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# Unsteady forces on an accelerating plate and application to hovering insect flight

Pullin, D. I. and Wang, Z. Jane (2004) Unsteady forces on an accelerating plate and application to hovering insect flight. Journal of Fluid Mechanics, 509 . pp. 1-21. ISSN 0021-1120. http://resolver.caltech.edu/CaltechAUTHORS:PULjfm04

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## Abstract

The aerodynamic forces on a flat plate accelerating from rest at fixed incidence in two-dimensional power-law flow are studied analytically and numerically. An inviscid approximation is made in which separation at the two plate edges is modelled by growing spiral vortex sheets, whose evolution is determined by the Birkhoff–Rott equation. A solution based on a similarity expansion is developed, valid when the scale of the separated vortex is much smaller than the plate dimension. The leading order is given by the well-known similarity growth of a vortex sheet from a semi-infinite flat plate, while equations at the second order describe the asymmetric sweeping effect of that component of the free-stream parallel to the plate. Owing to subtle cancellation, the unsteady vortex force exerted on the plate during the starting motion is independent of the sweeping effect and is determined by the similarity solution, to the order calculated. This gives a mechanism for dynamic stall based on a combination of unsteady vortex lift and pure added mass; the incidence angle for maximum vortex lift is $\arccos \sqrt{3/8}\,{\approx}\,52.2^\circ$ independent of the acceleration profile. Circulation on the flat plate makes no direct contribution. Both lift and drag force predictions from the unsteady inviscid theory are compared with those obtained from numerical solutions of the two-dimensional unsteady Navier–Stokes equations for an ellipse of high aspect ratio, and with predictions of Wagner's classical theory. There is good agreement with numerical results at high incidence and moderate Reynolds number. The force per unit span predicted by the vortex theory is evaluated for parameters typical of insect wings and is found to be in reasonable agreement with numerical simulations. Estimates for the shed circulation and the size of the start-up vortices are also obtained. The significance of this flow as a mechanism for insect hovering flight is discussed.

Item Type: Article "Reprinted with the permission of Cambridge University Press." Received August 7 2003, Revised November 10 2003, Published Online 7 June 2004 D. I. P. wishes to thank P. E. Dimotakis for helpful discussions. Z. J.W. wishes to acknowledge the support by NSF Early Career grant, ONR YIP, AFOSR and the Packard Foundation. CaltechAUTHORS:PULjfm04 http://resolver.caltech.edu/CaltechAUTHORS:PULjfm04 http://dx.doi.org/10.1017/S0022112004008821 No commercial reproduction, distribution, display or performance rights in this work are provided. 1313 CaltechAUTHORS Archive Administrator 10 Jan 2006 26 Dec 2012 08:43

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