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Relationship of roll and pitch oscillations in a fin flapping at transitional to high Reynolds numbers

Published online by Cambridge University Press:  29 May 2012

Promode R. Bandyopadhyay ,
David N. Beal ,
J. Dana Hrubes  and
Arun Mangalam
Promode R. Bandyopadhyay*
Affiliation:
Naval Undersea Warfare Center, Newport, RI 02841, USA
David N. Beal
Affiliation:
Naval Undersea Warfare Center, Newport, RI 02841, USA
J. Dana Hrubes
Affiliation:
Naval Undersea Warfare Center, Newport, RI 02841, USA
Arun Mangalam
Affiliation:
Tao Systems, Hampton, VA 23666, USA
*
†Email address for correspondence: [email protected]
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Abstract

Hydrodynamic effects of the relationship between the roll and pitch oscillations in low-aspect-ratio fins, with a laminar section and a rounded leading edge, flapping at transitional to moderately high Reynolds numbers, are considered. The fin is hinged at one end and its roll amplitude is large. Also examined is how this relationship is affected by spanwise twist, which alters the pitch oscillation amplitude and its phase relative to the roll motion. Force, efficiency and surface hot-film-anemometry measurements, and flow visualization are carried out in a tow tank. A fin of an abstracted penguin-wing planform and a NACA 0012 cross-section is used, and the chord Reynolds number varies from 3558 to 150 000 based on total speed. The fin is forced near the natural shedding frequency. Strouhal number and pitch amplitude are directly related when thrust is produced, and efficiency is maximized in narrow combinations of Strouhal number and pitch amplitude when oscillation of the leading-edge stagnation point is minimal. Twist makes the angle of attack uniform along the span and enhances thrust by up to 24 %, while maintaining high efficiency. Only 5 % of the power required to roll is spent to pitch, and yet roll and pitch are directly related. During hovering, dye visualization shows that a diffused leading-edge vortex is produced in rigid fins, which enlarges along the span; however, twist makes the vortex more uniform and the fin in turn requires less power to roll. Low-order phase maps of the measurements of force oscillation versus its derivative are modelled as due to van der Pol oscillators; the higher-order maps show trends in the sub-regimes of the transitional Reynolds number. Fin oscillation imparts a chordwise fluid motion, yielding a Stokes wave in the near-wall vorticity layer. When the roll and pitch oscillations are directly related, the wave is optimized: causing vorticity lift-up as the fin is decelerated at the roll extremity; the potential energy at the stagnation point is converted into kinetic energy; a vortex is produced as the lifted vorticity is wrapped around the leading edge; and free-stream reattachment keeps the vortex trapped. When the twist oscillation is phased along the span, this vortex becomes self-preserving at all amplitudes of twist, indicating the most stable (low-bandwidth) tuned nature.

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 702 , 10 July 2012 , pp. 298 - 331
Copyright
Copyright © Cambridge University Press 2012

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References

1. Albarède, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding and Chevron patterns in cylinder wakes. Phys. Fluids A 4, 744–756.CrossRefGoogle Scholar
2. 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
3. Bandyopadhyay, P. R. 1994 A low-dimensional structural model of a turbulent boundary layer separating intermittently in space. NUWC-NPT Tech. Rep. 10,400, Naval Undersea Warfare Center, Newport, Rhode Island, USA; also in Separated and Complex Flows – 1995, Proceedings of the ASME/JSME Fluids Engineering Conference and Exhibition, Hilton Head, SC, US; 13–18 August 1006, pp. 75–82.Google Scholar
4. Bandyopadhyay, P. R. 2009 Swimming and flying in nature – the route toward applications: the Freeman scholar lecture. Trans. ASME: J. Fluids Engng 131, 031801-29.Google Scholar
5. Bandyopadhyay, P. R., Beal, D. N. & Menozzi, A. 2008a Biorobotic insights into how animals swim. J. Expl Biol. 211, 206–214.CrossRefGoogle ScholarPubMed
6. Bandyopadhyay, P. R., Castano, J. M., Nedderman, W. H. & Donnelly, M. J. 2000 Experimental simulation of fish-inspired unsteady vortex dynamics on a rigid cylinder. Trans. ASME: J. Fluids Engng 122 (2), 219–238.Google Scholar
7. Bandyopadhyay, P. R. & Hrubes, J. D. A. 2009 Leading edge vortex in flapping fins. Video (captioned and edited), Naval Undersea Warfare Center, Newport, RI, USA, 180 s, QuickTime 9 MB (contact first author, or http://ecommons.library.cornell.edu/handle/1813/13736, or see~SI).Google Scholar
8. Bandyopadhyay, P. R., Singh, S. N., Thivierge, D. P., Annaswamy, A. M., Leinhos, H. A., Fredette, A. R. & Beal, D. N. 2008b Synchronization of animal-inspired multiple fins in an underwater vehicle using olivo-cerebellar dynamics. IEEE J. Ocean. Engng 33 (4), 563–578.CrossRefGoogle Scholar
9. Beal, D. N. & Bandyopadhyay, P. R. 2007 A harmonic model of hydrodynamic forces produced by a flapping fin. Exp. Fluids 43, 675–682.CrossRefGoogle Scholar
10. Bishop, R. E. D. & Hassan, A. Y. 1964 The lift and drag forces on a circular cylinder oscillating in a flowing fluid. Proc. R. Soc. Lond. A 277, 57–75.Google Scholar
11. Bomphrey, R. J., Lawson, N. J., Harding, N. J., Taylor, G. K. & Thomas, A. L. R. 2005 The aerodynamics of manduca sexta : digital particle image velocimetry analysis of the leading edge vortex. J. Expl Biol. 208, 1079–1094.CrossRefGoogle ScholarPubMed
12. Broze, G. & Hussain, F. 1994 Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J. Fluid Mech. 263, 93–132.CrossRefGoogle Scholar
13. Clark, R. P. & Smits, A. J. 2006 Thrust production and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 562, 415–429.CrossRefGoogle ScholarPubMed
14. Combes, S. A. & Daniel, T. L. 2003 Flexural stiffness in insect wings I. Scaling and the influence of wing venation. J. Expl Biol. 206, 2979–2987.CrossRefGoogle ScholarPubMed
15. Dickinson, M. H., Lehmann, F. & Sane, S. P. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960.CrossRefGoogle ScholarPubMed
16. Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78, 535–560.CrossRefGoogle Scholar
17. Dowling, A. P. & Ffowcs Williams, J. E. 1983 Sound and Sources of Sound. Ellis Harwood, John Wiley.Google Scholar
18. von Ellenrieder, K. D., Parker, K. & Soria, J. 2008 Fluid mechanics of flapping wings. Exp. Therm. Fluid Sci. 32, 1578–1589.CrossRefGoogle Scholar
19. Ellington, C. P. 1984 The aerodynamics of hovering insect flight, IV. Aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B 305, 79–113.Google Scholar
20. Ellington, C. P. 1999 The novel aerodynamics of insect flight: applications to micro-air vehicles. J. Expl Biol. 202, 3439–3448.CrossRefGoogle ScholarPubMed
21. Feldkamp, S. D. 1987 Swimming in the California sealion: morphometrics. J. Expl Biol. 131, 117–135.CrossRefGoogle ScholarPubMed
22. Fish, F. E. 1993 Power output and propulsive efficiency of swimming bottlenose dolphins (tursiops trunctus). J. Expl Biol. 185, 179–193.CrossRefGoogle Scholar
23. Fish, F. E. 1998 Origin of cetacean flukes. In The Emergence of Whales: Evolutionary Patterns in the Origin of Cetaceans (ed. Thewissen, J. G. M. ). Springer.Google Scholar
24. Garrick, I. E. 1936 Propulsion of a flapping and oscillating aerofoil. NACA Rep. 567.Google Scholar
25. Goman, M. & Khrabrov, A 1994 State-space representation of aerodynamic characteristics of an aircraft at high angles of attack. J. Aircraft 31 (5), 1109–1115.CrossRefGoogle Scholar
26. Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257–311.CrossRefGoogle Scholar
27. Heathcote, S., Wang, Z. & Gursul, I. 2008 Effect of spanwise flexibility on flapping wing propulsion. J. Fluids Struct. 24, 183–199.CrossRefGoogle Scholar
28. Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
29. Isogai, K., Shinmoto, Y. & Watanabe, Y. 1999 Effects of dynamic stall on propulsive efficiency and thrust of flapping aerofoils. AIAA J. 37, 1145–1151.CrossRefGoogle Scholar
30. Jones, K. D., Lund, T. C. & Platzer, M. F. 2001 Experimental and computational investigation of flapping wing propulsion for micro-air vehicles. In Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Application (ed. Mueller, T. J. ). Prog. Astro. Aero. , 195. pp. 307–339. AIAA.Google Scholar
31. von Kármán, T. & Burgers, J. M. 1934 General aerodynamic theory: perfect fluids. In Aerodynamic Theory, vol. II (ed. Durand, W. F. ). Springer.Google Scholar
32. Kazantsev, V. B., Nekorkin, V. I., Makarenko, V. I. & Llinas, R. 2004 Self-referential phase reset based on inferior olive oscillator dynamics. Proc. Natl Acad. Sci 101 (52), 18183–18188.CrossRefGoogle ScholarPubMed
33. Kern, S. & Koumoutsakos, P. 2006 Simulations of optimized anguilliform swimming. J. Expl Biol. 209, 4841–4857.CrossRefGoogle ScholarPubMed
34. Knight, K. 2009 Inside JEB: Coriolis keeps leading edge vortex in place. J. Expl Biol 212, i–ii.Google Scholar
35. Lentink, D. & Dickinson, M. H. 2009 Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Expl Biol 212, 2705–2719.CrossRefGoogle ScholarPubMed
36. Lighthill, J. 1975 Mathematical biofluiddynamics. CBMS-NSF Regional Conf. Series in Applied Mathematics, vol. 17. Society for Industrial Applied Mathematics.Google Scholar
37. Lovvorn, J. R., Liggins, G. A., Borstad, M. H., Calisal, S. M. & Mikkelsen, J. 2001 Hydrodynamic drag of diving birds: effects of body size, body shape and feathers at steady speeds. J. Expl Biol 204, 1547–1557.CrossRefGoogle ScholarPubMed
38. Mangalam, S. M. 2004 Real-time extraction of hydrodynamic flow characteristics using surface signatures. IEEE J. Ocean. Engng 29, 622–630.CrossRefGoogle Scholar
39. Menozzi, A., Leinhos, H. A., Beal, D. N. & Bandyopadhyay, P. R. 2008 Open-loop control of a multi-fin biorobotic underwater vehicle. IEEE J. Ocean. Engng 33 (2), 59–68.CrossRefGoogle Scholar
40. Miklosovic, D. S., Murray, M. M., Howle, L. E. & Fish, F. E. 2004 Leading edge tubercles delay stall on humpback whale (megaptera novaeangliae). Phys. Fluid 16, L39–L42.CrossRefGoogle Scholar
41. Miller, P. J. O., Johnson, M. P., Tyack, P. L. & Terray, E. A. 2004 Swimming gaits, passive drag and buoyancy of diving sperm whale Physeter macrocephalus . J. Expl Biol. 207, 1953–1967.CrossRefGoogle Scholar
42. Perry, A. E. & Chong, M. S. 1987 A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19, 125–155.CrossRefGoogle Scholar
43. Psaras, S. T. 2008 Comparison of linear and nonlinear processing with acoustic vector sensors. MS (Appl. Phys.) Thesis, Naval Postgraduate School, Monterey, CA.Google Scholar
44. Read, D. A., Hover, F. S. & Triantafyllou, M. S 2003 Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17, 163–183.CrossRefGoogle Scholar
45. Ramamurti, R. & Sandberg, W. 2001 Simulation of flow about flapping aerofoils using finite element incompressible flow solver. AIAA J. 39, 253–260.CrossRefGoogle Scholar
46. Rohr, J. J. & Fish, F. E. 2004 Strouhal numbers and optimization of swimming by odontocete cetaceans. J. Expl Biol. 207, 1633–1642.CrossRefGoogle ScholarPubMed
47. Sane, S. & Dickinson, M. 2001 The control of flight force by a flapping wing: lift and drag production. J. Expl Biol. 204, 2607–2626.CrossRefGoogle ScholarPubMed
48. Sato, K., Naito, Y., Kato, A., Niizuma, Y., Watanuki, Y., Charrassin, J. B., Bost, C.-A., Handrich, Y. & Maho, Y. L. 2002 Buoyancy and maximal diving depth in penguins: Do they control inhaling air volume? J. Expl Biol. 205, 1189–1197.CrossRefGoogle ScholarPubMed
49. Schlichting, H. 1979 Boundary Layer Theory, 7th edn. McGraw-Hill.Google Scholar
50. Sears, W. R. & Tellionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 28, 215–235.CrossRefGoogle Scholar
51. Shariff, K., Pulliam, T. H. & Ottino, J. M. 1991 A dynamical systems analysis of kinematics in the time-periodic wake of a circular cylinder. Lectures in Applied Mathematics 28, 613–646.Google Scholar
52. Shenk, A. L. 1977 Calculus and Analytical Geometry, 4th edn. Scott, Foresman and Company.Google Scholar
53. Skop, R. A. & Balasubramanian, S. 1997 A new twist on an old model for vortex-induced vibrations. J. Fluids Struct. 11, 395–412.CrossRefGoogle Scholar
54. Techet, A. H. 2008 Propulsive performance of biologically inspired flapping foils at high Reynolds numbers. J. Expl Biol. 211, 274–279.CrossRefGoogle ScholarPubMed
55. Theodorsen, T. 1935 General throery of aerodynamic instability and the mechanism of flutter. NACA TR 496.Google Scholar
56. Triantafyllou, M. S., Hover, F. S., Techet, A. H. & Yue, D. K. P. 2005 Review of hydrodynamic scaling laws in aquatic locomotion and fishlike swimming. Appl. Mech. Rev. 58, 226–237.CrossRefGoogle Scholar
57. Triantafyllou, M. S., Techet, A. H. & Hover, F. S 2004 Review of experimental work on biomimetic foils. IEEE J. Ocean. Engng 29, 585–594.CrossRefGoogle Scholar
58. Triantafyllou, M. S. & Triantafyllou, G. S. 1995 An efficient swimming machine. Sci. Am. 272, 64–70.CrossRefGoogle Scholar
59. Tuncer, I. & Kaya, M. 2005 Optimization of flapping aerofoils for maximum thrust and propulsive efficiency. AIAA J. 43 (11), 2329–2336.CrossRefGoogle Scholar
60. Tuncer, I. H., Walz, R. & Platzer, M. F. 1998 A computational study on the dynamic stall of a flapping aerofoil. AIAA Paper 98-2519.CrossRefGoogle Scholar
61. Usherwood, J. R. & Ellington, C. P. 2002 The aerodynamics of revolving wings I-II. J. Expl Biol. 205, 1547–1576.CrossRefGoogle Scholar
62. Vandenberg, C. & Ellington, C. E. 1997 The three-dimensional leading-edge vortex of a ‘hovering’ model hawkmoth. Phil. Trans. R. Soc., Lond. B 352, 329–340.CrossRefGoogle Scholar
63. Vandenberghe, N., Zhang, J. & Childress, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147–155.CrossRefGoogle Scholar
64. Van Dommelen, L. L. & Cowley, S. J. 1990 On the Lagrangian description of unsteady boundary layer separation. Part 1. General theory. J. Fluid Mech. 210, 593–626.CrossRefGoogle Scholar
65. Wakeling, J. M. & Ellington, C. P. 1997 Dragonfly flight. III. Lift and power requirements. J. Expl Biol. 200, 583–600.CrossRefGoogle ScholarPubMed
66. Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355–381.CrossRefGoogle Scholar
67. Wootton, R. J. 1992 Functional morphology of insect wings. Annu. Rev. Entomol. 37, 113–140.CrossRefGoogle Scholar
68. Wootton, R. J. 1993 Leading edge section and asymmetric twisting in the wings of flying butterflies (insecta, papilionoidea). J. Expl Biol. 180, 105–117.CrossRefGoogle Scholar
69. Young, J. & Lai, J. C. S. 2004 Oscillation frequency and amplitude effects on the wake of a plunging aerofoil. AIAA J. 42, 2042–2052.CrossRefGoogle Scholar

Bandyopadhyay et al. supplementary movie

The fin is driven by a van der Pol oscilaltor. The roll versus pitch angles are plotted for the case shown in Movie 2.

Download Bandyopadhyay et al. supplementary movie(Video)
Video 1 MB
Supplementary material: PDF

Bandyopadhyay et al. supplementary material

Supplementary data and figures

Download Bandyopadhyay et al. supplementary material(PDF)
PDF 1.8 MB

Bandyopadhyay et al. supplementary movie

The fin is driven by a van der Pol oscilaltor. The roll versus pitch angles are plotted for the case shown in Movie 2.

Download Bandyopadhyay et al. supplementary movie(Video)
Video 333.5 KB

Bandyopadhyay et al. supplementary movie

The fin is given a square pulse of external disturbance. The disturbance rejection property is demonstrated. See Movie 1 for pitch versus roll angles

Download Bandyopadhyay et al. supplementary movie(Video)
Video 2 MB

Bandyopadhyay et al. supplementary movie

The fin is given a square pulse of external disturbance. The disturbance rejection property is demonstrated. See Movie 1 for pitch versus roll angles

Download Bandyopadhyay et al. supplementary movie(Video)
Video 2.5 MB

Bandyopadhyay et al. supplementary movie

The formation of the LEV in twisted and untwisted fins is shown by injecting blue and red dyes near the leading edge from two sides of the fin. The nature of the wake vortices and thrust are also shown

Download Bandyopadhyay et al. supplementary movie(Video)
Video 32.2 MB

Bandyopadhyay et al. supplementary movie

The formation of the LEV in twisted and untwisted fins is shown by injecting blue and red dyes near the leading edge from two sides of the fin. The nature of the wake vortices and thrust are also shown

Download Bandyopadhyay et al. supplementary movie(Video)
Video 8.8 MB