Three-dimensional numerical simulations of viscoelastic flows – predictability and accuracy Original Research Article

In this paper, fully three-dimensional (3-D) numerical simulations of viscoelastic flows using an implicit finite volume method are discussed with the focus on the predictability and accuracy of the method. The viscoelastic flow problems involving the stress singularity, including plane stick–slip flow, the flow past a junction in a channel, and the 3-D edge flow, are used to test the ability of the method to predict the singularity features with accuracy. The accuracy of the numerical predictions is judged by comparing with the known asymptotic behaviour for Newtonian fluids and some viscoelastic fluids, and the investigations are extended to the viscoelastic cases with unknown singular behaviour. The Phan-Thien–Tanner (PTT) model, and in some cases, the upper-convected Maxwell (UCM) model, are used to describe viscoelastic fluids. The numerical results with mesh refinement show that the accuracy is quite satisfactory, especially for Newtonian flows. For viscoelastic flows, the asymptotic results for the flow around a re-entrant corner for the UCM as well as the PTT fluid are reproduced numerically. In the stick–slip flow, a Newtonian-like asymptotic behaviour is predicted for the UCM fluid. In edge flow, it is verified numerically that the kinematics are Newtonian for viscoelastic fluids described by models with a constant viscosity and a zero second normal stress difference. For viscoelastic fluids described by the models with a shear-thinning viscosity and zero second normal stress difference, the fluid behaves like a power-law fluid, and the difference from its Newtonian kinematics is localized in the region near the singularity, and to capture the asymptotic behaviour, a parameter-dependent mesh has to be used. With the 3-D simulations, it is confirmed that in edge flow, the flow around the edge could not be rectilinear, and some secondary flows on the plane normal to the primary flow direction are expected for viscoelastic fluids described by the models with a shear-dependent second normal stress difference, such as the full PTT model. The strength of the secondary flows will depend on the level of the departure of the second normal stress difference from a fixed constant multiple of viscosity of the fluid.