Fig 1 - A typical example of extravagant vortex enhancement for a Boger fluid flowing in an 8:1 axisymmetric contraction (From Ref. 2). Fig 2 - A schematic ...
Contraction Flows of Highly-Elastic Liquids : Experiment and Simulation K. Walters1, M. F. Webster2 2
1 Department of Mathematics, University of Wales Aberystwyth, UK Department of Computer Science, University of Wales Swansea, UK
Introduction It is well known that highly elastic liquids can exhibit flow characteristics which are markedly different from those found in Newtonian liquids. Nowhere is this better seen than in the case of Contraction Flows, whether axisymmetric or planar. Figure 1 shows the phenomenon of ‘vortex enhancement’, where the corner vortex (which is present in Newtonian liquids) is increased in size, often in an extravagant manner. Whether the enhancement is due to the growth of the salient corner (Newtonian) vortex or results from the appearance and ultimate domination of a so called ‘lip vortex’ (see Fig 2) is a very provocative question which has yet to be fully understood.
Salient corner
Lip vortex
Fig 2 - A schematic representation of a salient corner vortex and a lip vortex.
Rr Associated with vortex enhancement is an increase in the pressure drop required to pump the liquids through the contractions. This is often called the ‘Couette correction’ – it can reach extravagant proportions, as the typical example shown in Fig 3 makes clear. In the present lecture, we first highlight what is L already known experimentally about the flow of highly-elastic liquids in a contraction. We then discuss the distinctive challenges encountered in the numerical simulation of these flows, focusing on recent attempts within the University of Wales Fig 1 - A typical example of extravagant vortex Institute of non-Newtonian Fluid Mechanics to predict enhancement for a Boger fluid flowing in an 8:1 and understand the experimental observations. We axisymmetric contraction (From Ref. 2). limit the discussion to polymer solutions, where the viscoelastic phenomena are so apparent. We consider both shear-thinning solutions and also
constant-viscosity Boger fluids, flowing in planar and axisymmetric contractions. Experimental Due to limitations of space, we shall simply précis the main conclusions which can be drawn from the extensive experiments carried out by various research groups (particularly the trail-blazing work of Boger). Many of the phenomena are very adequately displayed in the book by Boger and Walters [1] and we also draw attention to the very recent work of Nigen and Walters [2] on contraction flows of Boger fluids.
Numerical Simulation The current authors have recently been involved in a detailed review of Computational Rheology [3], especially as this impacts on the Contraction-flow problem. To make headway, the Oldroyd B constitutive equation was used in an attempt to simulate the behaviour of the constant-viscosity Boger fluids and variants of the so-called Phan Thien-Tanner model were employed to take account of shear-thinning.
3 ) 2.5 s / g 2 ( 1.5 Q
enhancement can occur in both planar and axisymmetric contractions. Available experimental results point to the importance of solution type and contraction ratios in deciding whether growth of the salient corner vortex or the appearance and strength of a lip vortex is responsible for the phenomenon of vortex enhancement. Clearly, there are significant challenges confronting theoretical rheologists as they attempt to simulate numerically the behaviour found in the experiments.
Syrup Boger Fluid
1 0.5 0 0
0.5
1
1.5 5 P (10 Pa)
2
2.5
3
Planar
Axisymmetric
Fig 3 – Flow curve in a 16:1 axisymmetric contraction for a Boger fluid and a Newtonian fluid, both of viscosity 16.5Pa.s (From Ref. 2). To summarize the current situation, we remark that, for constant-viscosity Boger fluids, vortex enhancement is clearly evident in axisymmetric contractions, but is absent in the case of planar contractions. Indeed, large vorticies are as difficult to observe in the planar case as they are difficult to avoid in the axisymmetric case. In addition, in axisymmetric contractions, the Boger fluids exhibit the anticipated enhanced pressure losses. However, in the planar case, it is impossible to distinguish between Newtonian and Boger fluids of the same shear viscosity, so far as pressure drop/flow rate is concerned. In the case of shear-thinning polymer solutions, the situation is provocatively different. Here, vortex
Fig 4 – Simulations, Oldroyd-B, 4:1 contraction, sharp corner.
Planar
Axisymmetric
Planar
Axisymmetric
Fig 5 – Simulations, Oldroyd-B, 4:1 contraction, Fig 6 – Simulations, PTT, 4:1 contraction, sharp rounded-corner. corner Both sharp corners and rounded corners are of interest to experimentalists and these have been accommodated in the numerical work, which involves a hybrid finite element/finite volume scheme. This is a second-order accurate timestepping scheme, based upon a TaylorGalerkin/pressure-correction discretisation for velocity and pressure and a cell-vertex finite volume representation of stress [4-7]. Use is made of parent triangular finite element (fe) cells, finite volume (fv) triangular subcells and median dual cell subtended regions. Solution interpolation on the parent fe-cell is quadratic for velocity, linear for pressure; on the fvsubcells, stress is linearly interpolated. Finite volume discretisation calls upon Fluctuation Distribution (upwinding) and median-dual-cell approximation, with consistent treatment of flux, source and time terms. Non-conservative flux representation is adopted as well as discontinuous stress gradients from cell to cell. Discontinuity capturing is applied around the sharp-corner, in the form of reduce-corner integration sampling.
We present a sample of mesh-converged simulation results, across planar and axisymmetric configurations with sharp and rounded corners. These are taken under creeping flow conditions and increasing Weissenberg number, We=λ1U/d, with relaxation time λ 1, U and d characteristic velocity and length, respectively. For the Oldroyd-B results of Figure 4 (emulating Boger fluids), we observe that salient-corner vortex inhibition is apparent for planar flows, notably above We of unity. Conversely, for the axisymmetric configuration, vortex enhancement is the dominant feature, with vortex intensities some three-times larger than those of planar flow. Limiting Weissenberg numbers are relatively low, rising up to 4.4 for rounded-corner (Fig. 5) and 2.8 for sharpcorner (Fig. 4) planar flows. These are roughly halved for axisymmetric cases, where comparison is made on the basis of the same average downstream velocity and half-channel width (or radius). There is clear evidence of lip-vortex activity beyond a We of 2 in the planar sharp-corner flow. This is absent in rounded-corner or axisymmetric instances. Moving on to the shear-thinning PTT model, of exponential form (with governing parameter ε=0.02 and upper-convected derivative only), we may
consider again planar versus axisymmetric results for sharp and rounded corner geometries (see Figs. 6,7). This model adopts a uniaxial extensional viscosity that reflects significant strain-hardening at extension-rates of order unity, but strain-softens thereafter. These shear-thinning models, manifest significant vortex enhancement in both planar and axisymmetric contractions. Generally, the roundedcorner flow considerably expands the We range by an order of magnitude. Fingering to and around the corner is apparent around a value of We of 10 and above. One observes much larger vortices with axisymmetric over planar flows. The sharp-corner planar flow again gives rise to a lip-vortex. Here, vortex curvature structure of convex, concave and flat have all been observed. Planar
Axisymmetric
Fig 7 – Simulations, PTT, 4:1 contraction, rounded-corner. In summary, the detailed numerical predictions are in qualitative agreement with the experimental trends. The stage is now set for a determined attack on the quantitative prediction of observed behaviour. This will involve the use of more sophisticated constitutive models. Of particular interest will be the precise role of the lip vortex in vortex enhancement and how contraction ratio can affect this role.
References 1. Boger, D.V. and Walters, K. (1993). Rheological Phenomena in Focus, Elsevier. 2. Nigen, S. and Walters, K. (2002). Newtonian Fluid Mechanics 102, 343.
J. non-
3. Walters, K. and Webster, M.F. (2001). ECCOMAS Computational Fluid Dynamics Conference, Swansea. 4. Wapperom, P. and Webster, M.F. (1998). J. nonNewtonian Fluid Mechanics 79, 405. 5. Wapperom, P. and Webster, M.F. (1999). Comp. Meth. Appl. Mech. Eng. 180, 281. 6. Aboubacar, M. and Webster, M.F. (2001). non-Newtonian Fluid Mechanics 98, 83.
J.
7. Aboubacar, M., Matallah, H. and Webster, M.F. (2002). J. non-Newtonian Fluid Mechanics (in press).
