Consistent hybrid finite volume/element formulations: Model and complex viscoelastic flows

Abstract

The accuracy and consistency of a new cell-vertex hybrid finite element/volume scheme are investigated for viscoelastic flows. Finite element (FE) discretization is employed for the momentum and continuity equation, with finite volume (FV) applied to the constitutive law for stress. Here, the interest is to explore the consequences of utilizing conventional cell-vertex methodology for an Oldroyd-B model and to demonstrate resulting drawbacks in the presence of complex source terms on structured and unstructured grids. Alternative strategies worthy of consideration are presented. It is demonstrated how high-order accuracy may be achieved in steady state by respecting consistency in the formulation. Both FE and FV spatial discretizations are embedded in the scheme, with FV triangular sub-cells referenced within parent triangular finite elements. Both model and complex flow problems are selected to quantify and assess accuracy, appealing to analysis and experimental validation. The test problem is that of steady sink flow, a pure extensional flow, which reflects some of the numerical difficulties involved in solving more generalized viscoelastic flows, where both source and flux terms may contribute equally to stress propagation. In addition, a complex transient filament-stretching flow is chosen to compute the evolution of stress fields within liquid bridges. Shortcomings of the various stress upwinding schemes are discussed in this context, whilst dealing with such free-surface type problems. Here, stress fluctuation distribution alone is advocated, and a Lax-scheme is found to deliver accuracy and stability to the computational results, comparing well with the literature. © 2004 John Wiley and Sons, Ltd.