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 steadystate 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. (2016a) 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 abovementioned 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. †

We present a strongly-coupled immersed-boundary method for flow-structure interaction problems involving thin deforming bodies. The method is stable for arbitrary choices of solid-to-fluid mass ratios and for large body motions. As with many strongly-coupled immersed-boundary methods, our method requires the solution of a nonlinear algebraic system at each time step. The system is solved through iteration, where the iterates are obtained by linearizing the system and performing a block LU factorization. This restricts all iterations to small-dimensional subsystems that scale with the number of discretization points on the immersed surface, rather than on the entire flow domain. Moreover, the iteration procedure we propose does not involve heuristic regularization parameters, and has converged in a small number of iterations for all problems we have considered. We derive our method for general deforming surfaces, and verify the method with two-dimensional test problems of geometrically nonlinear beams undergoing large amplitude flapping behavior.

Many immersed boundary methods solve for surface stresses that impose the velocity boundary conditions on an immersed body. These surface stresses may contain spurious oscillations that make them ill-suited for representing the physical surface stresses on the body. Moreover, these inaccurate stresses often lead to unphysical oscillations in the history of integrated surface forces such as the coefficient of lift. While the errors in the surface stresses and forces do not necessarily affect the convergence of the velocity field, it is desirable, especially in fluid-structure interaction problems, to obtain smooth and convergent stress distributions on the surface. To this end, we show that the equation for the surface stresses is an integral equation of the first kind whose ill-posedness is the source of spurious oscillations in the stresses. We also demonstrate that for sufficiently smooth delta functions, the oscillations may be filtered out to obtain physically accurate surface stresses. The filtering is applied as a post-processing procedure, so that the convergence of the velocity field is unaffected. We demonstrate the efficacy of the method by computing stresses and forces that converge to the physical stresses and forces for several test problems.

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Modal decompositions such as proper orthogonal decomposition (POD), dynamic mode decomposition (DMD) and their variants are regularly used to educe physical mechanisms of nonlinear flow phenomena that cannot be easily understood through direct inspection. In fluid-structure interaction (FSI) systems, fluid motion is coupled to vibration and/or deformation of an immersed structure. Despite this coupling, data analysis is often performed using only fluid or structure variables, rather than incorporating both. This approach does not provide information about the manner in which fluid and structure modes are correlated. We present a framework for performing POD and DMD where the fluid and structure are treated together. As part of this framework, we introduce a physically meaningful norm for FSI systems. We first use this combined fluid-structure formulation to identify correlated flow features and structural motions in limit-cycle flag flapping. We then investigate the transition from limit-cycle flapping to chaotic flapping, which can be initiated by increasing the flag mass. Our modal decomposition reveals that at the onset of chaos, the dominant flapping motion increases in amplitude and leads to a bluff-body wake instability. This new bluff-body mode interacts triadically with the dominant flapping motion to produce flapping at the non-integer harmonic frequencies previously reported by Connell & Yue (2007). While our formulation is presented for POD and DMD, there are natural extensions to other data-analysis techniques. †