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A stable and accurate time-marching pressure-correction/Taylor-Galerkin finite element algorithm is presented to accommodate low Mach number compressible and incompressible viscoelastic liquid flows. The algorithm is based on an operator splitting constructive process that discloses three fractional stages. For the compressible regime, a piecewise-constant density interpolation with gradient recovery is employed, for which the background theory and consistency of approach are discussed. The scheme is applied to contraction flows for Oldroyd model fluids, covering entry-exit flows and high pressure-drop situations. Stability and performance characteristics of the new algorithmic implementation are highlighted. Solutions are provided for a range of compressible settings, tending to the incompressible limit at vanishing Mach number.

In this study, we analyse viscoelastic numerical solution for an Oldroyd-B model under incompressible and weakly-compressible liquid flow conditions. We consider flow through a planar four-to-one contraction, as a standard benchmark, throughout a range of Weissenberg numbers up to critical levels. At the same time, inertial and creeping flow settings are also addressed.Within our scheme, we compare and contrast, two forms of stress discretisation, both embedded within a high-order pressure-correction time-marching formulation based on triangles. This encompasses a parent-cell finite element/SUPG scheme, with quadratic stress interpolation and recovery of velocity gradients. The second scheme involves a sub-cell finite volume implementation, a hybrid fe/fv scheme for the full system.A new feature of this study is that both numerical configurations are able to accommodate incompressible, and low to vanishing Mach number compressible liquid flows. This is of some interest within industrial application areas. We are able to provide parity between the numerical solutions across schemes for any given flow setting. Close examination of flow patterns and vortex trends indicates the broad differences anticipated between incompressible and weakly-compressible solutions. Vortex reduction with increasing Weissenberg number is a common feature throughout. Compressible solutions provide larger vortices (salient and lip) than their incompressible counterparts, and larger stress patterns in the re-entrant corner neighbourhood. Inertia tends to reduce such phenomena in all instances. The hybrid fe/fvscheme proves more robust, in that it captures the stress singularity more tightly than the feform at comparable Weissenberg numbers, reaching higher critical levels. The sub-cell structure, the handling of cross-stream numerical diffusion, and corner discontinuity capturing features of the hybrid fe/fv-scheme, are all perceived as attractive additional benefits that give preference to this choice of scheme.

We introduce a high-resolution time-marching pressure-correction algorithm to accommodate weakly-compressible highly-viscous polymeric liquid flows at low Mach number. As the incompressible limit is approached ( 0 ≈ Ma ), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In retention, yet efficiency in implementation.