The stable and efficient numerical simulation of viscoelastic flows has been
a constant struggle due to the High Weissenberg Number Problem. While the
stability for macroscopic descriptions could be greatly enhanced by the
log-conformation method as proposed by Fattal and Kupferman, the application of
the efficient Newton-Raphson algorithm to the full monolithic system of
governing equations, consisting of the log-conformation equations and the
Navier-Stokes equations, has always posed a problem. In particular, it is the
formulation of the constitutive equations by means of the spectral
decomposition that hinders the application of further analytical tools.
Therefore, up to now, a fully monolithic approach could only be achieved in two
dimensions, as, e.g., recently shown in [P. Knechtges, M. Behr, S. Elgeti,
Fully-implicit log-conformation formulation of constitutive laws, J.
Non-Newtonian Fluid Mech. 214 (2014) 78-87].
The aim of this paper is to find a generalization of the previously made
considerations to three dimensions, such that a monolithic Newton-Raphson
solver based on the log-conformation formulation can be implemented also in
this case. The underlying idea is analogous to the two-dimensional case, to
replace the eigenvalue decomposition in the constitutive equation by an
analytically more "well-behaved" term and to rely on the eigenvalue
decomposition only for the actual computation. Furthermore, in order to
demonstrate the practicality of the proposed method, numerical results of the
newly derived formulation are presented in the case of the sedimenting sphere
and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found
that the expected quadratic convergence of Newton's method can be achieved.Comment: 21 pages, 9 figure
