Static instability of an inverted plate in channel flow: state-space representation and solution approximation

Abstract

Plate-like structures in channel flow are commonly found in engineering. This paper reports a theoretical study on the static aeroelastic instability of an inverted cantilevered plate in an inviscid channel flow through the state space. This study begins with the kernel function of the flow potential determined in the Fourier domain with the help of the mirror image method. Then, the instability equation is derived from the operator theory and transformed in the state space. Finally, with Glauert’s expansion, model functions, and error functions, the instability problem of such a plate has been modeled as a mathematical function approximation problem and solved by the least squares method. The derived instability equation is considered at the continuum level of description, and no approximation appears at the first equation level. The convergence and reliability of the proposed modeling and its solutions approximation are entirely tested, and it can successfully predict the instability boundary, behavior, and the channel effect. Numerical results show that the decreased channel height and asymmetric plate placement in the channel significantly decrease the critical flow velocity. The plate instability modes are close to the plate’s first natural ones and not sensitive to the channel parameters. This conclusion allows further theoretical exploration of a semi-analytical approximation of the instability boundary from the obtained instability equation. The current modeling strategy in a continuum sense may provide a new idea and essential reference for other instability problems.