Global modes and nonlinear analysis of inverted-flag flapping. (22nd October 2018) - Goza, Andres; Colonius, Tim; Sader, John E.
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, 2016 a ) 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.