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Scaling law for the lift force of autorotating falling seeds at terminal velocity

Published online by Cambridge University Press:  27 November 2017

Injae Lee  and
Haecheon Choi Open the ORCID record for Haecheon Choi [Opens in a new window]
Injae Lee
Affiliation:
Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 08826, Korea
Haecheon Choi*
Affiliation:
Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 08826, Korea
*
†Email address for correspondence: [email protected] Also at Institute of Advanced Machines and Design, Seoul National University, Seoul 08826, Korea
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Abstract

We provide a scaling law for the lift force of autorotating falling seeds at terminal velocity to describe the relation among the lift force, seed geometry and terminal descending and rotating velocities. Two theories, steady wing-vortex theory and actuator-disk theory, are examined to derive the scaling law. In the steady wing-vortex theory, the strength of a leading-edge vortex is scaled with the circulation around a wing and the lift force is modelled by the time derivative of vortical impulse, whereas the conservations of mass, linear and angular momentum, and kinetic energy across the autorotating falling seed are applied in the actuator-disk theory. To examine the validity of the theoretical results, an unsteady three-dimensional numerical simulation is conducted for flow around an autorotating seed (Acer palmatum) during free fall. The sectional lift coefficient predicted from the steady wing-vortex theory reasonably agrees with that from the numerical simulation, whereas the actuator-disk theory fails to provide an estimation of the sectional lift coefficient. The weights of 11 different species of autorotating falling seeds fall on the scaling law derived from the steady wing-vortex theory, suggesting that even a simple theoretical approach can explain how falling seeds support their weights by autorotation once the circulation from a leading-edge vortex is properly included in the theory.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 406 - 420
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Copyright
© 2017 Cambridge University Press 

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References

Augspurger, C. K. 1986 Morphology and dispersal potential of wind-dispersed diaspores of neotropical trees. Am. J. Bot. 73, 353–363.CrossRefGoogle Scholar
Azuma, A. & Yasuda, K. 1989 Flight performance of rotary seeds. J. Theor. Biol. 138, 23–53.CrossRefGoogle Scholar
Birch, J. M. & Dickinson, M. H. 2001 Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412, 729–733.CrossRefGoogle Scholar
Birch, J. M. & Dickinson, M. H. 2003 The influence of wing-wake interactions on the production of aerodynamic forces in flapping flight. J. Exp. Biol. 206, 2257–2272.CrossRefGoogle ScholarPubMed
Birch, J. M., Dickson, W. B. & Dickinson, M. H. 2004 Force production and flow structure of the leading edge vortex on flapping wings at high and low Reynolds numbers. J. Expl Biol. 207, 1063–1072.CrossRefGoogle Scholar
Dickinson, M. H., Lehmann, F.-O. & Sane, S. P. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960.CrossRefGoogle ScholarPubMed
Ellington, C. P., van den Berg, C., Willmott, A. P. & Thomas, L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626–630.CrossRefGoogle Scholar
Glauert, H. 1935 Aerodynamic Theory. Springer.Google Scholar
Hunsaker, D. F. & Phillips, W. F.2013 Momentum theory with slipstream rotation applied to wind turbines. AIAA Paper 2013-3161.CrossRefGoogle Scholar
Jameson, S., Fregene, K., Chang, M., Allen, N., Youngren, H. & Scroggins, J.2012 Lockheed Martin’s samarai nano air vehicle: challenges, research, and realization. AIAA Paper 2012-0584.CrossRefGoogle Scholar
Kim, D. & Choi, H. 2006 Immersed boundary method for flow around an arbitrarily moving body. J. Comput. Phys. 212, 662–680.CrossRefGoogle Scholar
Kweon, J. & Choi, H. 2010 Sectional lift coefficient of a flapping wing in hovering motion. Phys. Fluids 22, 071703.CrossRefGoogle Scholar
Lee, E.-J. & Lee, S.-J. 2016 Effect of initial attitude on autorotation flight of maple samaras (acer palmatum). J. Mech. Sci. Technol. 30, 741–747.CrossRefGoogle Scholar
Lee, I.2016 Numerical study of a freely-falling maple seed. PhD thesis, Seoul National University.Google Scholar
Lee, I. & Choi, H. 2017 Flight of a falling maple seed. Phys. Rev. Fluids 2, 090511.CrossRefGoogle Scholar
Lee, J., Choi, H. & Kim, H.-Y. 2015 A scaling law for the lift of hovering insects. J. Fluid Mech. 782, 479–490.CrossRefGoogle Scholar
Lee, J., Park, Y.-J., Jeong, U., Cho, K.-J. & Kim, H.-Y. 2013 Wake and thrust of an angularly reciprocating plate. J. Fluid Mech. 720, 545–557.CrossRefGoogle Scholar
Lee, S. J., Lee, E. J. & Sohn, M. H. 2014 Mechanism of autorotation flight of maple samaras (Acer palmatum). Exp. Fluids 55, 1718.CrossRefGoogle Scholar
Lentink, D., Dickson, W. B., van Leeuwen, J. L. & Dickinson, M. H. 2009 Leading-edge vortices elevate lift of autorotating plant seeds. Science 324, 1438–1440.CrossRefGoogle ScholarPubMed
Limacher, E. & Rival, D. E. 2015 On the distribution of leading-edge vortex circulation in samara-like flight. J. Fluid Mech. 776, 316–333.CrossRefGoogle Scholar
Nathan, R., Katul, G. G., Horn, H. S., Thomas, S. M., Oren, R., Avissar, R., Pacala, S. W. & Levin, S. A. 2002 Mechanisms of long-distance dispersal of seeds by wind. Nature 418, 409–413.CrossRefGoogle ScholarPubMed
Newman, J. N. 1977 Marine Hydrodynamics. MIT Press.CrossRefGoogle Scholar
Norberg, R. Å. 1973 Autorotation, self-stability, and structure of single-winged fruits and seeds (samaras) with comparative remarks of animal flight. Biol. Rev. 48, 561–596.CrossRefGoogle Scholar
Pandolfi, C. & Izzo, D. 2013 Biomimetics on seed dispersal: survey and insights for space exploration. Bioinspir. Biomim. 8, 025003.CrossRefGoogle ScholarPubMed
Phillips, W. F. 2002 Propeller momentum theory with slipstream rotation. J. Aircr. 39, 184–187.CrossRefGoogle Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.CrossRefGoogle Scholar
Rao, M., Hoysall, D. C. & Gopalan, J. 2014 Mahogany seed – a step forward in deciphering autorotation. Curr. Sci. 106, 1101–1109.Google Scholar
Rosen, A. & Seter, D. 1991 Vertical autorotation of a single-winged samara. Trans. ASME J. Appl. Mech. 58, 1064–1071.CrossRefGoogle Scholar
Salcedo, E., Treviño, C., Vargas, R. O. & Martínez-Suástegui, L. 2013 Stereoscopic particle image velocimetry measurements of the three-dimensional flow field of a descending autorotating mahogany seed (Swietenia macrophylla). J. Expl Biol. 216, 2017–2030.Google ScholarPubMed
Seter, D. & Rosen, A. 1992a Stability of the vertical autorotation of a single-winged samara. Trans. ASME J. Appl. Mech. 59, 1000–1008.CrossRefGoogle Scholar
Seter, D. & Rosen, A. 1992b Study of the vertical autorotation of a single-winged samara. Biol. Rev. 67, 175–197.CrossRefGoogle Scholar
Sunada, S., Ide, A., Hoshino, Y. & Okamoto, M. 2015 A study of autorotating plant seeds. J. Theor. Biol. 386, 55–61.CrossRefGoogle ScholarPubMed
Taha, H. E., Hajj, M. R. & Beran, P. S. 2014 State-space representation of the unsteady aerodynamics of flapping flight. Aerosp. Sci. Technol. 34, 1–11.CrossRefGoogle Scholar
Varshney, K., Chang, S. & Wang, Z. J. 2012 The kinematics of falling maple seeds and the initial transition to a helical motion. Nonlinearity 25, C1–C8.CrossRefGoogle Scholar
Varshney, K., Chang, S. & Wang, Z. J. 2013 Unsteady aerodynamic forces and torques on falling parallelograms in coupled tumbling-helical motions. Phys. Rev. E 87, 053021.Google ScholarPubMed
Wang, C. & Eldredge, J. D. 2013 Low-order phonomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27, 577–598.CrossRefGoogle Scholar
Wang, Q., Goosen, J. F. L. & van Keulen, F. 2016 A predictive quasi-steady model of aerodynamic loads on flapping wings. J. Fluid Mech. 800, 688–719.CrossRefGoogle Scholar
Wang, Z. J., Birch, J. M. & Dickinson, M. H. 2004 Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations versus robotic wing experiments. J. Expl Biol. 207, 449–460.CrossRefGoogle Scholar
Willmott, A. P., Ellington, C. P. & Thomas, L. R. 1997 Flow visualization and unsteady aerodynamics in the flight of the hawkmoth, Manduca sexta . Phil. Trans. R. Soc. B 352, 303–316.CrossRefGoogle Scholar
Wong, J. G., Kriegseis, J. & Rival, D. E. 2013 An investigation into vortex growth and stabilization for two-dimensional plunging and flapping with varying sweep. J. Fluids Struct. 43, 231–243.CrossRefGoogle Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25, 091901.CrossRefGoogle Scholar
Yasuda, K. & Azuma, A. 1997 The autorotation boundary in the flight of samaras. J. Theor. Biol. 185, 313–320.CrossRefGoogle Scholar