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


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).


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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


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)


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)


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).

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