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

doi:10.1016/S0377-0257(02)00015-0

Journal of Non-Newtonian Fluid Mechanics

Volume 104, Issues 2–3, 1 June 2002, Pages 125-164

https://doi.org/10.1016/S0377-0257(02)00015-0 Get rights and content

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.

Introduction

The 4:1 contraction problem, for incompressible isothermal viscoelastic flows, has emerged as a classical benchmark: a stringent testbed for stability and accuracy of numerical schemes. The relevance of such an entry flow is two-fold. From a practical point of view, it is used in several polymer processing operations, such as, in extrusion and injection molding. In rheometry, such flows are involved in the accurate measurement of material properties. From a numerical standpoint, the simplicity of the geometry, that generates complex deformation (both shear and extension), provides a well-suited framework to investigate complex viscoelastic flows. In the case of an abrupt contraction, the presence of a stress singularity at the re-entrant corner represents a stringent challenge to the stability properties of numerical schemes. This issue has attracted considerable attention over the last two decades. Associated with this aspect is the so-called “High Weissenberg number” problem: a challenging research area, that remains open in spite of significant progress to date. To alleviate the problem, it is possible to extend the range of elasticity attainable by rounding off the sharpness of the corner, and (or) by using constitutive models that display finite extensional-viscosity behaviour. An alternative approach is to incorporate the local form of stress singularity into the solution, known under certain flow assumptions for the Oldroyd-B model [1], [2], [3].

Today, a serious challenge is to establish quantitative comparison between numerical predictions and experimental observations. This may be represented in terms of pressure-drop, and vortex-activity in the salient-corner vicinity and near re-entrant corners of contraction geometries. Matching vortex patterns and trends with changing flow conditions is particularly crucial to judge scheme and model accuracy. Experimental evidence would indicate that viscoelastic fluids do not all support similar vortex patterns with increasing shear-rate, for a given contraction geometry. In particular, the importance of the extensional-viscosity and the role it plays in the development of vortices is currently highlighted, both experimentally [4], [5], [6] and numerically [7], [8], [9], [10]. With progress being made in reaching high levels of elasticity, researchers have been able to present qualitative agreement between numerical predictions and experimental flow visualisations for Boger fluids, by employing Oldroyd-B models [11]. Polymer melts and solutions have been studied, using both differential [12] and integral [9] constitutive equations to match the rheological data for experimental fluids. Although it is difficult to fit both the shear and extensional data simultaneously, qualitative comparisons are reasonably encouraging, and semi-quantitative agreement is promising. For example, Azaiez et al. [13] reported good quantitative agreement beyond creeping flow, against the experimental results of Quinzani et al. [14], between velocity and shear stress profiles at various sections upstream of a 4:1 planar contraction flow (Re=0.56, We=2.9). Comparisons on the downstream wall provided qualitative agreement only, for velocity, shear stress and normal stress differences. To account for these differences, these authors pointed to the single-mode constitutive equations employed (FENE-P, linear Phan-Thien/Tanner (PTT) and Giesekus). Despite this progress, the interaction between extensional-viscosity and the development of vortices is, as yet, not completely understood.

The aim of this paper is to provide a detailed study contrasting the flow patterns of four fluids with comparable shear-thinning properties, but with distinctly different extensional responses, in addition to the constant viscosity Oldroyd-B fluid. To this end, we choose the PTT class of constitutive models for its simplicity and ability to fulfill the above rheological requirements. Application is considered for both abrupt and smooth corner 4:1 contraction flows, in both axisymmetric and planar configurations. This article extends our previous work [10] on planar contraction geometries (sharp and rounded), and a review article [15] contrasting planar and axisymmetric flows. Here, we purposely restrict attention to creeping viscoelastic flows, since inertial effects would simply over-complicate what is already an intricate situation.

Typically, at low elasticity levels, the state of the contraction flow is steady and a weak recirculating salient-corner vortex is apparent. When elasticity is increased beyond a critical value, flow transition from steady, two-dimensional to time periodic, three-dimensional flow occurs. Vortex activity can vary, dependent upon a number of factors: fluid composition, shape of contraction (sharp, rounded-corner, or tapered-contraction), and precise contraction ratio. Several studies have been concerned with elastic instabilities, where elastic transitions are a major issue in relationship to polymer processing systems. In this overview, we restrict ourselves to stable axisymmetric flows (for steady two-dimensional flows). The reader is referred to our previous work [10] for a summary on planar flows, and to the extensive review of White et al. [16] on entry flows in general.

Starting with the elastic-constant-viscosity Boger fluids, we cite two experimental works of Boger et al. [4], [17] regarding axisymmetric 4:1 contractions. In the first study [4], these authors compared the flow of two fluids with essentially the same characteristic times, through an abrupt contraction. The observations revealed two distinct flow structure patterns with increasing shear-rate. The first fluid, a PAA/corn-syrup solution, showed continual salient-corner vortex growth, with the separation line changing from a concave to a convex shape. On the other hand, the second test fluid (a PIB/PB solution) exhibited a combination of lip- and salient-corner vortex: as shear-rate increased, the lip-vortex completely engulfed the decreasing salient corner vortex, leaving a single large recirculation region of convex shape. Note, in both experiments, the vortex stretched up to the re-entrant corner at high shear-rates. Boger et al. [4] concluded that knowledge of steady and dynamic shear properties alone is not sufficient to characterise elastic liquids in circular contractions. In the second study Boger and Binnington [17], examined the effect of rounding the re-entrant corner in a 4:1 contraction, again using two fluids with almost the same zero-shear-rate relaxation time. Not only did each fluid behave differently from a sharp to a rounded-corner contraction, but also, the two fluids showed distinctly different vortex development for a given geometry. For example, in the rounded-corner flow, the salient-corner vortex remained almost constant for the so-called M1 fluid (in the creeping flow regime), whilst lip-vortex activity was observed for the fluid referred to as P1. As for the abrupt contraction, the M1 fluid experienced marginal vortex growth, whereas vortex enhancement was more dramatic for the P1 fluid. Again, these major differences, between the flow of fluids with similar shear properties in the same geometries, leave their extensional properties as the underlying explanation for such diverse trends in vortex activity.

The experimental visualisation of various polymer solution and melt flows has added further weight to the view that extensional viscosity plays a major role in the manner vortices develop in contractions. Although these fluids are shear-thinning, a number of authors were able to establish experimentally the dominant influence of the extensional viscosity. For example, Tremblay [6] used a series of LLDPE/LDPE polymer blends to show that, at a given shear-rate, the higher the content of LDPE in the blend, the larger the vortices observed in 10.6:1 and 6.8:1 sudden contractions. Note, when used separately, the LDPE (strain-hardening, branched-structure) generated large vortices, whereas the LLDPE (no deviation in elongational stress growth experiments, linear-like structure with only short branches) exhibited only small vortices. Byars et al. [18] considered the flow of the strain-hardening test fluid S1 through 4:1 abrupt- and rounded-contractions. Noting that shear-thinning and inertial effects were minor in the range of shear-rates attempted, these authors demonstrated continual vortex growth with Weissenberg number (We) in both geometries. The shape of the vortex evolved from concave at low We, to convex at high We. Moreover, at each We, the vortex was larger in the abrupt-contraction than in the rounded-case. No lip-vortex actisvity was detected, and the flow remained stable for the range of Weissenberg numbers investigated (We≤5). Byars et al. also compared two differential constitutive models (PTT, Giesekus) and the Papanastasiou et al. [19] integral equation. They concluded that, among these models, the PTT model provided the best fit to the rheological characterisation of the fluid by Ooi and Sridhar [20].

A number of numerical studies have been published with the influence of extensional viscosity in contraction flows in mind, e.g. [7], [9], [11], [21], [22], [23]. Mitsoulis [9] considered the flow of an IUPAC–LDPE melt through 4:1 axisymmetric and 14:1 planar contractions (both geometries abrupt). Simulations were performed under creeping flow conditions, using an eight-mode K-BKZ integral constitutive model. For the axisymmetric flow, Mitsoulis demonstrated that the salient-corner vortex first grew, then decreased with increasing flow-rate, following the behaviour of the uniaxial extensional viscosity. In contrast, the same model failed to reproduce vortex enhancement in the planar contraction. Mitsoulis rectified this apparent discrepancy from the experiment observations of Wassner et al. [24], by modifying the damping function of the constitutive equation. This allowed for strain-hardening to emerge in the planar extensional viscosity as well.

The numerical study of Luo and Mitsoulis [7] went further in relating strong vortex activity to strain-hardening properties. These authors contrasted the flow of an LDPE and HPDE melt through axisymmetric sudden contractions (4:1 and 5.75:1). Both fluids were modelled using an eight-mode K-BKZ constitutive equation. The strain-thickening branched LDPE melt generated large upstream vortices, whilst the strain-softening linear HDPE melt produced no significant vortex activity. Retaining the shear properties of the HDPE fluid, whilst increasing its extensional viscosity to replicate the strain-hardening properties of LDPE, these authors generated salient-corner vortices similar to those of an LPDE melt. Similarly, retaining the shear properties of the LDPE melt and changing its extensional behaviour to reflect that of the HPDE fluid, produced vortex patterns similar to those observed with HPDE alone. Luo and Mitsoulis concluded, this was convincing evidence that strain-hardening was responsible for rapid upstream-vortex growth, whereas strain-softening stunted vortex growth with increasing flow-rate.

Section snippets

Governing equations

For an incompressible fluid of density ρ, the equation of motion and continuity may be expressed as $ρ ∂ u ∂t =∇· T −ρ u ·∇ u −∇p,$ and $∇· u =0,$ where p is the pressure, and $u$ the velocity vector. In this paper, the detailed rheology is represented via PTT models [25], [26], for which the extra-stress tensor $τ$ is defined by $T = τ +2μ_{2} D,$ $f τ +λ_{1} τ ∇ =2μ_{1} D,$ $f= 1+ ϵλ_{1} μ_{1} trace (τ), for linear version (LPTT), exp ϵλ_{1} μ_{1} trace (τ), for exponential version (EPTT),$ where λ₁ is the relaxation time, μ₁ and μ₂ the polymeric and solvent viscosity

Numerical scheme

Recently, we have developed a novel hybrid finite volume/element (fv/fe) algorithm. The momentum transport and continuity equations are approximated by a semi-implicit (fe) Taylor–Galerkin/pressure-correction scheme, whilst the stress constitutive equation is discretised with a cell-vertex fv algorithm. The combination results in a time-stepping process, with a fractional-staged formulation based upon each time-step. This second-order hybrid fv/fe method was first proposed by Wapperom and

Problem specification

Five model fluids are considered, Oldroyd-B and four PTT variants. Contraction ratio is selected as 4:1, with both sharp- and rounded-corner, in plane and axisymmetric flow configurations. Following Keiller [35] and Matallah et al. [33], the rounded-corner radius is taken equal to three-quarters of the downstream half-channel width L. A schematic representation of the geometry is provided in Fig. 2. The upstream and downstream lengths are selected as 27.5L and 49L, respectively. A

Flow structure

We contrast the streamline patterns of five different constitutive models, changing the rheology gradually in order to highlight the particular influence of extensional viscosity. We commence with the Oldroyd-B model, a constant shear viscosity fluid that displays extreme strain-hardening at modest extension-rates.² Subsequently, we pursue four PTT variants, all shear-thinning (Fig. 3a): EPTT (ϵ=0.02 or 0.25),

Pressure-drop

There are a number of ways to express pressure-drop across the contraction. For example, the experimental study of Rothstein and McKinley [41] compared the ratio of the total non-Newtonian pressure-drop (Δp) against the Newtonian counterpart (Δp_Newt) with increasing flow-rate. The flow considered was that of a Boger fluid in a 4:1:4 axisymmetric contraction/expansion under creeping flow conditions. Here, our concern is pressure variations with increasing fluid elasticity. Fig. 21 contrasts the

Conclusions

This work covers a depth of benchmarking, providing contrast over a range of different fluid rheological responses within abrupt and smooth 4:1 contraction flows, for both planar and axisymmetric configurations. This has been achieved through the implementation of a novel, accurate and efficient hybrid finite volume/element scheme. The influence of extensional properties on flow structure has been demonstrated clearly, by studying the Oldroyd-B model, and four PTT models with essentially the

Acknowledgements

In this study, the planar results were supported by an European grant FMRX-CT98-0210 within the TMR Network on “Dynamics of Polymeric Liquids”. The axisymmetric results were supported by work under EPSRC grant GR/-M82523. The authors also wish to acknowledge the fruitful discussions with our University of Wales Institute of Non-Newtonian Fluid Mechanics colleagues and, in particular, Prof. Ken Walters (FRS), and Prof. Tim Phillips.

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