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Universal scaling law for drag-to-thrust wake transition in flapping foils

Published online by Cambridge University Press:  07 June 2019

N. S. Lagopoulos Open the ORCID record for N. S. Lagopoulos [Opens in a new window] ,
G. D. Weymouth Open the ORCID record for G. D. Weymouth [Opens in a new window]  and
B. Ganapathisubramani Open the ORCID record for B. Ganapathisubramani [Opens in a new window]
N. S. Lagopoulos*
Affiliation:
Aerodynamics and Flight Mechanics Group, University of Southampton, UK
G. D. Weymouth
Affiliation:
Southampton Marine and Maritime Institute, University of Southampton, UK
B. Ganapathisubramani
Affiliation:
Aerodynamics and Flight Mechanics Group, University of Southampton, UK
*
†Email address for correspondence: N.Lagopoulos@soton.ac.uk
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Abstract

Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude $A$ and frequency $f$ because the transition also depends sensitively on other details of the kinematics. In this work we replace $A$ with the cycle-averaged swept trajectory ${\mathcal{T}}$ of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless ${\mathcal{T}}-f$ we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal $St_{{\mathcal{T}}}\rightarrow 1$. Physically, the product ${\mathcal{T}}f$ expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds $U_{\infty }$. The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.

JFM classification

Biological Fluid Dynamics: Propulsion Vortex Flows: Vortex shedding Wakes/Jets: Vortex streets
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , R1
Copyright
© 2019 Cambridge University Press 

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