M.F. Webster

Recovery and stress-splitting schemes for viscoelastic flows

Abstract Various recovery and stress-splitting schemes are investigated numerically for an Oldroyd-B model within the general framework of a time stepping fractional-staged finite element formulation, that of a Taylor-Galerkin/pressure-correction method with consistent streamline upwinding. Smooth and non-smooth planar flows are cited and both creeping and inertial conditions are considered. Problems addressed include flow through 4:1 contraction geometries, with rounded or sharp re-entrant corners and flow past a cylinder. Vortex behaviour and scheme performance is analysed. The recovery-based schemes are stability enhancing, being superior in higher De attenuation over conventional and EVSS alternatives. It is the recovery aspect and not the stress-splitting, that is the key element responsible for this improvement. Considerable care must be exercised with time-stepping schemes of the pressure-correction form to sustain accuracy and stability.

A taylor-petrov-galerkin algorithm for viscoelastic flow

Abstract A viscoelastic flow is solved using a generalised Taylor-Galerkin/pressure correction scheme that incorporates consistent Petrov-Galerkin streamline upwinding within the discretisation of the constitutive equations. The numerical approach is indirect, in the sense that fractional equation solution stages are introduced within the framework of a time-stepping scheme. The Oldroyd-B and Phan-Thien-Tanner constitutive models are considered and the proposed Taylor-streamline upwind/Petrov-Galerkin algorithm is used to simulate flow through a 4:1 planar contraction. For the Oldroyd-B model, the algorithm indicates the onset of instability as elasticity is increased and a limiting Weissenberg number is observed, with no lip vortices apparent for values less than five. For a particular class of Phan-Thien-Tanner models, it is found that the algorithm is stable at high Weissenberg numbers. In this case the solutions exhibit a lip-vortex mechanism for the establishment of recirculating regions as has been observed in some experiments. Results presented for various combinations of fluid parameters suggest that the extensional behaviour of the viscoelastic model is the single most important factor governing the stability and convergence of the algorithms for highly elastic fluids.

Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

Abstract A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers ( We ) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.

A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows

Abstract A stable and accurate implementation is presented of a cell-vertex hybrid finite volume/element scheme based upon triangular meshes. This scheme is novel to this domain and is applied to the numerical solution of Oldroyd model fluids in abrupt planar contraction flows. All important has been the use of non-conservative flux representation, consistency in treatment of transients, fluxes and sources, and a recovery technique for velocity gradients. Linear stress representation, with non-recovered stress gradients, has proved crucial to widening stability thresholds. Solution smoothness at the higher levels of elasticity results, so that converged solutions are attainable. We have also highlighted the importance of incorporating a reduced-integration local discontinuity capturing technique for the re-entrant corner solution. A diminishing lip vortex with mesh refinement is reported. With increasing elasticity, lip vortex growth and a diminishing salient corner vortex is noted. In addition, a trailing-edge vortex is found to accompany the onset of a lip vortex. Agreement with analytical theory is observed through the asymptotic behaviour of velocities and stresses near the re-entrant corner, a region where the loss of evolution of the discrete system is investigated.

On vortex development in viscoelastic expansion and contraction flows

Abstract Vortex structure and formation are investigated for both expansion and contraction flows of viscoelastic fluids concentrating on a class of constitutive models due to Phan-Thien and Tanner. Expansion flows are treated in both two and three dimensions, and viscoelasticity is observed to suppress vortex activity. Two-dimensional contraction flows are analysed for highly elastic solutions. Different geometric aspect ratios, levels of inertia and elasticity are considered for these various flows and lip vortices are observed under certain circumstances.

On Newtonian and Non-Newtonian Flow in Complex Geometries

A flow visualization technique by means of an expanded laser beam and trace amounts of particulate additives is used to study the behaviour of Newtonian and non-Newtonian elastic liquids in a number of complex geometries. Particular attention is paid to the effect of fluid elasticity on the flow characteristics. Attempts are made to simulate numerically the observed flows by using finite-difference techniques. The agreement between theory and experiment is very satisfactory.

Simulation for viscoelastic flow by a finite volume/element method

Stability of a second-order finite element/finite volume (FE/FV) hybrid scheme is investigated on the basis of flows with increasing Weissenberg number. FEs are used to discretise the balances of mass and momentum. For the stress equation a FV method is used, based on the recent development with fluctuation distribution schemes for pure convection problems. Examples considered include a start-up channel flow, flow past a cylinder and the non-smooth 4:1 contraction flow for an Oldroyd-B fluid. A considerable gain in efficiency per time step can be obtained compared to an alternative pure FE implementation. A distribution based on the flux terms is unstable for higher Weissenberg numbers, and this is also true for a distribution based on source terms alone. The instability is identified as being caused by the interaction of the balance equations and stress equation. A combination of distribution schemes based on flux and source terms, however, gives a considerable improvement to the hybrid FE/FV implementation. With respect to limiting Weissenberg number attenuation, the hybrid scheme is more stable than the pure FE alternative for the smooth flow past a cylinder, but less so for the non-smooth contraction flow. The influence of additional strain-rate stabilisation techniques is also analysed and found to be beneficial.

Numerical prediction of extensional flows in contraction geometries: hybrid finite volume/element method

Abstract We examine the flow of viscoelastic fluids with various shear and elongational properties in axisymmetric and planar 4:1 contractions, under creeping flow conditions. Particular attention is paid to the influence of elongational viscosity upon vortex enhancement/inhibition. Simulations are performed with a novel hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. Both abrupt and rounded-corner configurations are considered. The Oldroyd-B fluid exhibits vortex enhancement in axisymmetric flows, and vortex reduction in planar flows, qualitatively reproducing experimental observation for some Boger fluids. For shear-thinning fluids (Phan-Thien/Tanner models, PTT), both vortex enhancement and inhibition is observed. This follows trends in extensional viscosity. Lip-vortex activity has been observed in planar and sharp-corner instances, but not in axisymmetric or rounded-corner flows. Finally, cross-flow extensional-stress contours in the salient-corner neighbourhood reflect the size and curvature of the associated vortex structure.

On two- and three-dimensional expansion flows

Abstract This paper gives consideration to incompressible Newtonian flows through two- and three-dimensional expansions. Through a time-stepping scheme, steady-state solutions are sought for both small and large expansion ratios. The flow structure in the two-dimensional case can be significantly different from that of the three-dimensional equivalent, depending on the expansion ratio. Both lip and salient corner vortices are reported, and comparisons are made against available experimental data.

Experimental and numerical simulation of dough kneading in filled geometries

The main objective of this study is to counterpart numerical model and experimental studies for rotating flows associated with dough kneading, and validate the flow patterns generated. The flows considered are in a complex domain setting. Two types of cylindrical vessels are studied at various rotational speeds; one with one stirrer and a second with two. Laser Doppler anemometry is used to obtain the velocity vectors associated with the flow fields. Close agreement is obtained between numerical and experimental flow fields and the magnitudes of velocity vectors. Both sets of results show maximum shear-rates outside the stirring rods. The rate-of-work done also peaks in this region and this is an important quantity to dictate optimal mixer design.