Numerical Simulation of Contraction Flows for Boger Fluids using Finite Volume Methods M. Aboubacar1 , T. N Phillips2 , H. R. Tamaddon-Jahromi1 , M. F. Webster1 , and A. J. Williams3 1 Department of Computer Science, University of Wales, Swansea, UK. 2 Department of Mathematics, University of Wales, Aberystwyth, UK. 3 School of Computing and Mathematical Sciences, University of Greenwich, UK. RESULTS and DISCUSSION We consider two constant-viscosity viscoelastic fluids with the same total viscosity µ. The first is characterised by a high polymeric contribution (µ1 /µ = 0.89 and α = µ1 /µ2 = 8) to the total viscosity. This is commonly represented in the literature via the Oldroyd-B constitutive equation. In contrast, a second fluid possesses a high solvent-viscosity contribution (µ2 /µ = 0.89 and α = µ1 /µ2 = 0.1). This reflects the polymeric:solvent viscosity ratios typical of many Boger fluids. Our interest lies in charting the pressure/flow-rate behaviour for these elastic fluids against that of an equivalent Newtonian fluid of identical viscosity, in a manner following the experiments of Nigen and Walters [3]. The 4:1 abrupt contraction 600 500 400 ∆P INTRODUCTION Simulation of constant-viscosity memory-fluids flows is undertaken, considering single-mode differential models and two different numerical schemes. Comparisons are performed against experimental observations, for pressure drop (∆P ) and vortex cell-size (X). We interpret such issues through increasing flow-rate for a given fluid. Tests have demonstrated that this is equivalent to increasing relaxation time, throughout the creeping flow regime. To this end, we consider high and low solvent-fractions for Oldroyd-B models to approximate Boger fluids. Flows under consideration include planar and axisymmetric, abrupt, 4:1 geometric ratio, contraction flows. A series of meshes have been considered, covering triangular unstructured and rectangular structured forms. Numerical techniques employed are time-stepping algorithms, one of hybrid finite element/volume (fe/fv [1]) type, the other of pure finite volume form (SLFV2 [4]). In the former, a Taylor-Galerkin/pressure-correction finite element discretisation is used to solve for continuity and momentum balance equations, in combination with a second-order cell-vertex scheme for stress. Such a scheme has been developed for unstructured triangular meshes, that appeals to both fluctuation distribution and median-dual-cell nodal update contributions. The pure finite volume scheme is a staggeredgrid cell-centred-scheme, applied on rectangles (structured meshes). In this formulation the convection terms are treated in a semi-Lagrangian fashion, which has the effect of stabilising the calculations. The system of algebraic equations at each time step is solved using the SIMPLER methodology. 300 200 Axisym Newtonian Planar Newtonian Planar α = 8 Planar α = 0.1 Axisym α = 8 Axisym α = 0.1 100 0 0 0.5 1 1.5 2 2.5 3 λ U/ L Figure 1: Pressure-drop v W e. fe/fv data in figure 1 represent pressure-drop (loss) across the 4:1 abrupt contraction, plotted against increasing flow-rate. The pressure-drop is scaled µ by ( L Uref ), where Uref corresponds to the die-exit average-velocity at W e = 1. Here, W e = λUref /L, λ is a fluid relaxation-time and L the die-exit, halfwidth. These findings bear out experimental evidence, reported recently by Nigen and Walters [3] for the planar configuration. That is, typical Boger fluids (of high solvent-viscosity fraction) tend to emulate Newtonian pressure-drops, even on shorter die-exit-lengths. We note that, the polymericdominated models show departure in ∆P from the Newtonian level with increasing flow-rate, according to general simulation findings for such material properties. Couette correction (C) is the factor commonly reported for numerical studies. On this basis, there are no surprises. In all cases, C decreases monotonically, for these constant viscosity fluids (opposing trends may be observed for shearthinning fluids, see [1]. In fact, the C factor is a sensitive quantity, subject to estimation error. We increase in X, where results for the α = 0.4 model correspond to findings of Boger and Binnington [2] for fluid M1. Here, larger vortex enhancement is observed upon increasing α = 0.1 to α = 0.4. We note, however, these values are somewhat lower than those for the high-polymeric viscosity-fraction model (α = 8). We are mindful that these variations are not major and do lie within experimental scatter. It is the trends that are important. 4:1 abrupt axisymmetric contraction 0.2 X = Lv/ (2*Lu) take pains to ensure convergence in our results, on this factor. Once more, axisymmetric numerical predictions reflect lack of departure between Newtonian and solvent-viscosity dominated fluids, over the range of flow-rates covered. Again, this in agreement with experimental findings for long exit-dies (Nigen and Walters [3]). Experimentally, shorter dies do display some departure, but there, some doubt must be expressed over the lack of established fullydeveloped exit flow. Note, axisymmetric predictions for the polymeric viscosity-dominated fluid do not deviate from Newtonian levels. This may be due to the limited range of flow-rates attained in this instance. Convergence limitations are subject to the constitutive model itself, and to some degree, the fineness of the meshes employed [1, 4]. A further metric of comparison, across methods and experimentation, is that of salient-corner vortex cell-size, that provides information relating to vortex enhancement, or inhibition, with increasing Weissenberg number. Here, we restrict ourselves to the creeping flow regime, to avoid further complications due to inertial effects. We chart results 0.1 fe/fv, α = 8 fe/fv, α = 0.1 fe/fv, α = 0.4 Fluid M1, Boger et al. [2] 0 0 0.5 1 1.5 2 2.5 3 We 4:1 abrupt planar contraction Figure 3: Axisymmetric, vortex cell-size v W e. X = Lv/ (2*Lu) 0.2 0.1 fe/fv, α=8 fe/fv, α = 0.1 fe/fv, α = 0.4 SLFV2, α = 0.1 0 0 1 2 3 4 CONCLUSION We have achieved a degree of corroboration with two independent sources of experimental data for contraction flows and Boger fluids, Boger and Binnington [2] on vortex cell-size and Nigen and Walters [3] on pressure-loss. This validation evidence is encouraging, covering both axisymmetric and planar configurations. The two numerical schemes employed confirm our findings, in broad terms, for planar flows. The situation for more complex rheology awaits analysis. We Figure 2: Planar, vortex cell-size v W e in graphical form, following standard convention for Oldroyd-B fluids. This covers data for planar (Fig. 2) and axisymmetric flows (Fig. 3), with high and low polymeric viscosity-fraction models. For high polymeric viscosity-fraction models (α = 8), planar flows exhibit vortex inhibition, whilst vortex enhancement is observed for their axisymmetric counterparts. For high solvent-dominated models (α of O(0.1)), the situation is slightly different: now, vortex inhibition is not observed. In planar flows, there is little evidence of vortex enhancement. Data for fe/fv and SLFV2, show reasonable agreement, particularly around W e = 2. For axisymmetric instances and fe/fv data, there is marginal REFERENCES [1] M. Aboubacar, H. Matallah, H.R.TamaddonJahromi, and M.F. Webster. In press, J. NonNewtonian Fluid Mech., CSR1-2002, University of Wales, Swansea. [2] D.V. Boger and R.J. Binnington. 38(2):333–349, 1994. J. Rheol., [3] S. Nigen and K. Walters. J. Non-Newtonian Fluid Mech., 102:343–359, 2002. [4] T.N. Phillips and A.J. Williams. Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method. J. NonNewtonian Fluid Mech., 87:215–246, 1999.