a1 Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
The stroke-averaged lift forces due to various vortex rings and their mutual interactions are studied using a flapping flight vortex model (Rayner, J. Fluid Mech., vol. 91, 1979, p. 697; Ellington, Phil. Trans. R. Soc. Lond. B, vol. 305, 1984b, p. 115). The vortex system is decomposed into the wing plane (wing-linked) vortex ring, a loop closed by the bound vortex and (arc-shaped) trailing vortex and the wake (the vortex rings shed previously). Using the vorticity moment theory (Wu, AIAA J., vol. 19, 1981, p. 432) we are able to identify the roles of vortex rings in lift production or reduction and express the lift as function of areal contraction or expansion of vortex rings. The wake vortex rings induce areal contraction of the trailing vortex, which should decrease the lift, but this decrease is exactly compensated by the inducing effect of the trailing arc on the wake. The wake reduces the lift through inducing a downwash velocity on the wing plane. The lift force is shown to drop to a minimum at the second half stroke, and then increases to an asymptotic value slightly below the lift at the first half stroke, in such a way following the experimental observation of Birch & Dickinson (Nature, vol. 412, 2001, p. 729). The existence of the negative peak of lift is due to the first shed vortex ring which, just at the second half stroke, lies in the close vicinity to the wing plane, leading to a peak of the wing plane downwash velocity.
(Received April 20 2009)
(Revised January 31 2010)
(Accepted February 01 2010)
(Online publication April 23 2010)