Abstract
The free falling of a plate is an interesting and canonical problem in the study of fluid–body interactions. In this paper, experiments with a wide range of the dimensionless moment of inertia (\(I^*=0.05{-}1.6\)) and the Reynolds numbers (\(Re=10{-}350\)) were conducted to investigate the trajectory modes and the corresponding wake patterns of freely falling plates. Experimental results show that the freely falling plates exhibit four trajectory modes including vertical falling, fluttering, tumbling, and chaotic motions. Based on massive experiments, a detailed \(Re{-}I^*\) phase diagram is given and shows different trajectory regimes, which are distinguished by clear boundaries. Then, the wake patterns corresponding to different trajectory modes are investigated. When Re is rather small, plates fall vertically and the corresponding wake exhibits a stable and symmetrical pattern. With increasing Re, it is found that the plate falls vertically, whereas the corresponding wake has developed to be similar to the Kármán vortex street, thereby indicating that wake instability should be prior to the path instability. As Re and \(I^*\) further increase, both trajectory mode and wake pattern are unstable. The trajectory modes and the wake patterns of the freely falling plates are summarized in one \(Re{-}I^*\) phase diagram, and the onset of the wake instability is found to occur when Re is approximately 30.
Graphical abstract







Similar content being viewed by others
References
Andersen A, Persavento U, Wang ZJ (2005) Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J Fluid Mech 541:91–104
Andersen A, Persavento U, Wang ZJ (2005) Unsteady aerodynamics of fluttering and tumbling plates. J Fluid Mech 541:65–90
Assemat P, Fabre D, Magnaudet J (2012) The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study. J Fluid Mech 690:173–202
Auguste F, Magnaudet J, Fabre D (2013) Falling styles of disks. J Fluid Mech 719:388–405
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
Ern P, Magnaudet FR, Fernandes J (2005) Evolution of wake structure and wake-induced loads along the path of freely rising axisymmetric bodies. Phys Fluids 19:113302
Field S, Klaus M, Moore M, Nori F (1997) Chaotic dynamics of falling disks. Nature 388:252–254
Heisinger L, Newton P, Kanso E (2014) Coils falling in water. J Fluid Mech 742:243–253
Huang WT, Liu H, Wang FX, Wu JQ, Zhang HP (2013) Experimental study of a freely falling plate with an inhomogeneous mass distribution. Phys Rev E 88:053008
Lee CB, Su Z, Zhong HJ, Chen SY, Zhou MD, Wu JZ (2013) Experimental investigation of freely falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral. J Fluid Mech 732:77–104
Lentink D, Dickson WB, van Leeuwen JL, Dickinson MH (2009) Leading-edge vortices elevate lift of autorotating plant seeds. Science 324:1438–1440
Mahadevan L, Ryu WS, Samuel AT (1999) Tumbling cards. Phys Fluids 11:1–3
Maxwell J (1853) On a particular case of the descent of a heavy body in a resisting medium. Comb Dublin Math J 9:115–118
Mittal R, Seshadri V, U HS (2004) Flutter, tumble and vortex induced autorotation. Theor Comput Fluid Dyn 17:165–170
Smith EH (1971) Autorotating wings: an experimental investigation. J Fluid Mech 50:513–534
Spagnolie SE, Moret L, Shelley MJ, Zhang J (2010) Surprising behaviors in flapping locomotion with passive pitching. Phys Fluids 22:041903
Tanabe Y, Kaneko K (1994) Behavior of afalling paper. Phys Rev Lett 73:1372–1375
Vincent L, Shambaugh WS, Kanso E (2016) Holes stabilize freely falling coins. J Fluid Mech 801:250–259
Wang ZJ (2005) Dissecting insect flight. Annul Rev Fluid Mech 37:183
Williamson CHK, Govardhan R (2004) Vortex-induced vibrations. Annu Rev Fluid Mech 36:413–455
Willmarth WW, Hawk NE, Harvey RL (1964) Steady and unsteady motions and wakes of freely falling disks. Phys Fluids 7:197–208
Xia ZH, Connington KW, Rapaka S, Yue PT, Feng JJ, Chen SY (2009) Flow patterns in the sedimentation of an elliptical particle. J Fluid Mech 625:249–272
Zhong HJ, Chen SY, Lee CB (2011) Experimental study of freely falling thin disks: transition from planar zigzag to spiral. Phys Fluids 23(011):702–4
Zhong HJ, Lee CB, Su Z, Chen SY, Zhou MD, Wu JZ (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
Acknowledgements
Financial support from the State Key Development Program of Basic Research of China (2014CB744802) is gratefully acknowledged. Besides, this work was also supported by NSFC Projects (91441205 and 11372178).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiang, Y., Qin, S., Huang, W. et al. Trajectory modes and wake patterns of freely falling plates. J Vis 21, 433–441 (2018). https://doi.org/10.1007/s12650-017-0469-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12650-017-0469-8