An eigenvalue-free implementation of the log-conformation formulation

Journal of Non-Newtonian Fluid Mechanics, vol. 322, pp. 105133

Abstract

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

Authors 4

1. Florian Becker corresponding

   Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

   Affiliation as printed

   German Aerospace Center (DLR), Institute for Software Technology, High-Performance Computing, Cologne, Germany

2. Katharina Rauthmann

   Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

   Affiliation as printed

   German Aerospace Center (DLR), Institute for Software Technology, High-Performance Computing, Cologne, Germany

3. Lutz Pauli Aachen MAGMA Gießereitechnologie GmbH

   Magma (Germany)

   Affiliation as printed

   MAGMA Gießereitechnologie GmbH, Aachen, Germany

4. Philipp Knechtges

   Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

   Affiliation as printed

   German Aerospace Center (DLR), Institute for Software Technology, High-Performance Computing, Cologne, Germany