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Scaling laws for the thrust production of flexible pitching panels

  • Peter A. Dewey (a1), Birgitt M. Boschitsch (a1), Keith W. Moored (a2), Howard A. Stone (a1) and Alexander J. Smits (a1) (a3)...


We present experimental results on the role of flexibility and aspect ratio in bio-inspired aquatic propulsion. Direct thrust and power measurements are used to determine the propulsive efficiency of flexible panels undergoing a leading-edge pitching motion. We find that flexible panels can give a significant amplification of thrust production of $\mathscr{O}(100{\unicode{x2013}} 200\hspace{0.167em} \% )$ and propulsive efficiency of $\mathscr{O}(100\hspace{0.167em} \% )$ when compared to rigid panels. The data highlight that the global maximum in propulsive efficiency across a range of panel flexibilities is achieved when two conditions are simultaneously satisfied: (i) the oscillation of the panel yields a Strouhal number in the optimal range ( $0. 25\lt \mathit{St}\lt 0. 35$ ) predicted by Triantafyllou, Triantafyllou & Grosenbaugh (J. Fluid Struct., vol. 7, 1993, pp. 205–224); and (ii) this frequency of motion is tuned to the structural resonant frequency of the panel. In addition, new scaling laws for the thrust production and power input to the fluid are derived for the rigid and flexible panels. It is found that the dominant forces are the characteristic elastic force and the characteristic fluid force. In the flexible regime the data scale using the characteristic elastic force and in the rigid limit the data scale using the characteristic fluid force.


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Alben, S. 2008 Optimal flexibility of a flexible appendage in an inviscid fluid. J. Fluid Mech. 614, 355380.
Alben, S. 2010 Passive and active bodies in vortex streets. J. Fluid Mech. 642, 95125.
Alben, S., Witt, C., Baker, T. V., Anderson, E. & Lauder, G. V. 2012 Dynamics of freely swimming flexible foils. Phys. Fluids 24, 051901.
Allen, J. J. & Smits, A. J. 2001 Energy harvesting eel. J. Fluid Struct. 15, 629640.
Betz, A. 1912 Ein beitrag zur erklaerung des segelfluges. Z. Flugtech. Motorluftsch. 3, 269270.
Buchholz, J. H. J., Clark, R. P. & Smits, A. J. 2008 Thrust performance of unsteady propulsors using a novel measurement system, and corresponding wake patterns. Exp. Fluids 45, 461472.
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, 331365.
Clark, R. P. & Smits, A. J. 2006 Thrust production and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 562, 415429.
Dai, H., Luo, H., de Sousa, P. J. S. A. Ferreira & Doyle, J. F. 2012 Thrust performance of a flexible low-aspect-ratio pitching panel. Phys. Fluids 24, 101903.
Daniel, T. L. & Combes, S. A. 2002 Flexible wings and fins: bending by inertial or fluid-dynamic forces. Integr. Compar. Biol. 42, 10441049.
Dewey, P. A., Carriou, A. & Smits, A. J. 2012 On the relationship between efficiency and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 691, 245266.
Eldredge, J. F., Toomey, J. & Medina, A. 2010 On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94115.
Green, M. A. & Smits, A. J. 2008 Effects of three-dimensionality on thrust production by a pitching panel. J. Fluid Mech. 615, 211220.
Heathcote, S., Wang, Z. & Gursul, I. 2008 Effect of spanwise flexibility on flapping wing propulsion. J. Fluid Struct. 24, 183199.
Kang, C. K., Aono, H., Cesnik, C. E. S. & Shyy, W. 2011 Effects of flexibility on the aerodynamic performance of flapping wings. J. Fluid Mech. 689, 3274.
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an aerofoil with chordwise flexibility. J. Fluid Mech. 88, 485497.
Knoller, R. 1909 Die gesetze des luftwiderstandes. Flug Motortech. 3 (21), 17.
Lauder, G. V., Madden, P. G. A, Tangorra, J. L., Anderson, E. & Baker, T. V. 2011 Bioinspiration from fish for smart material design and function. Smart Mater. Struct. 20, 113.
Leftwich, M. C., Tytell, E. D., Cohen, A. H. & Smits, A. J. 2012 Wake structures behind a swimming robotic lamprey with a passively flexible tail. J. Exp. Biol. 215, 416425.
Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving aerofoil in a viscous flow. J. Fluid Mech. 492, 339362.
Masoud, H. & Alexeev, A. 2010 Resonance of flexible flapping wings at low Reynolds number. Phys. Rev. E 81, 15.
Michelin, S. & Smith, S. G Llewellyn 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21, 115.
Moored, K. W., Dewey, P. A., Haj-Hariri, H. & Smits, A. J. 2012 Hydrodynamic wake resonance as an underlying principle of efficient unsteady propulsion. J. Fluid Mech. 708, 329348.
Moored, K. W., Dewey, P. A., Leftwich, M. C., Bart-Smith, H. & Smits, A. J. 2011 Bioinspired propulsion mechanisms based on manta ray locomotion. Marine Technol. Soc. 45, 110118.
Ramananarivo, S., Godoy-Diana, R. & Thiria, B. 2011 Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. Proc. Natl Acad. Sci. 108, 59645969.
Spagnolie, S. E., Moret, L., Shelley, M. & Zhang, J. 2010 Surprising behaviours in locomotion with passive pitching. Phys. Fluids 22, 120.
Sunada, S. 2002 Optical measurements of the deformation motion, and generated force of the wings of a moth, Mythimna Separata (Walker). JSME Intl J. Ser. B 45, 836842.
Taylor, G. K., Nudds, R. L. & Thomas, A. L. R. 2003 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 435, 707711.
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496.
Thiria, B. & Godoy-Diana, R. 2010 How wing compliance drives the efficiency of self-propelled flapping flyers. Phys. Rev. E 82, 015303(R).
Timoshenko, S. 1974 Vibration Problems in Engineering. John Wiley and Sons.
Triantafyllou, M. S., Techet, A. H. & Hover, F. S. 2004 Review of experimental work in biomimetic foils. IEEE J. Ocean. Engng 29, 585594.
Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluid Struct. 7, 205224.
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, 95105.
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