Flutter and resonances of a flag near a free surface

Abstract

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 surface is characterized by the Froude number, corresponding to the ratio of the incoming flow velocity to that of the gravity waves. After presenting the simplified limit of infinite span (i.e. two-dimensional problem), the results are generalized to include finite-span and lateral confinement effects. In both cases, three unstable regimes are identified for varying Froude number. Rigidly-confined flutter is observed for low Froude number, i.e. when the free surface behaves as a rigid wall, and is equivalent to the classical problem of the confined flag. When the flow and wave velocities are comparable, a new instability is observed before the onset of flutter (i.e. at lower reduced flow speed) and results from the resonance of a structural bending wave and one of the fundamental modes of surface gravity waves. Finally, for large Froude number (low effect of gravity), flutter is observed with significant but passive deformation of the free surface in response of the flag's displacement.

Radiation-induced instability of a finite-chord Nemtsov membrane

“…which depends on the length Γ of the membrane. Indeed, since ω = ω n + iω i , where ω n = nπM w /Γ is the free membrane frequency (35) and ω i ∈ R is fixed, we have a clear numerical evidence that the zeros of (75) tend either to κ SW ± as Γ → ∞ or to κ DW ± as Γ → 0, see Fig. 8.…”

“…Indeed, this parameter acts as a linear factor at the nonlinear in ω operator and keeping it sensibly small prevents us from departing too far from the free membrane solution (corresponding to α = 0). This allows us to initiate our algorithm by choosing the eigenfrequency of the free membrane (35) as an initial starting point. With this first guess ω (0) = ω n , the method of successive linear problems is an iterative routine, where the p-th iteration requires to solve a linear eigenvalue problem…”

“…To estimate the critical value M = M c we use the idea of phase synchronisation between modes of the membrane and the surface of the flow [27,37] that in the infinite-chord membrane case gives a sufficient condition for the presence of an instability domain in the parameter space [26]. For the finite-chord membrane we consider the frequencies ω n of vibration modes of the free membrane defined by equation (35) and surface waves frequencies ω f that for simplicity we take from the shallow water dispersion relation of the decoupled system…”

“…where p 0 = ω/M w . Applying the residue theorem to the last integral in (63), we reproduce the eigenfrequencies ω n given by equation (35).…”

“…The need for light, inexpensive and rapidly deployable wave barriers requires taking into consideration submerged horizontal flexible plates and membranes [31,32]. Recent applications in energy harvesting exploit fluidstructure interaction, leading to the excitation (flutter) of an elastic plate or membrane, usually referred to as a flag [33,34], due to radiation of surface gravity waves when immersed in a moving flow with a free surface [35]. Another relevant setting comes from the problem of turbulent friction reduction in a boundary layer by using compliant coatings.…”