Aeroelastic instability of cantilevered flexible plates in uniform flow

Abstract

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 experimentally because hysteretical behaviour and three-dimensional effects appear for plates of large aspect ratio. The nature of the instability bifurcation (sub-or supercritical) is also discussed in the light of recent numerical results. IntroductionThe flutter of a flexible plate immersed in an axial flow is a canonical example of flowinduced vibrations. This instability can be experienced in everyday life when one observes a flag flapping in the wind. Because this phenomenon appears in many applications (paper industry, airfoil flutter, snoring), it has motivated a large literature which has been recently reviewed by Païdoussis (2004). This instability can be regarded as a competition between fluid forces and elasticity. Indeed, when the plate experiences a small lateral deflection, a destabilising pressure jump can appear across the plate, while the bending stiffness tends to bring the plate back to the stable planar state.This system can be studied by restricting the analysis to one-dimensional flutter modes as observed in most experiments. In this case, the plate motion obeys the Euler-Bernoulli beam equation with additional pressure forces which are calculated by assuming a potential flow. To simplify the problem further, Shelley et al. (2005) considered a plate infinite in both directions in a similar way to the stability analysis of a jet by Lord Rayleigh (1879) who already noted in his seminal paper the analogy with the problem of flag flutter.In other theoretical studies, the plate length L (or chord) takes a finite value while two asymptotic limits are considered for its span H. If H L, the fluid forces can be calculated using the slender body theory of Lighthill (1960) (e.g. Datta & Gottenberg 1975Lemaitre et al. 2005) and if H L the problem can be treated as two-dimensional (as done by Kornecki et al. 1976;Huang 1995;Watanabe et al. 2002a;Guo & Païdoussis 2000). In this latter case, the flow is entirely described by point-vortices which are distributed within the plate and possibly in its wake. It is known from airfoil theory that this problem does not admit a unique solution (intrinsically because the Laplace equation has to be solved on an open domain). Kornecki et al. (1976) used two approaches to arXiv:0804.0774v2 [physics.flu-dyn]

The origin of hysteresis in the flag instability

“…This system is solved by the same method as in our previous papers (Eloy et al 2007(Eloy et al , 2008, except that dissipative terms are retained in the analysis. The main steps of this linear stability analysis are as follow.…”

“…In large aspect ratio experiments, this hysteresis loop is typically of 20%. For aspect ratios smaller than one however, hysteresis can disappear (Eloy et al 2008). For small aspect ratios, and negecting the nonlinearities of the aerodynamical forces, Yadykin et al (2001) showed that elastic and inertial nonlinearities yield no hysteresis.…”

“…Even when bistability is present, the hysteresis loop is much smaller than in the experiments: Alben & Shelley (2008) and Michelin et al (2008) report an hysteresis loop of 2.5-4% and 4.5% while experiments in the same range of parameters exhibit loops of 20% or more. Several effects have been advanced to account for this discrepancy: blockage and confinement effects (Tang & Païdoussis 2007), planeity defects of the plate (Tang & Païdoussis 2007;Eloy et al 2008), or damping effects (Alben & Shelley 2008). But, until now, none of these hypotheses have been tested.…”

Effect of spanwise confinement on flag flutter: Experimental measurements

“…This discrepancy between different sets of parameters has already been observed by various authors, both numerically [15][16][17] and experimentally. 11 However, it was observed in the literature that even in the hysteretic case, the amplitude as function of the flow velocity is well fitted by a square root law, suggesting that the transition is supercritical. 11 In the experiments reported here, the hysteresis was rarely perceptible and was at most of 10%.…”

“…9 As these 2D works always underestimated the critical velocity when compared to experimental data, a 3D model for the flow was proposed by Eloy et al 10 Involving matching slender-body theory to 2D theory, this model evidenced the influence of the plate's aspect ratio on the critical velocity and was found to improve the flutter limit predictions. 11 It was, however, admitted that it is possible to approach the limit predicted by 2D models by adding horizontal walls near both edges of the plate. 1 A few experimental studies, 1,2,12 although not focused on this particular effect, investigated such geometries.…”