Nonlinear dynamics of slender inverted flags in axial flow

International audience; A nonlinear fluid-elastic continuum model of a slender cantilevered plate subjected to axial flow directed from the free end to the clamped one, also known as the inverted flag problem, is proposed. A nonlinear unsteady slender wing theory is employed to express the fluid-related forces acting on the plate. It is based on potential flow theory for inviscid fluids. The Euler-Bernoulli beam theory with exact kinematics and inextensibility is employed to derive the nonlinear partial integro-differential equation governing the dynamics of the plate. Discretization in space is carried out via a conventional Galerkin scheme using the linear modeshapes of the cantilevered beam. A Newton solver is implemented to obtain equilibrium solutions and integration in time is conducted using Gear's backward differentiation formula. A bifurcation diagram in terms of flow velocity is constructed in order to gain insight on the stability and post-critical behaviour of the system. It is shown that the undeflected static equilibrium is stable prior to a supercritical Hopf bifurcation giving rise to a flapping motion around the undeflected static equilibrium. By further increasing the flow velocity, the flag displays flapping motions around deflected static equilibria and change to fully-deflected static states at even higher flow velocities. These predictions are in excellent agreement with existing experimental data available in the literature.