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First published online December 16, 2008
Journal of Experimental Biology 212, 95-105 (2009)
Published by The Company of Biologists 2009
doi: 10.1242/jeb.016428

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Influence of flexibility on the aerodynamic performance of a hovering wing

Marcos Vanella, Timothy Fitzgerald, Sergio Preidikman, Elias Balaras and Balakumar Balachandran^*

Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA

View larger version (11K): [in this window] [in a new window]   Fig. 1. (A) The two-link model. The rigid links A and B (thick black line) are connected at hinge b by a torsion spring with stiffness k. The variables x(t), y(t), (t) and (t) are the generalized coordinates used to describe the wing's motion. In the hovering simulations, x(t), y(t) and (t) are prescribed and (t) is the only degree of freedom needed to define the system. (B) Decomposition of the wing's aerodynamic surfaces into rigid and deformable sections for the immersed-boundary scheme. The two rigid surfaces RSa and RSb are connected at points c1, c2, c3 and c4 by Hermite interpolating polynomials HS1 and HS2. mi is the total mass of the ith link (where i is A or B). i is the distance from the junction to the center of mass of bar i. N is the origin of the inertial reference frame, and the unit vectors i are fixed in this reference frame.

View larger version (16K): [in this window] [in a new window]   Fig. 2. Time histories of lift and drag force coefficients (CL, CD) for a symmetric harmonic hovering rigid link at Re=75 and two different grid resolutions. Blue line, rigid link, embedded boundary grid 1229x551; green line, embedded boundary grid 666x402; and red line, data from Wang and colleagues (Wang et al., 2004). t, time; T is the prescribed motion time period.

View larger version (17K): [in this window] [in a new window]   Fig. 3. (A) Variations of mean CL and CD with respect to the frequency ratio f/n; blue circle, CL at Re=75; red circle, CL at Re=250; black circle, CL at Re=1000; blue diamond, CD at Re=75; red diamond, CD at Re=250; and black diamond, CD at Re=1000. (B) Ratio of mean CL/CD versus f/n; blue circle, Re=75; red diamond, Re=250; and black triangle, Re=1000. (C) Mean lift coefficient per unit of driving power coefficient (CPW) versus f/n; same definitions as in B. The results obtained for the rigid wing are also plotted for comparison.

View larger version (31K): [in this window] [in a new window]   Fig. 4. Time histories of lift and drag coefficients for Re=75, 250 and 1000: (A) lift coefficient and (B) drag coefficient; blue dashed line, rigid wing; red dashed line, flexible wing with f/n=1/2; green line, flexible wing with f/n=1/3; black dash–dot line, flexible wing with f/n=1/4.

View larger version (72K): [in this window] [in a new window]   Fig. 5. Comparison of the rigid wing's performance during stroke reversal with respect to that of the flexible wing at a frequency ratio of f/n=1/3 at Re=75. (A–E) Vorticity contours for flexible wing with f/n=1/3 at five time instances: t/T=–0.0491, 0.0009, 0.0759, 0.1760 and 0.2260 (min=–10, max=10 and 80 contours). (F–J) Vorticity contours for the rigid wing at the same time locations. The white dashed lines indicate the end of stroke position; (K) lift coefficient history; and (L) histories of circulation of leading edge vortex (LEV), end of stroke vortex (ESV) and trailing edge vortex (TEV). , circulation.

View larger version (14K): [in this window] [in a new window]   Fig. 6. Averaged circulations as a function of time with respect to stroke reversal at Re=75; blue line, TEV; green dashed line, LEV; red dash–dot line, ESV. (A) Rigid wing, (B) flexible wing with f/n=1/2, (C) flexible wing with f/n=1/3, and (D) flexible wing with f/n=1/4.

View larger version (101K): [in this window] [in a new window]   Fig. 7. Instantaneous vorticity contours at Re=75. Contours range from –10 (blue) to 10 (red) with 80 intervals. Column A, rigid wing; Columns B–D, flexible wings with f/n=1/2, 1/3 and 1/4, respectively.

View larger version (112K): [in this window] [in a new window]   Fig. 8. Instantaneous vorticity contours at Re=250. Contours range from –10 (blue) to 10 (red) with 80 intervals. Column A, rigid wings; Columns B–D, flexible wings with f/n=1/2, 1/3 and 1/4, respectively.

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