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First published online February 20, 2004
Journal of Experimental Biology 207, 1137-1150 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00868

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Unsteady aerodynamic forces of a flapping wing

Jiang Hao Wu and Mao Sun^*

Institute of Fluid Mechanics, Beijing University of Aeronautics & Astronautics, Beijing 100083, People's Republic of China

View larger version (9K): [in a new window]   Fig. 1. Sketches of the reference frames and wing motion. OXYZ is an inertial frame, with the XY plane in the horizontal plane. oxyz is a frame fixed on the wing, with the x-axis along the wing chord and the y-axis along the wing span. , positional angle of the wing; , geometrical angle of attack of the wing; R, wing length.

View larger version (5K): [in a new window]   Fig. 2. The wing planform used.

View larger version (17K): [in a new window]   Fig. 3. Comparison of the calculated and measured lift (CL) and drag (CD) coefficients. The experimental data are taken from fig. 7 of Usherwood and Ellington (2002b). , angle of attack.

View larger version (19K): [in a new window]   Fig. 4. Effects of grid density on computed lift coefficient (CL) and vorticity field. (A) The time course of CL in one cycle. (B) Vorticity contour plots at half-wing length near the end of a stroke.

View larger version (14K): [in a new window]   Fig. 5. Time history of aerodynamic force coefficients of the typical case. (A) Non-dimensional angular velocity of pitching rotation () and azimuthal rotation (); (B) time courses of lift coefficient (CL) and (C) drag coefficient (CD) in one cycle. Reynolds number (Re)=200, stroke amplitude ()=150°, midstroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (26K): [in a new window]   Fig. 6. Vorticity plots at half-wing length at various times during one cycle. Solid and broken lines indicate positive and negative vorticity, respectively. The magnitude of the non-dimensional vorticity at the outer contour is 2 and the contour interval is 3. A–D, downstroke; E–H, upstroke. Reynolds number (Re)=200, stroke amplitude ()=150°, midstroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (24K): [in a new window]   Fig. 7. Time courses of the (A) lift coefficient (CL) and (B) drag coefficient (CD) in one cycle for various Reynolds number (Re). Stroke amplitude ()=150°, midstroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (12K): [in a new window]   Fig. 8. Vorticity plots at half-wing length near the end of a half-stroke at various Reynolds number (Re). Solid and broken lines indicate positive and negative vorticity, respectively. The magnitude of the non-dimensional vorticity at the outer contour is 2 and the contour interval is 3. Stroke amplitude ()=150°, midstroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (19K): [in a new window]   Fig. 9. Vorticity plots at half-wing length at various times during one cycle. Solid and broken lines indicate positive and negative vorticity, respectively. The magnitude of the non-dimensional vorticity at the outer contour is 2 and the contour interval is 3. A-D, downstroke; E-H, upstroke. Reynolds number Re=20, stroke amplitude =150°, midstroke angle of attack m=40° and non-dimensional duration of wing rotation r=1.87 (mean angular velocity of rotation =0.93); symmetrical rotation.

View larger version (9K): [in a new window]   Fig. 10. Mean lift (L) and drag (D) coefficients vs Reynolds number (Re). Stroke amplitude ()=150°, mid-stroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (21K): [in a new window]   Fig. 11. Mean lift (L) and drag (D) coefficients vs mid-stroke angle of attack (m) for various Reynolds number (Re). Stroke amplitude ()=150° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (18K): [in a new window]   Fig. 12. The effects of rotation duration (r) on force coefficients. (A) Non-dimensional angular velocity of pitching rotation () and azimuthal rotation (); (B) time courses of lift coefficient (CL) and (C) drag coefficient (CD) in one cycle. Reynolds number (Re)=200, stroke amplitude ()=150° and mid-stroke angle of attack (m)=40°; symmetrical rotation.

View larger version (21K): [in a new window]   Fig. 13. The effects of rotation timing on force coefficients. (A) Non-dimensional angular velocity of pitching rotation () and azimuthal rotation (); (B) time courses of lift coefficient (CL) and (C) drag coefficient (CD) in one cycle. Reynolds number (Re)=200, stroke amplitude ()=150°, mid-stroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93].

View larger version (21K): [in a new window]   Fig. 14. The effects of stroke amplitude () on force coefficients. (A) Non-dimensional angular velocity of pitching rotation () and azimuthal rotation (); (B) time courses of lift coefficient (CL) and (C) drag coefficient (CD) in one cycle. Reynolds number (Re)=200, mid-stroke angle of attack (m)=40° and non-dimensional duration of wing rotation (r)=1.87 [mean angular velocity of rotation ()=0.93]; symmetrical rotation.

View larger version (22K): [in a new window]   Fig. 15. The effects of stroke amplitude () on force coefficients when r/c is fixed (=20%). (A) Non-dimensional angular velocity of pitching rotation () and azimuthal rotation (); (B) time courses of lift coefficient (CL) and (C) drag coefficient (CD) in one cycle. Mid-stroke angle of attack (m)=40°; symmetrical rotation. r, non-dimensional duration of wing rotation; c, non-dimensional wingbeat period.

View larger version (15K): [in a new window]   Fig. 16. Mean lift (L) and drag (D) coefficients vs mid-stroke angle of attack (m; symmetrical rotation). Re, Reynolds number; , stroke amplitude; , mean non-dimensional angular velocity of wing rotation.