Journal of Non-Newtonian Fluid Mechanics, 42 (1992) 301-314 Elsevier Science Publishers B.V., Amsterdam

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Numerical simulation of circular entry flows of fluid Ml using an integral constitutive equation

H.J. Park and E. Mitsoulis

Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, KIN 9B4 (Canada)

(Received July 29, 1991; in revised form November 4, 1991)

Abstract

Numerical simulations have been undertaken for the entry flow of a well-characterized polymer solution (fluid Ml) through 4: 1 and 22: 1 circular contractions. The fluid has been modelled using an integral consti- tutive equation of the K-BKZ type with a spectrum of relaxation times. Numerical values for the constants appearing in the equation have been obtained from fitting shear and elongational viscosity data and normal stresses as measured in shear. The numerical solutions show that vortex growth occurs in a 4 : 1 circular contraction in agreement with experimental results for low to moderate shear rates. However, for high shear rates the simulations show a continuing increase in vortex size (but not in intensity), while experiments show a gradual elimination of the vortex even if the Reynolds number is less than one. In contrast, flow in a 22: 1 circular contraction exhibits a vortex growth always in agreement with experimental observations. The present results show that geometry and inertia effects play an important role in the flow of viscoelastic fluids through contractions and should not be neglected in the simulations.

Keywords: circular entry flows; fluid Ml; viscoelasticity; integral constitutive equations.

1. Introduction

Circular entry flows have received considerable attention in the litera- ture, particularly in recent years where viscoelastic behaviour accounts for unusual flow phenomena not observed for Newtonian fluids [l]. Early

Correspondence to: E. Mitsoulis, Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, KlN 9B4, Canada.

0377-0257/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

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attempts at modelling entry flows using relatively simple constitutive equa- tions, such as the second-order, upper-convected Maxwell and Oldroyd-B fluids, failed to converge at relatively low values of a dimensionless num- ber, characteristic of viscoelasticity (Weissenberg or Deborah number) [2]. Recent progress has seen the use of streamline-upwind (SU) finite ele- ments which have overcome problems of convergence [3]. However, the numerical results have failed to produce quantitative agreement on some of the key flow features observed experimentally [4]. This failure has brought into question the use of the Oldroyd-B equation for modelling even such simple viscoelastic fluids as Boger fluids. However, it is considered that since accurate solutions exist for the Oldroyd-B fluid, from a number of independent researchers, any numerical algorithm should be first tested against solutions of this equation to establish its validity.

The situation has prompted the search for more realistic models able to simulate the behaviour of polymer solutions and melts. It has been argued that integral constitutive equations with a spectrum of relaxation times are more capable of describing adequately the behaviour of viscoelastic fluids in simple shear, simple elongation and any combination of mixed flow [5]. Considerable success in modelling such flows has been achieved with an integral constitutive equation of the K-BKZ type, proposed by Papanasta- siou et al. [5]. This equation has been used to predict several well-known viscoelastic phenomena exhibited by polymer melts [6-121. Quite recently, it has also been used for the modelling of the Ml fluid by Chai and Yeow [13], who give the parameter constants for this fluid. They used this equation in the one-dimensional modelling of the gravity jet problem for fluid Ml with considerable success.

In the present paper we undertake the numerical simulation of the entry flow problem for fluid Ml through abrupt circular contractions using the K-BKZ integral equation proposed by Papanastasiou et al. [5] with the parameter constants determined by Chai and Yeow [13]. In particular, we first use the equation, as reduced to the integral equivalent of the Oldroyd-B model, to test the accuracy of the solution against well-established results in the literature [3,14]. Then we embark upon simulating the experiments by Boger and Binnington [15] for the Ml fluid and show the relative importance that geometry and inertia forces play in contraction flows of such viscoelastic polymer solutions.

2. Mathematical modelling

The flow is governed by the conservation equations of mass and momen- tum. For an incompressible fluid under isothermal flow conditions we have

v*v=o, (1) pv*vv= -vp+v*7, (2)

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where v is the velocity vector, T is the extra-stress tensor, and p is the scalar pressure. The constitutive equation that relates 7 to the deformation history is a K-BKZ equation proposed by Papanastasiou et al. [5] and is written as

T(t)=f c2 P --[ hkex i-Y)] ((.y - 3) +pIp-,+ (I -@I, c2(t’) dt”

(3) where A, and uk are the relaxation times and relaxation modulus coeffi- cients at a reference temperature T,,, respectively, cy and p are material constants, and I,, I,-1 are the first invariants of the Cauchy-Green tensor C, and its inverse C;‘, the Finger strain tensor.

The upper-convected Maxwell model of the integral type is readily obtained by setting LY + 03 (or equivalently a very large number, say 10 000) and p = 0. In this case the Maxwell integral model is given by

T(t) = It _l$exp (- y )]C;‘(t’) dt’,

where p is a constant viscosity and A a single relaxation time. The above model is of particular importance in the development of numerical tech- niques for integral models, because it has a simple differential counterpart and because some analytical solutions exist in a few simple cases.

The integral analogue of the Oldroyd-B model is given [16] by

/

f Pl r=p2j+ Fexp

--m 1

where + is the rate-of-strain tensor and

CL =I-9 +P2’

A,=A, p2 Pl+Pu,’

with A, and A, the relaxation and retardation times, respectively, and ,u the zero-shear viscosity. Commonly, p1 = 8~/9, p2 = p/9 and thus A, = A,/9. Equation (5) models solutions of viscoelastic liquid of relaxation time A, and constant viscosity pi in a Newtonian solvent of viscosity p2. Equation (5) reduces to the upper-convected Maxwell model when p2 = 0 and to the Newtonian law when A, = 0.

In the case of the Maxwell model it has been customary to define a dimensionless Deborah number by

De =A-jl,, (7)

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where i;, is the wall shear rate in fully-developed Poiseuille flow. In the case of the Oldroyd-B fluid, we have

De = A,+,. (8)

3. Method of solution

Special numerical schemes are required for the implementation of integral-type constitutive equations in finite elements. The streamline fi- nite-element method (SFEM) developed by Luo and Tanner [6] proved to work well for flows with open streamlines [7,9] (no recirculation present). Modifications made to that method to account for regions with recircula- tion (closed streamlines) have been presented quite recently by Luo and Mitsoulis [lO,ll]. In the new scheme the special streamline elements have been replaced by conventional elements for the sake of dealing with recirculations in the flow. Galerkin discretization has been maintained and the numerical algorithm for convergence is Picard iteration with an under- relaxation factor of 0.5 for acceleration of convergence as described in Ref. 11. The new scheme has been tested against simple test problems, such as simple shear and simple elongational flows, using the upper-convected Maxwell integral model of eqn. (4) with excellent results [lO,ll]. A para- metric study of mesh refinement and convergence has also been under- taken for a 4: 1 circular contraction flow of an LDPE melt modelled by the K-BKZ equation (3) and studied previously by DuPont and Crochet [8]. Convergent solutions have been obtained practically independent of mesh size, provided enough elements are used and that the solution procedure advances slowly from low flow rates (Newtonian behaviour) to higher ones either by using a flow rate increment scheme or an elasticity increment scheme [ll]. These results were also in close agreement with the ones obtained previously and independently by DuPont and Crochet [8].

4. Results and discussion

4.1 Oldroyd-B fluid

We first apply the newly developed numerical scheme to the calculation of the flow of an Oldroyd-B fluid through a 4: 1 abrupt contraction using its integral counterpart given by eqn. (5). Accurate solutions exist in the literature for its differential form, given by a number of researchers [3,4,14]. Thus, as a further test we pursue calculations for different De numbers and perform comparisons against solutions of this equation al- ready published.

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Vortex Detachment f Plane Vorlex ’

Contraction Plane

Center

Fig. 1. Schematic diagram of the circular contraction geometry and definition of the vortex length L,.

With regard to Fig. 1, let L,, L,,, be the lengths of the downstream and upstream channels, respectively, and let R, be the radius (half gap) of the downstream tube (channel). For our calculations, we have chosen the ratios L,/R, and Lres/Ro to be 30 and 16, respectively. Taking into account that we are not going to pursue calculations for very high De numbers, such lengths have been proven adequate [3,14]. For the boundary conditions, we assume there is no slip along the walls and impose a fully developed velocity profile at the entry and at the exit sections. The relative strain tensor upstream of the entry section is calculated on the basis of the fully developed velocity profile.

Figure 2 shows a partial view of the finite-element mesh near the die entry. The full mesh contains 300 quadrilateral elements, 1007 nodes and 2368 degrees of freedom for the velocity components and the pressure. Note that this mesh has also been used in earlier publications [lO,ll] and

-6.0 -40 -2.0 0.0 20 4.0 6.0

Fig. 2. Partial view of the finite-element mesh used in the computations for the 4: 1 abrupt contraction (the mesh extends upstream to - 16 R, and downstream to +30 R,).

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12.0

11.0

10.0

9.0

80

70

6.0

5 I",

3.0

20

-40 -20 00 2.0 40

Y/H,

Fig. 3. Streamline pattern in a 4: 1 abrupt planar contraction using the integral Oldroyd-B fluid (De = A &, = 2.0).

that coarser or finer meshes ranging from 200-400 elements have given virtually the same results [ll].

We present here the results for the Oldroyd-B fluid model in its integral form as given by eqn. (5) with the values of p1 = 8~./9, p2 = ~/9 and A, = hJ9. Figure 3 shows the streamline pattern at De = A,+, = 2 for the planar case. This result is intermediate between the solutions presented by Marchal and Crochet [3] for De = 0 and De = 7.6 (Fig. 24 of their paper), and it shows that there is no important vortex activity exhibited by Ol- droyd-B fluids in a 4 : 1 planar contraction. A more definite comparison is presented in Fig. 4, where we show the horizontal and vertical velocity components (made dimensionless by the average downstream velocity v,) along the y/H, = 0.9 line for the planar case at De = 2 together with those obtained by Marchal and Crochet [3] (their Fig. 21a) and Lt.10 and Tanner [14] (their Fig. 6). The agreement is very good. Figure 5 shows a compari- son of the dimensionless T,, stress component along the y/I-I,, = 1 line at De = 2 with the results of Refs. 3 (their Fig. 22a) and 14 (their Fig. 7a). Again, satisfactory agreement is obtained despite the fact that different codes and methods have been used. The wiggles appearing in our results are characteristic of the Gale&in formulation, while the streamline-upwind (SU) method used by the other researchers tends to smooth out the high peaks. Having thus established the validity of our code with this further

307

0.5 , I I I I I I 1

! De= 2.0 04 -

-2.5 -20 -15 -10 -0.5 0.0 0.5 10 15 2.0 2.5

Dlmenslonless Distance, X/H,

Fig. 4. Dimensionless horizontal (v, /ct,) and vertical (v, /vo) velocity components at De = 2.0 along the y/H,, = 0.9 line (this work, solid line), in comparison with results by Marchal and Crochet [3] (mixed SU 2~ 2 elements, - * -) and Luo and Tanner [14] (SU, --- ).

d 5 co

14 0

130 -

120 - I De=2.0 1

110 -

100 -

90 -

80 -

70 -

60 -

50 -

40 -

30 -

20 -

10 -

-50 -40 -30 -20 -1.0 00 1.0 2.0 3.0 40 50

Dlmensvxdess Distance. x/H,

Fig. 5. Dimensionless 7rX stress component at De = 2.0 along the y/Ho = 1 line (this work, solid line), in comparison with results by Marchal and Crochet [3] (mixed SU 2 X 2 elements, * * *> and Luo and Tanner [14] (SU, ---).

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TABLE 1

Material parameter values used in eqn. (3) for fitting data of fluid Ml according to Chai and Yeow [13] (a = 500, p = 0.001, p = 0.9 g crnd3)

k A, (s) a k (Pa)

1 1.04 x 100 5.50 x 10-l 2 8.90x 1o-2 5.46x 10’ 3 5.20x 1o-4 3.06 x lo3

test, we turn our attention to the flow simulation of the Ml fluid through circular contractions.

4.2 Ml fluid through a 4: 1 circular contraction

We now embark upon flow simulations of the test fluid Ml through circular contractions as done experimentally by Boger and Binnington [15]. The fluid Ml has been modelled previously using the K-BKZ integral eqn. (3) by Chai and Yeow [13], who also have determined the parameters of the model. These are reproduced in Table 1 (note error in Ref. 13 for the value of a,). With these parameters it is possible to fit the experimental data for shear and elongational viscosities and first normal stress difference as

lo-2 10-S lcr’

Fig. 6. Model predictions of shear viscosity qS, first normal stress difference N,, and elongational viscosity nn for fluid Ml using eqn. (3) with data given in Table 1. Symbols are experimental data given by Chai and Yeow [13].

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Fig. 7. Model prediction of stress ratio S, as a function of shear rate T for fluid Ml using eqn. (3) with data given in Table 1.

shown in Fig. 6. Note the good agreement obtained from the model for the shear viscosity by using three relaxation times. Note also that a value of p = 0.001 is more appropriate for passing approximately through the enve- lope of the experimental elongational viscosity data. The fitting of Ni data is less satisfactory, giving rise to a dimensionless stress ratio (or recoverable shear) S, defined by

Nl Nl S,=-=-

2Tv 2 %k (9)

and shown in Fig. 7 plotted vs. shear rate +. It is seen that S, passes through a maximum of 1.67 at about + = 100 s-l. This is particularly interesting since the range of simulations encloses this maximum. In this work, the elasticity level will be characterized by S, due to lack of a single relaxation’ time.

We started our simulations for the 4: 1 contraction as usual from the Newtonian creeping flow solution (S, = 0) and increased the shear rate +, to reach higher flow rates and thus elasticity levels. To separate inertia (Re # 0) from viscoelastic effects (S, # 01, we first proceed with the inertia- less flow patterns obtained by setting the left-hand side of eqn. (2) equal to zero. Figure 8 shows the flow patterns for Re = 0 and for increasing +,+,. It is seen that the corner vortex is increasing in size and intensity up to about 300 s-l, after which its intensity is reduced. This is in agreement with the general trend of S, vs. +, which also passes through a maximum, as shown in Fig. 7. We also note that the same behaviour is shown by qE, which passes through a maximum and then decreases for i > 100 s-r.

Fig. 8. Streamline patterns at different shear rates for the inertialess flow of fluid Ml in a 4 : 1 circular contraction (Re = 0, data given in Table 1).

Inclusion of the inertia terms produces the flow patterns of Fig. 9. It is seen that inertia reduces somewhat the vortex intensity but not the size for the same shear rate values. At this point it is interesting to compare the numerical solutions with the experimental observations [15]. In general, there is good agreement with the experiments as evidenced in Fig. 10, where the dimensionless vortex reattachment length X = LJD,,, is plotted vs. Reynolds number and in Fig. 11, where it is plotted vs. +,, but only for low values of Re (< 0.1) and i;, (< 50 s-l). For higher values, the experiments showed a decreasing vortex which was finally eliminated at Re = 1 (i/ = 500 s-i). The disagreement with the numerical simulation is puzzling, The experimental trend of vortex reduction was only reproducible

Fig. 9. Streamline patterns at different shear rates for the flow of fluid Ml in a 4: 1 circular contraction (data given in Table 1).

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y-exp.

16’ 10-l lo” Reynolds Number, Re

Fig. 10. Vortex length X vs. Reynolds number for fluid Ml compared with the experimental observations [15] (solid symbols).

for much higher Re values (1 < Re < 10). At this point, it is not clear why these discrepancies exist, although suggestions have been put forward that the material has already degraded at shear rates + > 50 s-l in the 4 : 1 contraction [17].

4.3 Ml jluid through a 22: 1 circular contraction

The simulations on a 22 : 1 circular contraction were carried out on a finite-element grid having 387 elements, 1280 nodes and 3007 degrees of freedom. Care was taken to have the same grid density and size as in Fig. 2 near the entry to the die, and the rest of the domain was filled with the extra elements. The entry upstream was set at -88 R, and the exit

05

00

lo” ld ld lo” Shear Rate, i, (l/s)

Fig. 11. Vortex length X vs. i; for fluid Ml compared with the experimental observations [15] (solid symbols).

-‘Zo-IIo 00 110 22sZPO-110 00 II 0 220-220-110 00 II 0 22 OS-22 0 -II 0 0 0 II 0 22 0

r/R, r/R, r/R, r/R,

Fig. 12. Streamline patterns at different shear rates for the flow of fluid Ml in a 22: 1 circular contraction (data given in Table 1).

downstream at 30 calculations.

The results from shear rate are also

R, to keep the same relative ratios with the 4: 1

the simulations as a function of Reynolds number and shown in Figs. 10 and 11. As found experimentally by

Boger and Binnington [15], no inertia-like phenomena are now observed and vortex growth is observed as both the Reynolds number and shear rate are increased. The streamline patterns of Fig. 12 reveal the same vortex activity found experimentally. For example, for low shear rates (+ < 100 s-r) we find an overall reattachment length of the cell very similar to the Newtonian value of 0.18. However, a close examination shows that a concave corner vortex and a convex lip vortex coexist. This coexistence continues for higher Re, but eventually the lip vortex is growing at the expense of the corner vortex and it finally takes over, eliminating the corner vortex at higher Re. The possibility that the lip vortex is a numerical artifact is excluded because of the very dense grid used (over 50 elements used in that area> and because it always appears in large contraction ratios [18]. A comparison between experiments and simulation presented in Figs. 10 and 11 shows that in the case of a 22: 1 contraction, the agreement is quite satisfactory. The monotonic increase of vortex size is well captured by the model both as a function of Re and of shear rate. This monotonic increase is not restricted only to the vortex size but to its intensity as well, as shown in Fig. 13. While for the 4: 1 contraction there is a decrease after a maximum value is reached, this is not the case for the high contraction ratio.

It is interesting to note that the above differences in vortex activity in 4 : 1 and 22 : 1 abrupt circular contractions have also been found earlier

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30.0

g 25.0

0.0 L

lo” ,‘,“,ar- Rate, -j_ (:,%

ld

Fig. 13. Vortex intensity vs. f, for the flow of fluid Ml in 4: 1 and 22: 1 circular contractions.

A Re:O

I 4:1 ?

o Re#O : q ReCO, 22:1

using a heuristic approach, called the Viscometric Approximation Theory [l&20]. There it was argued that geometry effects are more significant than believed earlier, especially in axisymmetric contractions, where the contraction ratio plays a much more important role than in planar contrac- tions [19]. The mitigating effect that inertia forces exert on vortex activity for small contraction ratios has also been found previously in annular contractions experimentally [21] and numerically [22].

5. Conclusions

A newly developed finite-element scheme for the solution of integral constitutive equations has been used to simulate the flow of test fluid Ml in abrupt circular contractions. To establish its validity the scheme has been first tested against results for the Oldroyd-B fluid using its integral counterpart. Good agreement has been found with previous simulations of its differential type.

The simulation of test fluid Ml has been carried out using an integral constitutive equation of the K-BKZ type with three relaxation times. In a 4 : 1 circular contraction, it was found that exclusion of inertia forces results in large vortices. Inclusion of the inertia terms reduces somewhat the vortex size and intensity and produces results close to experimental obser- vations for the low range of shear rates (+ < 50 s-l). However, the simulations fail to predict the drastic reduction and elimination of the vortices at high shear rates as was found experimentally.

In a 22 : 1 circular contraction, the vortex envelope always grows with Re, but during growth there are two vortices, one at the lip, the other at the

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reservoir corner. The lip vortex finally dominates and encompasses the corner vortex. Very good agreement was found between numerical results and experimental observations. The current results show that inertia effects may be significant for flow of polymer solutions, while geometry effects play a dominant role in flow through contractions, especially circular contractions with a large contraction ratio. Finally, more studies are needed to explain why sometimes there is elimination of vortices in small contractions at high shear rates.

Acknowledgements

Financial assistance from the Natural Sciences and Engineering Re- search Council of Canada (NSERC) is gratefully acknowledged.

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