Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T20:23:10.470Z Has data issue: false hasContentIssue false

Optimal flexibility of a flapping appendage in an inviscid fluid

Published online by Cambridge University Press:  16 October 2008

SILAS ALBEN*
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USAalben@math.gatech.edu

Abstract

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alben, S., Madden, P. G. & Lauder, G. V. 2007 The mechanics of active fin-shape control in ray-finned fishes. J. R. Soc. Interface 4 (13), 243256.CrossRefGoogle ScholarPubMed
Alben, S. & Shelley, M. J. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102 (32), 1116311166.CrossRefGoogle ScholarPubMed
Alben, S., Shelley, M. & Zhang, J. 2002 Drag reduction through self-similar bending of a flexible body. Nature 420, 479481.CrossRefGoogle ScholarPubMed
Alben, S., Shelley, M. & Zhang, J. 2004 How flexibility induces streamlining in a two-dimensional flow. Phys. Fluids 16, 1694.CrossRefGoogle Scholar
Alben, S. & Shelley, M. J. 2008 Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100, 74301.CrossRefGoogle Scholar
Antman, S. S. 1995 Nonlinear Problems of Elasticity. Springer.CrossRefGoogle Scholar
Bainbridge, R. 1963 Caudal fin and body movement in the propulsion of some fish. J. Exp. Biol. 40 (1), 2356.CrossRefGoogle Scholar
Bergou, A. J., Xu, S. & Wang, Z. J. 2007 Passive wing pitch reversal in insect flight. J. Fluid Mech. 591, 321.CrossRefGoogle Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
Childress, S., Vandenberghe, N. & Zhang, J. 2006 Hovering of a passive body in an oscillating airflow. Phys. Fluids 18, 117103.CrossRefGoogle Scholar
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 (17), 29792987.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 (17), 29892997.CrossRefGoogle ScholarPubMed
Didden, N. 1979 On the formation of vortex rings: Rolling-up and production of circulation. Z. Angew. Mathe. Phys. 30 (1), 101116.CrossRefGoogle Scholar
Ellington, C. P. 1984 The aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305, 4178.Google Scholar
Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech 38, 193224.CrossRefGoogle Scholar
Fish, F. E., Nusbaum, M. K., Beneski, J. T. & Ketten, D. R. 2006 Passive cambering and flexible propulsors: cetacean flukes. Bioinspiration and Biomimetics, 1 (4), S42S48.CrossRefGoogle ScholarPubMed
Gibb, A., Jayne, B. & Lauder, G. 1994 Kinematics of pectoral fin locomotion in the bluegill sunfish Lepomis Macrochirus. J. Exp. Biol. 189 (1), 133–61.CrossRefGoogle ScholarPubMed
Godoy-Diana, R., Aider, J. L. & Wesfreid, J. E. 2008 Transitions in the wake of a flapping foil. Phys. Rev. E 71, 016308.Google Scholar
Hauser, W. 1965 Introduction to the Principles of Mechanics. Addison-Wesley.Google Scholar
Jones, M. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–401.CrossRefGoogle Scholar
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech. 88 (3), 485497.CrossRefGoogle Scholar
Katz, J. & Weihs, D. 1979 Large amplitude unsteady motion of a flexible slender propulsor. J. Fluid Mech. 90 (4), 713723.CrossRefGoogle Scholar
Lauder, G. V., Madden, P. G. A., Mittal, R., Dong, H. & Bozkurttas, M. 2006 Locomotion with flexible propulsors: I. Experimental analysis of pectoral fin swimming in sunfish. Bioinspiration and Biomimetics, 1 (4), S25S34.CrossRefGoogle ScholarPubMed
Lian, Y., Shyy, W., Viieru, D. & Zhang, B. 2003 Membrane wing aerodynamics for micro air vehicles. Prog. Aero. Sci. 39 (6-7), 425465.CrossRefGoogle Scholar
Lighthill, J. M. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413446.CrossRefGoogle Scholar
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (2), 305317.CrossRefGoogle Scholar
Liu, P. & Bose, N. 1997 Propulsive performance from oscillating propulsors with spanwise flexibility. Proc. R. Soc. Lond. A 453, 17631770.CrossRefGoogle Scholar
Long, J., Hale, M., Mchenry, M. & Westneat, M. 1996 Functions of fish skin: flexural stiffness and steady swimming of longnose gar, Lepisosteus osseus. J. Exp. Biol. 199 (Pt 10), 2139–51.CrossRefGoogle ScholarPubMed
Mason, J. C. & Handscomb, D. C. 2003 Chebyshev polynomials. Chapman/Hall/CRC.Google Scholar
Miao, J. M. & Ho, M. H. 2006 Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil. J. Fluid. Struct. 22 (3), 401419.CrossRefGoogle Scholar
Muskhelishvili, N. I. 1953 Singular Integral Equations: Boundary Problems of Function Theory and their Application to Mathematical Physics. P. Noordhoff.Google Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139161.CrossRefGoogle Scholar
Prempraneerach, P., Hover, F. S. & Triantafyllou, M. S. 2003 The effect of chordwise flexibility on the thrust and efficiency of a flapping foil. In International Symposium on Unmanned Untethered Submersible Technology.Google Scholar
Pullin, D. I. & Perry, A. E. 1980 Some flow visualization experiments on the starting vortex. J. Fluid Mech. 97, 239255.CrossRefGoogle Scholar
Pullin, D. I. & Wang, Z. J. 2004 Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 121.CrossRefGoogle Scholar
Ralston, A. & Rabinowitz, P. 2001 A First Course in Numerical Analysis. Dover.Google Scholar
Saffman, P. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shadwick, R. E. & Lauder, G. V. 2006 Fish Biomechanics. Academic.Google Scholar
Sparenberg, J. A. 2002 Survey of the mathematical theory of fish locomotion. J. Eng. Math. 44 (4), 395448.CrossRefGoogle Scholar
Thwaites, B. 1987 Incompressible Aerodynamics: An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid past Aerofoils, Wings, and Other Bodies. Dover.Google Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3 (12), 28352837.CrossRefGoogle Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Ann. Rev. Fluid Mech. 32, 3353.CrossRefGoogle Scholar
Vandenberghe, N., Zhang, J. & Childress, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147155.CrossRefGoogle Scholar
Videler, J. J. 1993 Fish Swimming. Springer.CrossRefGoogle Scholar
Wainwright, S. A. 2000 The animal axis. Am. Zool. 40 (1), 1927.Google Scholar
Willmott, A. P. & Ellington, C. P. 1997 The mechanics of flight in the hawkmoth Manduca sexta. I. Kinematics of hovering and forward flight. J. Exp. Biol. 200 (21), 2705–22.CrossRefGoogle ScholarPubMed
Wu, T. Y. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321344.CrossRefGoogle Scholar