On the static aeroelastic instability of an inverted plate in wall effect: a continuum model and its solution approximation

• Published in: Archive of applied mechanics (1991) 2023-05, Vol.93 (5), p.1825-1840
• Main Authors: Li, Peng, Zhang, Dechun, Yin, Hong, Cui, Junzhe, Yang, Yiren
• Format: Article
• Language: English
• Subjects: Aeroelastic stability; Aeroelasticity; Air flow; Approximation; Cantilever plates; Classical Mechanics; Confinement; Continuum modeling; Critical flow; Differential equations; Engineering; Flow velocity; Integral equations; Low speed; Mathematical functions; Operators (mathematics); Original; Parameter sensitivity; Stability analysis; Theoretical and Applied Mechanics
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-022-02358-0

This paper aims at the static instability of a thin, flexible plate loaded by low-speed airflow in the wall effect. The present plate model has the flow impinging on its leading and free edge and is called an inverted cantilevered plate. A theoretical continuum model is first established, which mathematically presents such an instability problem as a mathematical function approximation problem within the framework of differential operators. The mirror image method considers the wall’s effect on fluids. The fluid force is presented as a Possio integral equation composed of Hilbert and Tricomi operators. We bring no approximation at the first equation level, and the derived instability equation is on the continuum. A convergent numerical scheme based on the Weierstrass theorem and the least square method is proposed to solve the instability equation. The results show that the current analysis strategy successfully predicts the plate instability in the wall effect compared with other theoretical and experimental studies. The confinement of the wall plays a destabilizing effect, and the critical flow velocity significantly decreases as the confinement is increased; however, the plate instability modes are not sensitive to wall confinement. The plate instability modes are close to the plate’s first natural ones and not sensitive to the channel character parameters. This conclusion allows further theoretical exploration of an approximation of the instability boundary from the obtained instability equation, a five-second scaling relation. The present plate aeroelasticity model on the continuum and its solution approximation may provide a new idea and essential reference for other instability problems.