1 2 Numerical Simulations and Experiments in a Double-Couette Flow … · a Double-Couette Flow Geometry Modeling of batch and continuous twin rotor mixers, widely used in polymer

M. Teverovskiy,1 I. Manas-Zloczower1*, P. Elemans2 and G. Rekers2

1Department of Macromolecular Science, Case Western Reserve University, Cleveland, OH, USA

2DSM Research, MD Geleen, The Netherlands

Numerical Simulations and Experiments ina Double-Couette Flow Geometry

Modeling of batch and continuous twin rotor mixers, widelyused in polymer processing, is a challenging computationalproblem. One approach to understand the mechanics of ¯owin twin rotor mixers is to simplify the geometry of the rotors byusing cylindrical rotors. This geometry is referred to as thedouble-Couette geometry. Advantages of the double-Couettegeometry are a simple symmetrical mesh design and no time-dependent ¯ow boundaries. In this work, we used the double-Couette geometry to study the mechanics of ¯ow and mixingef®ciency in laminar and turbulent ¯ow regimes. Flowvisualization experiments utilizing a ¯uorescent dye werecarried out in a transparent ¯ow cell. A ¯uid dynamicsanalysis package±FIDAP, based on the ®nite elementmethod, was used for the ¯ow simulations in laminar andturbulent ¯ow regimes. Numerical results showed goodagreement with the experimental data. We attempted aqualitative comparison for distributive mixing ef®ciency inlaminar and turbulent ¯ow regimes in light of the spreadingof a tracer line (dye) in the matrix. The analysis pointed out todifferences in the mixing mechanisms encountered indifferent ¯ow regimes.

1 Introduction

Batch and continuous twin rotor mixers are widely used inpolymer processing and therefore their modeling is of greatpractical signi®cance. However, the geometrically complexdesign and the transient character of the ¯ow, render hydro-dynamic analysis of such devices extremely dif®cult.

A conventional way for ¯ow simulations of complexdevices employs the ®nite element method (FEM). Thiswas the method of choice to simulate the ¯ow patterns in aBanbury mixer [1], the LCMAX mixer [2] and various twinscrew extruders [3 to 5]. The major dif®culty in solving these¯ow problems arises from the time dependent ¯ow bound-aries as the rotors rotate. The approach taken was to select anumber of sequential geometries to represent a complete

revolution of the rotors. For polymer processing operationswith laminar ¯ow of highly viscous materials, the overalleffect caused by a changing geometry can be analyzed fromthe results obtained separately in selected sequential geome-tries.

Yet another approach to understand the mechanics of ¯owin twin rotor mixers is to simplify the geometry of the rotorsby using cylindrical rotors. In this approach we approximatethe actual ¯ow in the mixer by the ¯ow generated betweentwo-adjacent cylinders rotating in a stationary, eight shaped,chamber. This geometry is referred to as the double-Couettegeometry. Advantages of the double-Couette geometry are asimple symmetrical mesh design and no time-dependent ¯owboundaries.

The double-Couette geometry was previously used inpolymer processing modeling for a number of differentpurposes: as a test geometry for the validation of someinnovative numerical techniques [6, 7] or as a model geo-metry allowing direct estimation of polymer melt rheologyfrom mixer speed and torque data [8]. Sernas and co-authors[9] considered the ¯ow in a double-Couette geometry as agood approximation for studying the ¯ow in the nip region ofa co-rotating twin screw extruder.

In this work, we employed the double-Couette geometryto study the mechanics of ¯ow and mixing ef®ciency inlaminar and turbulent ¯ow regimes. Flow visualizationexperiments using a ¯uorescent dye were carried out in atransparent ¯ow cell. A ¯uid dynamics analysis package±FIDAP [10], based on the ®nite element method, was used forthe ¯ow simulations in laminar and turbulent ¯ow regimes.We attempted a qualitative comparison for distributivemixing ef®ciency in laminar and turbulent ¯ow regimes byconsidering the spreading of a tracer line (dye) in the matrix.The analysis highlighted differences in mixing mechanismsencountered in different ¯ow regimes.

2 Laminar Flow

2.1 Formulation of the Laminar Flow Problem

The double-Couette ¯ow is the ¯ow between two rotatingcylindrical rotors and a barrel (Fig. 1). The main designparameters of the double-Couette device are the radius of therotors, R1, the radius of the barrel, R2 and the center line

* Mail address: Prof. Dr. I. Manas-Zloczower, Case Western ReserveUniversity, Cleveland, OH 44106, USA

242 # Hanser Publishers, Munich Intern. Polymer Processing XV (2000) 3

SCREW EXTRUSION =MIXING

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distance between the rotors, A. The rotors are co-rotatingwith the same angular velocity O (rpm). We analyzed theisothermal ¯ow of an incompressible, Newtonian ¯uid ofdensity r and viscosity m. Flow parameters were chosen suchas to maintain laminar conditions in the system i.e. theReynolds number Re � pR2

1Ory30m was always less than 1.A color tracer was placed as a vertical line in the left gap

(Fig. 1). The goal of the numerical modeling was to look atthe dynamics of the color tracer. The tracer motion isconsidered to be the motion of a discrete set of particles inthe viscous ¯uid (carrier phase). The ¯ow of the carrier phaseis a 2-D, steady state, isothermal ¯ow of a Newtonian ¯uid.For the particle tracking a Lagrangian approach is used. Theparticles (dispersed phase) are assumed to be massless points,such that their presence does not affect the ¯ow ®eld of thepure matrix. Interactions among particles such as Van derWaals attraction forces are also ignored. We solved theNavier-Stokes and continuity equations for the carrier phase:

v � Hv � ÿHp� 1

ReH2v; �1�

H � v � 0 �2�and ordinary differential equations for the trajectories of theparticles

drp

dt� v�rp�: �3�

In the equations above v � fvx; vyg is the vector velocity ofthe ¯uid; rp � fxp; ypg is the position vector of a particle andv�rp� is the ¯uid velocity along the particle's trajectory. Wechoose a Cartesian coordinate system whose origin is in themiddle of the distance between the two rotors.

Eqs. 1 to 3 are written in a dimensionless form. Velocitycomponents and distances are scaled with respect to the speedand radius of the rotor, respectively. For the characteristicspeed U, length L and time T we have employed the followingde®nitions:

U � p30

OR1; L � R1; T � L

U� 30

pOÿ1:

No-slip boundary conditions for the barrel and rotorsurfaces were employed. Fluid elements are stationary onthe barrel surface and move with the speci®ed angularvelocity on the rotor surfaces. Thus for Eqs. 1 and 2 thedimensionless boundary conditions have the following form:

at the barrel surface:

vx � vy � 0; �4�

at the rotor surfaces:

vx � yÿ y0; vy � x0 ÿ x;

where x0; y0

ÿ �are the coordinates of the center of a rotor.

The initial position of the tracer particles, rp0, wasspeci®ed and we have the following initial conditions forEq. 3:

t � 0 : rp � rp0�x; y�: �5�

We solved the ¯ow ®eld of the carrier phase (Eqs. 1 and 2)with boundary conditions (4) and used the results for thevelocity ®eld to obtain the particle trajectories. The FIDAPpackage using the ®nite element method was employed forthe mesh design, ¯ow simulations and particle tracking. Thefeatures of the numerical method are detailed in Appendix A.

2.2 Experimental Procedure

Flow visualization tests in the double-Couette geometry werecarried out using a Plexiglas ¯ow cell, which is part of a co-rotating twin-screw extruder device (diameter 30 mm, length62 mm). The deformation ®eld can be studied using glycerinwith a ¯uorescent dye as the tracer material. The ¯ow cellcontains two axes, on which screw elements can be mounted.In the present experiment, two tubes were mounted to obtainthe double- Couette geometry. The axes are hand-driven byrotating the knob at the bottom of the ¯ow cell. The axesrotate always simultaneously in the clockwise direction. Eachexperiment did not exceed 10 min in order to prevent visiblediffusion effects.

When placed in a dark room (under UV light), theexperiment allows for the visualization of the tracer line.Pictures are taken from above. Experiments were carried outin creeping ¯ow.

The ¯ow cell was put in the upright position and ®lled witha mixture of 95=5 glycerin=¯uorescin. The materials werepurchased from Acros Chimica NV (glycerin 99 %) andJanssen Chimica (water soluble ¯uorescin no. FW 376.28).Using a syringe (milliliter gas chromatography syringes fromSGE Microliters & Bonaduz, Switzerland), we positioned thetracer line just underneath the glycerin surface. The ¯ow cellwas lighted from above using a long wave UV lamp (modelB-100A from Ultra-Violet Products Inc., San Gabriel, CA).

The axes of the ¯ow cell were rotated and stopped atdifferent positions to take pictures of the deformed tracer line.We used a Nikon F3 camera with automatic timing andKodak Gold 2 Ektacolor 400 ®lm.Fig. 1. Double-Couette geometry

M. Teverovskiy et al.: Simulations and Experiments in a Double-Couette Flow Geometry

Intern. Polymer Processing XV (2000) 3 243© 2

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2.3 Comparison of Numerical Results with ExperimentalData

The 2D numerical simulations of particle tracking in thelaminar ¯ow were carried out with the aim of reproducing theexperimental results. The experimental conditions were asfollows: R1 � 1:065 cm, R2 � 1:55 cm, a � 2:6 cm, O �10 minÿ1 r � 1:3 g=cm3 and m � 15 g=(cm(s). The ®nite

element mesh comprises 3312 four nodes quadrilateral ele-ments (Fig. 2).

The streamline contour plot (Fig. 3) for the laminar ¯owshows a smooth ¯ow pattern. There are three families ofstreamlines in the double-Couette geometry resulting fromthe interaction of the two Couette ¯ows induced by eachrotor. Two of them follow closely the contours of the rotors,whereas the third family of streamlines encloses both rotors.

X

Y

O O O O O O OOOOO

OO

OO

OO

OOOOOOOOOOOOOOOO

OO

OO

OO

OO

OO

OO

OOOOOOOOOOOOOOOOOOO

OO

OO

OO

OO

O

OOOOOOO

OOOOOOOOOOOOOOOOOOOOOOOOOOOO

OO

OO

OO

OO

OO

OO

OOOOOOOOOOOOOOOO

OO

OO

OO

OOOOO O O O O O O O O O O O OOO

OO

OO

OO

OO

OO

OO

OO

OOOOOOOOOOOOOOOOO

-2D Double-Couette Flow, Mesh

FIDAP 7.6209/10/9817:08:23

ELEMENTMESH PLOT

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XMIN -.286E+01XMAX 0.286E+01YMIN -.254E+01YMAX 0.254E+01

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2-D Laminar Double-Couette Flow, Re=0.10

FIDAP 7.6209/10/9816:54:02

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-- 0.1250E-01-- 0.3751E-01-- 0.6251E-01-- 0.8752E-01-- 0.1125E+00-- 0.1375E+00-- 0.1625E+00-- 0.1875E+00-- 0.2125E+00-- 0.2375E+00

MINIMUM-0.28318E-13

MAXIMUM0.25004E+00

Fig. 2. Finite element mesh for the double-Couette geometry

Fig. 3. Streamline contour plot for the lami-nar ¯ow regime, Re � 0:1


244 Intern. Polymer Processing XV (2000) 3© 2

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In the case of co-rotating rotors the presence of a stagnationpoint in the inter-rotor region can be detected.

Experimental results are presented as a series of photo-graphs representing the sequential tracer positions at intervalsof 1=8 revolution of the rotors (Figs. 4, 5). To reproduce theexperimental results we tracked 3200 particles for 3 revolu-tions of the rotors. Figs. 6 and 7 illustrate the calculated shapeof the tracer at different times similar to the experimentalresults shown in Figs. 4 and 5.

Particles located close to the rotor surface move in acircular fashion following the motion of the rotor. A fractionof particles moves toward the right gap. Part of the particlesfrom this second fraction, going through the stagnation zone,experience some acceleration and on exit from the inter-rotorregion they move faster than the rest of the particles. Thiscauses the formation of a `̀ tongue'' directed toward the rightgap. After a revolution around the right rotor, this `̀ tongue''of particles is directed to the left rotor. This situation repeatsitself for the particles moving around the left rotor. After arevolution the second `̀ tongue'' directed toward the right gapappears in the inter-rotor region. There is good agreementbetween the experimental and the numerical results.

3 Turbulent Flow

3.1 Formulation of the Turbulent Flow Problem

In a simple Couette ¯ow geometry (¯ow between concentricrotating cylinders) turbulent ¯ow conditions were detectedwhen the Reynolds number exceeded 1000 [11, 12]. We usedthese results as guidelines and considered that, turbulent ¯owconditions in the double-Couette ¯ow geometry will prevailat Re � 1000.

The well-known kÿ e turbulence model [13], incorpo-rated into the FIDAP package, allowed for the numericalmodeling of the turbulent ¯ow of the carrier phase. In thismodel the turbulence ®eld is characterized in terms of twovariables, the kinetic energy k and the viscous dissipation rateof turbulent kinetic energy e which are de®ned as

k � 1

2�v̂2

x � v̂2y�; e � n

@v̂x

@x

� �2

� @v̂y

@y

� �2

;

where v̂i (i � 1; 2) are the ¯uctuating components of velocityand n is the kinematic viscosity of the ¯uid (overbars indicatetime averaged values). The components of the Reynoldsstress tensor are proportional to the mean velocity gradients,i.e.

ÿr0v̂iv̂j � mt�vi;j � vj;i�; �i � 1; 2�:

In this de®nition mt is the turbulent viscosity; index `̀ 0''denotes the laminar properties of ¯uid (molecular viscositym0, density r0); coma `̀ ,'' between the indices denotes

differentiation: vi;j � @viy@xj. The dimensionless governingequations for k and e are

@k

@t� vjk;j �

mt

sk

k;j

� �;j

ÿe� mtF;

@e@t� vje;j �

mt

see;j

� �;j

�c1

ekmtFÿ

c2e2

k; �6�

mt �cmk2

e;

where F � 12

dijdij, (dij � vi;j � vj;i) is a dissipation function.The equations are expressed in Cartesian tensor notation withan implied summation over repeated indices. The values kand e were scaled with respect to U2 and U3yL, respectively(here U and L are the characteristic speed and length as in thelaminar ¯ow); the turbulent viscosity mt and the dissipationfunction F were scaled with respect to r0UL and U2yL2,respectively. We will consider the steady-state form of theequations, i.e.

@k

@t� @e@t� 0: �7�

The closed system of the mean ®eld Eqs. 1 and 2 and Eq. 6with condition (7) de®nes the quasi steady-state kÿ e modelof turbulence. This system contains the following empiricalconstants [10]:

cm � 0:09; sk � 1:00; se � 1:30; c1 � 1:44; c2 � 1:92:

We applied no-slip boundary conditions for the componentsof ¯uid velocity at the barrel and rotor surfaces. The boundaryconditions for k and e require special consideration and theywill be described as features of the numerical method in thenext section.

As in the laminar ¯ow, we tracked particles using thevelocity ®eld of the carrier phase. A stochastic model [14] ofthe FIDAP package was used to analyze the in¯uence ofturbulence on particle trajectories. Instantaneous velocities ofthe carrier phase were used to solve the kinematic equationsfor the particles. These instantaneous values were computedby adding random ¯uctuations to the mean ¯ow solutionobtained from the kÿ e simulation. Eq. 3 becomes

dxp

dt� vx�rp� � lv 0x�rp�;

dyp

dt� vy�rp� � lv 0y�rp�; �8�

v 0x�rp� � v 0y�rp� � 23

k�rp�ÿ �1=2

;

where v�rp� is the mean velocity of the carrier phase alongparticle's trajectory; l is a random number betweenÿ1 and 1sampled from a normal distribution; �2

3k�rp��1y2 is the char-

acteristic turbulent eddy velocity scale. The initial conditionfor Eq. 8 remains the same as for particle tracking in laminar¯ow. The features of the numerical method for turbulence¯ow simulation are detailed in Appendix B.



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3.2 Results of Numerical Simulations for the Turbulent Flow

Numerical simulations for the ¯ow ®eld in a turbulent regimewere performed for the following Reynolds numbers:Re � 103, 104, 105. The convergence of the numericalalgorithm was reached for 13, 16 and 22 iterations, respec-tively. The relative error value of the solution and residualforce vectors are (9 � 10ÿ3, 7 � 10ÿ7), (5 � 10ÿ3, 10ÿ7) and(9 � 10ÿ3, 10ÿ8), respectively.

The calculations of the turbulent ¯ow ®eld were done forboth co- and counter-rotating rotors. In the case of co-rotatingrotors the structure of the mean ¯ow ®eld is similar to the onein the laminar ¯ow (Fig. 8). As in the laminar ¯ow, there arethree families of streamlines. The magnitude of the velocity®eld increases with the Reynolds number. We can also markthe presence of a stagnation zone in the inter-rotors region.

In the case of counter-rotating rotors we do not observe thethird family of the streamlines around both of the rotors (Fig.9). The intensity of the circular ¯ows around the rotors alsoincreases with the Reynolds number. There is no stagnationzone in the inter-rotor region. In fact, in this case we observe¯uid acceleration in the inter-rotor region, whereas in the caseof co-rotating rotors there was a deceleration of the ¯uid inthis region.

Figs. 10 and 11 show speed contours for co-rotating andcounter-rotating rotors, respectively. It is interesting to notethat by increasing the Reynolds number the region of relativehigh velocity seems to diminish in size (the speed contourplots show the dimensionless velocities for the ¯ow ®eld). AtRe � 104 and 105, the region of high-speed contours (dimen-sionless velocity ranging between 0.33 to 1) occupies lessthan 1=3 of the gap between the rotors and the barrel.

Fig. 4. Photographs of experimental resultsobtained at various time intervals during the ®rstrevolution of the rotors



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4 Distributive Mixing Ef®ciency in Laminar andTurbulent Flow Regimes

We tried to compare the ef®ciency of distributive mixing [15,16] in laminar and turbulent ¯ow regimes by considering thespreading of a tracer line (dye) in the matrix (continuum). Weused 300 particles for the tracer line and looked at thetrajectories marked by these particles. For clarity, particleshave different colors in a vertical sequence. Fig. 12 shows theregion marked by the tracer during 3 successive revolutionsin a laminar ¯ow regime. Figs. 13 and 14 display similar plotsfor turbulent ¯ows at Reynolds numbers of 1000 and 10 000respectively.

In the laminar ¯ow regime, the particles move alongsmooth trajectories, gradually ®lling the space. It is interestingto note that the structure of the tracer (sequence of color in theinitial tracer line) is preserved during the spreading process.

In the case of turbulent ¯ow (Figs. 13 and 14), the particlesin the tracer line do not move on smooth trajectories, butrather ¯uctuate around the streamlines of the mean ¯ow ®eld.This causes disturbances in the initial tracer structure (thesequence of colors in the initial tracer line is not preservedduring the spreading process). Also the trajectories of thetracer particles initially placed in the left Couette geometryseem not to spread well in the right Couette geometry,especially at the higher Reynolds number.

Fig. 5. Photographs of experimental resultsobtained at various time intervals during the thirdrevolution of the rotors



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Fig. 6. Tracer contour during the ®rst revolution of the rotors at successive time intervals similar to the experimental results

X

Y

2-D Laminar Double-Couette Flow, Re= 0.1, 5/8 Rev.

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Fig. 7. Tracer contour during the third revolution of the rotors at successive time intervals similar to the experimental results



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This qualitative analysis of distributive mixing in laminarand turbulent ¯ow regimes points out to the difference inthe mixing `̀ mechanisms'' encountered in the same ¯owgeometry by changing the ¯ow regime. Whereas a laminar¯ow regime preserves the initial structure of the minorcomponent, turbulent ¯ows enable cross mixing alongstreamlines, thus randomizing the arrangement=compositionof the system. We have shown only 3 revolutions in

each of the ¯ow regimes. Although distribution ofthe minor component in the right half of the Couettegeometry seems to progress better in the laminar ¯owregime, the dynamics of the mixing process (time evolutionfor the spatial distribution of the minor component) is notequivalent between the different ¯ow regimes (the dimen-sional time is very different when changing the Reynoldsnumber).

Fig. 9. Streamline contour plot for the turbulent ¯ow regime (counter-rotating rotors)Fig. 8. Streamline contour plot for the turbulent ¯ow regime (co-

rotating rotors)



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5 Summary

A ®nite element method was used for ¯ow simulations in adouble-Couette geometry. We analyzed the ¯ow ®eld inlaminar and turbulent ¯ow regimes. Particle tracking wasused for ¯ow visualization. Numerical results show good

agreement with experimental data. We attempted a qualita-tive comparison for distributive mixing ef®ciency in laminarand turbulent ¯ow regimes by considering the spreading of atracer line (dye) in the matrix. The analysis points out todifferences in mixing `̀ mechanisms'' encountered in differ-ent ¯ow regimes.

Fig. 10. Mean speed contour for turbulent ¯ow (co-rotating rotors) Fig. 11. Mean speed contour for turbulent ¯ow (counter-rotatingrotors)



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Fig. 12. Dye spreading in a laminar ¯ow regime, Re � 0:1 Fig. 13. Dye spreading in a turbulent ¯ow regime, Re � 103



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Appendix A

Features of the Numerical Method for Laminar FlowSimulation

For this problem the ®nite element method was not applieddirectly to the full system of governing equations of the ¯ow

[10]. It was applied to a perturbed system of equations, inwhich the continuity requirement (2) is weakened andreplaced by,

H � v � ÿep;

where e, the penalty parameter, is small ± typicallye � 10ÿ5 ÿ 10ÿ9. Physically this can be equated to simulat-ing the ¯ow of a slightly compressible ¯uid.

This approach, referred to as the penalty functionapproach, has the great advantage of eliminating theunknown pressure from the main system without losing anysigni®cant accuracy, providing that e is small enough. Thepressure can then be recovered from the velocity ®eld:pe � ÿH � veye (where ve; pe� � is the solution of the perturbedsystem). For brevity, the index `̀ e'' indicating the solution ofthe perturbed system will be dropped off from the symbols ofvelocity and pressure.

Application of the ®nite element procedure to the steady-state Navier-Stokes equations results in a set of algebraicequations that may be represented in matrix form as

K�v�v � F: (A-1)

Here K is the global system matrix, v is the global vector ofunknowns in the nodal points of the mesh (velocities,pressures, etc.) and F is a vector, which in the general caseincludes the effect of body forces and boundary conditions. Inparticular, for the penalty function approach and laminar¯ows, v is the vector of nodal velocities; for no-slip boundaryconditions F is zero.

Successive substitution and Newton-Raphson iterationmethods were used to solve the matrix equation A-1 for allthe problems in this work. Iterations were terminated whenthe two following convergence criteria were satis®ed simul-taneously

vi ÿ viÿ1

vi

� ev;R�vi�

R0

SeF:

Here k � k is the Euclidean norm; R�vi� � K�vi�vi ÿ F is theresidual force vector and R0 � R�v0� is a reference vectorevaluated using an initial solution vector, v0. FIDAP providesthe option to chose v0 between a constant (in particular, zeroinitial vector), a function of coordinates, or the solution of thelinear ¯ow problem (Stokes initial vector). In the latter case,FIDAP solves at ®rst the ®eld equations without the con-vective term and the obtained solution is used as the initialcondition ®eld for all variables in the computational domain.

For particle tracking we need to solve the kinematicequations for the particle trajectories using the ¯ow ®eld ofthe carrier ¯uid. The implicit backward Euler method with a`̀ double step'' technique was employed for both laminar andturbulent ¯ow. This implicit time integration algorithm, usedin FIDAP, is

rk�1p;n�1 � rp;n � Dtv�rk

p;n�1�;where rp;n � fxp�tn�; yp�tn�g is the particle's position at timetn and Dt � tn�1 ÿ tn is a time step. The reliance on informa-tion at time tn�1 dictates an iterative procedure fork � 1; 2; . . . to convergence, with the initial conditionr1

p;n�1 � rp;n.

Fig. 14. Dye spreading in a turbulent ¯ow regime, Re � 104



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The successive substitution method with Stokes initialvector was used for the ¯ow ®eld calculation (laminar andturbulent). The simulation was performed with the followingvalues of the penalty and accuracy parameters: e � 10ÿ8,ev � eF � 10ÿ4, ep � 10ÿ4 (where ep is accuracy for theparticle tracking).

Appendix B

Features of the Numerical Method for Turbulence FlowSimulation

The simulation was performed with the following numericalparameters: e � 10ÿ8, ev � eF � 10ÿ2; ep � 10ÿ4. The meshcomprised 5772 four nodes quadrilateral elements.

The behavior of a turbulent ¯ow in the immediate vicinityof the solid boundary requires special consideration. The ¯owin the near-wall region (region adjacent to the solid boundary)consists from a viscous sub-layer and a transitional or bufferlayer and has properties different from the ¯ow in the fullyturbulent region.

FIDAP [10] uses a special near-wall modeling methodol-ogy in order to obtain the correct solution in the vicinity of thesolid boundary. In this scheme, which is based on the use ofspecialized elements, the computational domain is extendedto the physical boundary and the full set of elliptic mean ¯owequations is solved all the way down to the wall. A one-element thick layer of special elements is then employed inthe near-wall region between the fully turbulent outer ¯ow®eld and the physical boundary. In these near-wall elements,specialized shape functions are used to accurately capture thesharp variations of the mean ¯ow variables (velocities,temperature). These specialized shape functions, which arebased on the universal near-wall pro®les, depend on thecharacteristic Reynolds numbers and adjust automaticallyduring the course of computation to accurately resolve thelocal ¯ow pro®les. Since use is still made of the standard highReynolds number kÿ e model, the k and e equations are notsolved in the layer of near-wall elements.

The computational domain for the k and e equationsextends to the `̀ top'' of the special near-wall elements. Aspart of the near-wall implementation, FIDAP applies thefollowing boundary conditions for k and e at this location:

@k

@n� 0; e �

c1=3m k

� �1:5

kd;

where k � 0:41 (Von Karman constant) and d is the actualheight of the element above the wall. These conditionsillustrate an `̀ equilibrium'' in the near-wall regions. Theturbulence length scale is de®ned as k1:5ye and varies linearlywith normal distance n from the wall (k is constant). TheNeumann boundary condition on k allows the level of k to`̀ ¯oat'' in response to turbulence processes occurring in bothlocal and neighboring regions. After each iteration this value

of k is used to calculate the characteristic turbulent velocityscale, k1y2, for the near-wall region. This value determines themagnitude of the Reynolds number in the local element,which in turn controls the degree of skewing in the specialbasis functions, thus enabling an accurate resolution of thelocal near-wall ¯ow pro®le.

The near-wall scheme described above functions properlyif the viscous and transitional sub-layers are fully containedwithin the special near-wall elements. In order to achieve thiscondition special methodical calculations were used tochoose an optimal mesh density in the vicinity of the solidboundaries: the barrel and rotor surfaces. The k and eboundary conditions were then applied automatically inFIDAP.

Separate calculations were carried out to select the initialvectors for the k and e equations in the computational domain.This procedure allows for good convergence of the solutionfor the mean ®eld equations.

References

1 Yang, H.H., Manas-Zloczower, I.: Int. Polym. Process. 7, p. 195(1992)

2 Yao, C.H., Manas-Zloczower, I.: Int. Polym. Process. 13, p. 334(1998)

3 Yang, H.H., Manas-Zloczower, I.: Polym. Eng. Sci. 32, p. 1411(1992)

4 Li, T., Manas-Zloczower, I.: Int. Polym. Process. 10, p. 314 (1995)5 Cheng, J. J., Manas-Zloczower, I.: Polym. Eng. Sci. 37, p. 1082

(1997)6 David, B., Sapir, T., Nir, A., Tadmor, Z.: Int. Polym. Process. 5, p.

155 (1990)7 David, B., Sapir, T., Nir, A., Tadmor, Z.: Int. Polym. Process. 7, p.

204 (1992)8 Bousmina, M., Ait-Kadi, A., Faisant, J.B.: J. Reol. 43, p. 415

(1999)9 Sastrohartono, T., Esseghir, M., Kwon, T.H., Sernas, V.: Polym.

Eng. Sci. 30 p. 1382 (1990)10 FIDAP Package, Theory Manual, Fluid Dynamics International,

Inc., Evanston, IL.11 Coles, D.: J. Fluid Mech. 21, p. 385 (1965)12 Andereck, C. D., Liu, S. S., Swinney, H. L.: J. Fluid Mech. 164, p.

155 (1986)13 Bradshaw, P., Cebeci, T., Whitelaw, J.: Engineering calculation

methods for turbulent ¯ow. Acad. Press, London (1981)14 Berlemont, A., Desjonqueres, P., Gouesbet, G.: Int. J. Multiphase

Flow 16, p. 19 (1990)15 Li, T., Manas-Zloczower, I.: Chem. Eng. Comm. 139, p. 223 (1995)16 Manas-Zloczower I.: Rheology Bulletin 66, p. 5 (1997)

Acknowledgements

The authors are gratefully acknowledging the ®nancialsupport of DSM Research in the Netherlands and the use ofcomputing services from Ohio Supercomputer Center.

Date received: September 26, 1999Date accepted: June 8, 2000



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