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Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency

Abstract

Dimensionless numbers are important in biomechanics because their constancy can imply dynamic similarity between systems, despite possible differences in medium or scale1. A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number1, St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U)2,3,4,5,6,7,8. St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions6,8. Propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < St < 0.4 (refs 3–8). Because natural selection is likely to tune animals for high propulsive efficiency, we expect it to constrain the range of St that animals use. This seems to be true for dolphins2,3,4,5, sharks3,4,5 and bony fish3,4,5, which swim at 0.2 < St < 0.4. Here we show that birds, bats and insects also converge on the same narrow range of St, but only when cruising. Tuning cruise kinematics to optimize St therefore seems to be a general principle of oscillatory lift-based propulsion.

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Figure 1: Wake structures for root-flapping and heaving hinged flat plates at varying St.
Figure 2: Strouhal number for 42 species of birds, bats and insects in unconfined, cruising flight.
Figure 3: Histogram from Monte Carlo analysis recalculating St for 50,000 iterations randomizing the order of residuals from regressing log(f), log(U) and log(A) against log(m).

References

  1. Alexander, R. M. Principles of Animal Locomotion (Princeton Univ. Press, Princeton, 2003)

    Book  Google Scholar 

  2. Rohr, J. J. et al. Observations of Dolphin Swimming Speed and Strouhal Number. Space and Naval Warfare Systems Center Technical Report No. 1769 (Space and Naval Warfare Systems Center, San Diego, 1998)

    Book  Google Scholar 

  3. Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3, 2835–2837 (1991)

    CAS  Article  Google Scholar 

  4. Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205–224 (1993)

    Article  Google Scholar 

  5. Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32, 33–53 (2000)

    MathSciNet  Article  Google Scholar 

  6. Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72 (1998)

    MathSciNet  Article  Google Scholar 

  7. Read, D. A., Hover, F. S. & Triantafyllou, M. S. Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17, 163–183 (2003)

    Article  Google Scholar 

  8. Wang, Z. J. Vortex shedding and frequency selection in flapping flight. J. Fluid Mech. 410, 323–341 (2000)

    Article  Google Scholar 

  9. Huang, R. F., Wu, J. Y., Jeng, J. H. & Chen, R. C. Surface flow and vortex shedding of an impulsively started wing. J. Fluid Mech. 441, 265–292 (2001)

    CAS  Article  Google Scholar 

  10. Motani, R. Scaling effects in caudal fin propulsion and the speed of ichthyosaurs. Nature 415, 309–312 (2002)

    Article  Google Scholar 

  11. Bandyopadhyay, P. R., Castano, J. M., Nedderman, W. H. & Donnelly, M. J. Experimental simulations of fish-inspired unsteady vortex-dynamics on a rigid cylinder. J. Fluids Eng. 122, 219–238 (2000)

    Article  Google Scholar 

  12. Wolfgang, M. J., Anderson, J. M., Grosenbaugh, M. A., Yue, D. K. P. & Triantafyllou, M. S. Near-body flow dynamics in swimming fish. J. Exp. Biol. 202, 2303–2327 (1999)

    Google Scholar 

  13. Gopalkrishnan, R., Triantafyllou, M. S., Triantafyllou, G. S. & Barrett, D. Active vorticity control in a shear flow using a flapping foil. J. Fluid Mech. 274, 1–21 (1994)

    Article  Google Scholar 

  14. Withers, P. C. & Timko, P. L. The significance of ground effect to the aerodynamic cost of flight and energetics of the black skimmer (Rhyncops nigra). J. Exp. Biol. 70, 13–26 (1977)

    Google Scholar 

  15. Baker, P. S. & Cooter, R. J. The natural flight of the migratory locust, Locusta migratoria L. I. Wing movements. J. Comp. Physiol. A 131, 79–87 (1979)

    Article  Google Scholar 

  16. Baker, P. S., Gewecke, M. & Cooter, R. J. The natural flight of the migratory locust, Locusta migratoria L. III. Wing-beat frequency, flight speed and attitude. J. Comp. Physiol. 141, 233–237 (1981)

    Article  Google Scholar 

  17. Scholey, K. D. Developments in Vertebrate Flight: Climbing and Gliding of Mammals and Reptiles and the Flapping Flight of Birds. Thesis, Univ., Bristol (1983)

    Google Scholar 

  18. Videler, J. J., Groenewegen, A., Gnodde, M. & Vossebelt, G. Indoor flight experiments with trained kestrels II. The effect of added weight on flapping flight kinematics. J. Exp. Biol. 134, 185–199 (1988)

    Google Scholar 

  19. Pennycuick, C. J. Span-ratio analysis used to estimate effective lift:drag ratio in the double-crested cormorant Phalacrocorax auritus from field observations. J. Exp. Biol. 142, 1–15 (1989)

    Google Scholar 

  20. Dudley, R. & DeVries, P. J. Flight physiology of migrating Urania fulgens (Uranidae) moths: kinematics and aerodynamics of natural free flight. J. Comp. Physiol. A 167, 145–154 (1990)

    Article  Google Scholar 

  21. Pennycuick, C. J. Predicting wingbeat frequency and wavelength of birds. J. Exp. Biol. 150, 171–185 (1990)

    Google Scholar 

  22. Tobalske, B. W. & Dial, K. P. Neuromuscular control and kinematics of intermittent flight in budgerigars (Melopsittacus undulatus). J. Exp. Biol. 187, 1–18 (1994)

    CAS  PubMed  Google Scholar 

  23. Tobalske, B. W. Neuromuscular control and kinematics of intermittent flight in the European starling (Sturnus vulgaris). J. Exp. Biol. 198, 1259–1273 (1995)

    CAS  PubMed  Google Scholar 

  24. Pennycuick, C. J. Wingbeat frequency of birds in steady cruising flight: new data and improved predictions. J. Exp. Biol. 199, 1613–1618 (1996)

    CAS  PubMed  Google Scholar 

  25. Tobalske, B. W. Scaling of muscle composition, wing morphology, and intermittent flight behavior in woodpeckers. Auk 113, 151–177 (1996)

    Article  Google Scholar 

  26. Tobalske, B. W. & Dial, K. P. Flight kinematics of black-billed magpies and pigeons over a wide range of speeds. J. Exp. Biol. 199, 263–280 (1996)

    CAS  PubMed  Google Scholar 

  27. Tobalske, B. W., Peacock, W. L. & Dial, K. P. Kinematics of flap-bounding flight in the zebra finch over a wide range of speeds. J. Exp. Biol. 202, 1725–1739 (1999)

    PubMed  Google Scholar 

  28. Pennycuick, C. J. Speeds and wingbeat frequencies of migrating birds compared with calculated benchmarks. J. Exp. Biol. 204, 3283–3294 (2001)

    CAS  PubMed  Google Scholar 

  29. Bullen, R. D. & McKenzie, N. L. Scaling bat wingbeat frequency and amplitude. J. Exp. Biol. 205, 2615–2626 (2002)

    CAS  PubMed  Google Scholar 

  30. Rayner, J. M. V. Form and function in avian flight. Curr. Ornithol. 5, 1–45 (1987)

    Google Scholar 

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Acknowledgements

We thank P. Johnson for comments on the statistical analysis. This work was funded by grants from the Biotechnology and Biological Sciences Research Council and a Christopher Welch Scholarship. G.K.T. is a Royal Commission for the Exhibition of 1851 Research Fellow and Weir Junior Research Fellow at University College, Oxford.

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Correspondence to Graham K. Taylor.

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Supplementary information

41586_2003_BFnature02000_MOESM1_ESM.pdf

Supplementary Figure 1: Regression and residual plots for log-log regressions of f, U, and A against m. These provide diagnostic plots for the regressions used in the the Monte Carlo analysis of whether f, U and A covary appropriately to constrain St. (PDF 79 kb)

41586_2003_BFnature02000_MOESM2_ESM.pdf

Supplementary Figure 2: Histograms and normal probability plots of St data. A. Plots of untransformed St data. B. Plots of data following square root transformation. (PDF 47 kb)

41586_2003_BFnature02000_MOESM3_ESM.pdf

Supplementary Figure 3: Regression and residual plots for log-log regression of St against m for the birds in our dataset. (PDF 66 kb)

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Taylor, G., Nudds, R. & Thomas, A. Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425, 707–711 (2003). https://doi.org/10.1038/nature02000

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