Optimal amplification of energy in fluid-flow induced flapping of a thin panel

The flapping of a thin streamlined and deformable body (panel) interacting with a fluid current is a physical phenomenon commonly observed in a variety of science and engineering situations1. In spite of a remarkable bulk of modern papers, the flow- induced flapping dynamics does not seem exhaustively studied from all the aspects. A prototype equation modelling the dynamical behaviour of a thin panel invested by a wind, in terms of its transverse position, may be written including inertia, bending elasticity, aerodynamic pressure and tension due to viscous shear stress effects. The control parameters are the ratio of the fluid density to that of the panel with which it interacts, and V, the ratio of the fluid velocity to the velocity of the so called bending waves. Various B.C.s may be enforced at leading and trailing edges for the cases of clamped and/or free edges. The non-normal character of the equation of motion has not yet investigated (apart from a two degrees-of-freedom finite-dimensional model), and this aspect constitutes, indeed, the subject of the present paper. For different values of the control parameters, as well as various B.C.s, it is shown that under subcritical conditions (i.e., before the occurrence of the flutter instability) the governing operator is non-normal. This peculiarity is described via the analysis of the pseudospectra curves. As a consequence, it is found that the subcritical flapping dynamics is crucially governed by the first two eigenvalues, i.e the first two natural frequencies that converge to each other due to the flow interaction and experience a collision followed by a subsequent splitting (leading to instability) at the critical conditions. The energy optimal amplification exhibits a transient growth characterized by significantly large time-periodic oscillations. The period of such oscillations is related to the topology of the spectrum4 and agrees strictly with the one of the periodic self-sustained global oscillations predicted via DNS of the unsteady dynamics of the panel. The physical relevance of the described oscillating behaviour is discussed as well