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Published August 15, 2009 | Published
Journal Article Open

Biofluiddynamic scaling of flapping, spinning and translating fins and wings


Organisms that swim or fly with fins or wings physically interact with the surrounding water and air. The interactions are governed by the morphology and kinematics of the locomotory system that form boundary conditions to the Navier–Stokes (NS) equations. These equations represent Newton's law of motion for the fluid surrounding the organism. Several dimensionless numbers, such as the Reynolds number and Strouhal number, measure the influence of morphology and kinematics on the fluid dynamics of swimming and flight. There exists, however, no coherent theoretical framework that shows how such dimensionless numbers of organisms are linked to the NS equation. Here we present an integrated approach to scale the biological fluid dynamics of a wing that flaps, spins or translates. Both the morphology and kinematics of the locomotory system are coupled to the NS equation through which we find dimensionless numbers that represent rotational accelerations in the flow due to wing kinematics and morphology. The three corresponding dimensionless numbers are (1) the angular acceleration number, (2) the centripetal acceleration number, and (3) the Rossby number, which measures Coriolis acceleration. These dimensionless numbers consist of length scale ratios, which facilitate their geometric interpretation. This approach gives fundamental insight into the physical mechanisms that explain the differences in performance among flapping, spinning and translating wings. Although we derived this new framework for the special case of a model fly wing, the method is general enough to make it applicable to other organisms that fly or swim using wings or fins.

Additional Information

© The Company of Biologists Ltd 2009. Accepted 5 February 2009. First published online July 31, 2009. We gratefully acknowledge Will Dickson, Koert Lindenburg and John Dabiri for proof reading preliminary versions of the manuscript. We thank Thomas Daniel for proof reading the final manuscript. We thank Johan van Leeuwen and GertJan van Heijst for hearty support, encouragement and proof reading of the various versions of the manuscript. This research is supported by NWO-ALW grant 817.02.012 to D.L. and by NSF grant IBN-0217229 to M.H.D.

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August 20, 2023
August 20, 2023