The energetic implications of using deforming reference descriptions to simulate the motion of incompressible, Newtonian fluids

• Published in: Computer methods in applied mechanics and engineering Vol. 180; no. 1; pp. 219 - 238
• Main Author: Childs, S.J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-01-1999; Elsevier
• Subjects: ALE; Arbitrary Lagrangian Eulerian; Completely general reference description; Computational techniques; Energy conservation; Exact sciences and technology; Finite elements; Finite-element and galerkin methods; Fluid dynamics; Free surface; Fundamental areas of phenomenology (including applications); General theory; Incompressible, Newtonian fluid; Mathematical methods in physics; New Poincaré inequality; Physics; Rigid body in a fluid; Incompressible, Newtonian fluid; Finite elements; Energy conservation; Completely general reference description; New Poincaré inequality; ALE; Free surface; Arbitrary Lagrangian Eulerian; Rigid body in a fluid; Numerical method; Computerized simulation; Implementation; Finite element method; Discretization; Dynamics; Motion study; Incompressible fluid; Newtonian fluid; Rigid material; Energy models; Navier-Stokes equations
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/S0045-7825(99)00076-6

In this work the issue of whether key energetic properties (nonlinear, exponential-type dissipation in the absence of forcing and long-term stability under conditions of time dependent loading) are automatically inherited by deforming reference descriptions is resolved. These properties are intrinsic to real flows and the conventional Navier–Stokes equations. A completely general reference description of an incompressible, Newtonian fluid, which reconciles the differences between opposing schools of thought in the literature is derived for the purposes of this investigation.
The work subsequently focusses on establishing a class of time discretisations which inherit these self-same energetic properties, irrespective of the time increment employed. 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. An `updated' approach as a means of avoiding ever burgeoning deformation gradients and a still further simplified implementation are further topics explored.