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Propulsion of a foil undergoing a flapping undulatory motion from the impulse theory in the linear potential limit

Published online by Cambridge University Press:  25 November 2019

J. Alaminos-Quesada  and
R. Fernandez-Feria Open the ORCID record for R. Fernandez-Feria [Opens in a new window]
J. Alaminos-Quesada
Affiliation:
Fluid Mechanics, Universidad de Málaga, Andalucía Tech, Dr Ortiz Ramos s/n, 29071Málaga, Spain
R. Fernandez-Feria*
Affiliation:
Fluid Mechanics, Universidad de Málaga, Andalucía Tech, Dr Ortiz Ramos s/n, 29071Málaga, Spain
*
†Email address for correspondence: ramon.fernandez@uma.es
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Abstract

We derive general analytical expressions for the aerodynamic force and moment on a flapping flexible foil undergoing a prescribed undulatory motion in a two-dimensional, incompressible and linearized potential flow from the vortical impulse theory. We consider a fairly broad class of foil motion, characterized by nine non-dimensional parameters in addition to the reduced frequency. Quite simple analytical expressions are obtained in the particular case when just a chordwise flexure mode is superimposed to a pitching or heaving motion of the foil, for which the optimal conditions generating a maximum thrust force and a maximum propulsion efficiency are mapped in terms of the reduced frequency and the relative amplitude and phase shift of the deflection of the foil. These results are discussed in relation to the optimal conditions for a pitching or heaving rigid foil. The present theoretical results are compared with available numerical data for some particular undulatory motions of the flexible foil, with good agreement for small amplitudes of the oscillations and sufficiently high Reynolds number.

JFM classification

Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A19
Copyright
© 2019 Cambridge University Press

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References

Alben, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.CrossRefGoogle Scholar
Breder, C. M. 1926 The locomotion of fishes. Zoologica 4, 159–291.Google Scholar
Butkov, E. 1968 Mathematical Physics. Addison-Wesley.Google Scholar
Chang, X., Zhang, L. & He, X. 2012 Numerical study of the thunniform mode of fish swimming with different Reynolds number and caudal fin shape. Comput. Fluids 68, 54–70.CrossRefGoogle Scholar
Dewey, P. A., Boschitsch, B. M., Moored, K. W., Stone, H. A. & Smits, A. J. 2013 Scaling laws for the thrust production of flexible pitching panels. J. Fluid Mech. 732, 29–46.CrossRefGoogle Scholar
Dong, G. J. & Lu, X. Y. 2007 Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids 19, 057107.CrossRefGoogle Scholar
Eloy, C. 2013 On the best design for undulatory swimming. J. Fluid Mech. 717, 48–89.CrossRefGoogle Scholar
Eloy, C. & Schouveiler, L. 2011 Optimisation of two-dimensional undulatory swimming at high Reynolds number. Intl J. Non-Linear Mech. 46, 568–576.CrossRefGoogle Scholar
Fernandez-Feria, R. 2016 Linearized propulsion theory of flapping airfoils revisited. Phys. Rev. Fluids 1, 084502.CrossRefGoogle Scholar
Fernandez-Feria, R. 2017 Note on optimum propulsion of heaving and pitching airfoils from linear potential theory. J. Fluid Mech. 826, 781–796.CrossRefGoogle Scholar
Fernandez-Feria, R. & Alaminos-Quesada, J. 2018 Unsteady thrust, lift and moment of a two-dimensional flapping thin airfoil in the presence of leading-edge vortices: a first approximation from linear potential theory. J. Fluid Mech. 851, 344–373.CrossRefGoogle Scholar
Floryan, D. & Rowley, C. W. 2018 Clarifying the relationship between efficiency and resonance for flexible inertial swimmers. J. Fluid Mech. 853, 271–300.CrossRefGoogle Scholar
Garrick, I. E.1936 Propulsion of a flapping and oscillating airfoil. NACA Tech. Rep. TR 567.Google Scholar
Heathcote, S. & Gursul, I. 2007 Flexible flapping airfoil propulsion at low Reynolds numbers. AIAA J. 45, 1066–1079.CrossRefGoogle Scholar
Hoover, A. P., Cortez, R., Tytell, E. D. & Fauci, L. J. 2018 Swimming performance, resonance and shape evolution in heaving flexible panels. J. Fluid Mech. 847, 386–416.CrossRefGoogle Scholar
Huera-Huarte, F. J. 2018 On the impulse produced by chordwise flexible pitching foils in a quiescent fluid. Trans. ASME J. Fluids Engng 140, 041206.Google Scholar
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, 32–74.CrossRefGoogle Scholar
von Kármán, T. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5, 379–390.Google Scholar
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an aerofoil with chordwise flexibility. J. Fluid Mech. 88, 485–497.CrossRefGoogle Scholar
Le, T. Q., Ko, J. H., Byun, D., Park, S. H. & Park, H. C. 2010 Effect of chord flexure on aerodynamic performance of a flapping wing. J. Bionic Engng 7, 87–94.CrossRefGoogle Scholar
Lighthill, M. J. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413–449.CrossRefGoogle Scholar
Lighthill, M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44, 265–301.CrossRefGoogle Scholar
Lighthill, M. J. 1975 Mathematical Biofluiddynamics. SIAM.CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21, 071902.CrossRefGoogle Scholar
Moore, M. N. J. 2014 Analytical results on the role of flexibility in flapping propulsion. J. Fluid Mech. 757, 599–612.CrossRefGoogle Scholar
Moore, M. N. J. 2015 Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing. Phys. Fluids 27, 091701.CrossRefGoogle Scholar
Moore, M. N. J. 2017 A fast Chebyshev method for simulating flexible-wing propulsion. J. Comput. Phys. 345, 792–817.CrossRefGoogle Scholar
Newman, J. N. 1977 Marine Hydrodynamics. The MIT Press.CrossRefGoogle Scholar
Olver, F. W. J. & Maximon, L. C. 2010 Bessel functions. In NIST Handbook of Mathematical Functions (ed. Olver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W.), pp. 215–286. National Institute of Standards and Technology.Google Scholar
Paraz, F., Schouvelier, L. & Eloy, C. 2016 Thrust generation by a heaving foil: resonance, nonlinearities, and optimality. Phys. Fluids 28, 011903.CrossRefGoogle Scholar
Pederzani, J. & Haj-Hariri, H. 2006 Analysis of heaving flexible airfoilsin viscous flow. AIAA J. 44, 2773–2779.CrossRefGoogle Scholar
Platzer, M., Jones, K., Young, J. & Lai, J. 2008 Flapping wing aerodynamics: progress and challenges. AIAA J. 46, 2136–2149.CrossRefGoogle Scholar
Polyanin, A. D. & Manzhirov, A. V. 1998 Handbook of Integral Equations. CRC Press.CrossRefGoogle Scholar
Prempraneerach, P., Hoover, F. S. & Triantafyllou, M. S. 2003 The effect of chordwise flexibility on the thrust and efficiency of a flapping foil. In Proceedings of the Thirteenth International Symposium on Unmanned Untethered Submersible Technology. The Autonomous Undersea Systems Institute.Google Scholar
Ramananarivo, S., Godoy-Diana, R. & Thiria, B. 2011 Rather than resonance, flapping wings flyers may play on aerodynamics to improve performance. Proc. Natl Acad. Sci. USA 108, 5964–5969.CrossRefGoogle ScholarPubMed
Shyy, W., Aono, H., Chimakurthi, S. K., Trizila, P., Kang, C. K., Cesnik, C. E. S. & Liu, H. 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aero. Sci. 46, 284–327.CrossRefGoogle Scholar
Shyy, W., Aono, H., Kang, C. K. & Liu, H. 2013 An Introduction to Flapping Wing Aerodynamics. Cambridge University Press.CrossRefGoogle Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep. TR 496.Google Scholar
Tytell, E. D. 2004 The hydrodynamics of eel swimming: I. Wake structure. J. Exp. Biol. 207, 1825–1841.CrossRefGoogle ScholarPubMed
Tzezana, G. A. & Breuer, K. S. 2019 Thrust, drag and wake structure in flapping compliant membrane wings. J. Fluid Mech. 862, 871–888.CrossRefGoogle Scholar
Vogel, S. 1994 Life in Moving Fluids: The Physical Biology of Flow. Princeton University Press.Google Scholar
Wang, S. & Zhang, L. J. 2016 Self-propulsion of flapping bodies in viscous fluids: recent advances and perspectives. Acta Mechanica Sin. 32, 980–990.CrossRefGoogle Scholar
Wu, J. C. 1981 Theory for the aerodynamic force and moment in viscous flows. AIAA J. 19, 432–441.CrossRefGoogle Scholar
Wu, T. Y. 1961 Swimming of a waving plate. J. Fluid Mech. 10, 321–344.CrossRefGoogle Scholar
Wu, T. Y. 1971a Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46, 521–544.CrossRefGoogle Scholar
Wu, T. Y. 1971b Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46, 337–355.CrossRefGoogle Scholar
Wu, T. Y. 2011 Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25–58.CrossRefGoogle Scholar
Wu, T. Y., Brokaw, C. J. & Brennen, C.(Eds) 1975 Swimming and Flying in Nature, vol. 1 and 2. Plenum Press.CrossRefGoogle Scholar
Zhang, D., Pan, G., Chao, L. & Zhang, Y. 2018 Effects of Reynolds number and thickness on an undulatory self-propelled foil. Phys. Fluids 30, 071902.CrossRefGoogle Scholar
Zhang, L. J. & Eldredge, J. D. 2009 A viscous vortex particle method for deforming bodies with application to biolocomotion. Intl J. Numer. Meth. Fluids 59, 1299–1320.CrossRefGoogle Scholar
Zhu, Q. 2007 Numerical simulation of a flapping foil with chordwise or spanwise flexibility. AIAA J. 45, 2448–2457.CrossRefGoogle Scholar