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First published online January 5, 2005
Journal of Experimental Biology 208, 195-212 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01376

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A computational fluid dynamics of `clap and fling' in the smallest insects

Laura A. Miller^1,* and Charles S. Peskin²

¹ Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA
² Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

View larger version (28K): [in a new window]   Fig. 1. `Clap and fling' (redrawn from Weis-Fogh, 1973). The three-dimensional motion (top) and the corresponding two-dimensional approximation (bottom). In this drawing, the insect flies with its body oriented almost vertically, and the wings move in a horizontal plane. At the beginning of the upstroke (A), the wings move from the ventral to the dorsal side of the body, and rotate together about the leading edges. At the beginning of the downstroke (B), the wings rotate apart about the trailing edges. Towards the end of rotation, the wings translate away from each other.

View larger version (13K): [in a new window]   Fig. 2. Dimensionless translational and angular velocities of the wing as a function of dimensionless time for one `clap and fling' stroke. The total motion was used for all `clap and fling' simulations. For `fling' simulations, the angular and translational velocities follow the second half of the graph. Note that the wing begins to rotate before the end of translation during the upstroke (first half-stroke). Translation during the downstroke (second half-stroke) also begins before wing rotation has ended.

View larger version (18K): [in a new window]   Fig. 3. Lift (blue) and drag (red) coefficients per wing during a two-winged fling half-stroke are plotted for two mesh widths. The coarser grid (dotted line) has a mesh width of about 8.33 x10–4 m (the same mesh width as the other simulations in this paper), and the finer grid (solid line) has a mesh width of about 4.17 x10–4 m. The two grid sizes show good agreement with small deviations occurring during the rotational part of fling.

View larger version (39K): [in a new window]   Fig. 4. Lift (A) and drag (B) coefficients for four sinusoidal strokes. The green lines show the results of the immersed boundary simulation, the blue lines represent numerical data for a two-dimensional elliptic wing (Wang et al., 2004), and the red lines describe the experimental data for a three-dimensional model wing (Wang et al., 2004). In all cases, the leading edge vortex did not appear to separate during translation. The wing in each case flaps in a sinusoidal motion with =0, ß=/2, and Ao/c=2.8, as defined in Eqns 17 and 18, at Re=75. This represents a 2.8 chord translation and a minimum angle of attack of 45°.

View larger version (14K): [in a new window]   Fig. 5. Rotational motion (t) of two wings during fling and the corresponding lift forces. (A) The broken line represents the motion used in the clap and fling experiments of Spedding and Maxworthy (1986). The solid line represents the rotational motion of the wings used in the immersed boundary simulation. The motion used by Sun and Yu (2003) is nearly identical. (B) The dotted line represents the lift forces measured over time in Spedding and Maxworthy's experiment at a Reynolds number (Re) of about 3 x103. The solid line represents the lift forces calculated by Sun and Yu at the same Re. The broken line represents the scaled lift forces calculated over time in the immersed boundary simulation at Re=128.

View larger version (62K): [in a new window]   Fig. 6. Flow visualization and streamline plots of rotational fling at five points in time. The wing motion used in each case is shown in Fig. 5A. (A) Flow visualization of fling captured by Spedding and Maxworthy (1986) at a Reynolds number (Re) 3 x103. (B) Streamline plots of fling calculated numerically for two rigid elliptic wings by Sun and Yu (2003) at Re 3 x103. (C) Streamline plots of fling calculated from immersed boundary simulations at Re 128. In all cases, two large leading vortices form and appear to remain attached to each wing during rotation.

View larger version (64K): [in a new window]   Fig. 7. Streamlines of the fluid flow around two wings (A) and one wing (B) at a Reynolds number of 8 and around two wings (C) at a Reynolds number (Re) of 128 for a fling half-stroke. The arrow on the left wing shows the direction of the normalized force acting on the wing at each time (i–vi). The wings begin at an angle of attack of 90° and rotate about the trailing edge to an angle of attack of 45°. (A) During rotation, attached leading edge vortices are formed on each wing and no trailing edge vortices are formed (i–iii). When translation begins, small attached trailing edge vortices begin to form (iii–v). As the trailing edge vortices grow in size relative to the leading edge vortices, lift is reduced. The leading edge vortices, however, remain larger than the trailing edge vortices for most of the half-stroke (v–vi). (B) In the one wing case, attached leading edge and trailing edge vortices are formed during rotation (i–iii). When translation begins, equally sized leading and trailing edge vortices are attached to the wing, creating substantially lower lift forces in comparison to the two-winged case (iii–vi). (C) At a Reynolds number (Re) of 128, attached leading edge vortices are formed on each wing and no trailing edge vortices are formed initially (i–iii). When translation begins, however, the leading edge vortices are shed, and trailing edge vortices are formed (v–vi). The trailing edge vortex grows in size and is subsequently shed from the wing as a new leading edge vortex begins to form. (D) Flow visualization of fling at Re=30 by Maxworthy (1979). Similar to case A, a pair of large leading edge vortices is formed and remains attached to the wing during rotation. A smaller pair of trailing edge vortices is formed and grows during translation. (E) Flow visualization of fling at Re=1.3 x104. Similar to case C, a pair of large leading edge vortices (1) forms during rotation and is shed during translation. A new pair of leading edge vortices forms during translation (2).

View larger version (24K): [in a new window]   Fig. 8. Lift coefficients per wing at a Reynolds number of 8 are plotted as functions of time for the fling half-strokes shown in Fig. 7A,B. The bar at the top of the graph shows the number of chord lengths traveled. The first peak in the lift coefficients corresponds to the large lift forces generated during wing rotation. The second peak in the lift coefficients corresponds to the period of translational acceleration. The lift forces per wing are on average about 35% greater during translation after clap and fling than during the steady translation of a single wing with no clap and fling (this average was taken over the fraction of the stroke from 0.37 to 1, after rotation had finished).

View larger version (32K): [in a new window]   Fig. 9. Lift coefficients are plotted as functions of time for a two-winged clap and fling half-stroke. The bar at the top of the graph shows the number of chord lengths traveled. The letters i–vi along the x axis correspond to the times the streamlined plots labelled i–vi in Fig. 7A,C were drawn. The angles of attack during pure translation were set to 45°. Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. The first peak corresponds to the lift generated during wing rotation, and the second peak corresponds to the lift generated during translational acceleration. The lift enhancing mechanisms of fling decrease with increasing Re. For Re=32 and below, lift coefficients decrease during translation after fling as the trailing edge vortex grows in strength. For Re=64 and above, lift coefficients fall as the leading edge vortices separate from the wings.

View larger version (33K): [in a new window]   Fig. 10. Lift coefficients are plotted as functions of time for a one-winged fling half-stroke. The letters i–vi along the x axis correspond to the times the streamlined plots labelled i–vi in Fig. 7B were drawn. The angle of attack during pure translation was set to 45°. The Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. The first peak in lift corresponds to the lift forces generated during wing rotation. The second peak corresponds to the lift forces generated during translational acceleration. For Re=64 and above, lift coefficients fall as the leading edge vortices separate from the wings.

View larger version (27K): [in a new window]   Fig. 11. Drag coefficients are plotted as functions of time for a two-winged fling half-stroke. The letters i–vi along the x axis correspond to the times the streamlined plots labelled i–vi in Fig. 7A,C were drawn. The angles of attack during pure translation were set to 45°. Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. The first large peak corresponds to drag generated during wing rotation. The second smaller peak corresponds to drag forces generated during translational acceleration. Drag coefficients increase with decreasing Re. This inverse relationship is particularly significant during the initial wing rotation.

View larger version (30K): [in a new window]   Fig. 12. Drag coefficients plotted as functions of time for a one-winged fling half-stroke. The letters i–vi along the x axis correspond to the times the streamlined plots labelled i–vi in Fig. 7B were drawn. The angles of attack during pure translation were set to 45°. Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. The first peak corresponds to the drag forces generated during translational acceleration. Note that the drag forces per wing generated during rotation in the one-winged case are significantly smaller than those generated per wing in the two-winged case (see Fig. 11). In general, drag coefficients increase with decreasing Re.

View larger version (10K): [in a new window]   Fig. 13. The average lift coefficients per wing during translation following two-winged fling divided by the corresponding average lift coefficients for one-winged fling vs Reynolds number (Re). The average lift coefficients following fling were calculated as the average lift coefficients generated after rotation and translational acceleration and during translation at a constant angle of attack. This value decreases with increasing Re, suggesting that the lift enhancing effects of clap and fling are more significant at lower Re.

View larger version (10K): [in a new window]   Fig. 14. The maximum drag coefficient during the rotation of the wings at the beginning of fling plotted against the Reynolds number (Re). The drag coefficient significantly increases for decreasing Re. This result suggests that relatively larger forces are needed for tiny insects to rotate their wings and perform a `fling'.

View larger version (77K): [in a new window]   Fig. 15. Streamlines of fluid flow around two wings (A) and around one wing (B) during a full clap and fling stroke at a Reynolds number (Re) of 8 and around two wings at Re=128 (C). The arrow on the left wing shows the direction of the normalized force acting on the wing. (A) During translation, leading and trailing edge vortices form and remain attached to the wing (i–iii). During `clap', the wings rotate together at the end of translation (iv–v). At this time, the leading and trailing edge vortices are shed. During `fling', the wings rotate apart forming two new leading edge vortices (vi). Towards the end of rotation, the wings are translated apart at a constant angle of attack and speed (vi–viii). During translation, the leading edge vortices remain attached to the wing, and weak trailing edge vortices are formed. (B) Large leading and trailing edge vortices are formed during the initial translation of the wing (i–iii). This pair of vortices is shed during rotation (iv–v), and a new pair of leading and trailing edge vortices is formed during the subsequent translation (vi–viii). Note that in the two winged case, no trailing edge vortices are formed during wing rotation, and much smaller trailing edge vortices are formed during the subsequent translation. (C) At Re=128, leading edge vortices are formed and the trailing edge vortices are shed (i–iii). After a translation of about 2.5 chord lengths, the leading edge vortices begin to separate from the wings (iii). During `clap', the wings rotate together at the end of translation (iv–v). At this time, the leading and trailing edge vortices are shed. During `fling', the wings rotate apart. Two large leading edge vortices are formed, and no trailing edge vortices are formed initially (v–vi). Towards the end of rotation, the wings are translated away from each other and the pair of leading edge vortices formed during rotation is shed. A second pair of leading edge vortices begins to form near the end of translation (vi–viii).

View larger version (35K): [in a new window]   Fig. 16. Lift coefficients as functions of time for a two-winged clap and fling stroke. The letters i–viii along the x axis correspond to the times the streamlined plots labelled i–viii in Fig. 15A,C were drawn. The angle of attack during pure translation was set to 45°. The Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. In general, lift coefficients were larger at higher Re during the initial upstroke. Lift coefficients, however, were smaller at higher Re during fling and subsequent translation. For Re=64 and higher, lift coefficients peak during translational acceleration and rotation. Lift coefficients drop when the leading edge vortices separate from the wings (vii–viii). For Re=32 and below, lift coefficients also peak during translational acceleration and rotation. Lift coefficients are relatively constant during translation in the first half-stroke (i–iii). Lift coefficients are transiently augmented during translation after fling (vi–viii).

View larger version (33K): [in a new window]   Fig. 17. Lift coefficients as functions of time for one wing moving with the same clap and fling motion as shown in Fig. 16. The letters i–viii along the x axis correspond to the times the streamlined plots labelled i–viii in Fig. 15B were drawn. The angle of attack during pure translation was set to 45°. The Reynolds number (Re) was varied by changing the translational velocity of the wing from 0.00375 to 0.06 m s–1. In general, lift coefficients increase with Re. The lift forces generated during the initial upstroke are very similar to those shown in the two-winged case (i–iii). During fling, lift coefficients during rotation and the subsequent translation (vi–viii) are significantly less than the two-winged case for Re=32 and below.

View larger version (30K): [in a new window]   Fig. 18. Drag coefficients as functions of time for a two-winged clap and fling stroke. The letters i–viii along the x axis correspond to the times the streamlined plots labelled i–viii in Fig. 15A,C were drawn. The angle of attack during pure translation was set to 45°. The 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 are larger for lower Re. This inverse relationship is most significant during periods of wing rotation (iv–vi). In general, drag forces peak during periods of acceleration and rotation and remain relatively constant during periods of pure translation.

View larger version (37K): [in a new window]   Fig. 19. Drag coefficients for one-winged fling as functions of time for a range of Reynolds numbers (Re). The letters i–viii along the x axis correspond to the times the streamlined plots labelled i–viii in Fig. 15B were drawn. The angle of attack during pure translation was set to 45°. In general, drag coefficients are smaller for higher Re. In comparison to the two-winged case, drag coefficients are significantly lower during `clap' (the end of the upstroke) for all Re. During wing rotation at the beginning of the downstroke, drag coefficients are substantially lower than the two-winged case for all Re. The difference between the one and two-winged case is greatest at lower Re.

View larger version (17K): [in a new window]   Fig. 20. The average lift and drag coefficients generated during fling for an entire fling half-stroke following a clap half-stoke and for an entire fling half-stroke started from rest as a function of the Reynolds number (Re). Force coefficients were averaged during wing rotation and a subsequent translation of about 3.5 chord lengths. Average lift coefficients increase slightly with decreasing Re. Average drag coefficients increase significantly as the Re is lowered. Force coefficients are higher for fling half-strokes that follow clap half-strokes as the wings move back through their wakes. This `wake capture' effect decreases with decreasing Re.

View larger version (26K): [in a new window]   Fig. 21. Regions of positive and negative vorticity during the translation of two wings following `clap and fling'. Rn1 and Rn2 denote regions of negative vorticity, Rp1 and Rp2 denote regions of positive vorticity. The two wings are initially clapped together and rotate apart along their trailing edges. This rotation creates two large leading edge vortices. Towards the end of rotation, the wings begin to translate apart. During translation, two weak trailing edge vortices begin to form. In this diagram, the wings are moving away from each other at a constant speed and angle of attack. The leading edge vortices (denoted by Rn1 and Rp1) are stronger than the trailing edge vortices (denoted by Rn2 and Rp2). This vortical asymmetry results in larger lift forces than in the symmetrical case without fling.