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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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References

Andersen, A., Bohr, T., Schnipper, T. & Walther, J. H. 2017 Wake structure and thrust generation of a flapping foil in two-dimensional flow. J. Fluid Mech. 812, R4.Google Scholar
Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.Google Scholar
Birnbaum, W. 1924 Das ebene Problem des schlagenden Flügels. Z. Angew. Math. Mech. 4 (4), 277–292.Google Scholar
Bohl, D. G. & Koochesfahani, M. M. 2009 MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. J. Fluid Mech. 620, 63–88.Google Scholar
Cleaver, D. J., Wang, Z. & Gursul, I. 2012 Bifurcating flows of plunging aerofoils at high Strouhal numbers. J. Fluid Mech. 708, 349–376.Google Scholar
Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224.Google Scholar
Godoy-Diana, R., Aider, J. L. & Wesfreid, J. E. 2008 Transitions in the wake of a flapping foil. Phys. Rev. E 77 (1), 016308.Google Scholar
Godoy-Diana, R., Marais, C., Aider, J. L. & Wesfreid, J. E. 2009 A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 23–32.Google Scholar
Koochesfahani, M. M. 1989 Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27 (9), 1200–1205.Google Scholar
Maertens, A. P. & Weymouth, G. D. 2015 Accurate Cartesian-grid simulations of near-body flows at intermediate Reynolds numbers. Comput. Meth. Appl. Mech. Engng 283, 106–129.Google Scholar
Mittal, R. & Balachandar, S. 1995 Effect of threedimensionality on the lift and drag of nominally two-dimensional cylinders. Phys. Fluids 7 (8), 1841–1865.Google Scholar
Platzer, M. & Jones, K. 2008 Flapping wing aerodynamics-progress and challenges. In 44th AIAA Aerospace Sciences Meeting and Exhibit, p. 500. AIAA.Google Scholar
Polet, D. T., Rival, D. E. & Weymouth, G. D. 2015 Unsteady dynamics of rapid perching manoeuvres. J. Fluid Mech. 767, 323–341.Google Scholar
Ramamurti, R. & Sandberg, W. 2001 Computational study of 3D flapping foil flows. In 39th Aerospace Sciences Meeting and Exhibit, p. 605. AIAA.Google Scholar
Read, D. A., Hover, F. S. & Triantafyllou, M. S. 2003 Forces on oscillating foils for propulsion and manoeuvering. J. Fluids Struct. 17 (1), 163–183.Google Scholar
Streitlien, K. & Triantafyllou, G. S. 1998 On thrust estimates for flapping foils. J. Fluids Struct. 12 (1), 47–55.Google Scholar
Taylor, G. K., Nudds, R. L. & Thomas, A. L. R. 2003 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425 (6959), 707.Google Scholar
Thiria, B., Goujon-Durand, S. & Wesfreid, J. E. 2006 The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123–147.Google Scholar
Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7 (2), 205–224.Google Scholar
Triantafyllou, M. S., Techet, A. H. & Hover, F. S. 2004 Review of experimental work in biomimetic foils. IEEE J. Ocean. Engng 29 (3), 585–594.Google Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3 (12), 2835–2837.Google Scholar
Vial, M., Bellon, L. & Hernández, R. H. 2004 Mechanical forcing of the wake of a flat plate. Exp. Fluids 37 (2), 168–176.Google Scholar
Von Kármán, T. 1935 General aerodynamic theory-perfect fluids. Aerodynamic Theory 2, 346–349.Google Scholar
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37 (1), 183–210.Google Scholar
Weymouth, G. D. & Yue, D. K. P. 2011 Boundary data immersion method for Cartesian-grid simulations of fluid–body interaction problems. J. Comput. Phys. 230 (16), 6233–6247.Google Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355–381.Google Scholar