A Galerkin formulation for the nonlinear analysis of a flexible beam in channel flow

Abstract

In this paper we propose a Galerkin formulation for the 1-D model describing the nonlinear flow-structure interaction of a flexible beam in confined flow. In broad terms, the system of PDEs is converted into a set of time-dependent equations (ODEs and algebraic) by developing all variables in terms of series of space-dependent orthogonal functions. The beam motion is developed in terms of its mode shapes while the flow pressure and velocity fields in each channel are developed in terms of Chebyshev polynomials. Additionally, a tau-variant of the Galerkin approach enables the enforcement of the nonlinear time-dependent boundary conditions in a well-posed manner. Ultimately, the resulting system is a set of nonlinear differential–algebraic equations that can be truncated at any suitable numbers of terms, leading to exploitable reduced formulations. Compared to CFD methods, this type of formulations is not only more computationally efficient, but also provide an easy discernment of the relevant parameters and often a more intuitive interpretation of results. Convergence studies, in terms of both the truncation of Chebyshev polynomials and beam modes, are performed to access to what extent reduced formulations are viable in different scenarios, including linear stability analysis as well as the calculation of limit-cycle oscillations with or without intermittent impacts between the beam and the side-walls. The presented framework can serve as a basis for a comprehensive analysis of the nonlinear dynamics of flexible beams in confined flow. Namely, it is well adapted to the use of bifurcation analysis tools for the continuation of periodic solutions, which will contribute to a richer understanding of the underlying physics occurring in this type of FSI systems. Moreover, the generic methodology presented here can also be adapted to different systems in the field of fluid–structure interaction, providing compact time-dependent formulations for nonlinear analysis.