tracking of immiscible interfaces in multiple … of immiscible interfaces in multiple-material...

Computational Materials Science 29 (2004) 103–118

www.elsevier.com/locate/commatsci

Tracking of immiscible interfaces in multiple-materialmixing processes

Hao Tang a,b,*, L.C. Wrobel a, Z. Fan a,b

a Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UKb Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

Received 19 February 2003; accepted 25 July 2003

Abstract

A numerical study is presented for tracking immiscible interfaces with piecewise linear (PLIC) volume-of-fluid

(VOF) methods on Eulerian grids in two and three-dimensional multiple-material processes. The method is coupled

with the continuum surface force (CSF) algorithm for surface force modelling, supported by a multi-grid solver that

enabled the resolution of large density ratio between the fluids and fine scale flow phenomena. A numerical modelling

coupled with experimental data is established and evaluated through various immiscible flow cases for maintaining

sharper interfaces between multiple fluids in the meso-/micro-scale, including the test symbol falling, collapsing cylinder

of water, and a viscous drop deformation. The immiscible binary metallurgical flow in a shear-induced mixing process is

investigated to study the fundamental mechanism of the twin-screw extruder (TSE) rheomixing process. It is observed

that the rupturing, interaction and dispersion of droplets are strongly influenced by shearing forces, viscosity ratio,

turbulence, and shearing time. Preliminary results show a good qualitative agreement with experimental results of a

rheomixing process.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Immiscible interface; VOF; Shear flow; Mixing process; Multi-material; Metallurgical flow

1. Introduction

Incompressible multi-material flows with sharp

immiscible interfaces occur in a large number of

natural and industrial processes. Casting, mold

filling, thin film processes, extrusion, spray depo-sition, and fluid jetting devices are just a few of the

* Corresponding author. Address: Department of Mechani-

cal Engineering, Brunel University, Uxbridge, Middlesex, UB8

3PH, UK. Tel.: +44-1895-274000; fax: +44-1895-256392.

E-mail address: [email protected] (H. Tang).

0927-0256/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.commatsci.2003.07.002

areas in material processing applications where

immiscible interfaces are the main feature and

dominate the whole process. In particular, casting

immiscible binary alloys is a typical interfacial

fluid flow problem, where evidence shows that the

solidified microstructure of cast immiscible alloysstrongly depends on the rheological behaviour

within the melt state during cooling [1]. There is an

increasing need to be able to control these complex

metallurgical processes and hence, an improved

capability to numerically simulate and study these

processes. Numerical simulations are, in principle,

ideally suited to study these complex immiscible

ed.

mail to: [email protected]

104 H. Tang et al. / Computational Materials Science 29 (2004) 103–118

interfacial flows and provide an insight into the

process that is difficult by experiments. However,

because of the limitation of numerical approaches,

there are still challenges in the development of

approaches for the simulation of flows with ma-terial interfaces of arbitrarily complex topology.

Several techniques exist for tracking immiscible

interfaces, each with its own strengths and weak-

nesses [2]. These techniques can be classified under

three main categories according to physical and

mathematical approaches: capturing (also known

as moving grid or Lagrangian approach), tracking

(also known as fixed grid or Eulerian approach)[3], as well as combined methods. Capturing

methods [4] include moving-mesh, particle-particle

scheme, and boundary integral method. Tracking

methods can be divided into two main approaches:

surface tracking and volume tracking, which in-

clude front-tracking [5], volume-of-fluid (VOF)

[6,7], marker and cell (MAC) method [8],

smoothed particle hydrodynamics (SPH) [9], levelset methods [10], and phase field [11]. Combined

methods include the mesh free/particle method

[12,13], coupled Eulerian–Lagrangian (CEL) [3,4]

and variants from the previously mentioned two

methods. Amongst these, an indicator function is

used which is a volume fraction (colour function)

for VOF methods or a level set for level-set

methods. The indicator function is a scalar stepfunction representing the space occupied by one of

the fluids in the VOF method, and a smooth ar-

bitrary function encompassing a prespecified iso-

surface which identifies the interface in level-set

methods. The VOF method is widely adopted by

in-house codes and built-in in commercial codes. It

is a popular interface tracking algorithm that has

proven to be a useful and robust tool since itsdevelopment decades ago. Recently, level-set

methods, originally introduced by [10], have been

applied to a wide variety of immiscible interfacial

problems. These methods use a level function u,with the 0 contour level defining the material in-

terface, much the same as the fractional volume

function C in the VOF method, to indicate the

shortest distance to the interface. This approachhas two inherent strengths. One is that the repre-

sentation of the interface as the level set of a

function u leads to convenient formulas for the

interface normal and curvature. Another advan-

tage (similar to the VOF method) is that no special

procedures are required in order to model topo-

logical changes of the front. However, due to flow

distortions, the deviations from the true distanceto the interface increase, and hence needs to be re-

initialised every couple of time-steps by the solu-

tion of a differential equation. Level set methods

have had problems with mass conservation,

though [14] claims that they can be overcome.

Multi-material problems can be mathematically

treated as two incompressible fluids separated by a

moving surface of discontinuity. The goal of thispaper is to incorporate a range of physical effects

to the VOF method with the application of various

existing numerical schemes to establish a numeri-

cal modelling toolkit for immiscible liquid alloy

flow in the TSE rheomixing process, for predicting

the rheological behaviour of immiscible liquid al-

loys in shear-induced mixing flow. To achieve this,

we have reviewed the VOF method used in ourstudy and also performed several numerical tests.

These simulations have provided some excellent

qualitative insight into multi-material problems.

The quantitative accuracy of these problems,

however, may be somewhat limited for reasons

which are made clear in later sections.

In the next section, a brief literature review is

presented for the development and applications ofVOF methods. Section 3 describes the mathe-

matical methods and physical model of the

piecewise linear interface construction (PLIC)-

VOF method by summarizing the numerical al-

gorithms used for solving the incompressible

multi-material interfacial flow. The next section

presents numerical tests to verify the performance

of algorithms, including the test symbol falling,collapsing cylinder of water with comparison to

experimental results, and a viscous drop defor-

mation. The simulation cases are conducted at the

meso- and micro-scale. The accuracy and effi-

ciency of the numerical modelling schemes are

verified with test cases. An application to the

rheological behaviour of immiscible binary alloy

flows in shear-induced mixing process is also pre-sented to demonstrate the potential of a numerical

modelling toolkit for immiscible liquid alloy flow

in the TSE rheomixing process.

H. Tang et al. / Computational Materials Science 29 (2004) 103–118 105

2. Review of VOF methods

Pioneering work on VOF methods goes back to

the early 1970s. The first three volume trackingmethods were introduced by DeBar�s method

(KRAKEN code, 1974) [15], Hirt and Nichols�VOF (volume-of-fluid, 1981) [6,7], Noh and

Woodward�s SLIC method (simple line interface

calculation, 1976) [16]. Meanwhile, Ramshaw and

Trapp [17], and Peskin [18] were also involved in

the pioneering stages. VOF methods became more

popular with Hirt and Nichols� D–A VOF method(donor–acceptor, 1981) [19] and their SOLA-VOF

code [20]. Significant development of volume

tracking methods was made by the new piecewise

linear schemes of Youngs� (PLIC, 1982) [21] andtheir hydrocode [22], and were subsequently

adopted in many high-speed hydrodynamics codes

involving material interfaces [23], such as Addessio

and coworkers� CAVEAT code (1984), Holian andcoworkers� MESA code (1991), Kothe and co-

workers� PAGOSA code (1992), Perry and co-

workers� Rhal code (1993). Many extensions and

enhancements to the work of Youngs have oc-

curred since its introduction. These versions are

now known as PLIC methods. Nowadays, the

VOF method has been adopted by some general

commercial CFD codes and casting process codes.Current development is geared towards applying

high-order time integration schemes to propaga-

tion algorithms and robust methods of polyhedral

truncation to 3D interface reconstruction.

Table 1

Development of VOF algorithms

Reconstruction interface geometry Tim

Piecewise linear, operator split PLIC On

Piecewise constant, operator split SLIC On

Piecewise constant, multi-dimensional FCT On

Piecewise constant, stair-stepped, multi-dimen-

sional

On

D–A

Piecewise linear, multi-dimensional PLIC On

Piecewise linear, operator split FLAIR On

Piecewise linear, multi-dimensional LVIRA M

PLIC

SS

PLIC––piecewise linear interface construction; SLIC––simplified line

corrected transport; FLAIR––flux line segment model for advection a

squares volume-of-fluid interface reconstruction algorithm.

2.1. Development of VOF algorithms

The essential concepts of VOF methods are

described as follows: at the beginning, an initialfluid volume is used to compute fluid volume

fractions in each computational cell from a speci-

fied interface topology. This requires the calcula-

tion of volumes truncated by the fluid interface in

each interface cell. Exact interface information is

then lost and instead discrete volume data is pro-

duced until an interface is reconstructed. The fluid

solver then generates a velocity field, and inter-faces are tracked by evolving fluid volumes in time

with the solution of an advection equation. At any

time in the solution, exact interfaces must be in-

ferred, based on local volume data and as-

sumptions of the particular algorithm. The

reconstructed interface is then used to compute the

volume fluxes necessary to integrate the volume

evolution equation. Therefore, the principal stepsof VOF methods are reconstructed interface ge-

ometry and time integration algorithms. There are

mainly three algorithms (piecewise constant,

piecewise constant stair-stepped, and piecewise

linear) for the reconstruction interface geometry

and two algorithms (1D or operator split, and

multi-dimensional) for time integration, as listed in

Table 1. However, many improvements and en-hancements have been developed subsequently to

these by a number of researchers.

These contributions are focused on the notable

improvement of algorithms for interface

e integration Author(s) and references Date

e dimensional DeBar [15] 1974

e dimensional Noh and Woodward [16] 1976

e dimensional Zaleski [24] 1979

e dimensional Chorin [25] 1980

Hirt and Nichols [7] 1981

e dimensional Youngs [21] 1982

e dimensional Ashgriz and Poo [26] 1991

ulti-dimensional Puckett et al. [27] 1997

Rider and Kothe [23] 1998

Harvie and Fletcher [28] 2000

ar interface construction; D–A––donor–acceptor; FCT––flux-

nd interface reconstruction; SS––stream scheme; LVIRA––least


reconstruction or time integration to achieve either

more accuracy or more efficiency. Youngs� for-

mula is adopted in many codes involving material

interfaces, as mentioned in Section 2.1.

The basic feature of piecewise constant, SLICand D–A methods is that the interfaces within cells

are assumed to be lines aligned with one of the

logical mesh coordinates, which is a 1D operator.

Since the interface normal follows from volume

differences based upon the current advection sweep

direction, improved methods use multi-dimen-

sional operators which are set on a 3 · 3 stencil in

2D to reconstruct the stair-stepped interfacewithin each cell. Its volume fluxes are formulated

algebraically by using flux-corrected transport

(FCT) methods. The piecewise constant method is

only a first-order scheme. Errors induced by its

algorithm result in unphysical interfaces, causing

submesh-size material bodies to separate from the

main material body tending to evict from inter-

faces. These are severely impacted on the overallinterfacial solution of flows with vorticity or shear

near the interface, where forces are significant.

This method is also difficult to apply for complex

topology multiple-material flows.

The piecewise linear method is different from

piecewise constant in that it reconstructs interface

lines with a slope, which is given by the interface

normal. The interface normal is determined with amulti-dimensional algorithm which does not rely

on the sweep direction. Recently, PLIC volume

tracking methods have been used successfully.

Several recent papers discussed this subject exten-

sively by introducing second-order time integra-

tion schemes or robust methods for truncation of

arbitrary polyhedra [23]. Obviously, multi-dimen-

sional schemes can be more accurate and efficientin calculating cell boundary fluxes compared to

operator split schemes, and are currently devel-

oped as described in [23,27,28].

The descriptions given by [23] on reconstruction

and advection algorithms of volume tracking

methods are provided in a clear and concise

manner. Comparisons with SLIC, D–A, FCT, and

Youngs� PLIC schemes have been reported by [29].Results have shown that Youngs� PLIC scheme

uses a more accurate interface reconstruction in

comparison to either SLIC and D–A or FCT.

Similar conclusions are also given by [30] after

comparing these and their CICSAM scheme. The

SS advection scheme coupled with Youngs� PLICpossibly provides more accuracy at potentially

greater computational expense [28]. Comparisonsof SLIC and PLIC with the level set method,

marker particles and piecewise parabolic method

(PPM) have been performed by [31]. Results show

that marker particles and PLIC methods allow the

robust calculation of difficult fluid flows with large

jumps in physical properties at the material inter-

face.

Following volume tracking methods, and vari-ous enhancements to interface reconstruction and

interface advection algorithms (named VOF-like

methods [32,33]), many methods are being cur-

rently developed for multi-material flows coupled

with other multi-phase methods, such as VOF-

DPM [34,35], VOF-two phase flow [36], VOF-

phase change (vapour or solidification) [37–39],

VOF-level set [40]. These algorithms are necessaryfor numerical simulations of more complex phe-

nomena.

2.2. Summary of VOF literature

Methods for tracking immiscible interfaces

have been reviewed during the last two decades.

General reviews of early tracking methods are gi-ven by [4,41] and more recent ones by [42,43].

Some general reviews of moving boundary meth-

ods are also discussed in [3,44]. Current reviews of

different algorithms of the VOF method are pre-

sented by [23,29,31,45], where detailed compari-

sons and error estimation are presented. A recent

review of numerical errors of the LVIRA-VOF

algorithm is given by [46], where an analysis ofeffects of the grid size on the numerical error of

interfacial reconstruction is presented. Such error,

which might significantly affect the description of

the physical phenomena, cannot be avoided by

applying better and more accurate front tracking

algorithms. The source of this error is the limita-

tion of the grid cell––the VOF model cannot sim-

ulate the portions of fluid which are smaller thanthe grid cell. One possibility for the reduction of

the numerical error is the adaptive grid refinement

of the mesh during the simulation. The first use of


adaptive mesh refinement (AMF) in a volume

tracking method can be found in [47]. A recent

report on AMF applications for bubble rising

problems is described in [48]. For tracking im-

miscible interfaces in multi-material problems,volume-tracking methods have been popularly and

successfully used since the mid-1970s. However,

several methods for sharper interfaces in multi-

phase flow are under development. A level set

method, for example, has been recently applied to

multi-phase problems [70].

2.3. Applications of VOF methods

Applications of VOF methods are found in

many industrial and biohydronamics areas, either

in the macro- or meso-/micro-scale, including

aero-/astro-/hydro-dynamics, metallurgical, vis-

cous, viscoelastic flows. A few special test cases

have benchmarks for the validation of interfacial

topology and propagation, and verification of ac-curacy and efficiency. They include static interface

reconstruction [21], Zaleski�s slotted solid disk

rotation [24,28,30], Rider–Kothe single vortex and

time reversed flows [23,28,30,31,46], Rudman�shollow square/circle [28–30], and Rayleigh–Taylor

instabilities [21,27,29,36,49,50]. Numerous papers

describe successful applications of VOF methods

in various fields. A few typical engineering areas ofmacro-scale flows include cast filling [38], coastal/

ocean wave flow [50], dam break flow [51], coating

process, liquid sloshing [52,53], liquid/air jet

[54,55], environment/fire fighting/HVAC area, and

material extrusion process. Meso-/micro-scale

flows include bubble rising, drop deformation and

rupturing [56,57], drop sediment/splash, drop in-

teraction [58], lubricating flow, two layer flows.Besides these, the VOF method is also applied

extensively in the biofield [3] area for plasma flow,

arterial blood flow, etc.

Examples of VOF codes [27] are KRAKEN,

SURFER, SOLA-VOF code and its descendants

(NASA-2D, NASA3D, RIPPLE, Tellurider (RIP-

PLE-3D version) and FLOW3D). SURFER

(originally by Zaleski) and RIPPLE (originally byKothe) are used bymany researchers since these are

free or public open source codes and further en-

hancements have been made [59]. Some examples

of general commercial CFD codes which use VOF

methods are FLOW3D, CFX, FLUENT, FIDAP,

PHOENICS, STAR-CD, as well as some CAE

codes for casting process, such as MAGMAsoft,

ProCAST, SIMULATOR, and CAST-Flow.

3. Numerical methods

The numerical methods adopted in the present

simulations are based on Hirt and Nichols� VOF

method [7] coupled with Youngs� PLIC scheme

[21], Brackbills� continuum surface force (CSF)model [49], and solved by algebraic multi-grid

(AMG) solver [60], as well as k-e turbulence model

[61], and the pressure-implicit with splitting of

operators (PISO) scheme for pressure–velocity

coupling [62]. A brief summary of the PLIC-VOF

methodology is provided in what follows.

3.1. The volume evolution equations

Immiscible metallic alloy flows are considered

here as multi-phase fluid systems in isothermal

state, with different density and viscosity. The

domain of interest contains an unknown free

boundary, which undergoes severe deformation

and separation.

In the VOF method, the motion of the interfacebetween multi-immiscible liquids of different den-

sity and viscosity is defined by a phase indicator––

the volume fraction function C, and the interface is

tracked by the following three conditions:

Ckðx; y; z; tÞ

¼0 ðoutside kth fluidÞ ð1Þ1 ðinside kth fluidÞ ð2Þ> 0; < 1 ðat the kth fluid interfaceÞ ð3Þ

8><>:

According to the local value of Ck, appropriate

properties and variables are assigned to each

control volume within the domain.

The volume fraction function Ck is governed bythe volume fraction equation

oCk

otþ u � rCk ¼ 0 ð4Þ

where u is the flow velocity.


The two-phase fluid flows are modelled with the

Navier–Stokes equation

qou

ot

�þ u � ru

�¼ �rp þ lr2uþ qg þ F ð5Þ

where F stands for body forces, g for gravity ac-

celeration, and p for pressure. The velocity field issubject to the incompressibility constraint,

r � u ¼ 0.

In a two-phase system, the properties appear-

ing in the momentum equation are determined by

the presence of the component phase in each

control volume. The average values of density

and viscosity are interpolated by the following

formulas:

qi;j ¼ q1 þ C2ðq2 � q1Þ ð6Þ

li;j ¼ q1 þ C2ðl2 � l1Þ ð7Þ

In multi-phase systems, the ‘‘onion skin’’ tech-

nique is used [21].

3.2. The interface tracking algorithm

The formulation of the VOF model requires

that the convection and diffusion fluxes throughthe control volume faces be computed and bal-

anced with source terms within the cell itself. The

interface will be approximately reconstructed in

each cell by a proper interpolating formulation,

since interface information is lost when the in-

terface is represented by a volume fraction field.

The geometric reconstruction PLIC scheme is

employed because of its accuracy and applica-bility for complex flows, compared to other

methods such as the donor–acceptor, Euler ex-

plicit, and implicit schemes. A VOF geometric

reconstruction scheme is divided into two parts: a

reconstruction step and a propagation step. The

key part of the reconstruction step is the deter-

mination of the orientation of the segment. This

is equivalent to the determination of the unitnormal vector n to the segment. Then, the normal

vector ni;j and the volume fraction Ci;j uniquely

determine a straight line. Once the interface has

been reconstructed, its motion by the underlying

flow field must be modelled by a suitable algo-

rithm.

3.2.1. The interface reconstruction algorithm

In the PLIC method, the interface is approxi-

mated by a straight line of appropriate inclination

in each cell. A typical reconstruction of the inter-face with a straight line in cell (i; j), which yields an

unambiguous solution, is perpendicular to an in-

terface normal vector ni;j and delimits a fluid vol-

ume matching the given Ci;j for the cell. A unit

vector n is determined from the immediate neigh-

bouring cells based on a stencil Ci;j of nine cells in

2D. The normal vector ni;j is thus a function of Ci;j,

ni;j ¼ rCi;j. Initially, a cell-corner value of thenormal vector ni;j is computed. An example at

iþ 1=2, jþ 1=2 in 2D is as follows:

nx;iþ1=2;jþ1=2 ¼1

2hðCiþ1;j � Ci;j þ Ciþ1;jþ1 � Ci;jþ1Þ

ð8Þ

ny;iþ1=2;jþ1=2 ¼1

2hðCi;jþ1 � Ci;j þ Ciþ1;jþ1 � Ciþ1;jÞ

ð9Þ

The required cell-centred values are given by av-

eraging

ni;j ¼1

4ðniþ1=2;j�1=2 þ ni�1=2;j�1=2 þ niþ1=2;jþ1=2

þ ni�1=1;jþ1=2Þ ð10Þ

The most general equation for a straight line on aCartesian mesh with normal ni;j is

nxxþ nyy ¼ a ð11Þ

The normal vector ni;j is defined by the vector

gradient of Ci;j, which can be derived from differ-

ent finite-difference approximations which directly

influence the accuracy of algorithms. These include

Green–Gauss, volume-average, least-squares,minimization principle, Youngs� gradients, as dis-cussed in [63]. It is noted that a wide, symmetric

stencil for ni;j is necessary for a reasonable esti-

mation of the interface orientation.

3.2.2. The fluid advection algorithm

During an advection step, the volume fraction

Ci;j is truncated by the formula

Ci;j ¼ min½1;maxðCfi;j; 0Þ� ð12Þ


at the (nþ 1) time step. Once the interface is re-

constructed, the velocity at the interface is

interpolated linearly and the new position of the

interface is calculated by the following formula:

xnþ1 ¼ xn þ uðDtÞ ð13Þ

The new Ci;j field is obtained according to the localvelocity field, and fluxes DC at each cell are

determined by algebraic or geometric approaches.

Here, the simplest operator split advection

(geometric) algorithm is used as proposed by

[21]

C_

i;j ¼ Cni;j þ

DtDx

½Fi�1=2;j � Fiþ1=2;j� ð14Þ

Cnþ1i;j ¼ C

_

i;j þDtDy

½G_

i;j�1=2 � G_

i;jþ1=2� ð15Þ

where Fi�1=2;j ¼ ðCuÞi�1=2;j denotes the horizontal

flux of the (i, j) cell, and Gi�1=2;j ¼ ðCvÞi;j�1=2 de-

notes the vertical flux of the (i, j) cell. That is,

volume fractions are updated at time level n fromCn

i;j to C_

i;j with an x sweep, then updated from C_

i;j

to Cnþ1i;j with a y sweep.

3.2.3. Surface force model

Surface tension along an interface arises as the

result of attractive forces between molecules in a

fluid. In a droplet surface, the net force is radially

inward, and the combined effect of the radialcomponents of forces across the entire spherical

surface is to make the surface contract, thereby

increasing the pressure on the concave side of the

surface. At equilibrium in this situation, the op-

posing pressure gradient and cohesive forces bal-

ance to form spherical drops. Surface tension acts

to balance the radially inward inter-molecular at-

tractive force with the radially outward pressuregradient across the surface.

Here, surface tension is applied using the CSF

scheme [49]. The addition of surface tension to the

VOF method is modelled by a source term in

the momentum equation. The pressure drop across

the surface depends upon the surface tension co-

efficient r

Dp ¼ r1

R1

�þ 1

R2

�ð16Þ

where R1 and R2 are the two radii, in orthogonal

directions, to measure the surface curvature. In the

CSF formulation, the surface curvature is com-

puted from local gradients in the surface normal atthe interface. The surface normal n is defined by

ni;j ¼ rCi;j ð17Þwhere Ci;j is the secondary phase volume fraction.

The curvature ji;j is defined in terms of the di-

vergence of the unit normal n̂n

j ¼ r � n̂n ¼ 1

jnjn

jnj � r� �

jnj�

� ðr � nÞ�

ð18Þ

where n̂n ¼ n

jnj ð19Þ

The surface tension can be written in terms of the

pressure jump across the interface, which is ex-

pressed as a volume force F added to the mo-

mentum equation

Fi;j ¼ r1;2ji;jqi;jrC

ðq1 þ q2Þ=2ð20Þ

where the volume-average density qi;j is given by

Eq. (6).

The CSF model allows for a more accurate

discrete representation of surface tension without

topological restrictions, and leads to surface ten-sion forces that induce a minimum in the free

surface energy configuration. This method has

been used by various researchers and is included in

most in-house, public and commercial codes such

as SURFER, RIPPLE, FLUENT, Star-CD,

Flow-3D, because of its simplicity of implemen-

tation. However, the solution quality of PLIC-

VOF and CSF is quite sensitive to n̂n ¼ rC=jrCj,so an accurate estimation of the normal vector

often dictates overall accuracy and performance.

CSF and CSF-based capillary force models are

in principle simple, robust and require only the

phase indicator C to be determined. In fact, both

are known to induce the so-called spurious cur-

rents near the interface, because once discretized,

the exact momentum jump condition at the inter-face is not always properly preserved, i.e. pressure

and viscous stress forces do not balance the cap-

illary forces. This is partly due to the lack of pre-

cision in solving the curvature, but it also results


from the way the surface term is discretized in the

momentum equation.

Fig. 1. Comparison of sharpness of static interface recon-

struction with different grid solutions, grid 32 · 128, 64 · 128,128· 512 from top.

Fig. 2. Illustration of sharpness of static interface reconstruc-

tion in 3D extrusion with grid 64 · 128.

4. Numerical experiments

Several numerical experiments are performed in

order to demonstrate the versatility of the VOF

method used in the present study. The study is

focused on establishing a fast process simulator for

analysing immiscible liquid alloys in rheomixing

process. Numerical experiments include static in-

terface reconstruction, moving interface topolo-gies, collapsing cylinder of water with comparison

to experimental results, and a 2D/3D viscous drop

deformation. The ability of representation of

complex topologies is scrutinized with different

grid sizes, numerical schemes and physical models.

The effectiveness of numerical methods based on

available general CFD codes is assessed for simu-

lating multiple-material flows.

4.1. Static interface reconstruction

The static interface test consists of a symbol

containing the fonts ‘‘test’’ followed by four

droplets of different sizes. They are reconstructed

in the xy-plane, and an outline of the symbol is

also extruded a small distance along the z-direc-tion. Within a 16 · 4 box, three grid sizes (32 · 128,64 · 256, 128 · 512) are tested though they are all

still coarse for the VOF method to reconstruct the

small droplets. The sharpness of the interface is

clearly identified in Fig. 1, where the coarsest grid

exhibits a ‘‘fuzzy’’ interface in the xy-plane and

clearly shows a sloping interface in the 3D extru-

sion graph of Fig. 2. The algorithms of the VOFmethod are fully dependent on mesh size and are

also influenced by the computation of the interface

normal ni;j.

4.2. Moving interface topologies

The moving interface cases based on the above

test symbol are set up to estimate the topology ofthe interface during time integrations. The test

symbol is initially assumed to be a fluid (water) in

air, which then falls into a shallow pool due to the

force of gravity. The computational domain is

16 · 4 with the same three grids as above, com-

puted by two interface reconstruction schemes:

PLIC and D–A. At time zero the test symbol is

allowed to fall, eventually splashing into the pool

within 0.24 s, as shown in Fig. 3. The test symbol is

not overly deformed and splashed due to its short

initial height that results in a relatively small free-fall velocity. The splashing characteristics can still

be tracked with a coarse mesh (Fig. 4).

4.3. Collapsing cylinder of water

To test the numerical procedures used in a more

realistic regime, we consider the problem of a

collapsing cylinder of water problem, for whichexperimental and numerical results are available in

[54,64]. In the experiment, a cylindrical column of

water of diameter 110 mm and height 200 mm was

released by suddenly lifting the tube which had

kept back the water. The water spreads radially on

the flat bottom to the sidewall of the pot, where it

sloshed upwards, falling back and collapsing to the

centre where a jet shot up. An axisymmetric

Fig. 3. Simulation results for test symbols falling into a pool at

time steps: t ¼ 0:0, 0.16, 0.18, and 0.20 s. Domain size 16 · 4,mesh size 64 · 256.

Fig. 4. Comparison of different grid solutions at time step

t ¼ 0:18 s. Grid 32 · 128 (top), 64· 256 (middle), 128· 512(bottom).


100 · 160 mesh, 220 · 355.2 domain is used. Twoschemes of pressure discretisation of the momen-

tum equation are employed: Body–Force–

Weighted (BFW) and PREssure STaggering

Option (PRESTO). The results are illustrated in

Fig. 5. Compared with experimental images, the

main features of the flow are shown to be well

simulated: collapse, radial spreading, sloshing on

side wall and secondary collapse. In comparisonwith the previous simulation, small-scale features

are blurred due mainly to the coarse grid, however

there is no unphysical thin central jet at t ¼ 0:38sas produced in [54,64], and the sloshing height is

closer to the experimental data. The parameters of

the main features, including characteristic times,

heights and run-out lengths were also well repro-

duced, as listed in Table 2.

4.4. Deformation of a 3D viscous drop

Further detailed investigations were performed

with a viscous drop deformation, in order to val-idate the performance of the interface evolution in

a 3D domain.

The deformation of a 3D viscous drop is shown

on the right side of Fig. 6. The simulation was

performed with a mesh size 96 · 32 · 32, compu-

tational domain size 3 · 1 · 1, time step

Dt ¼ 5:0e)4. Numerical results from [59] are

shown on the left side of Fig. 6. The spatial to-pologies of deformation are well reproduced. The

deformation of the viscous drop is in elliptical

form before t ¼ 10 s, and can be simply measured

by the Taylor deformation parameter. However,

the shape of the viscous drop changes to non-

elliptical after t ¼ 10 s, and it becomes difficult to

describe it with the Taylor deformation parameter.

The shape factor Kk for analysing the morphologyof drop can be defined as 36 times pi times ratio of

the drop area squared to the drop perimeter cubed:

Kk ¼ 36pðSdÞ2=ðPdÞ3. A perfect spherical drop has

a shape factor of 1 and a line has a shape factor

approaching 0.

5. Application of PLIC-VOF for immiscible liquidalloy flow

A novel twin-screw extruder (TSE) rheomixing

process has been successfully developed in our

laboratory for casting immiscible alloys [65]. The

solidified microstructure of cast immiscible alloys

strongly depends on the rheological behaviour of

the liquids during cooling. Here, we present anumerical analysis of the fundamental rheological

behaviour of an immiscible metallic drop in a

shear-induced turbulent flow, which is the main

flow feature in the TSE rheomixing process.

Numerical approaches described above are em-

ployed in the investigation and coupled with

simplified flow field for the TSE process. It is

noted that the differences of density ratio in im-miscible binary metallic alloys systems are not as

large as for air/water system. The viscosity ratio

of the system also changes substantially during

the process.

Fig. 5. Comparison of collapsing cylinder of water: left column are experimental images, middle column are graphs of simulation by

[54], right column are graphs of present simulation.

Table 2

Comparison of characteristic parameters of collapsing cylinder of water

t1 (s) t2 (s) h2 (mm) t3 (s) h3 (mm) Ref.

Experiment 0.20± 0.02 0.42± 0.02 160± 10 0.88± 0.04 400± 50 [64]

PLIC 100· 160 0.22 0.38 117 >0.8 >355.2 [64]

CFX4 (50· 80) 0.216 0.396 128 0.883 150 [64]

PLIC 100· 160 0.189 0.42 184.56 >0.8 >355.2 Present

t1––Time of arrival at the sidewall of the pot.

t2––Time of maximum sloshing height at the wall.

h2––Maximum sloshing height.

t3––Time of maximum collapse height.

h3––Maximum collapse height, height of simulation domain is 355.2 mm.


5.1. Overview of immiscible liquid alloy flow in

rheomixing process

A rheomixing process was developed based on

previous experience in the processing of semisolid

metal (SSM) slurry by a twin-screw extruder (TSE)

[66]. The flow field of the intermeshing co-rotating

twin-screw extruder undergoes cyclic stretching,folding, and reorienting [67]. Basically, the main

feature of a twin-screw extruder is a strong shearflow field produced by co-rotating intermeshing

screws [57]. Droplets are created in a microscopic

scale, and turbulent flow is enhanced by mixing,

swirling and pumping actions in a macroscopic

scale. Model experiments of parallel disks were

performed in order to study the fundamental

mechanism of immiscible polymeric materials in a

twin-screw extruder [68]. However, numerical

Fig. 7. Illustration of the sequences of a metallic drop defor-

mation in shear-induced flow with enhanced initial non-linear

shear rate near walls in 3D (top and middle, top graphic is a

cross-section in the x–z plane through the centre of the drop),

and in 2D domain (bottom).

Fig. 6. Comparison of interface evolution with the numerical

results in [59] as viewed from the side of the computational

domain, left column figures are for domain 3 · 1 · 2, Ca (cap-

illary number)¼ 0.42, k ¼ 1, equal density; right column figures

are for case 8, domain 3· 1 · 1, Ca ¼ 0:21, k ¼ 1, equal density.


simulations have provided advantages since vari-

ous shear rate profiles can be established for set-

ting up initial and boundary conditions, and more

complex forces can be easily imposed to reflect the

special operating conditions and screw configura-tion. Here, the essential micro-mechanism of im-

miscible Pb–Zn liquid alloys in rheomixing process

is presented. The rupturing, interaction and dis-

persion of droplets, the essential microscopic

mechanisms of the twin-screw extruder, are in-

vestigated to improve further our understanding of

the rheomixing process. Shear rate is estimated by

the equation _cc ¼ 2npðrs=d� 1Þ, where rs is thescrew radius, n is the screw rotation speed and d is

the gap between barrel and screw surface [69].

5.2. Comparison between 2D and 3D liquid metal

drop deformation

A metallic drop deformation in non-linear

double sided shear-induced flow is shown in Fig. 7.The deformation will lead to the drop break-up.

The 3D simulation adopted a grid 128 · 32 · 32,domain 16 · 4 · 4, viscosity ratio k ¼ 1, capillary

number Ca ¼ 0:45 and enhanced initial non-linear

shear rate near the walls. Comparing the 2D and

3D simulations, the rheological behaviour of the

drop deformation is very similar except for

shearing time. Therefore, the investigation of im-

miscible liquid metal alloys in shear-induced flow

will be conducted in 2D.

5.3. 2D Simulation of liquid metal drops in shear-

induced mixing process

The flow field within a twin-screw extruder in

rheomixing process is extremely complex as anal-

ysed in [58]. The deformation of Pb metallic drops

in the rheomixing process is evaluated in a sim-plified 2D computational domain as depicted in

Fig. 7. The immiscible metal Pb drops break up

into small droplets in a shear-induced flow, with

small daughter drops forming in areas of high

local shear. The initial break-up factor Kr is de-

fined as the ratio of the capillary number of

daughter drop to parent drop, Kr ¼ Cad=Cap ¼ð _ccrddlm=rÞ=ð _ccrdlm=rÞ, in which rdd denotes thedaughter drop radius.

Several simplified initial conditions are defined

in order to reflect the special operating conditions

and screw configuration, including case 1: one-side

initial shear rate with non-linear peak profile; case

2: two-side initial shear rate with non-linear peak

profile; case 3: two-side initial shear rate with lin-

ear profile; case 4: k ¼ 1, pure shear rate flow; case

Fig. 8. Comparison of maximum size of daughter droplet

during shearing time for cases 1–4. After full breakup of parent

drop, daughter droplets are in mixing stage with further re-

finement and coalescence occurring simultaneously. (See online

paper for colour version of the figure.)

05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45Shearing time (ms)

Km

ax m

axim

um s

cale

fact

or o

f dro

plet

case 6case 7case 6 average Kmaxcase 7 average Kmax

1λ =0.5λ =

10 15 20 25 30 35 40

Fig. 10. Comparison of maximum size of daughter droplet

during shearing time in turbulent flow for case 6 and case 7.

Shearing time indicated in the chart starts from the time of full

breakup of parent drop. (See online paper for colour version of

the figure.)


5: k ¼ 0:5, pure shear rate flow; case 6: k ¼ 1, one-

side shear-induced turbulent dynamic flow with

initial non-linear peak profile; case 7: k ¼ 0:5, one-side shear-induced turbulent dynamic flow with

initial non-linear peak profile.The evolution sequence includes drop defor-

mation, break-up, rupturing, mixing and interac-

tion. Fig. 8 shows Kmax, the ratio of the largest size

of daughter droplet to the diameter of the parent

drop, for cases 1–4. Fig. 9 summarises the shearing

time, from the time that the first daughter drop is

formed until full break-up. Fig. 10 illustrates a

comparison of maximum size of daughter droplets

10 12 14 16 18 20 22 24

shearing time (ms)

1

2

3

4

5

6

7

Cas

es

0 2 4 6 8

Fig. 9. Shearing time from first daughter drop formation to full

breakup for cases 1–7.

during shearing time in turbulent flow for case 6

and case 7. Shearing time indicated in the chartstarts from the time of full break-up of parent

drop. Fig. 11 illustrates the results of mixing and

interaction stages in comparison with experimental

results [71].

Fig. 11. Comparison of the morphology of droplet dispersion,

experimental image [71] (top left) and numerical simulation (top

right) using similarity principle approach. Comparison of the

droplet distribution at size range above 32 lm of diameter

(bottom chart).


5.4. Summary of rheological behaviour of metal

drop in 2D shear-induced flow

The rheological behaviour of a Pb metallic dropdeformation and break-up in various shear flow

fields was examined. It is noted that Pb metallic

drops can be broken up in thin viscous flow more

easily and get a more spherical shape in thick

viscous turbulent flow. The results show that in the

immiscible Pb–Zn binary alloy system, the Pb drop

will easily break up under equal viscosity or high

shear rate conditions. Turbulence will speed up thebreak-up process and will lead to the formation of

spherical droplets. Turbulence also leads to more

coalescence than in laminar flow, as shown by the

amplitude of oscillations in Fig. 10. But the am-

plitude of oscillation in a lower viscosity ratio

system is smaller than in an equal viscosity system,

meaning that viscosity helps resisting coalescence.

Increasing the viscosity of the matrix phase willdelay first daughter droplet forming and extending

the shearing time of full break-up. The rupturing,

interaction and dispersion of droplets are strongly

influenced by the shearing forces, viscosity ratio,

turbulence, and shearing time. Possible sugges-

tions for the rheomixing process maybe given as

follows: start shearing immiscible metallic binary

alloys under enhanced turbulence and temperatureabove Tm. Both phases would have the same vis-

cosity value, which might cause fast break-up and

fine droplets, shortening the shearing time of full

break-up, which means saving power consump-

tion. If the shearing remains at temperature Tm,this will result in spherical droplets as the viscosity

of the matrix phase is increased, droplets will also

be dispersed stably in a thick matrix phase. Thestudies reveal a wealth of interesting rheological

and microstructural features that provide qualita-

tive insights into rheomixing, which are consistent

with previous experimental work.

6. Conclusions

This paper shows that numerical methods are

capable of simulating the rheological behaviour of

an immiscible Zn–Pb binary alloy in shear-induced

mixing processes. The rheological behaviour of

immiscible metallic alloy flows in shear-induced

mixing process is demonstrated. Qualitative

agreements are achieved in comparison with ex-

perimental results. The simulation model can be

used to obtain an insight into shearing time, vis-cosity and shear force, thus providing a guide to

the operating condition of rheomixing process in

order to reduce trial and error experiments for

optimising parameters.

The effectiveness of a general CFD code is

studied for simulating multiple-material flows and

immiscible binary alloy flows in shear-induced

mixing process. Numerical procedures based onthe general CFD code are developed to account

for non-linear shear force profiles of the rheo-

mixing process, viscosity variation function, etc.

The main aim of numerical modelling coupled

with experimental database of rheomixing pro-

cessing is to establish a numerical procedure for

developing a fast process simulator for the analysis

of simultaneous mixing, filling and solidificationphenomena, needed for further improving current

prototypical rheomixing design. Numerical meth-

ods used in modelling are reviewed with various

interface test cases; the experimental database is

incorporated into a modelling toolkit, which may

be able to explore some interesting rheological

behaviour of immiscible liquid alloy in rheomixing

process with further development and investiga-tion. However, there still remain challenging

problems for establishing a robust and fast process

simulator, such as computer efficiency, develop-

ment of algorithm for time integration of interface

propagation, dynamic adaptive mesh technique,

transient database model for thermodynamic pa-

rameters and properties of immiscible alloy system.

Acknowledgements

We acknowledge financial support from EP-

SRC grant GN/N14033, Ford Motor Co., PRISM

(Lichfield, UK) and the Mechanical Engineering

Department at Brunel University. We are also

grateful to researchers in CFD group and BCAST(Brunel Centre for Advanced Solidification Tech-

nology) for helpful discussions on numerical ap-

proaches and the TSE rheomixing casting process.


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