The Lagrange-Galerkin method for fluid-structure interaction problems

Jorge San Martín,Loredana Smaranda,Jean-François Scheid

doi:10.1186/1687-2770-2013-246

Abstract

Abstract In this paper, we consider a Lagrange-Galerkin scheme to approximate a two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.

Highlights

In this article, we present a modified characteristics method for the discretization of the equations modelling the motion of a solid immersed in a cavity filled by a viscous incompressible fluid
We are interested in rigid and deformable solids modelling some particulate flows in the case of rigid solid and the swimming of slender, neutrally buoyant fish, for the deformable structure
In [, ], the authors have introduced a convergent numerical method based on finite elements with a fixed mesh for a two-dimensional fluid-rigid body problem, where the densities of the fluid and the solid are equal

Summary

Introduction

We present a modified characteristics method for the discretization of the equations modelling the motion of a solid immersed in a cavity filled by a viscous incompressible fluid. In [ , ], the authors have introduced a convergent numerical method based on finite elements with a fixed mesh for a two-dimensional fluid-rigid body problem, where the densities of the fluid and the solid are equal. In [ , ], we have introduced crucial modifications on the characteristic function, and we have proposed a convergent numerical scheme for a two dimensional fluid-rigid body problem where the densities of the fluid and the solid are different.

Results

Conclusion