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Influence of aspect ratio on tumbling plates

Published online by Cambridge University Press:  26 September 2013

W. B. Wang ,
R. F. Hu ,
S. J. Xu  and
Z. N. Wu
W. B. Wang
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, PR China
R. F. Hu
Affiliation:
School of Electro-Mechanical Engineering, Xidian University, Xi’an, 710071, PR China
S. J. Xu
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, PR China
Z. N. Wu*
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, PR China
*
†Email address for correspondence: ziniuwu@tsinghua.edu.cn
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Abstract

The tumbling motion of freely falling plates with aspect ratio ranging from 2 to 10 is studied using both experimental measurement and simplified lifting line theory. The trajectories of plates are recorded by high-speed video camera and analysed to obtain instantaneous and average kinematic and aerodynamic parameters, including the descent angle, rotating speed, descent velocities, lift and drag coefficients, and their fluctuation spectrums. A double period rotation with period doubling is observed here for some range of aspect ratio and the double frequency is determined from a Fourier analysis. By adding a correction from the inducing effect of trailing vortices and wing-tip vortices to the corresponding two-dimensional force and torque expressions, a simplified kinematic model is obtained which successfully predicts the qualitative influence of aspect ratio on the averaged kinematics, that the descent angle and the vertical descent velocity component are decreasing functions of the aspect ratio, while the rotation speed and horizontal velocity component are increasing functions of the aspect ratio. For decreasing aspect ratio, the induced drag due to the trailing and tip vortices reduces the lift to drag ratio and thus increases the descent angle, while the dissipative torque due to translational induced drag decreases the rotation speed.

JFM classification

Aerodynamics: Aerodynamics Aerodynamics: Flow-structure interactions Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 650 - 679
Copyright
©2013 Cambridge University Press 

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References

Andersen, A., Pesavento, U. & Wang, Z. J. 2005a Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91–104.Google Scholar
Andersen, A., Pesavento, U. & Wang, Z. J. 2005b Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90.CrossRefGoogle Scholar
Anderson, J. D. 2001 Fundamentals of Aerodynamics, 3rd edn. McGraw-Hill.Google Scholar
Augspurger, C. K. 1986 Morphology and dispersal potential of wind-dispersed diaspores of neotropical trees. Am. J. Bot. 73, 353–363.Google Scholar
Auguste, F., Magnaudet, J. & Fabre, D. 2013 Falling styles of disks. J. Fluid Mech. 719, 388–405.Google Scholar
Bai, C. Y. & Wu, Z. N. 2014 Generalized Kutta–Joukowski Theorem for multi-vortex and multi-airfoil flow (lumped vortex model). Chinese J. Aeronautics 27 (1) (in press), doi:10.1016/j.cja.2013.07.022.Google Scholar
Belmonte, A., Eisenberg, H. & Moses, E. 1998 From flutter to tumble: inertial drag and froude similarity in falling paper. Phys. Rev. Lett. 81, 345–348.Google Scholar
Bönisch, S. & Heuveline, V. 2007 On the numerical simulation of the unsteady free fall of a solid in a fluid: I. The Newtonian case. Comput. Fluids 36, 1434–1445.CrossRefGoogle Scholar
Bustamante, A. C. & Stone, G. W. 1969 The autorotation characteristics of various shapes for subsonic and hypersonic flows. In Proceedings of the AIAA 7th Aerospace Sciences Meeting. New York, NY.Google Scholar
Chrust, M., Bouchet, G. & Dûsek, J. 2013 Numerical simulation of the dynamics of freely falling discs. Phys. Fluids 25, 044102.Google Scholar
Davis, R. H. & Acrivos, A. 1985 Sedimentation of noncolloidal particles at low Reynolds-numbers. Annu. Rev. Fluid Mech. 17, 91–118.Google Scholar
Dickson, W. B., Straw, A. D., Poelma, C. & Dickinson, M. H. 2006 An integrative model of insect flight control. Proceedings of the AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV.Google Scholar
Dupleich, P. 1941 Rotation in free fall of rectangular wings of elongated shape. Tech. Rep. 1201. NACA Tech. Mem.Google Scholar
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121.Google Scholar
Fabre, D., Assemat, P. & Magnaudet, J. 2011 A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid. J. Fluid Struct. 27, 758–767.Google Scholar
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2008 Dynamics of axisymmetric bodies rising along a zigzag path. J. Fluid Mech. 606, 209–223.CrossRefGoogle Scholar
Field, S. B., Klaus, M., Moore, M. G. & Nori, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252–254.Google Scholar
Hirata, K., Shimizu, K., Fukuhara, K., Yamauchi, K., Kawaguchi, D. & Funaki, J. 2009 Aerodynamic characteristics of a tumbling plate under free flight. J. Fluid Sci. Tech. 4, 168–187.Google Scholar
Horowitz, M. & Williamson, C. H. K. 2010a The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251–294.CrossRefGoogle Scholar
Horowitz, M. & Williamson, C. H. K. 2010b Vortex-induced vibration of a rising and falling cylinder. J. Fluid Mech. 662, 352–383.CrossRefGoogle Scholar
Iversen, J. D. 1979 Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number. J. Fluid Mech. 92, 327–348.Google Scholar
Jenny, M., Duŝek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201–239.Google Scholar
Jin, C. Q. & Xu, K. 2007 Numerical study of the unsteady aerodynamics of freely falling plates. Commun. Comput. Phys. 3, 834–851.Google Scholar
Jones, M. & Shelley, M. J. 2005 Falling cards. J. Fluid Mech. 540, 393–425.Google Scholar
Kolomenskiy, D. & Schneider, K. 2010 Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization. Theor. Comput. Fluid Dyn. 24, 169–173.Google Scholar
de Langre, E. 2008 Effects of wind on plants. Annu. Rev. Fluid Mech. 40, 141–168.Google Scholar
Leweke, T., Thompson, M. C. & Hourigan, K. 2009 Motion of a Möbius band in free fall. J. Fluids Struct. 25, 687–696.Google Scholar
Lua, K. B., Lim, T. T. & Yeo, K. S. 2010 A rotating elliptic aerofoil in fluid at rest and in a parallel free stream. Exp. Fluids 49, 1065–1084.Google Scholar
Lugt, H. J. 1983 Autorotation. Annu. Rev. Fluid Mech. 15, 123–147.Google Scholar
Lugt, H. J. & Ohring, S. 1977 Rotating elliptic cylinders in a viscous fluid at rest or in a parallel stream. J. Fluid Mech. 79, 127–156.CrossRefGoogle Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.Google Scholar
Mahadevan, L. 1996 Tumbling of a falling card. C. R. Acad. Sci. Paris 323, 729–736.Google Scholar
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 1999 Tumbling cards. Phys. Fluids 11, 1–3.Google Scholar
Manga, M. & Stone, H. A. 1993 Buoyancy-driven interactions between deformable drops at low Reynolds numbers. J. Fluid Mech. 256, 647–683.Google Scholar
Maxwell, J. C. 1854 On a particular case of the descent of a heavy body in a resisting medium. Camb. Dublin Math. J. 9, 145–148.Google Scholar
McCutchen, C. W. 1977 The spinning rotation of ash and tulip tree samaras. Science 197, 691–692.CrossRefGoogle ScholarPubMed
Michelin, S. & Smith, S. G. L. 2009 An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23, 127–153.Google Scholar
Michelin, S. & Smith, S. G. L. 2010 Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model. Theor. Comput. Fluid Dyn. 24, 195–200.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1973 Theoretical Aerodynamics, 4th edn. Dover.Google Scholar
Mittal, R., Seshadri, V. & Udaykumar, H. S. 2004 Flutter, tumble, and vortex induced autorotation. Theor. Comput. Fluid Dyn. 17, 165–170.Google Scholar
Pesavento, U. & Wang, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and centre of mass elevation. Phys. Rev. Lett. 93, 144501.Google Scholar
Rayner, J. M. V. 1979 A vortex theory of animal flight. Part 1. The vortex wake of a hovering animal. J. Fluid Mech. 91, 697–730.Google Scholar
Sedov, L. I. 1965 Two-dimensional Problems in Hydrodynamics and Aerodynamics. Interscience.Google Scholar
Shenoy, A. R. & Kleinstreuer, C. 2010 Influence of aspect ratio on the dynamics of a freely moving circular disk. J. Fluid Mech. 653, 463–487.Google Scholar
Skews, B. W. 1990 Autorotation of rectangular plates. J. Fluid Mech. 217, 33–40.CrossRefGoogle Scholar
Skews, B. W. 1991 Autorotation of many-sided bodies in an airstream. Nature 352, 512–513.Google Scholar
Smith, E. H. 1971 Autorotating wings: an experimental investigation. J. Fluid Mech. 50, 513–534.Google Scholar
Tam, D., Bush, J. W. M., Robitaille, M. & Kudrolli, A. 2010 Tumbling dynamics of passive flexible wings. Phys. Rev. Lett. 104, 184504.Google Scholar
Tanabe, Y. & Kaneko, K. 1994 Behavior of a falling paper. Phys. Rev. Lett. 73, 1372–1375.Google 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.Google 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 Scholar
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.Google Scholar
Wang, X. X. & Wu, Z. N. 2010 Stroke-averaged lift forces due to vortex rings and their mutual interactions for a flapping flight model. J. Fluid Mech. 654, 453–472.Google Scholar
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208.CrossRefGoogle Scholar
Xia, Z. H., Connington, K. W., Rapaka, S., Yue, P. T., Feng, J. J. & Chen, S. Y. 2009 Flow patterns in the sedimentation of an elliptical particle. J. Fluid Mech. 625, 249–272.Google Scholar
Zhong, H. J., Chen, S. Y. & Lee, C. B. 2011 Experimental study of freely falling thin disks: transition from planar zigzag to spiral. Phys. Fluids 23, 011702.Google Scholar
Zhong, H. J., Lee, C. B., Su, Z., Chen, S. Y., Zhou, M. D. & Wu, J. Z. 2013 Experimental investigation of freely falling thin disks. Part 1. The flow structures and Reynolds number effects on the zigzag motion. J. Fluid Mech. 716, 228–250.Google Scholar