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Global modes and nonlinear analysis of inverted-flag flapping

Published online by Cambridge University Press:  22 October 2018

Andres Goza Open the ORCID record for Andres Goza [Opens in a new window] ,
Tim Colonius  and
John E. Sader Open the ORCID record for John E. Sader [Opens in a new window]
Andres Goza*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540, USA
Tim Colonius
Affiliation:
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA
John E. Sader
Affiliation:
ARC Centre of Excellence in Exciton Science, School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: ajgoza@gmail.com
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Abstract

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.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 312 - 344
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Copyright
© 2018 Cambridge University Press 

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