Tracking of immiscible interfaces in multiple-material mixing processes Hao Tang a,b, * , L.C. Wrobel a , Z. Fan a,b a Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK b 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. Ó 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 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 Computational Materials Science 29 (2004) 103–118 www.elsevier.com/locate/commatsci * 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 Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2003.07.002
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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
The volume fraction function Ck is governed bythe volume fraction equation
oCk
otþ u � rCk ¼ 0 ð4Þ
where u is the flow velocity.
108 H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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
H. Tang et al. / Computational Materials Science 29 (2004) 103–118 109
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
110 H. Tang et al. / Computational Materials Science 29 (2004) 103–118
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).
H. Tang et al. / Computational Materials Science 29 (2004) 103–118 111
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