We give an explanation for the onset of wind-induced flutter in a flag. Our theory accounts for the various physical mechanisms at work: the finite length and the small but finite bending stiffness of the flag, the unsteadiness of the flow, the added mass effect and vortex shedding from the trailing edge. Our analysis allows us to predict a critical speed for the onset of flapping as well as the frequency of flapping. We find that in a particular limit corresponding to a low density fluid flowing over a soft high density flag, the flapping instability is akin to a resonance between the mode of oscillation of a rigid pivoted airfoil in a flow and a hinged-free elastic filament vibrating in its lowest mode.
Using nonlinear vortex-sheet simulations, we determine the region in parameter space in which a straight flag in a channel-bounded inviscid flow is unstable to flapping motions. We find that for heavier flags, greater confinement increases the size of the region of instability. For lighter flags, confinement has little influence. We then compute the stability boundaries analytically for an infinite flag, and find similar results. For the finite flag we also consider the effect of channel walls on the large-amplitude periodic flapping dynamics. We find that multiple flapping states are possible…
We investigate the effects of a nearby free surface on the stability of a flexible plate in axial flow. Confinement by rigid boundaries is known to affect flag flutter thresholds and fluttering dynamics significantly, and this work considers the effects of a more general confinement involving a deformable free surface. To this end, a local linear stability is proposed for a flag in axial uniform flow and parallel to a free surface, using one-dimensional beam and potential flow models to revisit this classical fluid-structure interaction problem. The physical behaviour of the confining free sur…
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…
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 structur…
There have been many studies of the instability of a flexible plate or flag to flapping motions, and of large-amplitude flapping. Here we use inviscid simulations and a linearized model to study more generally how key quantities -- mode number (or wavenumber), frequency, and amplitude -- depend on the two dimensionless parameters, flag mass and bending stiffness. In the limit of small flag mass, flags perform traveling wave motions that move at nearly the speed of the oncoming flow. The flag mode number scales as the -1/4 power of bending stiffness. The flapping frequency has the same scaling,…
The stability of a cantilevered elastic sheet in a uniform flow has been studied extensively due to its importance in engineering and its prevalence in natural structures. Varying the flow speed can give rise to a range of dynamics including limit cycle behaviour and chaotic motion of the cantilevered sheet. Recently, the "inverted flag" configuration - a cantilevered elastic sheet aligned with the flow impinging on its free edge - has been observed to produce large-amplitude flapping over a finite band of flow speeds. This flapping phenomenon has been found to be a vortex-induced vibration, a…
We investigate the effect of piezoelectric (PZT) material on the flutter speed, vibration mode and frequency, and energy harvesting power and efficiency of a flexible flag in various fluids. We develop a fully coupled fluid-solid-electric model by combining the inviscid vortex sheet model with a linear electro-mechanical coupling model. A resistance only circuit and a resonant resistance-inductance (RL) circuit are considered. For a purely resistive circuit, an increased electro-mechanical coupling factor results in an increased flutter speed, vibration frequency, averaged electric power and e…
We perform high-fidelity, two-dimensional (2D), fluid-structure interaction (FSI) simulations at a Reynolds number of $Re=200$ of uniform flow past an inverted flag (i.e., clamped at its trailing edge). The inverted flag system can exhibit large-amplitude flapping motions (on the order of the flag length) that can be converted to electricity via, e.g., piezoelectric materials. We investigate the effect of structural nonuniformity in altering the FSI dynamics compared with the uniform-stiffness scenario that has been thoroughly characterized. We consider linear, quadratic, and cubic stiffness d…
A fully-resolved direct-numerical-simulation (DNS) approach for investigating flexible bodies forced by a turbulent incoming flow is designed to study the flapping motion of a flexible flag at moderate Reynolds number. The incoming turbulent flow is generated by placing a passive grid at the inlet of the numerical domain and the turbulence level of the flow impacting the flag can be controlled by changing its downstream distance from the grid. The computational framework is based on the immersed boundary method for dealing with arbitrary geometries and implemented using a graphics-processing-u…
We present a fluid dynamics video showing simulations of flexible bodies flapping and falling in an inviscid fluid. Vortex sheets are shed from the trailing edges of the bodies according to the Kutta condition. For interacting flags, a sampling of synchronous and asynchronous states are shown. For falling flexible sheets, the basic behavior is a repeated series of accelerations to a critical speed at which the sheet buckles, and rapidly decelerates, shedding large vortices. Examples of persistent circling, quasi-periodic flapping, and more complex trajectories are shown.
The interactions and synchronization of two parallel and slender flags in a uniform axial flow are studied in the present paper by generalizing Lighthill's Elongated Body Theory (EBT) and Lighthill's Large Amplitude Elongated Body Theory (LAEBT) to account for the hydrodynamic coupling between flags. The proposed method consists in two successive steps, namely the reconstruction of the flow created by a flapping flag within the LAEBT framework and the computation of the fluid force generated by this nonuniform flow on the second flag. In the limit of slender flags in close proximity, we show t…
The spontaneous flapping of a flag can be used to produce electrical energy from a fluid flow when coupled to a generator. In this paper, the energy harvesting performance of a flag covered by a single pair of PVDF piezoelectric electrodes is studied both experimentally and numerically. The electrodes are connected to a resistive-inductive circuit that forms a resonant circuit with the piezoelectric's intrinsic capacitance. Compared with purely resistive circuits, the resonance between the circuit and the flag's flapping motion leads to a significant increase in the harvested energy. Our exper…
Linear stability analysis of an elastically anchored wing in a uniform flow is investigated both analytically and numerically. The analytical formulation explicitly takes into account the effect of the wake on the wing by means of Theodorsen's theory. Three different parameters non-trivially rule the observed dynamics: mass density ratio between wing and fluid, spring elastic constant and distance between the wing center of mass and the spring anchor point on the wing. We found relationships between these parameters which rule the transition between stable equilibrium and fluttering. The shape…
The flapping flag instability occurs when a flexible cantilevered plate is immersed in a uniform airflow. To this day, the nonlinear aspects of this aeroelastic instability are largely unknown. In particular, experiments in the literature all report a large hysteresis loop, while the bifurcation in numerical simulations is either supercritical or subcritical with a small hysteresis loop. In this paper, this discrepancy is addressed. First weakly nonlinear stability analyses are conducted in the slender-body and two-dimensional limits, and second new experiments are performed with flat and curv…
The spontaneous flapping of a flag in a steady flow can be used to power an output circuit using piezoelectric elements positioned at its surface. Here, we study numerically the effect of inductive circuits on the dynamics of this fluid-solid-electric system and on its energy harvesting efficiency. In particular, a destabilization of the system is identified leading to energy harvesting at lower flow velocities. Also, a frequency lock-in between the flag and the circuit is shown to significantly enhance the system's harvesting efficiency. These results suggest promising efficiency enhancements…
Fluttering piezoelectric plates may harvest energy from a fluid flow by converting the plate's mechanical deformation into electric energy in an output circuit. This work focuses on the influence of the arrangement of the piezoelectric electrodes along the plate's surface on the energy harvesting efficiency of the system, using a combination of experiments and numerical simulations. A weakly non-linear model of a plate in axial flow, equipped with a discrete number of piezoelectric patches is derived and confronted to experimental results. Numerical simulations are then used to optimize the po…
The flow-induced flutter has a significant role in aircraft stability, renewable energy extraction, animal locomotion, among many other applications. While being a ubiquitous phenomenon, the control of the flutter response has been primarily limited to simplified systems and, often, with the help of linear inviscid flow theories. In this paper, we numerically investigate how the plunging response of a foil can be regulated using an active flap to improve structural safety or enhance the energy extraction efficiency of the foil with a tightly coupled fluid-structure interaction algorithm. A bro…
We address the flutter instability of a flexible plate immersed in an axial flow. This instability is similar to flag flutter and results from the competition between destabilising pressure forces and stabilising bending stiffness. In previous experimental studies, the plates have always appeared much more stable than the predictions of two-dimensional models. This discrepancy is discussed and clarified in this paper by examining experimentally and theoretically the effect of the plate aspect ratio on the instability threshold. We show that the two-dimensional limit cannot be achieved experime…
In this article the energy transfer between a flow and a fluttering piezoelectric plate is investigated. In particular, the benefits of the use of a Synchronized Switch Harvesting on Inductor (SSHI) circuit are studied. Both wind tunnel experiments and numerical simulations are conducted in order to analyse the influence of the switching process on the dynamics and the efficiency of the system. Numerical simulations consist of a weakly non-linear model of a plate in axial flow equipped with a single pair of piezoelectric patches, discretized using a Galerkin method where basis functions are th…
The fluid-solid-electric dynamics of a flexible plate covered by interconnected piezoelectric patches in an axial steady flow are investigated using numerical simulations based on a reduced-order model of the fluid loading for slender structures. Beyond a critical flow velocity, the fluid-solid instability results in large amplitude flapping of the structure. Short piezoelectric patches positioned continuously along the plate convert its local deformation into electrical currents that are used within a single internal electrical network acting as an electric generator for the external output c…
Self-sustained oscillations resulting from fluid-solid instabilities, such as the flutter of a flexible flag in axial flow, can be used to harvest energy if one is able to convert the solid energy into electricity. Here, this is achieved using piezoelectric patches attached to the surface of the flag that convert the solid deformation into an electric current powering purely resistive output circuits. Nonlinear numerical simulations in the slender-body limit, based on an explicit description of the coupling between the fluid-solid and electric systems, are used to determine the harvesting effi…
