Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-20T14:06:18.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic theory of a flapping wing of a circular cross-section

Published online by Cambridge University Press:  27 April 2022

Artem N. Nuriev Open the ORCID record for Artem N. Nuriev [Opens in a new window]  and
Andrey G. Egorov
Artem N. Nuriev*
Affiliation:
Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal University, 18, Kremlyovskaya St., Kazan, Republic of Tatarstan 420008, Russia
Andrey G. Egorov
Affiliation:
Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal University, 18, Kremlyovskaya St., Kazan, Republic of Tatarstan 420008, Russia
*
†Email address for correspondence: nuriev_an@mail.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper is devoted to the study of the propulsive motion of a flapping wing of a circular cross-section performing translational–rotational oscillations in a viscous incompressible fluid. To describe the flow past the wing, the unsteady Navier–Stokes equation is solved. Using the method of asymptotic expansions for the case of small amplitudes of oscillations, an analytical solution of the problem is constructed in the first two terms. It is shown that the nonlinear interaction of time harmonics of translational and rotational oscillations causes secondary flows (steady streaming) that make the wing move in the direction perpendicular to the axis of translational oscillations. For the case of cruising motion, when the average hydrodynamic force acting on the wing is equal to zero, the dependence of the average speed on the dimensionless oscillation parameters is found. The results show that for relatively large angles of rotation the cruising speed of a flapping wing can be as high as the velocity amplitude of translational oscillations. The limits of applicability of the asymptotic theory are investigated using direct numerical simulations. Numerical data demonstrate that the theory well describes the flow past the wing in a wide range of dimensionless amplitudes, frequencies and angles of rotation. In conclusion, the efficiency of the propulsion system is evaluated. It is shown that in terms of relative energy consumption, a cylindrical flapping wing can be attributed to the most efficient propulsors in the range of Reynolds numbers $Re \sim 10^2\text {--}10^3$.

JFM classification

Aerodynamics: Flow-structure interactions Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 941 , 25 June 2022 , A23
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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. 2021 a Collective locomotion of two-dimensional lattices of flapping plates. Part 1. Numerical method, single-plate case and lattice input power. J. Fluid Mech. 915, A20.CrossRefGoogle Scholar
Alben, S. 2021 b Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency. J. Fluid Mech. 915, A21.CrossRefGoogle Scholar
Alben, S. & Shelley, M. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. 102 (32), 11163–11166.CrossRefGoogle ScholarPubMed
Becker, L.E., Koehler, S.A. & Stone, H.A. 2003 On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer. J. Fluid Mech. 490, 15–35.CrossRefGoogle Scholar
Bhosale, Y., Parthasarathy, T. & Gazzola, M. 2020 Shape curvature effects in viscous streaming. J. Fluid Mech. 898, A13.CrossRefGoogle Scholar
Birnbaum, W. 1924 Der schlagflügelpropeller und die kleinen schwingungen elastisch befestigter tragflügel. Z. Flugtech. Motorluftschiffahrt 15, 128–134.Google Scholar
Blackburn, H.M., Elston, J.R. & Sheridan, J. 1999 Bluff-body propulsion produced by combined rotary and translational oscillation. Phys. Fluids 11 (1), 4–6.CrossRefGoogle Scholar
Dynnikova, G.Y. & Andronov, P.R. 2018 Expressions of force and moment exerted on a body in a viscous flow via the flux of vorticity generated on its surface. Eur. J. Mech. (B/Fluids) 72, 293–300.CrossRefGoogle Scholar
Dynnikova, G.Y., Dynnikov, Y.A., Guvernyuk, S.V. & Malakhova, T.V. 2021 Stability of a reverse Kármán vortex street. Phys. Fluids 33 (2), 024102.CrossRefGoogle Scholar
Egorov, A.G. & Nuriev, A.N. 2021 Steady streaming generated by low-amplitude oscillations of a cylinder. Lobachevskii J. Math. 42 (9), 2102–2108.CrossRefGoogle Scholar
Ferziger, J.H. & Perić, M. 1999 Computational Methods for Fluid Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Garrick, I.E. 1936 Propulsion of a flapping and oscillating airfoil. NACA Rep. 567. National Advisory Committee for Aeronautics.Google Scholar
Giesing, J.P. 1968 Nonlinear two-dimensional unsteady potential flow with lift. J. Aircraft 5 (2), 135–143.CrossRefGoogle Scholar
Glauert, H. 1930 The Force and Moment on an Oscillating Aerofoil, pp. 88–95. Springer.CrossRefGoogle Scholar
Golubev, V.V. 1949 Lektsii po teorii kryla. Lectures on the Theory of Wing. Gos Izd. Tekhn.-Teor. Lit. p. 482. Moscow, Russian.Google Scholar
Golubev, V.V. 1951 On some questions of theory of a flapping wing. Uch. Zap. MGU 152, 23–31.Google Scholar
Gradshteyn, I.S. & Ryzhik, I.M. 2007 Table of Integrals, Series, and Products, 7th edn. Academic Press.Google Scholar
Greenshields, C. 2019 OpenFOAM v7 User Guide. https://cfd.direct/openfoam/user-guide-v7/.Google Scholar
Hancock, G.J. & Newman, M.H.A. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 217 (1128), 96–121.Google Scholar
Holtsmark, J., Johnsen, I., Sikkeland, T. & Skavlem, S. 1954 Boundary layer flow near a cylindrical obstacle in an oscillating, incompressible fluid. J. Acoust. Soc. Am. 26 (1), 26–39.CrossRefGoogle Scholar
Isogai, K., Shinmoto, Y. & Watanabe, Y. 1999 Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil. AIAA J. 37 (10), 1145–1151.CrossRefGoogle Scholar
Issa, R.I. 1986 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 40–65.CrossRefGoogle Scholar
Keldysh, M.V. & Lavrentiev, M.A. 1935 On the theory of the oscillating wing. Tekhnicheskiye zametki TsAGI im. Prof. N.E. Zhukovskogo, vol. 35, pp. 48–52.Google Scholar
Koehler, C., Beran, P., Vanella, M. & Balaras, E. 2015 Flows produced by the combined oscillatory rotation and translation of a circular cylinder in a quiescent fluid. J. Fluid Mech. 764, 148–170.CrossRefGoogle Scholar
Lavrentyev, M.A. & Lavrentyev, M.M. 1962 On one principle of creating the thrust force in motion. J. Appl. Mech. Tech. Phys. 4, 6–9.Google 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. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (2), 305–317.CrossRefGoogle Scholar
Lighthill, M.J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44 (2), 265–301.CrossRefGoogle Scholar
Liu, H., Ellington, C.P., Kawachi, K., van den Berg, C. & Willmott, A.P. 1998 A computational fluid dynamic study of hawkmoth hovering. J. Exp. Biol. 201 (4), 461–477.CrossRefGoogle ScholarPubMed
Logvinovich, G.V. 1970 Hydrodynamics of a thin flexible body (assessment of hydrodynamics of fish). Bionika 4, 5–11.Google Scholar
Lua, K.B., Dash, S.M., Lim, T.T. & Yeo, K.S. 2016 On the thrust performance of a flapping two-dimensional elliptic airfoil in a forward flight. J. Fluids Struct. 66, 91–109.CrossRefGoogle Scholar
Maertens, A.P., Triantafyllou, M.S. & Yue, D.K.P. 2015 Efficiency of fish propulsion. Bioinspir. Biomim. 10 (4), 046013.CrossRefGoogle ScholarPubMed
Nekrasov, A.I. 1947 Theory of Wings of Unsteady Flow. Izd-vo Akademii nauk SSSR.Google Scholar
Newman, J.N. & Wu, T.Y. 1975 Hydromechanical Aspects of Fish Swimming, pp. 615–634. Springer US.CrossRefGoogle Scholar
Nuriev, A.N. & Egorov, A.G. 2019 Asymptotic investigation of hydrodynamic forces acting on an oscillating cylinder at finite streaming Reynolds numbers. Lobachevskii J. Math. 40, 794–801.CrossRefGoogle Scholar
Nuriev, A.N., Egorov, A.G. & Kamalutdinov, A.M. 2021 Hydrodynamic forces acting on the elliptic cylinder performing high-frequency low-amplitude multi-harmonic oscillations in a viscous fluid. J. Fluid Mech. 913, A40.CrossRefGoogle Scholar
Nuriev, A.N., Egorov, A.G. & Zaitseva, O.N. 2018 Numerical analysis of secondary flows around an oscillating cylinder. J. Appl. Mech. Tech. Phys. 59 (3), 451–459.CrossRefGoogle Scholar
Nuriev, A.N., Kamalutdinov, A.M. & Egorov, A.G. 2019 A numerical investigation of fluid flows induced by the oscillations of thin plates and evaluation of the associated hydrodynamic forces. J. Fluid Mech. 874, 1057–1095.CrossRefGoogle Scholar
Pedro, G., Suleman, A. & Djilali, N. 2003 A numerical study of the propulsive efficiency of a flapping hydrofoil. Intl J. Numer. Meth. Fluids 42 (5), 493–526. Available at: https://onlinelibrary.wiley.com/doi/pdf/10.1002/fld.525.CrossRefGoogle Scholar
Perry, A.E. & Chong, M.S. 1987 A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19 (1), 125–155.CrossRefGoogle Scholar
Prandtl, L. 1924 Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben, pp. 18–33. Springer.CrossRefGoogle Scholar
Purcell, E.M. 1977 Life at low Reynolds number. Am. J. Phys. 45 (1), 3–11.CrossRefGoogle Scholar
Riley, N. 1967 Oscillatory viscous flows. Review and extension. IMA J. Appl. Maths 3 (4), 419–434.CrossRefGoogle Scholar
Riley, N. 1975 The steady streaming induced by a vibrating cylinder. J. Fluid Mech. 68, 801–812.CrossRefGoogle Scholar
Riley, N. & Watson, E.J. 1993 Eccentric oscillations of a circular cylinder in a viscous fluid. Mathematika 40, 187–202.CrossRefGoogle Scholar
Rozhdestvensky, K.V. & Ryzhov, V.A. 2003 Aerohydrodynamics of flapping-wing propulsors. Prog. Aerosp. Sci. 39 (8), 585–633.CrossRefGoogle Scholar
Schlichting, H. 1932 Berechnung ebener periodischer grenzschichtströmungen. Phys. Zeit. 33, 327–335.Google Scholar
Sedov, L.I. 1966 Two-Dimensional Problems of Hydrodynamics and Aerodynamics. Nauka.CrossRefGoogle Scholar
Spagnolie, S.E., Moret, L., Shelley, M.J. & Zhang, J. 2010 Surprising behaviors in flapping locomotion with passive pitching. Phys. Fluids 22 (4), 041903.CrossRefGoogle Scholar
Stokes, G.G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Taha, H.E. 2020 Geometric nonlinear control of the lift dynamics of a pitching-plunging wing. In AIAA Scitech 2020 Forum, AIAA Paper 2020-0824.Google Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA R-496. National Advisory Committee for Aeronautics.Google Scholar
von Kármán, T. & Burgers, J.M. 1935 Aerodynamic Theory: General Aerodynamic Theory: Perfect Fluids. Springer.Google Scholar
Wagner, H. 1925 Über die entstehung des dynamischen auftriebes von tragflügeln. Z. Angew. Math. Mech. 5 (1), 17–35. Available at: https://onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.19250050103.CrossRefGoogle Scholar
Wang, C.-Y. 1968 On high-frequency oscillatory viscous flows. J. Fluid Mech. 32 (1), 55–58.CrossRefGoogle Scholar
Wieselsberger, C. 1921 New data on the laws of fluid resistance. Phys. Zeit. 22, 321–328.Google Scholar
Wu, T.Y.-T. 1971 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 (2), 337–355.CrossRefGoogle Scholar
Wu, X., Zhang, X., Tian, X., Li, X. & Lu, W. 2020 A review on fluid dynamics of flapping foils. Ocean Engng 195, 106712.CrossRefGoogle Scholar
Zhang, J., Liu, N.-S. & Lu, X.-Y. 2010 Locomotion of a passively flapping flat plate. J. Fluid Mech. 659, 43–68.CrossRefGoogle Scholar

Nuriev and Egorov supplementary movie 1

See pdf file for movie caption

Download Nuriev and Egorov supplementary movie 1(Video)
Video 1.7 MB

Nuriev and Egorov supplementary movie 2

See pdf file for movie caption

Download Nuriev and Egorov supplementary movie 2(Video)
Video 2.9 MB

Nuriev and Egorov supplementary movie 3

See pdf file for movie caption

Download Nuriev and Egorov supplementary movie 3(Video)
Video 1.8 MB

Nuriev and Egorov supplementary movie 4

See pdf file for movie caption

Download Nuriev and Egorov supplementary movie 4(Video)
Video 2.5 MB
Supplementary material: PDF

Nuriev and Egorov supplementary material

Captions for movies 1-4

Download Nuriev and Egorov supplementary material(PDF)
PDF 11.5 KB