Velocity overshoots in gradual contraction flows
Introduction
Experimental velocity measurements of the flow of a high-molecular weight flexible polymer solution through planar gradual contraction–sudden expansion geometries [1], [2] have revealed an interesting fluid-dynamic effect. Spanwise1 profiles of the streamwise velocity in the XZ-centreplane exhibited extreme velocity overshoots close to the sidewalls, up to three times the centreline velocity in magnitude, that due to their appearance were called “cat's ears”. More recent experiments, without the sudden expansion component, have confirmed that the appearance of the “cat's ears” profiles are a sole consequence of the smooth contraction [3]. Representative velocity profiles are reproduced in Fig. 1(a), together with a schematic of the contraction geometry used in the experiments, in which a representative velocity profile along the spanwise (neutral) direction is illustrated (Fig. 1(b)).
Three-dimensional viscoelastic calculations using the Phan-Thien–Tanner (PTT) model [4] have been attempted to match the experimental conditions of Ref. [2] with a few limited simulations reported in Poole et al. [2] and an extended systematic study reported in Afonso and Pinho [5]. Although some modest success in predicting velocity overshoots was achieved, the magnitude of the overshoots – at most about 10% higher than the centreline velocity – was always much lower than that observed in the experiments. To capture these weak overshoots the full PTT model was required (ξ ≠ 0 producing N2 ≠ 0 in steady simple shear flow) together with strong strain hardening (low values of ɛ) and some inertia. In these simulations it was speculated that the presence of the geometric singularity due to the sudden expansion prevented convergence at higher Deborah numbers and that, if convergence could be achieved, a non-zero second normal-stress difference may not be required for the effect to be observed (i.e. ξ ≠ 0 just allowed the De–Re space to be reached where “cat's ears” occur).
Our interest in the current study is to revisit the problem in an attempt to capture the extreme nature of the “cat's ears” effect and to try to reveal the mechanism for their appearance. To do so our approach, in contrast to the simulations of Refs. [2], [5], is to concentrate on modelling a gradual contraction section alone, as in the recent experiments of Keegan et al. [3]. Furthermore we set aside the goal of trying to exactly match the experimental conditions of Refs. [1], [2] by selecting a related, but simplified, 3D-geometry and by varying the Re and De numbers in a systematic way. Using such a methodology we are able to show that, even for the rheologically “simple” UCM model, extreme velocity overshoots can be predicted even in the absence of inertia, i.e. the velocity overshoots are a purely elastic effect. Thus “cat's ears” profiles appear to be an inherent feature of viscoelastic flow through gradual contractions provided certain conditions, which we identify based on our numerical results, are satisfied.
The rest of this paper is organised as follows; in Section 2 we briefly describe the equations to be solved, the numerical method, the geometry and the meshes used; in Section 3 we discuss the results for a Newtonian fluid followed by the results of the viscoelastic models in Section 4; in Section 5, based on our numerical results, we discuss a possible mechanism for the “cat's ears” effect before summarising our findings in Section 6.
Section snippets
Governing equations, numerical method, geometry and computational meshes
We are concerned with the isothermal flow of an incompressible viscoelastic fluid through a gradual three-dimensional planar contraction geometry. The equations to solve are those of conservation of mass:and of momentum:
For reasons of rheological simplicity most of the simulations we report here are for the well known upper-convected Maxwell (UCM) model [6]; in addition some simulations are conducted for the simplified version of the Phan-Thien and Tanner model (PTT) [4]
Newtonian simulations
In classical Newtonian fluid mechanics gradual contractions are, at least at relatively high contraction ratios, often used to produce “uniform” velocity profiles; the most obvious exploitation of which is in wind-tunnel design in aerodynamics [11], [12], although they are also used in pipe-flow studies as inlet conditioners [13], [14] and elsewhere [15].
Fig. 4 shows 3D contours of the velocity development in the XZ-centreplane for a modest contraction ratio (in this case CR = 4) for different
Viscoelastic simulations
In order to identify the driving mechanism for the onset of “cat's ears”, we undertook a systematic study of the influence of De and CR on the observed flow patterns and on the local spanwise velocity profiles within the contraction using the UCM model. In Fig. 6 we show the effect of De on the observed velocity profiles along the XZ centreplane for a high contraction ratio (CR = 8) and in Fig. 7 we present similar plots for a low contraction ratio (CR = 2). For the higher contraction ratio, one
Origin of “cat's ears” phenomena
From the foregoing it is clear that the “cat's ears” phenomenon is due to elastic effects and can be predicted under creeping-flow conditions. Although, as we have shown, inertia enhances the phenomenon, we will restrict the present analysis to creeping flow of constant shear-viscosity fluids. However it can be expected that the underlying mechanism remains the same regardless of the level of inertia.
We shall focus our attention on the central XZ plane (y = 0), as this is the plane where the
Conclusions
We have conducted a systematic numerical study of viscoelastic flow, modelled using both the UCM and PTT models, through three-dimensional gradual planar contractions with the aim of simulating the experimentally observed “cat's ears” effect [1], [2], [3]. We have been able to reproduce the phenomenon using the UCM model, even for creeping-flow conditions, and our results thus show that neither inertia, shear-thinning shear viscosity nor a second-normal-stress difference are required for these
Acknowledgements
The authors would like to thank the British Council/Conselho de Reitores das Universidades Portuguesas for support through the ‘Treaty of Windsor Programme’ (Acção n B-11/07). MAA acknowledges funding by Fundação para a Ciência e a Tecnologia (FCT, Portugal) under projects PTDC/EQU-FTT/71800/2006 and REEQ/928/EME/2005. An illuminating roundtable discussion between the authors, Professor Radhakrishna “Suresh” Sureshkumar (University of Washington St Louis), Dr. Alexander Morozov (University of
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