Finite Element Simulation of the Motion of a Rigid Body in a Fluid with Free Surface

Abstract

In this work a finite element simulation of the motion of a rigid body in a fluid, with free surface, is described. A completely general referential description (of which both Lagrangian and Eulerian descriptions are special cases) of an incompressible, Newtonian fluid is used. Such a description enables a distorting finite element mesh to be used for the deforming fluid domain. A new scheme for the linearised approximation of the convective term is proposed and the improved, second order accuracy of this scheme is proved. A second theorem, which provides a guideline for the artificial adjustment of the Reynolds number when applying a continuation technique, is also proved. The most effective means of eliminating pressure as a variable and enforcing incompressibility are reviewed. A somewhat novel method to generate finite element meshes automatically about included rigid bodies, and which involves finite element mappings, is described. The approach taken when approximating the free surface, is that it may be treated as a material entity, that is, the material derivative of the free surface is assumed zero. Euler's equations and conservation of linear momentum are used to determine the motion of the rigid body. A predictor-corrector method is used to solve the combined sub-problems. The resulting model is tested in the context of a driven cavity flow, a driven cavity flow with various, included rigid bodies, a die-swell problem, and a Stokes second order wave.