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Locomotion of a passively flapping flat plate

Published online by Cambridge University Press:  28 June 2010

JIE ZHANG ,
NAN-SHENG LIU  and
XI-YUN LU
JIE ZHANG
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
NAN-SHENG LIU
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
XI-YUN LU*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
*
†Email address for correspondence: [email protected]
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Abstract

Locomotion of a passively flapping flat plate has been studied numerically by means of a multiblock lattice Boltzmann method. A flexible plate is modelled by a rigid plate with a torsion spring acting about the pivot at the leading edge of the plate. A dynamic model of this kind is called a lumped-torsional-flexibility model. When the leading edge is forced to heave sinusoidally, the plate pitches passively and propels itself in the horizontal direction as a result of the fluid–plate interaction. We have investigated various aspects of the mechanics behind the behaviour of the flapping plate, including the periodic- and non-periodic-flow states, the spontaneous motion of the plate, vortical structure and how they compare to similar propulsion systems in animals. In the periodic-flow regime, two dynamical responses of the passively pitching plate (forward and backward movements) are observed. Which movement will occur depends only on the frequency ratio F of the natural frequency of the system and the heaving frequency associated with the lumped torsional flexibility. It is found that the plate will select the forward movement when F > 1 and the backward movement when F ≤ 1. In the forward-movement regime, analysis of the dynamical behaviours and propulsive properties of the passively pitching plate indicates that the torsional flexibility can remarkably improve the propulsive performance. In addition, four kinds of vortex structures in the near wake are identified, which mainly depend on the forward speed of the plate. Finally the forward movement is compared to the flapping-based locomotion of swimming and flying animals. The results obtained in this study are consistent with the observations and measurements of swimming and flying animals; thus, they may provide physical insights into understanding of the propulsive mechanisms of the flapping wings and fins of animals.

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 43 - 68
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Aidun, C. K. & Ding, E. J. 2003 Dynamics of particle sedimentation in a vertical channel: period-doubling bifurcation and chaotic state. Phys. Fluids 15, 1612–1621.CrossRefGoogle Scholar
Aidun, C. K., Lu, Y. N. & Ding, E. J. 1998 Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech. 373, 287–311.CrossRefGoogle Scholar
Alben, S. & Shelly, M. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102, 11163–11166.CrossRefGoogle ScholarPubMed
Alexander, R. M. 1993 Principles of Animal Locomotion. Princeton University Press.Google Scholar
Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.CrossRefGoogle Scholar
Azuma, A. & Watanabe, T. 1988 Flight performance of a dragonfly. J. Exp. Biol. 137, 221–252.CrossRefGoogle Scholar
Birch, J. M. & Dickinson, M. H. 2001 Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412, 729–733.CrossRefGoogle Scholar
Blake, R. W. 1983 Fish Locomotion. Cambridge University Press.Google Scholar
Brodsky, A. K. 1994 The Evolution of Insect Flight. Oxford University Press.CrossRefGoogle Scholar
Bouzidi, M., Firdaouss, M. & Lallemand, P. 2001 Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys. Fluids 13, 3452–3459.CrossRefGoogle Scholar
Buchholz, J. H. J. & Smits, A. J. 2008 The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel. J. Fluid Mech. 603, 331–365.CrossRefGoogle ScholarPubMed
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
Combes, S. A. & Daniel, T. L. 2001 Shape, flapping and flexion: wing and fin design for forward flight. J. Exp. Biol. 204, 2073–2085.CrossRefGoogle ScholarPubMed
Combes, S. A. & Daniel, T. L. 2003 a Flexural stiffness in insect wings. I. Scaling and the influence of wing venation. J. Exp. Biol. 206, 2979–2987.CrossRefGoogle ScholarPubMed
Combes, S. A. & Daniel, T. L. 2003 b Flexural stiffness in insect wings. II. Spatial distribution and dynamic wing bending. J. Exp. Biol. 206, 2989–2997.CrossRefGoogle ScholarPubMed
Connington, K., Kang, Q. J., Viswanathan, H., Abdel-Fattah, A. & Chen, S. Y. 2009 Peristaltic particle transport using the lattice Boltzmann method. Phys. Fluids 21, 053301.CrossRefGoogle Scholar
Dickinson, M. H., Lehmann, F. O. & Sane, S. P. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960.CrossRefGoogle ScholarPubMed
Dong, H., Mittal, R. & Najjar, F. M. 2006 Wake topology and hydrodynamic performance of low aspect-ratio flapping foils. J. Fluid Mech. 566, 309–343.CrossRefGoogle Scholar
Drucker, E. G. & Lauder, G. V. 2001 Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J. Exp. Biol. 204, 2943–2958.CrossRefGoogle ScholarPubMed
von Ellenrieder, K. D., Parker, K. & Soria, J. 2003 Flow structures behind a heaving and pitching finite-span wing. J. Fluid Mech. 490, 129–138.CrossRefGoogle Scholar
Ellington, C. P. 1984 a The aerodynamics of hovering insect flight. II. Morphological parameters. Phil. Trans. R. Soc. Lond. B 305, 17–40.Google Scholar
Ellington, C. P. 1984 b The aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305, 41–78.Google Scholar
Ellington, C. P., van den Berg, C., Willmott, A. P. & Thomos, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626–630.CrossRefGoogle Scholar
Ennos, A. R. 1987 A comparative study of the flight mechanism of Diptera. J. Exp. Biol. 127, 355–372.CrossRefGoogle Scholar
Ennos, A. R. 1988 a The importance of torsion in the design of insect wings. J. Exp. Biol. 140, 137–160.CrossRefGoogle Scholar
Ennos, A. R. 1988 b The inertial cause of wing rotation in Diptera. J. Exp. Biol. 140, 161–169.CrossRefGoogle Scholar
Ennos, A. R. 1989 Inertial and aerodynamic torques on the wings of Diptera in flight. J. Exp. Biol. 142, 87–95.CrossRefGoogle Scholar
Fang, H. P., Wang, Z. W., Lin, Z. F. & Liu, M. R. 2002 Lattice Boltzmann method for simulating the viscous flow in large distensible blood vessels. Phys. Rev. E 65, 051925.CrossRefGoogle ScholarPubMed
Fish, F. E. 1993 Power output and propulsive efficiency of swimming bottlenose dolphins (Tursiops truncatus). J. Exp. Biol. 185, 179–193.CrossRefGoogle Scholar
Fish, F. E. 2004 Structure and mechanics of non-piscine control surfaces. IEEE J. Ocean. Engng 29, 605–621.CrossRefGoogle Scholar
Gao, T. & Lu, X.-Y. 2008 Insect normal hovering flight in ground effect. Phys. Fluids 20, 087101.CrossRefGoogle Scholar
Gao, T., Tseng, Y.-H. & Lu, X.-Y. 2007 An improved hybrid Cartesian/immersed boundary method for fluid-solid flows. Intl J. Numer. Methods Fluids 55, 1189–1211.CrossRefGoogle Scholar
Godoy-Diana, R., Marais, C., Aider, J.-L. & Wesfreid, J. E. 2009 A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 23–32.CrossRefGoogle Scholar
Gopalkrishnan, R., Triantafyllou, M. S., Triantafyllou, G. S. & Barrett, D. 1994 Active vorticity control in a shear flow using a flapping foil. J. Fluid Mech. 274, 1–21.CrossRefGoogle Scholar
Ishihara, D., Horie, T. & Denda, M. 2009 A two-dimensional computational study on the fluid–structure interaction cause of wing pitch changes in dipteran flapping flight. J. Exp. Biol. 212, 1–10.CrossRefGoogle Scholar
von Kármán, T. & Burgers, J. 1935 General aerodynamic theory-perfect fluids. In Aerodynamic Theory. Julius Springer.Google Scholar
Kern, S. & Koumoutsakos, P. 2006 Simulations of optimized anguilliform swimming. J. Exp. Biol. 209, 4841–4857.CrossRefGoogle ScholarPubMed
Lallemand, P. & Luo, L. S. 2003 Lattice Boltzmann method for moving boundaries. J. Comput. Phys. 184, 406–421.CrossRefGoogle Scholar
Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech. 492, 339–362.CrossRefGoogle Scholar
Lighthill, M. J. 1975 Mathematical Biofluiddynamics. SIAM.CrossRefGoogle Scholar
Liu, H., Ellington, C. P., Kawachi, K., van den Berg, C. & Willmott, A. P. 1998 A computational fluid dynamics study of hawkmoth hovering. J. Exp. Biol. 201, 461–477.CrossRefGoogle ScholarPubMed
Lu, X.-Y. & Liao, Q. 2006 Dynamic responses of a two-dimensional flapping foil motion. Phys. Fluids 18, 098104.CrossRefGoogle Scholar
Mei, R. W., Yu, D. Z., Shyy, W. & Luo, L. S. 2002 Force evaluation in the lattice Boltzmann method involving curved geometry. Phys. Rev. E 65, 041203.CrossRefGoogle ScholarPubMed
Norberg, R. A. 1975 Hovering flight of the dragonfly Aeschna juncea. In Swimming and Flying in Nature (ed. Wu, Y.-T., Brokaw, C. J. & Brennen, C.), vol. 2. Plenum.Google Scholar
Peng, Y., Shu, C., Chew, Y. T., Niu, X. D. & Lu, X.-Y. 2006 Application of multi-block approach in the immersed boundary-lattice Boltzmann method for viscous fluid flows. J. Comput. Phys. 218, 460–478.CrossRefGoogle Scholar
Qian, Y. H., d'Humieres, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–484.CrossRefGoogle Scholar
Ramamurit, R. & Sandberg, W. C. 2002 A three-dimensional computation study of the aerodynamic mechanisms of insect flight. J. Exp. Biol. 205, 1507–1518.CrossRefGoogle Scholar
Richard, B. 1958 The speed of swimming of fish as related to size and to the frequency and amplitude of the tail beat. J. Exp. Biol. 35, 109–133.Google Scholar
Sato, M. & Azuma, A. 1997 The flight performance of a damselfly Ceriagrion melanurum selys. J. Exp. Biol. 200, 1765–1779.CrossRefGoogle ScholarPubMed
Schnipper, T., Andersen, A. & Bohr, T. 2009 Vortex wakes of a flapping foil. J. Fluid Mech. 633, 411–423.CrossRefGoogle Scholar
Schultz, W. W. & Webb, P. W. 2002 Power requirements of swimming: do new methods resolve old questions? Integr. Comp. Biol. 42, 1018–1025.CrossRefGoogle ScholarPubMed
Shelley, M., Vandenberghe, N. & Zhang, J. 2005 Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett. 94, 094302.CrossRefGoogle ScholarPubMed
Spagnolie, S. E. 2008 Flapping, ratcheting, bursting, and tumbling: a selection of problems in fluid–body interaction dynamics. PhD dissertation, New York University.Google Scholar
Sun, M. & Tang, J. 2002 Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J. Exp. Biol. 205, 55–70.CrossRefGoogle Scholar
Taylor, G., Nudds, R. & Thomas, A. 2003 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425, 707–711.CrossRefGoogle Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids 3, 2835–2837.CrossRefGoogle Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32, 33–53.CrossRefGoogle Scholar
Vandenberghe, N., Childress, S. & Zhang, J. 2006 On unidirectional flight of a free flapping wing. Phys. Fluid 18, 014102.CrossRefGoogle Scholar
Vandenberghe, N., Zhang, J. & Childress, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147–155.CrossRefGoogle Scholar
Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Exp. Biol. 212, 95–105.CrossRefGoogle ScholarPubMed
Videler, J. J. 1993 Fish Swimming. Chapman & Hall.CrossRefGoogle Scholar
Vogel, S. 1994 Life in Moving Fluids. Princeton University Press.Google Scholar
Wang, Z. J. 2000 Two-dimensional mechanism for insect hovering. Phys. Rev. Lett. 85, 2216–2219.CrossRefGoogle Scholar
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.CrossRefGoogle Scholar
Wang, Z. J., Birch, J. M. & Dickinson, M. H. 2004 Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computational vs robotic wing experiments. J. Exp. Biol. 207, 449–460.CrossRefGoogle ScholarPubMed
Wootton, R. J. 1999 Invertebrate paraxial locomotory appendages: design, deformation and control. J. Exp. Biol. 202, 3333–3345.CrossRefGoogle ScholarPubMed
Wootton, R. J., Herbert, R. C., Young, P. G. & Evans, K. E. 2003 Approaches to structural modeling of insect wings. Phil. Trans. R. Soc. Lond. B 358, 1577–1587.CrossRefGoogle ScholarPubMed
Wu, J.-Z., Lu, X.-Y. & Zhuang, L.-X. 2007 Integral force acting on a body due to local flow structures. J. Fluid Mech. 576, 265–286.CrossRefGoogle Scholar
Xia, Z.-H., Connington, K. W., Rapaka, S., Yue, P., Feng, J. J. & Chen, S.-Y. 2009 Flow patterns in the sedimentation of an elliptical particle. J. Fluid Mech. 625, 249–272.CrossRefGoogle Scholar
Yu, D. Z., Luo, L. S, Mei, R. W. & Shyy, W. 2003 Viscous flow computations with the method of lattice Boltzmann equation. Prog. Aerosp. Sci. 39, 329–367.CrossRefGoogle Scholar
Yu, D. Z., Mei, R. W. & Shyy, W. 2002 A multi-block lattice Boltzmann method for viscous fluid flows. Intl J. Numer. Methods Fluids 39, 99–120.CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for flags in a two-dimensional wind. Nature 408, 835–839.CrossRefGoogle Scholar
Zhang, J., Liu, N.-S. & Lu, X.-Y. 2009 Route to a chaotic state in fluid flow past an inclined flat plate. Phys. Rev. E 79, 045306(R).CrossRefGoogle Scholar
Zhang, J. & Lu, X.-Y. 2009 Aerodynamic performance due to forewing and hindwing interaction in gliding dragonfly flight. Phys. Rev. E 80, 017302.CrossRefGoogle ScholarPubMed