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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 scale. A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number, St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U). St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions. 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 dolphins, sharks and bony fish, 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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“Previous work has demonstrated that bony fish swimming tends to fall in a significantly lower range of 0 . 2 < St < 0 . 4 [27].”
From Measurements and modeling of swimming speed dependence on stroke frequency in scyphozoan jellyfish (2026) · cites this paper as [27]
In the citing paper
“Boletzky [36] describes how juvenile squid use cilia to propel themselves out of the viscous egg sac and use jet propulsion once they are free in the water.”
From Reynolds number limits for jet propulsion: a numerical study of simplified jellyfish (2011) · cites this paper as [36]
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Metadata
- Authors
- Taylor, G. K., Nudds, R. L., & Thomas, A. L. R.
- Affiliations
- University of Oxford (GB)
- University of Leeds (GB)
- Corresponding author
- Taylor, Graham K.
- Venue
- Nature
- Published
- 2003-10
- Volume
- 425
- Issue
- 6959
- Pages
- 707-711
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- OpenAlex
- W2003172346
- Cited by
- 994
- References
- 42
- Citations by year
- 2026: 53 · 2025: 45 · 2024: 71 · 2023: 56 · 2022: 43 · 2021: 70 · 2020: 69 · 2019: 50 · 2018: 50 · 2017: 56 · 2016: 55 · 2015: 51 · 2014: 43 · 2013: 44 · 2012: 32
- FWCI
- 40.63
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Research Explorer (The University of Manchester) (submittedVersion)
- Topics
- Physical Sciences › Engineering › Aerospace Engineering › Biomimetic flight and propulsion mechanisms
- Physical Sciences › Engineering › Computational Mechanics › Fluid Dynamics and Vibration Analysis
- Physical Sciences › Engineering › Computational Mechanics › Fluid dynamics and aerodynamics studies
- Keywords
- Strouhal number, propulsive efficiency, vortex shedding, birds, bats, insects, flying animals, dynamic similarity, natural selection
- MeSH terms
- Animals, Biomechanical Phenomena, Birds, Chiroptera, Flight, Animal, Insecta, Models, Biological, Monte Carlo Method, Swimming, Wings, Animal
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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