Computers & Fluids 44 (2011) 68–78

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Computers & Fluids

journal homepage: www.elsevier .com/locate /compfluid

Numerical prediction of three-dimensional time-dependent viscoelasticextrudate swell using differential and algebraic models

G. Mompean a,b,⇑, L. Thais a,b, M.F. Tomé c, A. Castelo c

a University of Lille Nord de France, F-59000 Lille, Franceb LML, CNRS, UMR 8107, F-59655 Villeneuve d’Ascq, Francec Department of Applied Mathematics and Statistics, University of São Paulo, São Carlos, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 May 2010Received in revised form 1 October 2010Accepted 9 December 2010Available online 15 December 2010

Keywords:Extrudate swellThree-dimensional free surface flowsViscoelastic fluidsAlgebraic viscoelastic modelFinite difference method

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.12.010

⇑ Corresponding author at: University of Lille NoFrance. Tel.: +33 328 767464; fax: +33 328 767301.

E-mail addresses: gilmar.mompean@polytech-lillthais@polytech-lille.fr (L. Thais), murilo@icmc.usp.brusp.br (A. Castelo).

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic freesurface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicitmodel. This investigation was carried out to develop a simplified approach that can be applied to theextrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper andan algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraicrepresentation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaleddyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equationfor a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and theconstitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependentfinite difference method. The free surface of the fluid was modeled using a marker-and-cell approach.The algebraic model was validated by comparing the numerical predictions with analytic solutions forpipe flow. In comparison with the classical UCM model, one advantage of this approach is that compu-tational workload is substantially reduced: the UCM model employs six differential equations whilethe algebraic model uses only one. The results showed stable flows with very large extrudate growthsbeyond those usually obtained with standard differential viscoelastic models.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

When a viscoelastic fluid is extruded from a pipe, the diameterof the extruded jet usually increases, in some cases up to two orthree times the diameter of the pipe. Several theories have beenproposed to understand the phenomenon of die swell, but a com-pletely satisfactory explanation has not yet been proposed. How-ever, a simple proposition to explain the extrudate swell wasintroduced by Weissenberg. He proposed that viscoelastic fluidflows have an extra tension along the streamlines. Once the fluidhas passed the end of the pipe the extra stress can not be sup-ported, and the fluid contracts axially and swells radially (see[5,19]). Following this proposition, the first normal stress differ-ence seems to govern the elastic recovery and causes the increasein the diameter of the jet. In the literature several works have beendevoted to the experimental investigation of the extrudate swell(see e.g. [12,13,10]). For example, Huang and White [13] presented

ll rights reserved.

rd de France, F-59000 Lille,

e.fr (G. Mompean), laurent.(M.F. Tomé), castelo@icmc.

experimental investigations of the extrudate swell of polystyreneand polypropylene melts and observed extrudate ratios over 200%.

The investigation of the extrudate swell problem, and moregenerally of viscoelastic free surface flows, is a research topic ofgrowing interest within the non-Newtonian fluid mechanics com-munity. Viscoelastic fluids are largely employed in industrial pro-cesses, and they frequently flow in or around three-dimensionalgeometries. For specific applications of extrusion, the modelingof viscoelastic materials and the free surface shape prediction arestill challenges of Computational Fluid Dynamics (CFD). The inher-ent difficulties are related to the constitutive equation, the particletracking scheme and the very large number of degrees of freedomof the problem.

The numerical simulation of the extrudate swell of non-Newto-nian fluids has already been addressed by other investigators,mostly using a finite element formulation. For instance, Brasseuret al. [6] showed results for time-dependent axisymmetric Poiseu-ille flow of an Oldroyd-B fluid. Crochet and Keunings [9] consideredthe extrudate swell of an Oldroyd-B fluid flowing through both acircular and slit dies. The constitutive multi-mode Phan–Thien–Tanner model was applied by Béraudo et al. [4] to simulate thedie swell for two-dimensional and axisymmetric geometries. Thememory-integral model was used by Normandin et al. [18] and

G. Mompean et al. / Computers & Fluids 44 (2011) 68–78 69

Huang and Lu [14]; these latter authors reported die swell ratiosover 100%. Standard viscoelastic models (Upper-Convected Max-well (UCM), Oldroyd-B, Phan–Thien–Tanner (PTT)) are based uponpartial differential equations, one for each component of the extra-stress tensor [5]. For the specific application of unsteady three-dimensional extrusion, Tomé et al. [24] showed that simulationswith the Oldroyd-B model were very intensive and could becomeunstable at high swell rates.

The objective of this work is to present numerical results from thesimulation of the three-dimensional time-dependent extrudateswell using the UCM constitutive equation and a newly devised alge-braic model. Previous works dealing with free surface tracking ofnon-Newtonian fluids involved either Second Order Fluids or stan-dard differential models. In this work, we focus on a new algebraicconstitutive model that can also handle viscoelastic free surfaceflows.

This investigation has been performed to develop a simplifiedapproach that can be applied to the extrudate swell problem, tak-ing into account the relevant physics of this flow phenomenon. Thenew model is based on an algebraic explicit formulation involvingthe scaled dyadic product of the velocity field. The elasticity of thefluid is modeled by a single transport equation to calculate a dy-namic scalar quantity which measures a generalized strain rate.This approach was motivated by previous works using tensorialalgebraic formulations to derive explicit algebraic extra-stressmodels [16,17,20].

The governing equations were embedded in the unsteady three-dimensional particle tracking algorithm GENSMAC3D developedby Tomé et al. [22] which is based on the simplified marker-and-cell (SMAC) method of Amsden and Harlow [2] (see also [11]). A re-cent review on this topic can be found in McKee et al. [15]. Thisalgorithm has already been successfully applied to three-dimen-sional viscoelastic free surface flows using the Oldroyd-B [24],UCM [25] and the PTT [26] models.

The layout of this paper is as follows. In Section 2, we presentthe governing equations, the algebraic extra-stress model, and itsproperties for steady uniaxial shear flows. The numerical methodis briefly described in Section 3. In Section 4, the algebraic modelis validated for a laminar pipe flow for which an analytic solutionexists. Results for the extrudate swell problem are then presentedin Section 5 and conclusions are drawn in the closing section.

2. Mathematical formulation

We consider the isothermal three-dimensional laminar free sur-face flow of an incompressible viscoelastic fluid whose constitutivelaw is given by the UCM constitutive equation or by an algebraicmodel. We first proceed by describing the governing equations,and then the properties of the algebraic model are derived.

2.1. Mass and momentum conservation equations

The flow is governed by the mass and momentum conservationequations coupled with a constitutive relation for the non-Newto-nian extra-stress tensor. The basic conservation equations govern-ing the flow are:

(i) Mass conservation:

r � v ¼ 0; ð1Þ

(ii) Momentum conservation:

q@v@tþr � ðvvÞ

� �¼ �rpþr � Tþ qg; ð2Þ

where v denotes the velocity field, q is the fluid density, p is thepressure, T is the extra-stress tensor and g is the gravity field.

2.2. Constitutive equations

In this work the extra-stress tensor T is modeled by the UCMdifferential model or by the algebraic model proposed inSection 2.2.2.

2.2.1. Upper-Convected Maxwell modelThe UCM constitutive equation is given by

Tþ k T5¼ 2gD: ð3Þ

D ¼ 12 ½rv þ ðrvÞT � is the rate of deformation tensor, g is the fluid

viscosity, k is the relaxation-time of the fluid and T5

is the upper-convected derivative of T defined by

T5¼ @T@tþr � ðvTÞ � ðrvÞT � T� T � ðrvÞ:

2.2.2. Algebraic model equationsFor the algebraic model the extra-stress tensor is split according

to

T ¼ 2gDþ s; ð4Þ

where the tensor s represents the elastic part of the extra-stresstensor. Taking into account that the extrudate swell phenomenonis dominated by the first normal stress difference (N1), the secondnormal stress difference (N2) can be neglected (see [5]). Then, letus propose the following equation for the elastic part:

s ¼ j _mlð _mÞ vvT

v2

� �; ð5Þ

where j is a positive non-dimensional parameter, _m is a parameterwith dimension of strain rate [s�1] and lð _mÞ is a parameter havingdimension of viscosity [Pa s]. More details about these parametersare given next.

This equation is quadratic in velocity as it explicitly involves thescaled dyadic product of v (where vT denotes the transpose of v)and v2 is the squared norm of v. This definition of s was motivatedto give a proper orientation of the elastic part of the extra-stresstensor in viscoelastic shear flows. In this case, the first normal com-ponent is given by,

sxx ¼ j _mlð _mÞ vxvx

v2

� �; ð6Þ

where the subscript x denotes the main direction of the flow. There-fore, in viscoelastic shear flows this component of the elastic tensoris important and it is responsible for the extrudate swell phenome-non, as it was suggested by Weissenberg (see [5, p. 72]).

The frame-invariance and the limitations of this model are dis-cussed in Appendix A. Nonlinear explicit models such as second or-der fluids usually depend on squared velocity gradients, whichmakes them more prone to numerical instabilities. We shall seein the results section that this algebraic model has the doubleadvantage of being simple and numerically stable.

The variable _m is governed by the following differentialequation:

_mþ k@ _m@tþ v � r _m

� �¼ _c: ð7Þ

Note that this formulation has memory, with the elastic recovery ofthe fluid being driven by the relaxation-time k. We can see that forviscometric flows, Eq. (7) is such that _m tends towards the general-

ized shear rate _c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2trfD2g

q, (tr{�} denoting the trace operator).

The other two parameters, lð _mÞ and j, are discussed below:

70 G. Mompean et al. / Computers & Fluids 44 (2011) 68–78

(i) lð _mÞ has dimension of viscosity. This parameter is a functionof the viscoelastic quantity _m through a Bird-Carreau typerelation (8) with exponent n, cut-off frequency s0, andzero-shear value l0:

lð _mÞ ¼ l0½1þ ðs0 _mÞ2�n�1

2 : ð8Þ

(ii) j is a positive dimensionless scaling parameter. The normalstress difference is linearly proportional to this parameter,thus, it is expected that viscoelastic effects in the flowshould increase with j.

The ‘‘viscosity’’ l0 and the dimensionless parameter j drive themean value of the elastic stresses, the higher their values, the high-er the elastic response of the fluid. The parameter s0 corresponds toa cut-off frequency which determines the level of rate-of-strain be-yond which the elastic stresses depend on _m. The exponent n de-fines the steepness of this dependence. For instance, n = 2 willcorrespond to a roughly linear growth of the elastic response as afunction of _m.

2.3. Application of the algebraic model to steady shear flow

As the extrudate swell is dominated by shear, we investigatethe algebraic model properties for steady planar shear flow givenby v = (u(y),0,0), where _c ¼ j@u=@yj is the shear rate. Straightfor-ward algebra shows that,

N1 ¼ Txx � Tyy ¼ jlð _cÞ _c; ð9aÞN2 ¼ Tyy � Tzz ¼ 0; ð9bÞTxy ¼ 2g _c: ð9cÞ

Since j > 0, Eq. (9a) ensures positive first normal stress difference.Moreover, the present model can be adjusted to reproduce the

same flow produced by an UCM fluid in steady shear flow. For thisproperty to hold, the algebraic model parameters must be chosenas j = 2k/s0, n = 2, l0 = g, and s0 �maxf _c�1g, in which case theabove expressions revert to a quadratic first normal stress differ-ence and a linear shear stress matching the UCM fluid expressions,i. e.

N1 ¼ 2kg _c2; ð10aÞN2 ¼ 0; ð10bÞTxy ¼ 2g _c: ð10cÞ

It is also worth noting that the first normal stress difference N1 in-creases with the model constants j and l0.

3. Numerical solution of the governing equations

Details about the numerical solution of the UCM model can befound in Tomé et al. [25]. For this reason they are not presentedhere. In the following we present the numerical solution for thealgebraic model equations only.

To solve the continuity and the momentum conservation Eqs.(1) and (2), either coupled with the UCM constitutive Eq. (3), orcoupled with the algebraic model Eqs. (4), (5), (7) and (8), we em-ploy the following non-dimensionalisation:

�t ¼ UL

t; �x ¼ xL; �v ¼ v

U; �p ¼ p

qU2 ; �s ¼ s

qU2 ;

�l0 ¼l0

g; �_c ¼ L

U_c; �_m ¼ L

U_m; �s0 ¼

LU

s;ð11Þ

where L and U are, respectively, length and velocity scales.The continuity, momentum and algebraic equations can then be

written (omitting the over-bars for convenience) as

r � v ¼ 0; ð12aÞ

@v@tþr � ðvvÞ ¼ �rpþ 1

Rer2v þr � sþ 1

F2r

g; ð12bÞ

s ¼ 1Re

j _ml0 1þ ðs0 _mÞ2h in�1

2 vvT

v2

� �ð12cÞ

_mþWi@ _m@tþ ðv � rÞ _m

� �¼ _c; ð12dÞ

where the parameters Re = qU L/g, Wi = kU/L and Fr ¼ U=ffiffiffiffiffigL

pare the

Reynolds, Weissenberg and Froude numbers, respectively.

3.1. Boundary conditions

On rigid boundaries the no-slip condition is applied while atfluid entrances (inflows) the velocity is prescribed. At fluid exits(outflows) the homogeneous Neumann condition ð@v

@n ¼ 0Þ is em-ployed. We shall be concerned with viscoelastic flow of a fluid pos-sessing a moving free surface. If we neglect surface tension, thenon the free surface the correct boundary conditions are given by(see [3])

n � r � n ¼ 0; ð13aÞm1 � r � n ¼ 0; ð13bÞm2 � r � n ¼ 0; ð13cÞ

where n is the outward unit normal vector to the free surface, m1,m2 are unit tangential vectors and r is the total Cauchy stress ten-sor r = �pI + 2D/Re + s.

3.2. Numerical method

Eqs. (12b)–(12d) together with the mass conservation Eq. (12a)will be solved within the framework of the GENSMAC3D method ofTomé et al. [22]. In summary, we solve the equations by the follow-ing algorithm.

It is supposed that at time tn, the solenoidal-velocity field v(x, tn)and the elastic stress s (x, tn) are known and suitable boundaryconditions for the velocity and pressure are given. The updatedvelocity field v(x, tn+1), the pressure field p(x, tn+1) and the elasticstress tensor s(x, tn+1) at time tn+1 = tn + dt are calculated by the fol-lowing steps:

Step 1: Let ~pðx; tnÞ be the correct pressure field satisfying the nor-mal stress condition on the free surface (see Eq. (13a)).

Step 2: Compute the intermediate velocity field, ~vðx; tnþ1Þ from

@~v@t¼ �r � ðvvÞ � r~pþ 1

Rer2v þr � sþ 1

F2r

g ð14Þ

with ~vðx; tnÞ ¼ vðx; tnÞ using the correct boundary condi-tions for v(x, tn).

Step 3: Solve the Poisson equation

r2w ¼ r � ~vðx; tnþ1Þ: ð15Þ

The appropriate boundary conditions for this equation are[21]

@w@n¼ 0 on rigid boundaries and

w ¼ 0 on the free surface:

Step 4: Compute the final velocity field

vðx; tnþ1Þ ¼ ~vðx; tnþ1Þ � rwðx; tnþ1Þ: ð16Þ

Step 5: Compute the updated pressure field

G. Mompean et al. / Computers & Fluids 44 (2011) 68–78 71

pðx; tnþ1Þ ¼ ~pðx; tnÞ þwðx; tnþ1Þ

dt: ð17Þ

Step 6: Calculate _mðx; tnþ1Þ from a discretization of

@ _m@t¼ � @ðu _mÞ

@xþ @ðv

_mÞ@y

þ @ðw_mÞ

@z

� �þ 1

Wið _c� _mÞ: ð18Þ

Step 7: Compute the extra-stress tensor from the explicit alge-braic equation

s ¼ 1Re

j _ml0½1þ ðs0 _mÞ2�n�1

2vvT

v2

� �: ð19Þ

Step 8 Update the positions of the marker particles by solving:

_x ¼ vðx; tnþ1Þ: ð20Þ

3.3. Finite difference approximations

We consider three-dimensional Cartesian flows where

v ¼u

vw

264

375 and s ¼

sxx sxy sxz

sxy syy syz

sxz syz szz

264

375: ð21Þ

The equations presented in the previous section are solved by a fi-nite difference method on a staggered grid (see Fig. 1b). The compo-nents of the velocity u, v and w are located at the centre of cell faceswhile the pressure field, the components of the elastic stress tensorand the structural scalar variable _m are positioned at the centre of acell. We shall be concerned with flows where the fluid is continu-ously moving and a Lagrangian procedure to track the fluid regionis employed. Fig. 1a illustrates the cell classification for a two-dimensional situation (for clarity). The three-dimensional free sur-face is approximated by a piecewise linear surface composed of tri-angles and quadrilaterals that contain marker particles on theirvertices. It may happen that two vertices become too far away ortoo close to each other so that a scheme for inserting and deletingparticles (creating/deleting (new) triangles and quadrilaterals) isemployed, see Castelo et al. [7] for further details.

We observe that the equations of Step 1 to Step 5 and Step 8together with the equations for the boundary conditions on thefree surface (see Eqs. (13a)–(13c)) are the same equations thatare solved for an Oldroyd-B fluid (see [24]). Therefore, the corre-sponding approximations for Eqs. (14)–(17) and (20) using finitedifferences can be found in Tomé et al. [22]. For reasons of spacethey are not presented here.

a b

Fig. 1. (a) Types of cells used by GENSMAC3D. F, fluid; S, surface; E, empty;

The time derivative in the transport equation for _m (Eq. (18)) isapproximated by a forward derivative while the convective termsare calculated using the high order upwind CUBISTA method devel-oped by Alves et al. [1]. Therefore, Eq. (18) is approximated at thecell centres by

_mðnþ1Þi;j;k ¼ _mðnÞi;j;k�dt Cðu _mÞi;j;kþCðv _mÞi;j;kþCðw _mÞi;j;kþ

1Wi

_ci;j;k� _mi;j;k

� �ðnÞ;

ð22Þ

where the shear rate _ci;j;k is approximated by

_ci;j;k ¼ 2@u@x

� �2

i;j;kþ 2

@v@y

� �2

i;j;kþ 2

@w@z

� �2

i;j;kþ @u

@yþ @v@x

� �2

i;j;k

þ @u@zþ @w@x

� �2

i;j;kþ @v

@zþ @w@y

� �2

i;j;k

!1=2

: ð23Þ

The derivatives in Eq. (23) are calculated by second order differ-ences, see Tomé et al. [23]. The terms Cðu _mÞi;j;k; Cðv _mÞi;j;k; Cðw _mÞi;j;kin Eq. (23) represent the finite difference discretisations of the con-vective terms of the transport equation for _m by the CUBISTA meth-od. Details of the finite difference equations involved can be foundin Costacurta [8].

After the calculation of _m everywhere in the fluid, we computethe components of the elastic stress tensor by

si;j;k ¼1Re

j _mi;j;kl0 1þ ðs0 _mi;j;kÞ2h in�1

2 vi;j;kvTi;j;k

v2i;j;k

" #: ð24Þ

4. Validation results for fully developed pipe flow

In order to simulate free surface flows governed by the algebraicconstitutive equation presented in this paper, the numerical meth-od described in Section 3.2 was implemented into the Freeflow3Dcode of Castelo et al. [7]. To validate the numerical method we con-sidered fully developed flow inside a 3D-pipe of radius R = 1 andlength 10. On the pipe end an outflow boundary was imposedand at the pipe entrance we imposed a steady-state fully devel-oped parabolic flow given by

u ¼ v ¼ 0; wðx; yÞ ¼ 2 1� ½ðx� xcÞ2 þ ðy� ycÞ2�

n o: ð25Þ

Here (u,v,w) are the velocity components in a Cartesian coordinatesystem (x,y,z) and (xc,yc) denote the coordinates of the centre of thepipe, with z aligned along the axis of the pipe. With this imposed

B, boundary; I, inflow; O, outflow. (b) Three-dimensional staggered grid.

Table 1Algebraic model parameters used for pipe flow validation.

j l0 s0 n

0.05 1.8 5 2

Fig. 2. Numerical simulation of the flow inside a three-dimensional pipe using thealgebraic model. Fluid flow visualization at selected times.

72 G. Mompean et al. / Computers & Fluids 44 (2011) 68–78

velocity field we obtain _m � _c ¼ 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xcÞ2 þ ðy� ycÞ

2q

(see Eq.(12d)) and the only non-zero extra-stress components are

Txz ¼ � 4Reðx� xcÞ; Tyz ¼ � 4

Reðy� ycÞ; Tzz ¼ jlð _cÞ _c: ð26Þ

The algebraic model parameters employed are displayed in Table 1.For this flow, gravity was neglected, and the Reynolds and Weiss-enberg numbers were Re = Wi = 1 (based on the radius of the pipeand on the bulk velocity). We started with an empty pipe and in-jected fluid at the pipe entrance until the pipe was full and stea-dy-state had been established. To simulate this problem weemployed four meshes, denoted by M0, M1, M2 and M3, in increas-ing order of refinement (see Table 2). Fig. 2 displays the fluid flowinside the pipe at selected times.

These plots were obtained using mesh (M2).Fig. 3 displays both the numerical and the analytic values of w,

Tzz, Txz and Tyz at the line 0 < x < 2, y� = dy/2, z� = 5 (the data formesh M3 was omitted to avoid confusion due to too manysymbols).

We can see that the agreement between the numerical solu-tions obtained on the three meshes and the analytic values is verygood. Moreover, we can conclude from Fig. 3 that as the mesh isrefined the numerical solutions approach the analytic solution. Inparticular, the shear component Tyz uniformly converges towardszero as the mesh is refined, in agreement with Eq. (26). Addition-ally, we computed the relative l2-norm errors between the exactanalytic solution and the numerical results by,

EðXÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi;jðXEx � XNumÞ2P

i;jðXExÞ2

vuut ; ð27Þ

where X stands for any of the variables which have been monitored(w, Tzz, Txz and Tyz). Table 2 displays the errors using the fourmeshes. We can see that they all decrease as the mesh is refined.These results demonstrate the convergence of the numerical meth-od for solving the algebraic constitutive model given by Eqs. (12c)and (12d). An additional validation study was carried out employ-ing the Upper-Convected Maxwell (UCM) analytic solution for pipeflow. We compared the UCM analytic solution [25] with the resultsobtained with the algebraic model, both flows having the same Rey-nolds and Weissenberg numbers (Re = 0.5, Wi = 0.3), while the alge-braic model parameters were chosen to be n = 2, l0 = 2, s0 = 100 andj = 2Wi/s0 = 0.006. These parameters were chosen such that thealgebraic model would reproduce the UCM steady shear flow prop-erties (see Section 2.3). For this study the pipe dimension was thesame as above and the mesh used was M1. Fig. 4 shows the normalstress component Tzz along a selected cross section after steady-state had been reached. The predicted solution obtained with the

Table 2Meshes used for pipe flow validation of the algebraic model (dx = dy = dzvelocity component and the extra-stress components.

Mesh M0 M1

Number of cells 12 12 60 16 16dx = dy = dz 0.166667 0.125000E(w) 102 0.50560 0.26484E(Txz) 102 1.1751 0.56025E(Tyz) 102 1.1744 0.55758E(Tzz) 102 2.3209 1.0312

algebraic model is superposed onto the UCM exact solution asexpected.

5. Numerical simulation of viscoelastic time-dependentextrudate swell

To demonstrate that the algebraic model presented in this pa-per can cope with unsteady viscoelastic free surface flows we ap-plied it to simulate the time-dependent extrudate swell. A firsttest case was computed to compare the results of the algebraicmodel with the standard UCM model. Then, the extrudate swellof highly elastic fluids was simulated using the algebraic model.

5.1. Comparison with the UCM model

We considered the time-dependent flow of an axisymmetric jetflowing inside an empty tube with radius R = 1 which, after it hadbeen filled, the jet was extruded from the end of the tube of lengthL = 10. To simulate this problem the total domain size was4 4 20. The mesh used had 32 32 160 cells, giving a cellspacing of dx = dy = dz = 0.125. On the tube wall a no-slip imperme-able condition was imposed. A fully developed flow was assumedat the tube entrance (see Eq. (25)). As for outflow, homogeneousNeumann boundary conditions (@u/@z = @v/@z = @w/@z = 0) wereassigned at a distance of Le = 10 from the tube exit. For low Rey-nolds number flows, these boundary conditions should not affectthe extrudate swell if the exit distance Le is large enough. In ourprevious works, using Oldroyd-B and Phan–Thien–Tanner (PTT)models in similar flow conditions, we performed various testsvarying this distance and we found that a value of 10 was enoughto produce extrudate swell ratios independent of Le.

On the fluid free surface, the full free surface stress conditionswere applied (see Eqs. (13a)–(13c)). For this flow, the Reynoldsnumber was Re = 0.5, the Weissenberg number was Wi = 0.3, and

is the uniform cell spacing) and l2-norm errors obtained for the w

M2 M3

80 20 20 100 24 24 1200.100000 0.0833330.15806 0.105610.35019 0.246100.35149 0.243200.67512 0.46745

0

0.5

1

1.5

2

0 0.5 1 1.5 2

velo

city

W

x

M0M1M2

Exact solution

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2

Tzz

x

M0M1M2

Exact solution

-4

-3

-2

-1

0

1

2

3

4

0 0.5 1 1.5 2

Txz

x

M0M1M2

Exact solution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2Ty

zx

M0M1M2

Exact solution - M0Exact solution - M1Exact solution - M2

a b

c d

Fig. 3. Comparison between the analytic and the algebraic model solutions obtained for pipe flow (slice along 0 6 x 6 2, y� = yc + dy/2, z� = 5) using meshes M0, M1 and M2. (a)Velocity w(x,y�,z�), (b) Tzz (x,y�,z�), (c) Txz(x,y�,z�), (d) Tyz(x,y�,z�).

0

2

4

6

8

10

12

14

16

18

20

22

0 0.5 1 1.5 2

Tzz

x

τ0 = 100UCM (Exact)

Fig. 4. Comparison between the UCM exact solution and the algebraic modelnumerical solution (using mesh M1) for the normal stress component Tzz(x,y�,z�) ina pipe flow (slice along 0 6 x 6 2, y� = yc + 2.5dy, z� = 5).

Table 3Algebraic model parameters used for numerical simulation of extrudate swell andcomparison with UCM model (Re = 0.5, Wi = 0.3).

l0 n s0 j = 2Wi/s0

2 2 10, 50, 100 0.060, 0.012, 0.006

Fig. 5. Three-dimensional flow visualization at different times of unsteady extru-date swell (Re = 0.5, Wi = 0.3). Comparison of UCM model (left) and algebraic modelwith increasing values of s0 from right to left.

G. Mompean et al. / Computers & Fluids 44 (2011) 68–78 73

gravity was neglected. For comparison, the die swell flow was firstsimulated with the UCM solver of Tomé et al. [25] and subse-quently with the algebraic model. The three sets of algebraic modelparameters used were the same as those used previously in thevalidation section (see Table 3). We point out that these algebraicmodel parameters were chosen to ensure that the algebraic andthe UCM models exhibit the same steady pipe flow behaviour.

Fig. 5 displays successive snapshots of the free surface shapesobtained with the UCM and the algebraic models while Fig. 6shows a frontal view of the results at time t = 30 and t = 34. We ob-serve that at time t = 7.5 the jets have not yet left the tube, and thatthe free surface predicted by the UCM model is slightly ahead of

Table 4Extrudate swell ratio obtained at Re = 0.5, Wi = 0.3 with the UCM and the algebraicmodels.

Model UCM Algebraic(s0 = 10)

Algebraic(s0 = 50)

Algebraic(s0 = 100)

Swell ratio 0.286 0.406 0.402 0.406

74 G. Mompean et al. / Computers & Fluids 44 (2011) 68–78

the three algebraic model jets. At later times, all the jets reachedthe tube exit and were extruded into the air. The free jets of thealgebraic model have a mushroom-shaped head, which later ondisappears when the jets reach the outflow. This peculiar shapewas also reported by Tomé et al. [26] using a PTT model. The extru-date swell ratios Sr = (Dmax � D)/D, where D is the pipe diameterand Dmax is the maximum diameter of the jet, obtained in thesesimulations are given in Table 4. For this moderately elastic flow,the UCM model predicted a swell ratio of 30% while the threealgebraic flow cases gave a swell ratio of approximately 40%.

5.2. Effect of increasing elasticity

In this section, we shall demonstrate that the algebraic modelcan simulate extrudate swell of highly elastic fluids. To allow forhigher die swell ratios, we employed a computational domain sizeof 6.75 6.75 15.75. A cell spacing of dx = dy = dz = 0.125 wasused which produced a mesh with 54 54 126 cells. The pipe ra-dius was R = 1 while its length was L = 5. The outflow boundarywas set at a distance 10 from the pipe exit, gravity was neglectedand the Reynolds number was Re = 1.

5.2.1. The effect of the parameter jA first possibility to increase the swelling ratio is to increase the

value of the parameter j appearing in the algebraic model (Eq. (5)).We performed a total of three simulations where the parameter as-sumed the values 0.01, 0.05 and 0.1, with a constant Weissenbergnumber Wi = 0.5. The other model parameters were assigned thevalues l0 = 1.8, s0 = 5 and n = 2.

Fig. 6. Frontal view visualization of unsteady extrudate swell (Re = 0.5, Wi = 0.3).Comparison of UCM model (left) and algebraic model with increasing values of s0

from right to left.

Fig. 7 displays the three-dimensional visualization of the jetflow obtained in these three simulations while Fig. 8 shows thefront view of the results. We can see in Fig. 7 the jet flowing insidethe tube and then being extruded into the atmosphere. At timet = 5 the jets are just leaving the tube and the differences betweenthe three simulations are not too noticeable. However, at latertimes when the jet has already undergone extrusion the differ-ences between the simulations become more visible, for exampleat times t = 7.5 (second row) and t = 10 (third row). At time t = 20(fourth row) we can observe that the jet corresponding toj = 0.01 flows faster and has already entered the outflow regionwhereas the cases with higher values of j have not yet reachedthe outflow. Fig. 9 shows the final stage of the computation forthe flow case j = 0.1. This flow case presents the typical mush-room-shaped jet head mentioned above and exhibits a very highdie swell ratio.

Finally, Table 5 displays the swelling ratios obtained togetherwith the jet velocity downstream the tube, W1. The marked differ-ence between the jet velocities confirms why the jets for the twosmallest value of j reach the outflow before the jet with j = 0.1.We observe that the results obtained with j = 0.01 presented amuch larger swelling than the results from the other two simula-

Fig. 7. Three-dimensional flow visualization at selected times of unsteady extru-date swell using the algebraic model with increasing elasticity from left to right(Re = 1, Wi = 0.5).

Fig. 7 (continued)

Fig. 8. Same flows as in Fig. 7, front view.

G. Mompean et al. / Computers & Fluids 44 (2011) 68–78 75

tions. Indeed, the swelling ratios for the three simulations were25.6% for j = 0.01, 48.9% for j = 0.05 and 140.0% for j = 0.1. Thefact that the swelling ratio increases with j, all other parametersbeing held constant, is consistent with the algebraic model sincethe elastic stress contribution is proportional to j. This is also inagreement with Eq. (6) which shows that the first normal stressdifference is an increasing function of j for shear flows.

5.2.2. The effect of the Weissenberg number (Wi)To show that the Weissenberg number has a strong influence on

the extrudate swell, we performed four simulations for the follow-ing values of Wi: 1, 1.5, 2 and 3. We considered the same input dataemployed to obtain the results shown in Fig. 7 for j = 0.05 and car-ried out the simulations until t = 45. Fig. 10 displays the three-dimensional view of the results obtained at selected times whileFig. 11 shows the respective frontal view of the results. The valueof the extrudate swell ratio obtained for each value of Wi is dis-played in Fig. 12 where it can be observed that the extrudate swellratio is an increasing function of Wi. This result was expected sincethe first normal stress difference which drives the swelling phe-nomenon is supposed to increase the fluid elasticity.

6. Final remarks and conclusions

In this paper, we investigated the numerical simulation ofthree-dimensional time-dependent extrudate swell using theUpper Convected Maxwell (UCM) constitutive equation and an

algebraic explicit model. This study was carried out to develop asimplified approach that can be applied to the extrudate swellproblem. The algebraic model is based on an explicit algebraic rep-resentation of the elastic part of the extra-stress tensor. The elas-ticity is governed by a single transport equation to compute ascalar quantity which has dimension of strain rate and relaxes to-wards the generalized shear rate for viscometric flows. The modelis characterized by a positive first normal stress difference for stea-dy shear flow. The UCM constitutive equation and the algebraicmodel were embedded in the free surface flow solver GENSMAC3Ddeveloped by Tomé et al. [22] and validation results were providedby comparison with analytic solutions for pipe flow. The parame-ters of the algebraic model were tuned to reproduce the UCMbehaviour in shear flows so that a first series of extrudate swellsimulations using these parameters was carried out. The resultsobtained with the algebraic model were similar to those obtainedwith the UCM model. The main difference was a higher die swellratio predicted by the algebraic model. In order to observe theinfluence of the parameter j on the extrudate swell ratio threesimulations with j = 0.01, 0.05, 0.1 were performed. The resultsshowed that the extrudate swell ratio increases with j. The maxi-mum extrudate swell ratio was Sr = (Dmax � D)/D = 140%. Moreover,simulations of the extrudate swell for Weissenberg numbers in therange [0.5,3] were presented. The results showed that the extru-date swell ratio is an increasing function of Wi. The computationsshowed stable flows with very large extrudate growths well be-yond die swell ratios usually attained with standard differentialviscoelastic models.

Finally, one should mention that the algebraic model is compu-tationally faster than differential models. For the three-dimen-sional flow in a 4:1 contraction using a finite volume confinedflow solver, the saving in computing time with respect to the Old-royd-B model was a factor of 2. However, for the present extrudate

Fig. 8 (continued)

Fig. 9. Flow visualization up to the final time of the transient extrudate swell usingthe algebraic model with j = 0.10 (Re = 1, Wi = 0.5).

76 G. Mompean et al. / Computers & Fluids 44 (2011) 68–78

swell flows simulated with GENSMAC3D, we observed a smallergain of order 30% with respect to the UCM model. This more mod-est gain is a consequence of the free surface particle tracking algo-rithm which occupies a significant part of the workload,irrespectively of the fluid model used. In summary, the presentalgebraic model has proven to be robust and effective in computingthree-dimensional unsteady extrudate swell problems.

Future studies should involve tests of the present model forconfined flows in complex geometries. Preliminary computationsfor the benchmark 4:1 planar contraction flow showed interestingresults at high level of elasticity. In analogy with the Phan–Thien–Tanner model extending the Oldroyd-B differential model [5], a

Table 5Time-dependent extrudate swell results obtained with the algebraic model forvarious values of j. The other model parameters were l0 = 1.8, s0 = 5 and n = 2.

j 0.01 0.05 0.10

W1 0.755 0.509 0.146Swell ratio 0.256 0.489 1.400

Fig. 10. Three-dimensional flow visualization at selected times of unsteadyextrudate swell using the algebraic model with increasing Weissenberg number.Re = 1, j = 0.05, l0 = 1.8, s0 = 5, n = 2.

Fig. 11. Same flow as in Fig. 10, front view.

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3 3.5

Sr

Wi

Fig. 12. Dieswell ratio as a function of Wi. Re = 1, j = 0.05, l0 = 1.8, s0 = 5, n = 2.

G. Mompean et al. / Computers & Fluids 44 (2011) 68–78 77

possible extension of the present model would be to introducenon-linearities through modification of the linear term in Eq. (7).

Acknowledgments

The authors are grateful to Dr. Debbaut for many helpful discus-sions concerning the development of the algebraic model. Weacknowledge the financial support from Universidade de São Paulo(USP, Brasil) and the Comité Français d’Evaluation de la Coopéra-tion Universitaire avec le Brésil (COFECUB, France) under ContractUc Ph 112/08.

The Brazilian authors would like to thank the support given bythe Brazilian funding agencies: FAPESP – Fundação de Amparo apesquisa do Estado de São Paulo (Grants Nos. 04/16064-9, 07/07038-2); CNPq – Conselho Nacional de Desenvolvimento

Cientı́fico e Tecnológico (Grants Nos. 304422/2007-0, 470764/2007-4); CAPES/FCT Grant No. 226/09. This work was carried outin the framework of the Instituto Nacional de Ciência e Técnologiaem Medicina Assistida por Computação Cientı́fica (CNPq, Brazil).

Appendix A. Frame-invariance of the algebraic model

The frame-invariance of the algebraic Eq. (5) depends on theframe-change properties of the orientation tensor S which is thedyadic product of the velocity field:

S ¼ vvT : ðA:1Þ

As the velocity field is a frame-dependent vector, the objective ofthis appendix is to determine under which conditions the presentmodel will be frame-invariant.

Any general transformation from one reference frame R to an-other reference frame R� implies a translation vector c and anorthogonal rotation matrix Q, i. e.

x� ¼ Qxþ c; ðA:2Þ

where x is a vector position in R, and x� its counterpart in R�.Considering the case of translations, the present model can be

made frame-invariant by substituting in the expression for tensorS the relative velocity field

vr ¼ v � _c; ðA:3Þ

where the superscript dot denotes the time derivative. Note thatthis also applies if R� is a translating non-inertial frame of reference( _c is not necessarily a constant).

Since the norm of vr is frame-invariant, the frame-change prop-erties of the model under rotation reverts to the frame-changeproperties of the relative ’unscaled’ dyadic product

Sr ¼ vrvTr : ðA:4Þ

Straightforward algebra gives

S�r ¼ _Q xxT _Q T þ _Q xvT Q T þ QvxT _Q T þ QSrQT : ðA:5Þ

Therefore, the only chance for Sr to be frame-invariant under rota-tion would be that the sum of the first three terms in the right-handside of Eq. (A.5) would vanish, in which case the frame-invariantproperty of a second order tensor (S�r ¼ QSrQ

TÞ would be satisfied.This of course cannot happen in general, unless _Q ¼ 0, whichamounts to no rotation.

The practical implication of this result is that the present alge-braic model should not be used in a non-inertial rotating frame ofreference. However, it can be used in any inertial or non-inertialtranslating frame of reference under the condition that the correc-tion of Eq. (A.3) be applied.

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