The Energetic Implications of the Time Discretisation in Implementations of the A.L.E. Equations

Abstract

A class of A.L.E. time discretisations which inherit key energetic properties (nonlinear dissipation in the absence of forcing and long-term stability under conditions of time dependent loading), irrespective of the time increment employed, is established in this work. These properties are intrinsic to real flows and the conventional Navier-Stokes equations. A description of an incompressible, Newtonian fluid, which reconciles the differences between the various schools of A.L.E. thought in the literature is derived for the purposes of this investigation. The issue of whether these equations automatically inherit the afore mentioned energetic properties must first be resolved. In this way natural notions of nonlinear, exponential-type dissipation in the absence of forcing and long-term stability under conditions of time dependent loading are also formulated. The findings of this analysis have profound consequences for the use of certain classes of finite difference schemes in the context of deforming references. It is significant that many algorithms presently in use do not automatically inherit the fundamental qualitative features of the dynamics. The main conclusions are drawn on in the simulation of a driven cavity flow, a driven cavity flow with various, included rigid bodies, a die-swell problem, and a Stokes second order wave. The improved, second order accuracy of a new scheme for the linearised approximation of the convective term is proved for the purposes of these simulations. A somewhat novel method to generate finite element meshes automatically about included rigid bodies, and which involves finite element mappings, is also described.