A Mesh-Free Compliant-Wall Fluid-Structure Interaction Model

Abstract

This paper presents the development of a numerical algorithm for the simulation of closely coupled fluid-structure interaction (FSI) systems. The particular FSI system investigated in this work involves a high-Reynolds number flow over a single-sided compliant wall section between rigid baffles upstream and downstream. This system is a fundamental analogue of many complex FSI systems found in nature ranging from biomedical applications to drag-reduction using compliant coatings. The present study compares the efficacy of various numerical techniques to resolve the fully-coupled, non-linear FSI dynamics. Of particular interest is the resolution of coupled dynamics at fluid-structure density ratios of approximately unity where typical segmented solution techniques tend to have difficulties. Numerical techniques for resolving these tightly coupled dynamics are crucial to the development of generalized workable grid-free computational methods based on boundary-element and discrete vortex formulations. The flow in this study is represented numerically as an ideal or potential axial flow, however it is important to note that the numerical schemes developed are equally applicable to rotational and viscous flow fields. The flow over the non-linear deforming surface is handled by a boundary-element method formulation of the Laplace equation. The structural dynamics are represented numerically by a finite-difference formu- lation of the Euler-Bernoulli beam equation. Various algorithms for the coupling of the fluid and structure equations will be tested for their numerical efficiency, stability and overall accuracy. The particular al- gorithms of note involve the semi-implicit, the linearised fluid inertia and the fully-implicit coupling methods. The compliant-wall is modelled using a one-dimensional (1D), non-linear, Euler-Bernoulli beam model, with the non-linearity captured through an induced tension term. We look at the transient response ob- tained from the initial value problem, with the solution obtained numerically through an implicit time stepping scheme and the finite difference method (FDM). In all cases, the O(n 2 ) computational complex- ity that is typical with the numerical solution of a boundary-element formulation is mitigated through the use of a fast-multipole method (FMM) that reduces the complexity to O(n logn). Thus, the numerics are handled in such a way that system matrices are not explicitly formed and thereby avoiding issues of associated memory storage. The results validate well against previously published experimental and numerical work. They show that the semi-implicit method is an efficient computational technique for the solution of low density-ratio FSI problems, however it fails to achieve convergence at high density ratios. The fully-implicit coupling method achieved a good convergence and efficiency in the case of high density ratio models, however it's computational cost was higher than the semi-implicit method, but still lower than the coupling of the linearised fluid inertia term. Further work in this area will involve using these results to facilitate modelling fluid-structure systems that incorporate the dynamics of full viscous and rotational flow.