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First published online July 26, 2004
Journal of Experimental Biology 207, 3073-3088 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01138

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When vortices stick: an aerodynamic transition in tiny insect flight

Laura A. Miller^* and Charles S. Peskin

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

View larger version (19K): [in a new window]   Fig. 1. A two-dimensional approximation of a three-dimensional stroke. The motion of the wing is divided here into three stages: downstroke (A), rotation (B) and upstroke (C). In reality, the rotational phase overlaps with the downstroke and the upstroke. The wing moves approximately along a horizontal plane. The center of rotation is 0.2 chord lengths from the leading edge of the wing.

View larger version (14K): [in a new window]   Fig. 2. Translational velocity and the angular velocity of the wing as a function of dimensionless time for one stroke cycle. This motion was used for all simulations unless otherwise stated. Note that the wing begins to rotate during the first half stroke (or downstroke). Since most of the rotation occurs at the end of the downstroke, this motion describes the case of `advanced rotation'.

View larger version (34K): [in a new window]   Fig. 3. This diagram illustrates the numerical method used for these simulations: the immersed boundary method. The fluid domain is represented as a Cartesian grid. The boundary (wing) points are represented as red squares. These points interact with the fluid and move at the local fluid velocity. The green springs represent the bending and stretching stiffness of the boundary. The desired motion of the wing is prescribed by the target points, which are shown as blue circles. These points do not interact with the fluid and they move according to the desired motion of the wing. They also apply a force to the actual boundary via the target springs (shown in purple). The further the actual boundary is from its target boundary, the larger the force applied to the actual boundary.

View larger version (37K): [in a new window]   Fig. 4. Drag coefficients are plotted as functions of time for one stroke cycle. The arrows along the axis show the times at which streamline plots in Figs 9, 10 were drawn. The angles of attack were chosen to produce a symmetric stroke. In all cases, the angle of attack was 45°. Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s-1. In general, drag coefficients increase with decreasing Re. Maximum drag forces occur during acceleration from rest at the beginning of the downstroke and rotation at the end of the downstroke and at the beginning of the upstroke.

View larger version (11K): [in a new window]   Fig. 5. Drag coefficient averaged during periods of steady translation at a constant angle of attack of 45° for the downstroke and upstroke plotted against log10 Re. Mean drag coefficients are higher during the upstroke and increase with decreasing Re. Filled symbols represent numerical data, and open symbols represent experimentally determined values reported in the literature. Open circles denote drag coefficients measured by Thom and Swart (1940) for a wing held at an angle of attack of 45° in a steady flow. Open diamonds represent drag coefficients measured by Dickinson and Götz (1993) averaged over a distance of seven chord lengths for a wing translated from rest. Open squares represent drag coefficients measured by Dickinson (1994) during translation at an angle of attack of 45° following one half stroke at an angle of attack of 76° and wing rotation with a center of rotation at 0.2 chord lengths from the leading edge.

View larger version (29K): [in a new window]   Fig. 6. Lift coefficients are plotted as functions of time for one stroke cycle. The arrows along the axis show the times at which streamline plots in Figs 9, 10 were drawn. The angles of attack were chosen to produce a symmetric stroke. In all cases, the angle of attack was 45°. Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s-1. Lift coefficients fall into two patterns. For Re64, lift peaks during the initial acceleration of the wing and oscillate during pure translation for the downstroke. Large lift coefficients are generated during wing rotation. During the upstroke, lift coefficients show strong oscillations during pure translation. For Re32, lift coefficients peak during acceleration and drop to a constant value during pure translation. Lift coefficients peak again during the acceleration and rotation of the wing. The lift coefficients then drop again to relatively constant values during pure translation.

View larger version (10K): [in a new window]   Fig. 7. Peak and mean lift coefficients during periods of steady translation at a constant angle of attack of 45° for the downstroke only plotted against log10 Re. In general, lift coefficients increase with increasing Re. Filled symbols represent numerical data, and open symbols represent experimentally determined values reported in the literature. Open circles denote lift coefficients measured by Thom and Swart (1940) for a wing held at an angle of attack of 45° in a steady flow. Open diamonds represent lift coefficients measured by Dickinson and Götz (1993) averaged over a distance of seven chord lengths for a wing translated from rest.

View larger version (8K): [in a new window]   Fig. 8. The diagram above shows lift/drag averaged during steady downstroke translation at a constant angle of attack of 45° plotted against log10 Re. Lift/drag increases with increasing Re.

View larger version (35K): [in a new window]   Fig. 9. The plots show the streamlines of fluid flow around a flapping wing for one stroke cycle starting from rest, at an Re of 128. The arrow on the wing shows the direction of the normalized aerodynamic forces acting on the wing. The angle of attack during pure translation was 45° for both the downstroke and the upstroke. The maximum translational velocity was 0.06 m s-1. The colors of the streamline reflect the value of the stream function, , along the streamlines. Red denotes the most positive values and blue denotes the most negative values. During the downstroke, an attached leading edge vortex (LEV) is initially formed while the trailing edge vortex is shed (A,B). This corresponds to a growth in lift forces. In C, the LEV is being shed while a new trailing edge vortex is formed. This corresponds to a drop in lift. During rotation (D,E), the leading and trailing edge vortices are shed. At the beginning of the upstroke (F), a new LEV is formed and a new trailing edge vortex is formed and shed. This corresponds to an increase in lift. In G, the LEV is shed and a new trailing edge vortex is formed. This results in a drop in lift. Finally, a second leading edge vortex is formed and the trailing edge vortex is shed, resulting in another lift peak (H,I).

View larger version (40K): [in a new window]   Fig. 10. Streamline plots of fluid velocity around a flapping wing for one stroke cycle starting from rest, at an Re of 8. The arrow on the wing shows the direction of the normalized aerodynamic forces acting on the wing. The angle of attack during pure translation was 45° for both the downstroke and the upstroke. The maximum translational velocity was 0.00375 m s-1. The colors of the streamline reflect the value of the stream function, , along the streamlines. Red denotes the most positive values and blue denotes the most negative values. Note that both the leading and the trailing edge vortices remain attached to the wing except during stroke reversal (A-D and F-I). This differs from the higher Re case, where lift forces oscillate due to the alternate shedding of leading and trailing edge vortices. Similar vortex dynamics were observed for Re up to 32.

View larger version (32K): [in a new window]   Fig. 11. Drag coefficients as functions of time are plotted for five angles of attack for an Re of 128. For each simulation, the same angle of attack was used on the downstroke and upstroke. The angles of attack for the five simulations were 10°, 20°; 30°; 40° and 50°. The maximum translational velocity of the wing was 0.06 m s-1. Drag coefficients increase with increasing angle of attack. Maximum drag forces occur during acceleration from rest at the beginning of the stroke and during rotation at the end of the downstroke and at the beginning of the upstroke. Drag forces are larger during rotation at lower angles of attack because the distance over which the wing moves and its angular velocity are larger.

View larger version (34K): [in a new window]   Fig. 12. Drag coefficients as functions of time are plotted for five angles of attack for an Re of 8. For each simulation, the same angle of attack was used on the downstroke and upstroke. The angles of attack for the five simulations were 10°, 20°, 30°, 40° and 50°. The maximum translational velocity of the wing was 0.00375 m s-1. Drag coefficients increase with increasing angle of attack. Maximum drag forces occur during acceleration from rest at the beginning of the stroke and during rotation at the end of the downstroke and at the beginning of the upstroke. Drag forces are larger during rotation at lower angles of attack because the distance over which the wing moves and its angular velocity are larger.

View larger version (32K): [in a new window]   Fig. 13. Lift coefficients as functions of time are plotted for five angles of attack for an Re of 128. For each simulation, the same angle of attack was used on the downstroke and upstroke. The angles of attack for the five simulations were 10°, 20°, 30°, 40° and 50°. The maximum translational velocity of the wing was 0.06 m s-1. Lift coefficients are greatest for an angle of attack near 40°. Maximum lift forces occur during acceleration from rest at the beginning of each half stroke and during rotation at the end of the downstroke and at the beginning of the upstroke.

View larger version (29K): [in a new window]   Fig. 14. Lift coefficients as functions of time are plotted for five angles of attack for an Re of 8. For each simulation, the same angle of attack was used on the downstroke and upstroke. The angles of attack for the five simulations were 10°, 20°, 30°, 40° and 50°. The maximum translational velocity of the wing was 0.00375 m s-1. During translation, lift coefficients increase with increasing angle of attack in the range of 10-40°, and lift coefficients for angles of attack of 40° and 50° are quite similar. Maximum lift forces occur during acceleration from rest at the beginning of each half stroke and during rotation at the end of the downstroke and beginning of the upstroke. Lift forces are larger during rotation at lower angles of attack because the distance over which the wing moves and its angular velocity are larger.

View larger version (27K): [in a new window]   Fig. 15. Drag coefficients for two mesh widths. The 600x600 grid size was used for other simulations in this paper. Both simulations used the stroke kinematics described in Fig. 2 at an Re of 128 and with an angle of attack of 45°.

View larger version (27K): [in a new window]   Fig. 16. Lift coefficients for two mesh widths. The 600x600 mesh was used in all other simulations in this paper. Both simulations used the stroke kinematics described in Fig. 2 at an Re of 128 and with an angle of attack of 45°.

View larger version (27K): [in a new window]   Fig. 17. Drag coefficients as a function of distance traveled for a wing started from rest and translated seven chord lengths at a 45° angle of attack. Dotted lines represent data collected during an experiment by Dickinson and Götz (1993), and solid lines represent the results of our two-dimensional simulation. Oscillations in drag are smaller than those measured in lift and correspond to the alternate shedding of the leading and trailing edge vortices.

View larger version (26K): [in a new window]   Fig. 18. Lift coefficients as a function of distance traveled for a wing started from rest and translated seven chord lengths at a 45° angle of attack. Dotted lines represent data collected during an experiment of Dickinson and Götz (1993), and solid lines represent the results of our two-dimensional simulation. Lift coefficients in both cases oscillate with the shedding of the leading and trailing edge vortices. Note that lift forces in our simulation have stronger oscillations than the forces measured by Dickinson and Götz.

View larger version (30K): [in a new window]   Fig. 19. Regions of positive and negative vorticity for `high' and `low' Re. Rn denotes regions of negative vorticity, Rp denotes regions of positive vorticity, and Ro denotes regions of negligible vorticity. (A) For a wing in a fluid moving from left to right at Re>64, an attached leading edge vortex with negative vorticity is formed. A trailing edge vortex of positive vorticity is formed and shed from the wing. This asymmetry in the time rate of change of the first moment of positive and negative vorticity produces lift. (B) For a wing in a fluid moving from left to right at an Re between 8 and 32, an attached leading edge vortex with negative vorticity and an attached trailing edge vortex of positive vorticity are formed. This `near-symmetry' in the time rate of change of the first moment of positive and negative vorticity reduces lift.