Computational Materials Science 29 (2004) 103–118
www.elsevier.com/locate/commatsci
Tracking of immiscible interfaces in multiple-material
mixing processes
Hao Tang
b
a,b,*
, L.C. Wrobel a, Z. Fan
a,b
a
Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK
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 deposition, and fluid jetting devices are just a few of the
*
Corresponding author. Address: Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8
3PH, UK. Tel.: +44-1895-274000; fax: +44-1895-256392.
E-mail address: [email protected] (H. Tang).
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 alloys
strongly 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
0927-0256/$ - see front matter 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2003.07.002
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 material interfaces of arbitrarily complex topology.
Several techniques exist for tracking immiscible
interfaces, each with its own strengths and weaknesses [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 include front-tracking [5], volume-of-fluid (VOF)
[6,7], marker and cell (MAC) method [8],
smoothed particle hydrodynamics (SPH) [9], level
set 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 step
function representing the space occupied by one of
the fluids in the VOF method, and a smooth arbitrary function encompassing a prespecified isosurface 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 its
development 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 interface, much the same as the fractional volume
function C in the VOF method, to indicate the
shortest distance to the interface. This approach
has two inherent strengths. One is that the representation of the interface as the level set of a
function u leads to convenient formulas for the
interface normal and curvature. Another advantage (similar to the VOF method) is that no special
procedures are required in order to model topological changes of the front. However, due to flow
distortions, the deviations from the true distance
to the interface increase, and hence needs to be reinitialised every couple of time-steps by the solution 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 this
paper is to incorporate a range of physical effects
to the VOF method with the application of various
existing numerical schemes to establish a numerical modelling toolkit for immiscible liquid alloy
flow in the TSE rheomixing process, for predicting
the rheological behaviour of immiscible liquid alloys in shear-induced mixing flow. To achieve this,
we have reviewed the VOF method used in our
study 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 of
VOF methods. Section 3 describes the mathematical methods and physical model of the
piecewise linear interface construction (PLIC)VOF method by summarizing the numerical algorithms 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 deformation. The simulation cases are conducted at the
meso- and micro-scale. The accuracy and efficiency 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 presented 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
2.1. Development of VOF algorithms
Pioneering work on VOF methods goes back to
the early 1970s. The first three volume tracking
methods were introduced by DeBarÕs method
(KRAKEN code, 1974) [15], Hirt and NicholsÕ
VOF (volume-of-fluid, 1981) [6,7], Noh and
WoodwardÕs 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 NicholsÕ 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 YoungsÕ (PLIC, 1982) [21] and
their hydrocode [22], and were subsequently
adopted in many high-speed hydrodynamics codes
involving material interfaces [23], such as Addessio
and coworkersÕ CAVEAT code (1984), Holian and
coworkersÕ MESA code (1991), Kothe and coworkersÕ PAGOSA code (1992), Perry and coworkersÕ Rhal code (1993). Many extensions and
enhancements to the work of Youngs have occurred 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 propagation algorithms and robust methods of polyhedral
truncation to 3D interface reconstruction.
The essential concepts of VOF methods are
described as follows: at the beginning, an initial
fluid volume is used to compute fluid volume
fractions in each computational cell from a specified interface topology. This requires the calculation of volumes truncated by the fluid interface in
each interface cell. Exact interface information is
then lost and instead discrete volume data is produced until an interface is reconstructed. The fluid
solver then generates a velocity field, and interfaces 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 inferred, based on local volume data and assumptions 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 steps
of VOF methods are reconstructed interface geometry 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 enhancements have been developed subsequently to
these by a number of researchers.
These contributions are focused on the notable
improvement of algorithms for interface
Table 1
Development of VOF algorithms
Reconstruction interface geometry
Piecewise
Piecewise
Piecewise
Piecewise
sional
Piecewise
Piecewise
Piecewise
linear, operator split
constant, operator split
constant, multi-dimensional
constant, stair-stepped, multi-dimenlinear, multi-dimensional
linear, operator split
linear, multi-dimensional
PLIC
SLIC
FCT
D–A
PLIC
FLAIR
LVIRA
PLIC
SS
Time integration
Author(s) and references
Date
One
One
One
One
DeBar [15]
Noh and Woodward [16]
Zaleski [24]
Chorin [25]
Hirt and Nichols [7]
Youngs [21]
Ashgriz and Poo [26]
Puckett et al. [27]
Rider and Kothe [23]
Harvie and Fletcher [28]
1974
1976
1979
1980
1981
1982
1991
1997
1998
2000
dimensional
dimensional
dimensional
dimensional
One dimensional
One dimensional
Multi-dimensional
PLIC––piecewise linear interface construction; SLIC––simplified linear interface construction; D–A––donor–acceptor; FCT––fluxcorrected transport; FLAIR––flux line segment model for advection and interface reconstruction; SS––stream scheme; LVIRA––least
squares volume-of-fluid interface reconstruction algorithm.
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H. Tang et al. / Computational Materials Science 29 (2004) 103–118
reconstruction or time integration to achieve either
more accuracy or more efficiency. YoungsÕ formula is adopted in many codes involving material
interfaces, as mentioned in Section 2.1.
The basic feature of piecewise constant, SLIC
and 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-dimensional operators which are set on a 3 · 3 stencil in
2D to reconstruct the stair-stepped interface
within 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 interfaces. These are severely impacted on the overall
interfacial 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 a
multi-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 extensively by introducing second-order time integration schemes or robust methods for truncation of
arbitrary polyhedra [23]. Obviously, multi-dimensional schemes can be more accurate and efficient
in calculating cell boundary fluxes compared to
operator split schemes, and are currently developed 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
YoungsÕ PLIC schemes have been reported by [29].
Results have shown that YoungsÕ 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 YoungsÕ PLIC
possibly provides more accuracy at potentially
greater computational expense [28]. Comparisons
of 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 interface.
Following volume tracking methods, and various enhancements to interface reconstruction and
interface advection algorithms (named VOF-like
methods [32,33]), many methods are being currently developed for multi-material flows coupled
with other multi-phase methods, such as VOFDPM [34,35], VOF-two phase flow [36], VOFphase change (vapour or solidification) [37–39],
VOF-level set [40]. These algorithms are necessary
for numerical simulations of more complex phenomena.
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 given by [4,41] and more recent ones by [42,43].
Some general reviews of moving boundary methods are also discussed in [3,44]. Current reviews of
different algorithms of the VOF method are presented by [23,29,31,45], where detailed comparisons and error estimation are presented. A recent
review of numerical errors of the LVIRA-VOF
algorithm is given by [46], where an analysis of
effects 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 limitation of the grid cell––the VOF model cannot simulate the portions of fluid which are smaller than
the 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
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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 immiscible 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 multiphase 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, viscous, viscoelastic flows. A few special test cases
have benchmarks for the validation of interfacial
topology and propagation, and verification of accuracy and efficiency. They include static interface
reconstruction [21], ZaleskiÕs slotted solid disk
rotation [24,28,30], Rider–Kothe single vortex and
time reversed flows [23,28,30,31,46], RudmanÕs
hollow 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 of
macro-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 interaction [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 (RIPPLE-3D version) and FLOW3D). SURFER
(originally by Zaleski) and RIPPLE (originally by
Kothe) are used by many researchers since these are
free or public open source codes and further enhancements have been made [59]. Some examples
107
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 NicholsÕ VOF
method [7] coupled with YoungsÕ PLIC scheme
[21], BrackbillsÕ 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 interface
between multi-immiscible liquids of different density and viscosity is defined by a phase indicator––
the volume fraction function C, and the interface is
tracked by the following three conditions:
Ck ðx; y; z; tÞ
8
>
< 0 ðoutside kth fluidÞ
¼ 1 ðinside kth fluidÞ
>
:
> 0; < 1 ðat the kth fluid interfaceÞ
ð1Þ
ð2Þ
ð3Þ
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 by
the volume fraction equation
oCk
þ u rCk ¼ 0
ot
where u is the flow velocity.
ð4Þ
108
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
The two-phase fluid flows are modelled with the
Navier–Stokes equation
ou
q
ð5Þ
þ u ru ¼ rp þ lr2 u þ qg þ F
ot
where F stands for body forces, g for gravity acceleration, and p for pressure. The velocity field is
subject to the incompressibility constraint,
r u ¼ 0.
In a two-phase system, the properties appearing 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 C2 ðq2 q1 Þ
ð6Þ
li;j ¼ q1 þ C2 ðl2 l1 Þ
ð7Þ
In multi-phase systems, the ‘‘onion skin’’ technique is used [21].
3.2.1. The interface reconstruction algorithm
In the PLIC method, the interface is approximated by a straight line of appropriate inclination
in each cell. A typical reconstruction of the interface with a straight line in cell (i; j), which yields an
unambiguous solution, is perpendicular to an interface normal vector ni;j and delimits a fluid volume matching the given Ci;j for the cell. A unit
vector n is determined from the immediate neighbouring 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 the
normal vector ni;j is computed. An example at
i þ 1=2, j þ 1=2 in 2D is as follows:
nx;iþ1=2;jþ1=2 ¼
1
ðCiþ1;j Ci;j þ Ciþ1;jþ1 Ci;jþ1 Þ
2h
ð8Þ
ny;iþ1=2;jþ1=2 ¼
1
ðCi;jþ1 Ci;j þ Ciþ1;jþ1 Ciþ1;j Þ
2h
ð9Þ
3.2. The interface tracking algorithm
The formulation of the VOF model requires
that the convection and diffusion fluxes through
the control volume faces be computed and balanced 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 interface is represented by a volume fraction field.
The geometric reconstruction PLIC scheme is
employed because of its accuracy and applicability for complex flows, compared to other
methods such as the donor–acceptor, Euler explicit, 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 determination of the orientation of the segment. This
is equivalent to the determination of the unit
normal 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 algorithm.
The required cell-centred values are given by averaging
1
ni;j ¼ ðniþ1=2;j1=2 þ ni1=2;j1=2 þ niþ1=2;jþ1=2
4
þ ni1=1;jþ1=2 Þ
ð10Þ
The most general equation for a straight line on a
Cartesian mesh with normal ni;j is
nx x þ ny y ¼ a
ð11Þ
The normal vector ni;j is defined by the vector
gradient of Ci;j , which can be derived from different finite-difference approximations which directly
influence the accuracy of algorithms. These include
Green–Gauss, volume-average, least-squares,
minimization principle, YoungsÕ gradients, as discussed in [63]. It is noted that a wide, symmetric
stencil for ni;j is necessary for a reasonable estimation 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
f
Ci;j ¼ min½1; maxðCi;j
; 0Þ
ð12Þ
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
at the (n þ 1) time step. Once the interface is reconstructed, 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 local
velocity 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]
_
n
C i;j ¼ Ci;j
þ
_
Dt
½Fi1=2;j Fiþ1=2;j Dx
nþ1
Ci;j
¼ C i;j þ
_
Dt _
½Gi;j1=2 Gi;jþ1=2 Dy
ð14Þ
ð15Þ
where Fi1=2;j ¼ ðCuÞi1=2;j denotes the horizontal
flux of the (i, j) cell, and Gi1=2;j ¼ ðCvÞi;j1=2 denotes the vertical flux of the (i, j) cell. That is,
volume_fractions are updated at time level n from
_
n
Ci;j
to C i;j with an x sweep, then updated from C i;j
nþ1
to Ci;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 radial
components 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 opposing pressure gradient and cohesive forces balance to form spherical drops. Surface tension acts
to balance the radially inward inter-molecular attractive force with the radially outward pressure
gradient 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 coefficient r
1
1
Dp ¼ r
þ
ð16Þ
R1 R2
109
where R1 and R2 are the two radii, in orthogonal
directions, to measure the surface curvature. In the
CSF formulation, the surface curvature is computed from local gradients in the surface normal at
the interface. The surface normal n is defined by
ni;j ¼ rCi;j
ð17Þ
where Ci;j is the secondary phase volume fraction.
The curvature ji;j is defined in terms of the divergence of the unit normal ^n
1
n
r jnj ðr nÞ
ð18Þ
j ¼ r ^n ¼
jnj
jnj
where
^n ¼
n
jnj
ð19Þ
The surface tension can be written in terms of the
pressure jump across the interface, which is expressed as a volume force F added to the momentum equation
Fi;j ¼ r1;2 ji;j
qi;j rC
ðq1 þ q2 Þ=2
ð20Þ
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 tension 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 implementation. However, the solution quality of PLICVOF and CSF is quite sensitive to ^n 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 currents near the interface, because once discretized,
the exact momentum jump condition at the interface is not always properly preserved, i.e. pressure
and viscous stress forces do not balance the capillary forces. This is partly due to the lack of precision in solving the curvature, but it also results
110
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
from the way the surface term is discretized in the
momentum equation.
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 interface reconstruction, moving interface topologies, 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 simulating 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-direction. 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 extrusion graph of Fig. 2. The algorithms of the VOF
method 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 of
the 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
Fig. 1. Comparison of sharpness of static interface reconstruction with different grid solutions, grid 32 · 128, 64 · 128,
128 · 512 from top.
Fig. 2. Illustration of sharpness of static interface reconstruction in 3D extrusion with grid 64 · 128.
force of gravity. The computational domain is
16 · 4 with the same three grids as above, computed 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 freefall 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 which
experimental 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
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
111
4.4. Deformation of a 3D viscous drop
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.
Further detailed investigations were performed
with a viscous drop deformation, in order to validate 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, computational 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 topologies 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 nonelliptical after t ¼ 10 s, and it becomes difficult to
describe it with the Taylor deformation parameter.
The shape factor Kk for analysing the morphology
of drop can be defined as 36 times pi times ratio of
the drop area squared to the drop perimeter cubed:
Kk ¼ 36pðSd Þ2 =ðPd Þ3 . A perfect spherical drop has
a shape factor of 1 and a line has a shape factor
approaching 0.
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).
5. Application of PLIC-VOF for immiscible liquid
alloy flow
100 · 160 mesh, 220 · 355.2 domain is used. Two
schemes of pressure discretisation of the momentum 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 comparison
with 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 0:38s
as 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 reproduced, as listed in Table 2.
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 a
numerical 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 employed in the investigation and coupled with
simplified flow field for the TSE process. It is
noted that the differences of density ratio in immiscible 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.
112
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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
Experiment
PLIC 100 · 160
CFX4 (50 · 80)
PLIC 100 · 160
t1 (s)
t2 (s)
h2 (mm)
t3 (s)
h3 (mm)
Ref.
0.20 ± 0.02
0.22
0.216
0.189
0.42 ± 0.02
0.38
0.396
0.42
160 ± 10
117
128
184.56
0.88 ± 0.04
>0.8
0.883
>0.8
400 ± 50
>355.2
150
>355.2
[64]
[64]
[64]
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 shear
flow 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
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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 (capillary 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 various shear rate profiles can be established for setting up initial and boundary conditions, and more
complex forces can be easily imposed to reflect the
special operating conditions and screw configuration. Here, the essential micro-mechanism of immiscible Pb–Zn liquid alloys in rheomixing process
is presented. The rupturing, interaction and dispersion of droplets, the essential microscopic
mechanisms of the twin-screw extruder, are investigated to improve further our understanding of
the rheomixing process. Shear rate is estimated by
the equation c_ ¼ 2npðrs =d 1Þ, where rs is the
screw 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-
113
Fig. 7. Illustration of the sequences of a metallic drop deformation 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).
miscible liquid metal alloys in shear-induced flow
will be conducted in 2D.
5.3. 2D Simulation of liquid metal drops in shearinduced mixing process
The flow field within a twin-screw extruder in
rheomixing process is extremely complex as analysed in [58]. The deformation of Pb metallic drops
in the rheomixing process is evaluated in a simplified 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 defined as the ratio of the capillary number of
daughter drop to parent drop, Kr ¼ Cad =Cap ¼
ðc_ rdd lm =rÞ=ðc_ rd lm =rÞ, in which rdd denotes the
daughter 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 linear profile; case 4: k ¼ 1, pure shear rate flow; case
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
K max maximum scale factor of droplet
114
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
case 6 λ = 1
case 7 λ = 0.5
0.2
case 6 average Kmax
0.1
case 7 average Kmax
0
5
10
15
30
25
20
Shearing time (ms)
35
40
45
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 refinement and coalescence occurring simultaneously. (See online
paper for colour version of the figure.)
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, oneside shear-induced turbulent dynamic flow with
initial non-linear peak profile; case 7: k ¼ 0:5, oneside shear-induced turbulent dynamic flow with
initial non-linear peak profile.
The evolution sequence includes drop deformation, break-up, rupturing, mixing and interaction. 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
during shearing time in turbulent flow for case 6
and case 7. Shearing time indicated in the chart
starts 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].
7
Cases
6
5
4
3
2
1
0
2
4
6
8
10
12
14
16
18
20
22
24
shearing time (ms)
Fig. 9. Shearing time from first daughter drop formation to full
breakup for cases 1–7.
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).
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
5.4. Summary of rheological behaviour of metal
drop in 2D shear-induced flow
The rheological behaviour of a Pb metallic drop
deformation 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 the
break-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 amplitude 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 will
delay 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 suggestions for the rheomixing process maybe given as
follows: start shearing immiscible metallic binary
alloys under enhanced turbulence and temperature
above Tm . Both phases would have the same viscosity value, which might cause fast break-up and
fine droplets, shortening the shearing time of full
break-up, which means saving power consumption. 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. The
studies reveal a wealth of interesting rheological
and microstructural features that provide qualitative 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
115
immiscible metallic alloy flows in shear-induced
mixing process is demonstrated. Qualitative
agreements are achieved in comparison with experimental results. The simulation model can be
used to obtain an insight into shearing time, viscosity 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 on
the general CFD code are developed to account
for non-linear shear force profiles of the rheomixing process, viscosity variation function, etc.
The main aim of numerical modelling coupled
with experimental database of rheomixing processing is to establish a numerical procedure for
developing a fast process simulator for the analysis
of simultaneous mixing, filling and solidification
phenomena, needed for further improving current
prototypical rheomixing design. Numerical methods 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 investigation. However, there still remain challenging
problems for establishing a robust and fast process
simulator, such as computer efficiency, development of algorithm for time integration of interface
propagation, dynamic adaptive mesh technique,
transient database model for thermodynamic parameters and properties of immiscible alloy system.
Acknowledgements
We acknowledge financial support from EPSRC 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 Technology) for helpful discussions on numerical approaches and the TSE rheomixing casting process.
116
H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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