Modal decomposition of fluid-structure interaction with application to flag flapping

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

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

Accurate computation of surface stresses and forces with immersed boundary methods

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

A strongly-coupled immersed-boundary formulation for thin elastic structures

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 flags undergoing large amplitude flapping behavior

Bio-inspired variable-stiffness flaps for hybrid flow control, tuned via reinforcement learning

A bio-inspired, passively deployable flap attached to an airfoil by a torsional spring of fixed stiffness can provide significant lift improvements at post-stall angles of attack. In this work, we describe a hybrid active-passive variant to this purely passive flow control paradigm, where the stiffness of the hinge is actively varied in time to yield passive fluid-structure interaction (FSI) of greater aerodynamic benefit than the fixed-stiffness case. This hybrid active-passive flow control strategy could potentially be implemented using variable stiffness actuators with less expense compared with actively prescribing the flap motion. The hinge stiffness is varied via a reinforcement learning (RL)-trained closed-loop feedback controller. A physics-based penalty and a long-short-term training strategy for enabling fast training of the hybrid controller are introduced. The hybrid controller is shown to provide lift improvements as high as 136\% and 85\% with respect to the flap-less airfoil and the best fixed-stiffness case, respectively. These lift improvements are achieved due to large-amplitude flap oscillations as the stiffness varies over four orders of magnitude, whose interplay with the flow is analyzed in detail

Numerical Methods for Fluid-Structure Interaction, and their Application to Flag Flapping

This thesis is divided into two parts. Part I is devoted to the development of numerical techniques for simulating fluid-structure interaction (FSI) systems and for educing important physical mechanisms that drive these systems’ behavior; part II discusses the application of many of these techniques to investigate a specific FSI system. Within part I, we first describe a procedure for accurately computing the stresses on an immersed surface using the immersed-boundary method. This is a key step to simulating FSI problems, as the surface stresses simultaneously dictate the motion of the structure and enforce the no-slip boundary condition on the fluid. At the same time, accurate stress computations are also important for applications involving rigid bodies that are either stationary or moving with prescribed kinematics (e.g., characterizing the performance of wings and aerodynamic bodies in unsteady flows or understanding and controlling flow separation around bluff bodies). Thus, the method is first formulated for the rigid-body prescribed-kinematics case. The procedure described therein is subsequently incorporated into an immersed boundary method for efficiently simulating FSI problems involving arbitrarily large structural motions and rotations. While these techniques can be used to perform high-fidelity simulations of FSI systems, the resulting data often involves a range of spatial and temporal scales in both the structure and the fluid and are thus typically difficult to interpret directly. The remainder of part I is therefore devoted to extending tools regularly used for understanding complex flows to FSI systems. We focus in particular on the application of global linear stability analysis and snapshot-based data analysis (such as dynamic mode decomposition and proper orthogonal decomposition) to FSI problems. To our knowledge, these techniques had not been applied to deforming-body problems in a manner that that accounts for both the fluid and structure leading up to this work. Throughout part I, our methods are derived in the context of fairly general FSI systems and are validated using results from the literature for flapping flags in both the conventional configuration (in which the flag is pinned or clamped at its leading edge with respect to the oncoming flow) and the inverted configuration (in which the flag is clamped at its trailing edge). In part II, we apply many of the techniques developed in part I to uncover new physical mechanisms about inverted-flag flapping. We identify the instability-driving mechanism responsible for the initiation of flapping and further characterize the large-amplitude and chaotic flapping regimes that the system undergoes for a range of physical parameters.</p

A global mode analysis of flapping flags

We perform a global stability analysis of a flapping flag in the conventional configuration, in which the flag is pinned or clamped at its leading edge, and in the inverted configuration, in which the flag is clamped at its trailing edge. Specifically, we consider fully coupled fluid-structure interaction for two-dimensional flags at low Reynolds numbers. For the conventional configuration, we show that the unstable global modes accurately predict the onset of flapping for a wide range of mass and stiffness ratios. For the inverted configuration, we identify a stable deformed equilibrium state and demonstrate that as the flag becomes less stiff, this equilibrium undergoes a supercritical Hopf bifurcation in which the least damped mode transitions to instability. Previous stability analyses of inverted flags computed the leading mode of the undeformed equilibrium state and found it to be a zero-frequency (non-flapping) mode, which does not reflect the inherent flapping behavior. We show that the leading mode of the deformed equilibrium is associated with a non-zero frequency, and therefore offers a mechanism for flapping. We emphasize that for both configurations the global modes are obtained from the fully-coupled flow-flag system, and therefore reveal both the most dominant flag shapes and the corresponding flow structures that are pivotal to flag flapping behavior