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First published online November 5, 2004
Journal of Experimental Biology 207, 4269-4281 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01266

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The effect of advance ratio on the aerodynamics of revolving wings

William B. Dickson^* and Michael H. Dickinson

California Institute of Technology, Mail Code 138-78, Pasadena, CA 91125, USA

View larger version (27K): [in a new window]   Fig. 1. (A) Diagram of robotic apparatus. The wing assembly is shown mounted on linear translation rails above the 1 mx2.4 mx1.2 m towing tank. (B) Coordinate system for the mechanical wing. Three angles are used to specify the position of the wing: , and . The instantaneous stroke position, (t), is defined as the angular position of the projection of the wing axis in the stroke plane. The instantaneous stroke deviation, (t), is defined as the angle between the wing axis and the stroke plane. The instantaneous angle of attack, (t), is defined as the angle between the wing's chord and the tangent to its trajectory.

View larger version (17K): [in a new window]   Fig. 2. Diagram of sectional flow velocities. The wing is travelling through the fluid at forward velocity Vf. (A) Downstroke. The wing is sweeping into the incident flow. The magnitude sectional flow velocity at wing position r is given by r +Vfcos(). (B) Upstroke. The wing is sweeping with the incident flow. The flow velocity at wing position r is given by –r –Vfcos().

View larger version (53K): [in a new window]   Fig. 3. Instantaneous lift traces. The regions where the stroke position of wing is between –90° and 90° are highlighted in gray and roughly approximate the phase of an upstroke, =–72 deg. s–1, or downstroke, =72 deg. s–1, between wing rotations. Angle of attack is held constant in each trail and varied from –10° to 100° in steps of 10° for each advance ratio. (Ai–Di) =72 deg. s–1, advance ratio equal to 1/8, 1/4, 3/8 and 1/2, respectively. (Aii–Dii) =–72 deg. s–1, advance ratio equal to 1/8, 1/4, 3/8 and 1/2, respectively.

View larger version (56K): [in a new window]   Fig. 4. Instantaneous drag traces. The regions where the stroke position of wing is between –90° and 90° are highlighted in gray and roughly approximate the phase of an upstroke or downstroke between wing rotations. Angle of attack varied from –10° to 100° in steps of 10° for each advance ratio. (Ai–Di) =72 deg. s–1, advance ratio equal to 1/8, 1/4, 3/8 and 1/2, respectively. (Aii–Dii) =–72 deg. s–1, advance ratio equal to 1/8, 1/4, 3/8 and 1/2, respectively.

View larger version (14K): [in a new window]   Fig. 5. Added mass component of the measured force as a function of acceleration for all 96 trials (black circles). The theoretical estimate from Sedov model is shown in red. Values are means ± 1 S.D.

View larger version (12K): [in a new window]   Fig. 6. Mean angle between the wing and the net force vector as a function of angle of attack. Values are means ± 1 S.D.

View larger version (16K): [in a new window]   Fig. 7. Normalized lift and drag coefficients for all 96 trials. (Ai,Bi) Normalized lift and drag coefficients as a function of angle of attack, , respectively (black circles). The functions sin()cos() and sin()sin() are shown for comparison (red). (Aii,Bii) Normalized lift and drag coefficients as a function of 2sin()cos() and sin()sin(), respectively. The identity function is shown for comparison in red. Normalized coefficients were computed from the data highlighted in gray in Figs 3 and 4 using equations 13 and 14. Values are means ±1 S.D.

View larger version (18K): [in a new window]   Fig. 8. Lift coefficient CL versus drag coefficient CD as a function of angle of attack for various tip velocity ratios (µ): 0.5 (red), 0.25 (blue), 0 (green), –0.25 (orange), –0.5 (black). A fit of equations 15 and 16 to hovering data is shown for comparison (gray).

View larger version (14K): [in a new window]   Fig. 9. Lift and drag coefficient amplitude and offset functions determined via least-squares fit of equations 15 and 16. Lift amplitude function K0(µ) (blue circles), Drag amplitude function K1(µ) (red circles), and drag offset function K2(µ) (green circles). Fit of equation 27 to the amplitude and offset functions. Lift amplitude fit (blue line) k0,2=1.38, k0,1=1.65 and k0,0=2.01. Drag amplitude fit (red line) k1,2=1.38, k1,1=1.44 and k1,0=1.38. Drag offset fit (green line) k2,2=0.15, k2,1=0.24 and k2,0=0.32.

View larger version (9K): [in a new window]   Fig. 10. Lift coefficient CL and drag coefficient CD as a function of angle of attack for steadily translating non-revolving wing (black circles). Predicted lift and drag coefficients from equations 25 and 26 in the limit as µ approaches infinity (red line).