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.
“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]
Juvenile squid use cilia to propel themselves out of the viscous egg sac and use jet propulsion once they are free in the water.
“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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- 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
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