The dynamics of compressible herschel-bulkley fluids in die-swell flows

Abstract

In a variety of industrial applications, modelling compressible inelastic free-surface flows remains a numerical challenge. This is largely due to the physical phenomena involved and the computational cost associated with the simulations of such flows. In particular, the die-swell benchmark problem is characterised by specific features. These are related to the presence of a sharp separation point at the die-exit, the location and the shape of the free-surface, and additionally the consideration of fluid compressibility under various forms of material modelling. In this article, a time-marching pressure-correction scheme is considered to solve both incompressible and compressible inelastic flows. This is achieved via a pressure-based approach within a finite element framework employing efficient high-order time-stepping schemes. A Tait-type model is utilised to express the equation of state that links density to pressure, so that pressure is retained as a primary variable. Various material models are considered in this numerical study for the die-swell problem, where the material rheological characteristics have a direct impact upon the location and form of the free-surface. Initially, unyielded material is considered through Newtonian and power-law assumptions. Further complication is then introduced through the Bingham model, where fluid yield stress is taken into account. More general rheological modelling is constructed via the Herschel-Bulkley model, combining inelastic behaviour with yield stress presence. This is complimented by relaxing incompressible assumptions, allowing the effects of compressibility to enter the problem. Results are presented for steady and transient flow scenarios and numerical solutions are validated against published data. There is Focus upon on the effect of variation in compressibility parameter setting, inertia level, power-law index and yield stress level, with regard to the evolving shape/location of the free-surface and the response in extrudate swell. Extrudate swell is observed to decline with decrease in power-law index. With increase in Reynolds number, extrudate swell decreases before finally reaching a plateau at high Reynolds number, in agreement with experimental results. Swelling also decreases with rise in yield stress levels. The combination of these parameters within the compressible Herschel-Bulkley model renders it difficult to predict, a priori, the outcome in terms of die-swell behaviour. © Springer Science + Business Media B.V. 2009.