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First published online December 28, 2007
Journal of Experimental Biology 211, 234-238 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.013797
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Aerodynamic efficiency of flapping flight: analysis of a two-stroke model

Z. Jane Wang

Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA


Figure 1
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  		Fig. 1. Wing motions. (A) Steady forward flight, (B) hovering using a pair of motion in part a, (C) gliding (rotation of A), (D) gliding followed by a vertical upstroke, (E) gliding followed by a lift-generating upstroke, and (F) hovering using a mirror pair of E. Parameters: {alpha}d,u are the angle of attack in down- and upstrokes, respectively, βd,u the angle of the stroke plane, Ud,u the velocity, and {epsilon} the fraction of time spent on the upstroke, i.e. Uu/Ud=[(1–{epsilon})/{epsilon}]/(sinβd/sinβu).

 

Figure 2
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  		Fig. 2. Lift and drag coefficients. (A) An airfoil (NACA2414) at Re~3x105. Circles are experimental data (Selig, 2002Go) and solid lines are given by CL=2.77sin2({alpha}+0.03), CD=0.0086+0.24[1–cos2({alpha}–0.02)]. (B) A low Reynolds number plate at Re~103, CL=1.5sin2{alpha}, CD=1.1–cos2{alpha}.

 

Figure 3
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  		Fig. 3. Two-stroke flapping motions near the optimum. (A) 1/P* vs {alpha}d for steady motion (black), two-stroke motion composed of a gliding downstroke followed by a vertical upstroke (green), and two-stroke motion composed of the same gliding stroke followed by a near optimal lift-generating upstroke (red). (B) The near optimal down- and upstrokes. Ai, Bi, airfoil NACA2414 at Re~3x105; Aii, Bii, plate at Re~103.

 

Figure 4
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  		Fig. 4. The isosurface of 1/P* as a function of {alpha}d, βu, {epsilon}. It has a cylindrical-like shape whose longitudinal direction corresponds to the multiple solutions shown in Fig. 3. Starting from the innermost surface, the iso-surface values are 1/P*=58.5, 58, 57.5, 57 (A) and 4.7, 4.6, 4.5, 4.4 (B).

 

Figure 5
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  		Fig. 5. Strokes A and B generate almost the same amount of force. The only difference occurs near the end of the stroke when the wing reverses its pitch. In A the leading edge remains the same, and in B it switches. The wing pitching in stroke A can be facilitated by wing inertia and aerodynamic torque as the wing decelerates (Berman and Wang, 2007Go; Bergou et al., 2007Go).

 

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© The Company of Biologists Ltd 2008