Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-01T04:49:51.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamic wake resonance as an underlying principle of efficient unsteady propulsion

Published online by Cambridge University Press:  08 August 2012

K. W. Moored ,
P. A. Dewey ,
A. J. Smits  and
H. Haj-Hariri
K. W. Moored*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
P. A. Dewey
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
A. J. Smits
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
H. Haj-Hariri
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A linear spatial stability analysis is performed on the velocity profiles measured in the wake of an actively flexible robotic elliptical fin to find the frequency of maximum spatial growth, that is, the hydrodynamic resonant frequency of the time-averaged jet. It is found that: (i) optima in propulsive efficiency occur when the driving frequency of a flapping fin matches the resonant frequency of the jet profile; (ii) there can be multiple wake resonant frequencies and modes corresponding to multiple peaks in efficiency; and (iii) some wake structures transition from one pattern to another when the wake instability mode transitions. A theoretical framework, termed wake resonance theory, is developed and utilized to explain the mechanics and energetics of unsteady self-propulsion. Experimental data are used to validate the theory. The analysis, although one-dimensional, captures the performance exhibited by a three-dimensional propulsor, showing the robustness and broad applicability of the technique.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Instability: Instability Biological Fluid Dynamics: Swimming/flying Wakes/Jets: Wakes/Jets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 329 - 348
Copyright
Copyright © Cambridge University Press 2012

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

1. 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
2. Bridges, T. J. & Morris, P. J. 1984a Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55 (3), 437–460.CrossRefGoogle Scholar
3. Bridges, T. J. & Morris, P. J. 1984b Spectral calculations of the spatial stability of non-parallel boundary layers, AIAA Paper 84-0437, AIAA 22nd Aerospace Sciences Meeting.Google Scholar
4. 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 (3), 461–472.CrossRefGoogle ScholarPubMed
5. 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, 331–366.CrossRefGoogle ScholarPubMed
6. Chan, A. S., Dewey, P. A., Jameson, A., Liang, C. & Smits, A. J. 2011 Vortex suppression and drag reduction in the wake of counter rotating cylinders. J. Fluid Mech. 679, 343–382.CrossRefGoogle Scholar
7. Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
8. 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
9. 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, 245–266.CrossRefGoogle Scholar
10. Gantmacher, F. R. 1959 The Theory of Matrices, vol. 1. Chelsea.Google Scholar
11. Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.CrossRefGoogle Scholar
12. Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87 (1), 33–54.CrossRefGoogle Scholar
13. Haj-Hariri, H. 1988 Transformations reducing the order of the parameter in differential eigenvalue problems. J. Comput. Phys. 77 (2), 472–484.CrossRefGoogle Scholar
14. Helmholtz, H. 1868 On discontinuous movements of fluids. Phil. Mag 36 (4), 337–346.CrossRefGoogle Scholar
15. Hultmark, M., Leftwich, M. & Smits, A. J. 2007 Flowfield measurements in the wake of a robotic lamprey. Exp. Fluids 43 (5), 683–690.CrossRefGoogle ScholarPubMed
16. Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
17. Jiménez, J. M. 2002 Low Reynolds number studies in the wake of a submarine model using particle image velocimetry. Master’s thesis, Princeton University, Princeton, NJ, USA.Google Scholar
18. von Kármán, T. & Burgers, J. M. 1935 General aerodynamic theory-perfect fluids. In Aerodynamic Theory II, (ed. W. F. Durand), pp. 280–310. Dover Publications, 1963.Google Scholar
19. Karniadakis, G. & Triantafyllou, G. 1989 Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199, 441–469.CrossRefGoogle Scholar
20. Kelvin W. T., Lord 1871 Hydrokinetic solutions and observations. Phil. Mag. 42, 362–377.Google Scholar
21. Koochesfahani, M. M. 1989 Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27, 1200–1205.CrossRefGoogle Scholar
22. Lauder, G. V. & Drucker, E. G. 2002 Forces, fishes, and fluids: hydrodynamic mechanisms of aquatic locomotion. Physiology 17 (6), 235–240.CrossRefGoogle ScholarPubMed
23. Lentink, D., Muijres, F. T., Donker-Duyvis, F. J. & Van Leeuwen, J. L. 2008 Vortex-wake interactions of a flapping foil that models animal swimming and flight. J. Expl Biol. 211 (2), 267–273.CrossRefGoogle ScholarPubMed
24. Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving aerofoil in a viscous flow. J. Fluid Mech. 492, 339–362.CrossRefGoogle Scholar
25. Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73 (3), 497–520.CrossRefGoogle Scholar
26. Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51 (2), 233–272.CrossRefGoogle Scholar
27. Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689–703.CrossRefGoogle Scholar
28. Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
29. Stuart, J. T. 1987 Instability, three-dimensional effects, and transition in shear flows. In Perspectives in Turbulence Studies (ed. Meier, H. U. & Bradshaw, P. ). pp. 1–25. Springer.Google Scholar
30. 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 425 (6959), 707–711.CrossRefGoogle Scholar
31. Triantafyllou, G. S., Triantafyllou, M. S. & Chryssostomidis, C. 1986 On the formation of vortex streets behind stationary cylinders. J. Fluid Mech. 170, 461–477.CrossRefGoogle Scholar
32. Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7 (2), 205–224.CrossRefGoogle Scholar
33. Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3, 2835.CrossRefGoogle Scholar
34. Tytell, E. D., Borazjani, I., Sotiropoulos, F., Baker, T. V., Anderson, E. J. & Lauder, G. V. 2010 Disentangling the functional roles of morphology and motion in the swimming of fish. Integr. Compar. Biol. 50 (6), 1140.CrossRefGoogle ScholarPubMed
35. 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