Skip to main content Accessibility help

Global modes and nonlinear analysis of inverted-flag flapping

  • Andres Goza (a1), Tim Colonius (a2) and John E. Sader (a3) (a4)


An inverted flag has its trailing edge clamped and exhibits dynamics distinct from that of a conventional flag, whose leading edge is restrained. We perform nonlinear simulations and a global stability analysis of the inverted-flag system for a range of Reynolds numbers, flag masses and stiffnesses. Our global stability analysis is based on a linearisation of the fully coupled fluid–structure system of equations. The calculated equilibria are steady-state solutions of the fully coupled nonlinear equations. By implementing this approach, we (i) explore the mechanisms that initiate flapping, (ii) study the role of vorticity generation and vortex-induced vibration (VIV) in large-amplitude flapping and (iii) characterise the chaotic flapping regime. For point (i), we identify a deformed-equilibrium state and show through a global stability analysis that the onset of small-deflection flapping – where the oscillation amplitude is significantly smaller than in large-amplitude flapping – is due to a supercritical Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of Sader et al. (J. Fluid Mech., vol. 793, 2016a) that classical VIV exists when the flag is sufficiently light with respect to the fluid. We also show that for heavier flags, large-amplitude flapping persists (even for Reynolds numbers ${<}50$ ) and is not classical VIV. Finally, with respect to point (iii), chaotic flapping has been observed experimentally for Reynolds numbers of $O(10^{4})$ , and here we show that chaos also persists at a moderate Reynolds number of 200. We characterise this chaotic regime and calculate its strange attractor, whose structure is controlled by the above-mentioned deformed equilibria and is similar to a Lorenz attractor.


Corresponding author

Email address for correspondence:


Hide All
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750756.
Colonius, T. & Taira, K. 2008 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Meth. Appl. Mech. Engng 197 (25), 21312146.
Connell, B. S. & Yue, D. K. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 3367.
Criesfield, M. A. 1991 Non-Linear Finite Element Analysis of Solids and Structures, vol. 1. Wiley.
Degroote, J., Bathe, K.-J. & Vierendeels, J. 2009 Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction. Comput. Struct. 87 (11), 793801.
Goza, A. & Colonius, T. 2017 A strongly-coupled immersed-boundary formulation for thin elastic structures. J. Comput. Phys. 336, 401411.
Gurugubelli, P. S. & Jaiman, R. K. 2015 Self-induced flapping dynamics of a flexible inverted foil in a uniform flow. J. Fluid Mech. 781, 657694.
Gurugubelli, P. S. & Jaiman, R. K.2017 On the mechanism of large amplitude flapping of inverted foil in a uniform flow. arXiv:1711.01065.
Khalak, A. & Williamson, C. H. 1999 Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13, 813851.
Kim, D., Cossé, J., Cerdeira, C. H. & Gharib, M. 2013 Flapping dynamics of an inverted flag. J. Fluid Mech. 736, R1.
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.
Mittal, S. & Singh, S. 2005 Vortex-induced vibrations at subcritical Re . J. Fluid Mech. 534, 185194.
Roshko, A.1954 On the drag and shedding frequency of two-dimensional bluff bodies. Tech. Rep. National Advisory Committee for Aeronautics, Washington, DC, United States.
Ryu, J., Park, S. G., Kim, B. & Sung, H. J. 2015 Flapping dynamics of an inverted flag in a uniform flow. J. Fluids Struct. 57, 159169.
Sader, J. E. 1998 Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 84 (1), 6476.
Sader, J. E., Cossé, J., Kim, D., Fan, B. & Gharib, M. 2016a Large-amplitude flapping of an inverted flag in a uniform steady flow: a vortex-induced vibration. J. Fluid Mech. 793, 524555.
Sader, J. E., Huertas-Cerdeira, C. & Gharib, M. 2016b Stability of slender inverted flags and rods in uniform steady flow. J. Fluid Mech. 809, 873894.
Sarpkaya, T. 2004 A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct. 19, 389447.
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449465.
Shen, N., Chakraborty, D. & Sader, J. E. 2016 Resonant frequencies of cantilevered sheets under various clamping configurations immersed in fluid. J. Appl. Phys. 120, 144504.
Shoele, K. & Mittal, R. 2016 Energy harvesting by flow-induced flutter in a simple model of an inverted piezoelectric flag. J. Fluid Mech. 790, 582606.
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Perseus.
Taneda, S. 1968 Waving motions of flags. J. Phys. Soc. Japan 24 (2), 392401.
Tian, F.-B., Dai, H., Luo, H., Doyle, J. F. & Rousseau, B. 2014 Fluid–structure interaction involving large deformations: 3D simulations and applications to biological systems. J. Comput. Phys. 258, 451469.
Williamson, C. H. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica D 16 (3), 285317.
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