On the Numerical Implementation of Nonlinear Viscoelastic Models in a Finite-Volume Method

Abstract

Numerical aspects of the implementation of nonlinear viscoelastic uid models in a nite volume method are investigated: (1) decomposition of the discretized stress equations in such a way that diagonal dominance is maximized, in order to promote numerical stability with iterative solvers; (2) imposition of boundary conditions for the stress components normal to a wall plane, and for pressure. These issues are investigated in relation to the Giesekus constitutive equation and illustrative ow examples showing the bene ts of the proposed methodology are given.

Uniform flow of viscoelastic fluids past a confined falling cylinder

“…It is a simpler alternative to Lagrangian approaches whose implementation in conjunction with finite element methods is addressed in the review of Baaijens (1998). Additional details of the present method regarding the implementation of the Giesekus model, so that numerical stability is improved, can be found in Oliveira (2001).…”

“…All the calculations were carried out with a finite-volume method (for details see Oliveira et al 1998and Alves et al 2000, 2001 for the limiting case of Re=0, which was imposed numerically by neglecting the convective terms of the momentum equation. Accurate representation of the convective terms in the constitutive equation is of extreme importance in viscoelastic simulations, and the CUBISTA high-resolution scheme developed by Alves et al (2003) was applied for this purpose.…”

“…Figure 4 also shows a comparison between our predictions of τ xx along the rear wake centreline with those of Dou and Phan-Thien (2003) on their mesh M4; a significant difference of the order of 50% is observed due to the high level of azimuthal refinement of M120 WR mesh. Previous works have shown mesh refinement not to be so crucial with constitutive models exhibiting shearthinning or bounded extensional viscosity (Oliveira et al 1998, Oliveira 2001, Oliveira and Miranda 2005and Alves et al 2000, 2001, such as FENE-CR, PTT or Giesekus models, and therefore, we decided not to use the finer mesh M120 WR and restrict the simulations with these models to meshes M60 and M60 WR . This decision is corroborated by the comparisons shown in Those authors also employed a "high resolution" FEM, and the matching against our data is excellent.…”

The Finite Volume Method in Computational Rheology

“…It is usual practice in CFD to extrapolate linearly the pressure to the wall from the two nearest neighbour cells (Ferziger and Perić, 2002) and this practice also works well for some viscoelastic fluids. However, in viscoelastic flow with fluids exhibiting strong normal stresses perpendicular to the wall ( 2 0 N  ), pressure extrapolation is not satisfactory and a better formulation can be derived from the momentum equation normal to the wall at the interior point P, as explained in detail in Oliveira (2001), leading to the following corrected extrapolation formula ( P a is the central coefficient in the momentum equation)…”

“…(46), and the other due to normal stress at the face which is null for constitutive models with N 2 = 0 as those used for the computations in Section 4. For constitutive models with 2 0 N  , such as the Giesekus or the full PTT, the interested reader is referred to Oliveira (2001) for the determination of n i T .…”