Nonlinear static response of low-aspect-ratio inverted flags subjected to a steady flow. (November 2018) - Tavallaeinejad, Mohammad; Païdoussis, Michael P.; Legrand, Mathias
Abstract: The nonlinear static response of inverted flags, otherwise known as cantilevered plates in a reverse steady axial flow, is investigated. Recent studies have shown that the stability of low-aspect-ratio inverted flags in fluid flow is different to that of wide inverted flags. The latter undergo a supercritical divergence with increasing flow velocity, while a low-aspect-ratio inverted flag buckles abruptly within a wide bistable region. This paper aims to understand the effect of aspect ratio on the stability and bifurcation characteristics of the multiple equilibria of low-aspect-ratio inverted flags. To this end, the inverted flag is idealized as a thin Euler–Bernoulli beam coupled with the time-invariant form of the fluid forces obtained from a potential flow theory. A Hamiltonian framework is employed to derive the nonlinear integro-differential equation governing the motion expressed in terms of the mid-line rotation angle of the flag along its longitudinal axis. This allows for reliable predictions even at very large deflection amplitudes. The presence of an initial curvature along the length of the flag is also accounted for in the modelling. The Galerkin modal decomposition technique along with a Newton method is employed and the fixed-points of the governing equation are found. Bifurcation diagrams of the response with respect to the flow velocity are constructed. It is shown that the system with small aspect ratio undergoes a static divergence via a subcritical pitchfork bifurcation followed by a saddle–node bifurcation. Supercritical pitchfork bifurcations are expected for sufficiently large aspect ratios. In addition, numerical simulations are performed to investigate the effect of initial curvature amplitude on the nonlinear response of the system. The numerical predictions stemming from the proposed model compare well with existing experimental measurements available in the literature. However, it should be noted that some of the reported static responses of the inverted flag might violate the assumption of potential flow theory.