Global modes and nonlinear analysis of inverted-flag flapping

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 vortex shedding 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 flapping is due to a supercritical Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of Sader et al. (2016) that for a range of parameters this regime is a VIV. We also show that there are other flow regimes for which large-amplitude flapping persists and is not a VIV. Specifically, flapping can occur at low Reynolds numbers ($<50$), albeit via a previously unexplored mechanism. 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. These results are contextualised with bifurcation diagrams that depict the different equilibria and various flapping regimes.