An analytical tool is presented to compute the parametric regions of flutter instabilities of a two-dimensional flexible foil elastically mounted. It is based on a new analytical formulation of the unsteady fluid-estructure interaction valid for small-amplitude oscillations and deformations of the foil immersed in an inviscid fluid. The formulation extends a previous analysis by including the effects of gravity and a second flexural mode, increasing its validity range to much smaller rigidities. The analytical results are validated with available numerical results, capturing the first two natural flexural modes down to values of the stiffness parameter $S$ of order $10^{-1}$. When only passive heave, or only passive pitch, is allowed, the rigid foil is stable, existing an upper stiffness bound for the flexural instabilities, wich become coupled with the spring instability mode for small spring constant increasing the growth rate. These coupled spring (linear or torsional) and flexural instability modes occur below a threshold value of $S$ and above a threshold value of $R$, both depending on the corresponding spring constant. Coupled pitch-heave flutter instabilities of a rigid foil occur in a region below a curve of the parametric plane of the two springs constants that depends on $R$, which shrinks to zero as $R$ decreases. For a flexible foil, the flexural unstable modes become coupled with the springs unstable mode as $S$ decreases from infinity, enlarging the mass ratio range for flutter instability and increasing its growth rate, the more so the smaller the springs constants. The parametric regions for flutter instabilities are easily characterized with the present analytical tool, providing the corresponding frequency and critical flutter velocity. The present results can be useful as a guide in the design of future turbines based on flexible oscillating foils.
Characterizing the onset of flutter instability of a flexible plate in uniform flow is both a classical theoretical problem in fluid-structure interaction and a relevant problem of interest in many engineering fields. Flutter instabilities, arising above a critical free stream fluid velocity from the dynamic interaction between aerodynamic forces, structural elasticity, and inertia, can lead to self-excited oscillations that grow in amplitude and potentially result in structural failure. While flutter of flexible foils or plates has traditionally been treated as a limiting instability in aeroelastic design, specially in aeronautics [1,2] and some manufacturing processes [3,4], it also serves to understand some physiological phenomena [5,6], and recent research has demonstrated its potential as an effective mechanism for flow-induced energy harvesting [7][8][9]. In particular, two-dimensional foils provide a canonical and well-controlled configuration for investigating the fundamental physics of flutter-driven energy extraction through fully passive flapping foil turbines [10][11][12][13][14].
Owing to their mathematical tractability and physical clarity, two-dimensional (2-D) foil models have long been used as the basis for analytical aeroelastic theories, numerical simulations and laboratory experiments enabling systematic investigation of flutter onset and post-critical behavior [5,[15][16][17][18][19][20][21][22][23][24][25][26]. Within the vast literature on flutter instabilities in 2-D foils, this work is focused on analytical theories that take into account foil flexibility to characterize its effect on pitch-heave coupled flutter of a flexible plate.
Classical analytical descriptions of foil flutter are rooted in linear potential-flow theory. Theodorsen’s theory, in particular, provides a closed-form expression for the lift and moment acting on an oscillating airfoil undergoing ramon.fernandez@uma.es (R. Fernandez-Feria) ORCID(s): 0000-0001-9873-1933 (R. Fernandez-Feria) harmonic pitching and plunging motions [15]. When coupled with linear structural models, this formulation yields an eigenvalue problem whose solution predicts flutter speed, frequency, and mode coupling. Despite its simplifying assumptions, this approach remains a cornerstone of flutter analysis and continues to inform the design of flutter-based energy harvesting systems, allowing parametric studies of mass ratio, stiffness, and energy extraction mechanisms. More recently, data-driven aerodynamic models are also being commonly used for aeroelastic simulations and flutter stability analyses [27]. They are based on high fidelity CFD simulations and usually provide precise models valid even for large amplitudes of deformations and oscillations. However, they require costly numerical simulations for their development and are only valid for a specific fluid-structure configuration, without reliable information about their validity when the configuration changes. Models based on linear potential theory, although valid only for small amplitudes, adapt easily to different configurations and it is always possible to quantify their validity range when additional simplifications are made because the theory provides a precise physical meaning of each contribution to the model. In any case, descriptions based on linear potential flow theory always provide a first direct analytical estimate of the onset and the physics of flutter instability.
Theodorsen’s flutter theory is for a rigid foil with passive pitch and heave. To account for the effect of the flexibility of the foil, one needs to solve the fluid-structure interaction (FSI) equations coupling the linearized inviscid flow equations with the equation for the plate dynamics for appropriate boundary conditions. The resulting complex eigenvalue problem has been formulated and solved numerically in a number of works, mostly using Galerkin decomposition methods, but limited to cases with a fixed leading edge, specially for a clamped-free plate (clamped at its leading edge and free at its trailing edge) [e.g., 20,25] or a pinned-free plate [e.g., 26], though other boundary conditions have also been considered [e.g. , 17]. To obtain closed analytical descriptions one needs further simplifying approximations for the FSI. Still for a plate with a fixed leading edge, Kornecki et al. [16] used a quasi-steady, non-circulatory aerodynamic theory and solved the resulting simplified eigenvalue problem using two modes only to obtained an approximation of the critical parameters for the flutter instability of a clamped-free plate. A simpler approach was considered by Shelley et al. [18] for a heavy foil in water, assuming a superposition of sinusoidal traveling waves for the foil and the inviscid fluid and considering a linear stability analysis, finding a simple expression for the flutter velocity. For the same problem with a clamped-free foil, Argentina and Mahadevan [19] included the circulatory part
This content is AI-processed based on open access ArXiv data.