Skip to main content Accessibility help

Progression of heavy plates from stable falling to tumbling flight

  • Edwin M. Lau (a1), Wei-Xi Huang (a1) and Chun-Xiao Xu (a1)


This study examines the transition of stable falling to tumbling flight for freely falling heavy plates in a two-dimensional viscous fluid, solved via direct numerical simulation with the immersed boundary method. The simulations are performed at a range of Reynolds number ( $Re$ ) of up to 500 and a dimensionless moment of inertia ( $I^{\ast }$ ) up to 10. It is found that a plate may settle to stable falling or develop into tumbling descent depending on the initial angle of release $\unicode[STIX]{x1D703}_{0}$ . The characteristics and performance that distinguish two flight states are investigated. This bistability is analysed with phase portraits and the region mapped across the regime of $I^{\ast }$ and $Re$ at a specific thickness ratio. In determining the flight state, the respective critical $\unicode[STIX]{x1D703}_{0}$ is found to follow a power law through $I^{\ast }$ and $Re$ . It is suggested that the changing slope of the lift curve that the plate undergoes sets the two flight states apart. Flow fields also reveal that the recirculation behind the plate is confined by the vortex structures and provides an additional rotation to the plate. An experiment is performed suggesting that bistability also occurs at Re ${\sim}O(10^{4})$ . Other shapes are also simulated and the different bistable effects are discussed.


Corresponding author

Email address for correspondence:


Hide All
Andersen, A., Pesavento, U. & Wang, Z. J. 2005a Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91104.
Andersen, A., Pesavento, U. & Wang, Z. J. 2005b Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 6590.
Auguste, F., Magnaudet, J. & Fabre, D. 2013 Falling styles of disks. J. Fluid Mech. 719, 388405.
Belmonte, A., Shelley, M. J., Eldakar, S. T. & Wiggins, C. H. 2001 Dynamic patterns and self-knotting of a driven hanging chain. Phys. Rev. Lett. 87 (11), 345348.
Chrust, M., Bouchet, G. & Dušek, J. 2013 Numerical simulation of the dynamics of freely falling discs. Phys. Fluids 25 (4), 044102.
Dupleich, P.1941 Rotation in free fall of rectangular wings of elongated shape. NASA Tech. Mem. 1201.
Dušek, J., Chrust, M. & Bouchet, G. 2016 Transitional dynamics of freely falling discs. In Advances in Fluid-Structure Interaction, vol. 133, pp. 105116.
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 (1), 97121.
Field, S. B., Klaus, M. & Moore, M. G. 1997 Chaotic dynamics of falling disks. Nature 388, 252254.
Fonseca, F. & Herrmann, H. J. 2005 Simulation of the sedimentation of a falling oblate ellipsoid. Physica A 345 (3–4), 341355.
Heisinger, L., Newton, P. & Kanso, E. 2014 Coins falling in water. J. Fluid Mech. 742, 243253.
Horowitz, M. & Williamson, C. H. K. 2010 The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251294.
Hu, R. & Wang, L. 2014 Motion transitions of falling plates via quasisteady aerodynamics. Phys. Rev. E 90 (1), 013020.
Huang, W.-X., Shin, S. J. & Sung, H. J. 2007 Simulation of flexible filaments in a uniform flow by the immersed boundary method. J. Comput. Phys. 226 (2), 22062228.
Kim, J., Kim, D. & Choi, H. 2001 An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comput. Phys. 171 (1), 132150.
Kolomenskiy, D. & Schneider, K. 2009 Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization. Theor. Comput. Fluid Dyn. 24 (1–4), 169173.
Kuznetsov, S. P. 2015 Plate falling in a fluid: regular and chaotic dynamics of finite-dimensional models. Regular Chaotic Dyn. 20 (3), 345382.
Lamb, H. 1945 Hydrodynamics. Dover.
Mathai, V., Zhu, X., Sun, C. & Lohse, D. 2017 Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders. Phys. Rev. Lett. 133, 054501.
Maxwell, J. C. 1852 Draft of paper on the descent of a body in a resisting medium. In The scientific letters and papers of James Clerk Maxwell (ed. Harman, P. M.), pp. 213218. Cambridge University Press.
Mittal, R., Seshadri, V. & Udaykumar, H. S. 2004 Flutter, tumble and vortex induced autorotation. Theor. Comput. Fluid Dyn. 17 (3), 165170.
Okamoto, M. & Azuma, A. 2011 Aerodynamic characteristics at low Reynolds numbers for wings of various planforms. AIAA J. 49 (6), 054501.
Paidoussis, M., Price, S. & Langre, E. 2011 Fluid-Structure Interactions Cross-Flow-Induced Instabilities. Cambridge University Press.
Pesavento, U. & Wang, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93 (14), 144501.
Peskin, C. S. 2002 The immersed boundary method. Acta Numer. 11, 479517.
Smith, E. H. 1971 Autorotating wings: an experimental investigation. J. Fluid Mech. 50, 513534.
Sun, Q. & Boyd, I. D. 2004 Flat-plate aerodynamics at very low Reynolds number. J. Fluid Mech. 502, 199206.
Wan, H., Dong, H. & Liang, Z. 2012 Vortex formation of freely falling plates. In 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2012-1079.
Wang, W. B., Hu, R. F., Xu, S. J. & Wu, Z. N. 2013 Influence of aspect ratio on tumbling plates. J. Fluid Mech. 733, 650679.
Wang, Y., Shu, C., Teo, C. J. & Yang, L. M. 2016 Numerical study on the freely falling plate: effects of density ratio and thickness-to-length ratio. Phys. Fluids 28 (10), 103603.
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7 (2), 197208.
Wu, R.-J. & Lin, S.-Y. 2015 The flow of a falling ellipse: numerical method and classification. J. Mech. 31 (06), 771782.
Zhong, H., Lee, C., Su, Z., Chen, S., Zhou, M. & Wu, J. 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, 228250.
Zhou, W., Chrust, M. & Dušek, J. 2017 Path instabilities of oblate spheroids. J. Fluid Mech. 833, 445468.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed