doi:10.1016/j.commatsci.2003.07

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: hao.tang@brunel.ac.uk (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

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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.

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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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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

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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

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

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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 ½Fi 1=2;j Fiþ1=2;j Dx

nþ1 Ci;j ¼ C i;j þ

_ Dt _ ½Gi;j 1=2 Gi;jþ1=2 Dy

ð14Þ 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 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

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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

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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

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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.

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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-

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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

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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

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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.

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