Successes and Failures of Discontinuous Galerkin Methods in Viscoelastic Fluid Analysis

Abstract

To date, the more successful numerical methods in viscoelastic fluid dynamics are based upon the so called Discrete Elastic Viscous Stress Splitting (DEVSS) algorithm [6] together with a suitable form of upwinding of the hyperbolic part of the constitutive equation. An elegant way to perform upwinding on the viscoelastic stress tensor can be found in Discontinuous Galerkin techniques [4]. In particular the recently developed DEVSS/DG version [1], has proven to be successful in analyzing viscoelastic fluid flow problems in both smooth and non-smooth geometries. A particularly attractive feature of DG-based methods is that they allow for an efficient resolution of flow problems with multiple relaxation times, as was demonstrated in Baaijens et al. [1] which has recently been extended to three dimensional flows [2].However, one of the key issues in simulations of viscoelastic flows remains the assessment of temporal stability of the computational method. Especially, increasing elasticity beyond critical values of the Weissenberg number can give rise to numerical instabilities in flows that are otherwise mathematically stable.KeywordsViscoelastic FluidDiscontinuous Galerkin MethodNormal Stress DifferenceWeissenberg NumberMixed Finite Element MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.