Global modes and nonlinear analysis of inverted-flag flapping

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.