Abstract

We give an explanation for the onset of fluid-flow-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 plate vibrating in its lowest mode. T he flutter of a flag in a gentle breeze and the flapping of a sail in a rough wind are commonplace and familiar observations of a rich class of problems involving the interaction of fluids and structures, of wide interest and importance in science and engineering (1). Folklore attributes this instability to some combination of (i) the Bénard-von Kármán vortex street that is shed from the trailing edge of the flag and (ii) the KelvinHelmholtz problem of the growth of perturbations at an interface between two inviscid fluids of infinite extent moving with different velocities (2). However, a moment's reflection makes one realize that neither of these is correct. The frequency of vortex shedding from a thin flag (with an audible acoustic signature) is much higher than that of the observed flapping, while the lack of a differential velocity profile across the flag and its finite flexibility and length make it qualitatively different from the Kelvin-Helmholtz problem. After the advent of highspeed flight, these questions were revisited in the context of aerodynamically induced wing flutter by Theodorsen (3-5). While this important advance made it possible to predict the onset of flutter for rigid plates, these analyses are not directly applicable to the case of a spatially extended elastic system such as a flapping flag. Recently, experiments on an elastic filament flapping in a flowing soap film (6) and of paper sheets flapping in a breeze (ref. 7 and references therein) have been used to further elucidate aspects of the phenomena such as the inherent bistability of the flapping and stationary states, and a characterization of the transition curve. In addition, numerical solutions of the inviscid hydrodynamic (Euler) equations using an integral equation approach (8) and of the viscous (NavierStokes) equations (9) have shown that it is possible to simulate the flapping instability. However, the physical mechanisms underlying the instability remain elusive. In this paper, we remedy this in terms of the following picture: For a given flag, there is a critical flow velocity above which the fluid pressure can excite a resonant bending instability, causing it to flutter. In fact, we show that in the limit of a heavy flag in a fast-moving light fluid the instability occurs when the frequency associated with the lowest mode of elastic bending vibrations of the flag becomes equal to the frequency of aerodynamic oscillations of a hinged rigid plate immersed in a flow. Physically, the meaning of this result is as follows: For a heavy flag in a rapid flow, the added mass effect due to fluid motion is negligible so that the primary effect of the fluid is an inertial pressure forcing on the plate. For a plate of length L weakly tilted at an angle , the excess fluid pressure on it scales as f U 2 , where f is the fluid density and U is the fluid velocity. 1/2 . As we will see in the following sections, this simple result arises naturally from the analysis of the governing equations of motion of the flag and the fluid. In particular, our analysis is capable of accounting for the unsteady nature of the problem in terms of the added mass of the fluid and the vortex shedding from the trailing edge in terms of the seminal ideas of Theodorsen (3). Equations of Motion Elasticity. We consider the dynamics of an inextensible twodimensional elastic plate b of length L, width l, and thickness h Ͻ Ͻ L Ͻ Ͻ l, made of a material of density s and Young's modulus E embedded in a three-dimensional parallel flow of an ambient fluid with a density f and kinematic viscosity , shown schematically in [1] Here, and elsewhere A b ϵ ѨA͞Ѩb, m ϭ s hl is the mass per unit length of the flag, B ϭ Eh 3 l͞12(1 Ϫ 2 ) is its flexural rigidity (here is the Poisson ratio of the material), ⌬P is the pressure difference across the plate due to fluid flow, and T is the tension in the flag induced by the flow. In deriving Eq. 1, we have assumed that the slope of the plate is small so that we can neglect the effect of any geometrical nonlinearities; these become important in determining the detailed evolution of the instability but are not relevant in understanding the onset of flutter. For the case when the leading edge of the flag is clamped and the trailing edge is free, the boundary conditions associated with Eq. 1 are (10): Y͑t, 0͒ ϭ 0, Y x ͑t, 0͒ ϭ 0, [2] Y xx ͑t, L͒ ϭ 0, Y xxx ͑t, L͒ ϭ 0. To close the system of Eqs. 1 and 2, we must evaluate the fluid pressure ⌬P by solving the equations of motion for the fluid in the presence of the moving plate. a To whom correspondence should be addressed. E-mail: lm@deas.harvard.edu. b Our analysis also carries over to the case of an elastic filament in a two-dimensional parallel flow.