
In the range of amplitudes studied here, 3–5 chords, the force coefficients have a weak dependence on stroke amplitude. As expected, the forces are sensitive to the phase between stroke angle and angle of attack, a result that can be explained by the orientation of the wing at reversal. This dependence on amplitude and phase suggests a simple maneuver strategy that could be used by a flapping wing device.
In all cases the unsteady forces quickly reach an almost periodic state with continuous flapping. The fluid forces are dominated by the pressure contribution. The force component directly proportional to the linear acceleration is smaller by a factor proportional to the ratio of wing thickness and stroke amplitude; its net contribution is zero in hovering. The ratio of wing inertia and fluid force is proportional to the product of the ratio of wing and fluid density and the ratio of wing thickness and stroke amplitude; it is negligible in the robotic wing experiment, but need not be in insect flight.
To identify unsteady effects associated with wing acceleration, and coupling between rotation and translation, as well as wake capture, we examine the difference between the unsteady forces and the estimates based on translational velocities, and compare them against the estimate of the coupling between rotation and translation, which have simple analytic forms for sinusoidal motions. The agreement and disagreement between the computed forces and experiments offer further insight into when the 3D effects are important.
A main difference between a 3D revolving wing and a 2D translating wing is the absence of vortex shedding by a revolving wing over a distance much longer than the typical stroke length of insects. No doubt such a difference in shedding dynamics is responsible in part for the differences in steady state force coefficients measured in 2D and 3D. On the other hand, it is unclear whether such differences would have a significant effect on transient force coefficients before the onset of shedding. While the 2D steady state force coefficients underpredict 3D forces, the transient 2D forces measured prior to shedding are much closer to the 3D forces. In the cases studied here, the chord is moving between 3 to 5 chords, typical of hovering insect stroke length, and the flow does not appear to separate during each stroke in the cases of advanced and symmetrical rotation. In these cases, the wing reverses before the leading edge vortex would have time to separate even in 2D. This suggests that the time scale for flow separation in these strokes is dictated by the flapping frequency, which is dimensionally independent. In such cases, the 2D unsteady forces turn out to be good approximations of 3D experiments.
Key words: insect flight, computational fluid dynamics, biofluid dynamics, vorticity field, two-dimensional force, three-dimensional force
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