INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids (2012)Published online in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3723
Two-dimensional non-Newtonian injection molding with a newcontrol volume FEM/volume of fluid method
Carlos Salinas1, Diego A. Vasco2 and Nelson O. Moraga3,*,†
1Departamento de Ingeniería Mecánica, Universidad del Bío-Bío, Av. Collao 1202, Concepción, Chile2Departamento de Ingeniería Mecánica, Universidad de Santiago de Chile, Av. Lib. Bdo. O’Higgins 3363,
Santiago, Chile3Departamento de Ingeniería Mecánica, Universidad de La Serena, Av. Benavente 980, La Serena, Chile
Several studies have been focused on the development of efficient methods and algorithms for theaccurate prediction of the position of an interface in biphasic (gas–liquid) fluid flow. Although theapplications may be numerous, the reported mainly deal with industrial processes such as polymerinjection molding [1–3] and metal casting [4–6].
The methods employed for interface tracking are classified according to the domain discretization.In the variable grid methods, known and Lagrangian methods, the interface coincides with thefront of the moving grid, which has to be redefined after each time step [7, 8]. Although theimplementation of these methods does not require an explicit equation that describes the movementof the interface, the high computational cost for remeshing and the numerical instabilities producedby complex interfaces are two major disadvantages. For the fixed grid methods or Eulerians [9, 10],the grid is unique and remains constant during the whole calculation process. The implementation offixed grid methods requires lower computational effort; nevertheless, the intrinsic characteristic ofthe interface being represented by a discontinuous function is missed. On the contrary, the interfaceis described by a continuous function in a region of the domain.
*Correspondence to: Nelson O. Moraga, Departamento de Ingeniería Mecánica, Universidad de La Serena,Av. Benavente 980, La Serena, Chile.
The volume of fluid (VOF) method proposed by Hirt and Nichols [11] is a fixed grid techniquebroadly used mainly because of the simplicity of its formulation. Broadly speaking, the VOFformulation is a conservation equation that describes the convective transport of the volume of fluidfraction or simply the volume fraction. The numerical solution of the VOF differential equationleads to the numerical dispersion of the interface, normally known as numerical smearing. Severalvariations to the VOF method have been proposed to avoid the numerical smearing of the interface[12–14]. The present work is mainly oriented to this purpose.
Several pioneering works have implemented alternatives to avoid the dispersion of the interface.Ramshaw and Trapp [15] proposed the donor–acceptor scheme for the study of biphasic flows. Thisapproach established that the mixture properties in a cell are calculated according to the transportproperties of each phase corresponding to the donor cells; meanwhile, the volume fractions stand forthe values on the acceptor cells. This scheme ensured that a cell contained initially by gaseous phasewill accept only this phase until gas will be evacuated from the donor cells. Van Leer [16] proposeda fully implicit scheme in terms of finite differences for the solution of the continuity equationfor incompressible fluids. The Van Leer scheme has been implemented and a suitable strategy foravoiding interface dispersion has been presented.
Estacio and Mangiavacchi [17] proposed an approach for the numerical solution of the VOFconvective equation that restricts the fluid flow only from the completely filled control volumes.The effect of numerical smearing of the interface is diminished by calculation of a variable timestep, which assures that the control volumes next to the interface will be completely filled.
Kim and Lee [18] implemented an explicit formulation of the VOF equation in terms of theactual fractional volume of fluid, which is a measure of the wetted area of the boundary of a controlvolume. In this method, the effective volume of a cell is calculated as a function of the orientationvector of the free surface and the value of the volume fraction in such a cell. On the other hand, thefluid flow through the boundary of a cell is calculated as a function of the fractional volume of fluidinstead of the volume fraction.
Cruchaga et al. [19] proposed that the discontinuous function, which represents the interfaceposition, could be approximated through a continuous derivable function. Nevertheless, in theirwork the authors emphasized that the discontinuous function, which represents the positionof the interface, should be recovered from the continuous function by means of minimumsquares projection. The calculation of the mixture properties, for instance, is made by using thediscontinuous function.
A modification of the VOF equation was proposed by Swaminathan and Voller [20] through theimplementation of a variable (G/ instead of the volume fraction (F / in the convective term. Thetransport properties for each phase are calculated according to the F andG values obtained in thedomain in such a way that the flow of the liquid phase takes place only from the completely filledcontrol volumes. A detailed description of the algorithm for the implementation of this approachhas been presented by Maliska and de Vasconcellos [21].
The control volume finite element method (CVFEM) settled by Baliga and Patankar [22], is usedfor the numerical integration of the partial differential equations. This method may be consideredas a hybrid method made from combining the FEM and finite-volume method (FVM). CVFEMpossess the variational analysis advantages of FEM and the conservativeness properties of FVM.Yu et al. [23] implemented CVFEM for the integration of the momentum equations in the fillingprocess of a two-dimensional domain; Estacio et al. [24] applied the method for the integration ofthe Hele–Shaw equation to calculate the pressure in a nonisothermal injection molding process.
In the present work a new model based on VOF is proposed for the simulation of the filling processof two-dimensional cavities. The dispersion of the interface is nullified through the implementationof a scheme performed in two main stages denominated accumulation and distribution. In the firststage the VOF equation with a noninterfacial flux condition is numerically solved, while in thesecond stage the accumulated volume of fluid is dispersed. The implementation of this method isconditioned to a nonsmear displacement of the interface. The CVFEM method is used as a numericalintegration method for the transport equations according to an Euler implicit formulation, whichincludes central difference and a potential interpolation scheme for the diffusive and convectiveterms, respectively. The implemented method is verified through reported numerical data and it is
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
applied to two-dimensional nonisothermal injection molding process of a non-Newtonian fluid. Theresults obtained are represented as transient interface positions, isotherms, isobars, and during theinjection molding.
2. MATHEMATICAL AND NUMERICAL MODELING
The mathematical model consists of a set of second-order nonlinear partial differential equations thatrepresent (1) mass conservation, (2) Navier–Stokes, (3) heat transfer, and (4) volume fraction. Thisset of equations settled for a two-dimensional two-phase space xi (i D 1, 2) describes the transientbehavior of the pressure, velocity field, and the distribution of the temperature (T / and the volumefraction (F /. An incompressible non-Newtonian fluid of viscosity (�) and constant density (�),thermal conductivity (k/, and heat capacity (cp/ is considered.
@vi
@xiD 0 (1)
@.�vi /
@tC vj
@.�vi /
@xj�
@
@xj
�� . P�/
@vi
@xj
�C@p
@xiC �gi D 0 (2)
@��cpT
�@t
C vi@.�cpT /
@xi�
@
@xj
�k@T
@xj
�D 0 (3)
@F
@tC vi
@F
@xiD 0. (4)
The non-Newtonian viscosity is a function of the deformation rate represented in terms of thesecond invariant of the deformation rate tensor .�/
P� D
r1
2.� W�/, (5)
in which the deformation rate tensor components are given by
�ij D�@vi=@xj
�C�@vj =@xi
�.
The mathematical model expressed by Equations (1)–(4) can be represented by a generalconvective–diffusive transport Equation (6), according to the properties given in Table I.
@�
@tCr
*
V � �r.�r�/� S D 0. (6)
The integration of Equation (6) in a two-dimensional space of volume V discretized by finitevolumes (FV) of volume V n and contour Sn with nD 1 : : : N using the Green’s Theorem givesZ
V n
@�
@tdV C
ZSn
���*
V ��r�
�� On
�ds �
ZV n
F dV D 0, (7)
Table I. Diffusion coefficients and sourceterms for the general transport equation.
where each FV of volume V n consists of the partial contributions Ve.e D 1, ..,E/ of the finiteelements (FE) (Figure 1), according to
V D
NXnD1
V l D
EXeD1
Ve (8a)
and the segments Sn are defined as
Sn D
EXeD1
V n \ V e .„ ƒ‚ …Sne
(8b)
Each segment Sne is comprised of two line segments. The FV is centered in the local vertex 1,SnkD Sn
kaC Sn
kc, with the vectors .na,nb/ normal to the line segments drawn from side midpoints
.a, c/ to the centroid .G/ of the FE, as shown in Figure 2.Integration of the convective and diffusive terms of Equation (7) requires a function that represents
the variation of the variable � in each FE. A linear function (9a) is adopted for the diffusive term torepresent the �-variation while an exponential function (9b, 10) is adopted for the convective term
�.x,y/D AxCBy CC (9a)
�.Z,Y /D AZCBY CC , (9b)
Figure 1. Discretized representation of a two-dimensional domain by CVFEM.
Figure 2. Local system of coordinates (Y ,Z) represented on a finite element.
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
where the coefficients A, B , and C are defined as a function of the nodal values of the dependentvariable � and the coordinates of the vertices of the FE (local nodes 1, 2, and 3). The interpolated�-value in Equation (9a) is evaluated as a function of the global coordinate values .x,y/.
For the linear interpolation of the Equation (9b), the coordinates of the vertices of each FE aredefined according to a local system of coordinates .Z,Y /, where the Z-axis is parallel to the vectorvelocity Vi with module U (Figure 2). Z is expressed as an exponential function of the globalcoordinate x and the Peclet number
Z D�
�U
�exp
�Pe.x � xmax/
xmax � xmin
�� 1
�. (10)
An implicit Euler scheme is added to the previous description for the spatial integration tointegrate the transient term of Equation (6). In this way a linear system of algebraic equations isgenerated and then solved through the successive overrelation (SOR) Gauss–Seidel iterative method.More details about the computational implementation can be found in the work of Salinas et al. [25].
3. PROPOSED VOLUME OF FLUID MODEL
In this section, the proposed VOF model is described. First, a variable .G/ is introduced in theconvective term of the VOF equation; this variable gathers the discontinuous nature of the functionthat describes the position of the interface. The implementedG-variable has mainly a restrictive roleof the fluid flow through the boundaries of the FVs adjacent to the interface. Unlike the method ofSwaminathan and Voller [20], the implementation of this new variable in the VOF equation does notrequire an iterative process for the determination of the interface position.
The calculation of the volume fraction and therefore the position of the interface are obtainedthrough the implementation of two main stages denominated accumulation and distribution. In thefirst stage, the VOF equation with a noninterfacial flux condition is solved. As a consequence, for aset of FV the condition that F > 1 will be obtained. In the second stage the accumulated volume offluid is dispersed between the adjacent FVs with F < 1. The intrinsic discontinuous nature of thefunction that describes the position of the interface is met by the distribution stage; therefore, thenumerical smearing of the interface is avoided. Both stages will be better described through theirimplementation in the following formulations.
3.1. One-dimensional formulation
An initially empty duct of length L and constant transversal section is considered. The duct is beingfilled with a fluid that enters at a constant velocity U in x D 0 (Figure 3). In this formulation, theVOF equation is modified with the parameter (G/ as
@F
@tCU
@G
@xD 0. (11)
The convective term of the Equation (11) is integrated according to CVFEM for the domain oflength L discretized by FV (V n with nD 1: : :N ) introducing the upwind interpolation. Meanwhile,
Figure 3. Scheme of the filling process for one-dimensional duct.
the transient term is integrated through an explicit Euler scheme. In this way, for an FV centered onP the following equation is obtained:
FP �FoP CCr.GP �GW /D 0I Cr D
U ��t
�x, (12)
where Cr is the nondimensional Courant number and the subscript W refers to the G-value of theFV centered on a node at the right of the FV centered on P .
The following three-step procedure allows the solution of the filling process for the describedone-dimensional duct from the initial time t D t0 to t D t0Cdt . Initially, the values of F and G areconsidered null in the entire domain but in the inlet position where F DG D1.
Step 1: Calculation of the present value of F according to
FP D FoP CCr
�GoW �G
oP
�, (13)
where the values of G are obtained through
GP D 0.0 if FP < 1.0
GP D 1.0 if FP > 1.0.(14)
This step is considered the accumulation stage because it will be found that there is an FVwith F >1.
Step 2: Calculation of the correction factor of the Courant number .�/ according to the correspond-ing F -value
� D
8̂<:̂0.0 if FP < 1.0
1.0� 1.0FP�F
oP
if FP > 1.0.
1.0 if FP D 1.0
(15)
This factor modifies Cr allowing for the distribution of the excess volume of fluid to theadjacent FV located at the right. The Courant number is now defined as
Cr D � Cro. (16)
The calculations (13) and (15) are performed for each FV of volume V n with nD 1, : : : ,N .Step 3: In the last step, the values of F are updated according to
FP D 1.0 if FP > 1.0. (17)
Through the implementation of the described algorithm for a duct of unit length andCr D 1.2, theresults for the volume fraction shown in Table II are calculated. These results are exactly the sameas those obtained by Swaminathan and Voller [20], by using an iterative algorithm. It is evident thatthe proposed method avoids the dispersion of the interface noticed when the classic VOF methodis implemented.
3.2. Two-dimensional formulation
Here, a two-dimensional discretized domain according to the formulation of the CVFEM methodis treated. In Figure 4 a part of an FV conformed by an FE, which represents the three possibleF -values, is shown. Let EJa, EJb , and EJc be the fluxes of F through the line segments drawn from themidpoints of the sides of the FE (a, b, and c/ to the centroid G. The condition that only the fluid
Figure 4. Scheme of the transport of fluid from a VF with F D 1 centered on the node (2) through theinterfaces according to the conditions of Equation (18).
flows from filled FV is accomplished by the implementation of the conditions in Equation (18).According to this condition, in Figure 4 only EJa and EJb would be non-null fluxes of F .
EJa ¤ 0.0 if G1 ¤G2
EJb ¤ 0.0 if G2 ¤G3
EJc ¤ 0.0 if G3 ¤G1
(18)
with G D 0 if F < 1 or G D 1 if F >1The discretized form of Equation (4) modified with the parameter G, considering an implicit
Euler scheme to calculate the transient term and the upwind scheme for the evaluation of the fluxesof the fraction of volume, is given by
FP �FoP
�tC
NXnD1
EXeD1
GPVnSne D 0.P D 1,N/. (19)
The boundary conditions for F corresponds to null values in � and F D1 at the inlet position ofthe fluid. The flux of F at the wall and the interface is nullified according to Equation (18).
The restrictions expressed by (18) present an implicit condition of null flux through the interfacewhere 0 < F < 1. Such a restriction produces an accumulation of the volume fraction in the FVsadjacent to the interface; therefore, this stage is called accumulation stage. The excess of F assignedto some FVs is distributed according to a diffusive criterion that governs the transport of F : F isdiffused to the neighbors FVs with F > 1, keeping in mind that only those FVs with G D 1
are allowed to transfer fluid to the empty FVs. This is described in more detail in the followingcalculation scheme.
The calculation scheme is described for the determination of the volume fraction over the wholedomain. The pressure, velocity field and heat transfer are evaluated by implementing the PressureImplicit Momentum Explicit algorithm [26]. Given the initial values for the velocity, pressure,
temperature, and volume fraction, the calculation of the interface position from an initial time t D t0
to t D t0C�t is performed following the next procedure:
(i) The solution of the linear system of equations for F , which is obtained after the evaluationof Equation (19) under the restrictions given by (18) to each of the nodes. The solution ofthis linear system of equations has implicit accumulation stage.
(ii) Determination of the volume of fluid in excess for the FVs with FP > 1.0 according to
V ep D .Fp � 1.0/Vp (20)
(iii) The available empty volume V di of each FV around the FVs with FP > 1.0 and the total
volume V tp to be filled are calculated as (pure-distribution of the volume fraction)
V di D
�1.0�F 0i
�Vi IV
tp D
nbXiD1
V di (21)
or according to the fluxes calculated at the interface (flux-based distribution of the volumefraction). For instance for F2 > 1.0 (see Figure 4), Ja
�J d1�
and Jb�J d3�
are calculated as
Ja D nxaVxG C nyaVyG IJb D nxbVxG C nybVyG (22)
while the total flux J d from the nodes with FP > 1.0 is calculated as the summation of thenodal fluxes.
(iv) The distribution of V ep could be carried out proportionately to the available empty volume
of each neighbor FV i (pure-distribution of the volume fraction approach)
Fi D F0i C V
ep
V di
Vi � Vtp
(23a)
or through the flux-based distribution approach
Fi D F0i C V
ep
J di
J d(23b)
(v) The steps (ii) to (iv) are repeated in such a way that the condition Fp 6 1 will be fulfilledfor all nodes.
(vi) A macroscopic correction of the mass conservation is made to avoid numerical errors thatcould arise because of the raised algorithm. This step is equivalent to the determination ofthe macroscopic divergence of F (DF / and its distribution between FVs adjacent to theinterface in a similar way to the performed in the step (iv)
DF D
ZS
F On � ds�ZV
.F �F 0/dV (24)
(vii) Updating of the G values according to
GP D 0 if FP < 0.0
GP D 1 if FP > 1.0.(25)
3.3. Fluid injection in a rectangular duct
The proposed algorithm for the calculation of the position of the interface is performed in the
injection of an isothermal Newtonian fluid at constant velocity�*u in D 1.0m=s
, to a duct of
constant transversal section (0.03m). The dimensions of the duct allow that the wall effects in
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
Figure 5. Transient (t D 0.05, 0.1, and 0.15 s) interface location for the problem of a fluid that enters atconstant velocity to a duct of rectangular constant region (G, continuous lines; F dashed lines).
the ´-direction could be assumed to be negligible, for this reason a two-dimensional modelis appropriate.
Frictionless conditions at the walls are considered and fully developed flow at the outlet.The physical properties of the inlet fluid
�l D 4.705Ns=m2I �l D 1350kg=m3
�and the air�
g D 1.254e � 5Ns=m2I �g D 1.205kg=m3�
are considered constant. In the region where F D 0the pressure is considered null, which is equivalent to a null traction boundary condition at theinterface. The physical properties at the interface are calculated using the mixture rule using theF -values.
According to the conditions of the problem, the macroscopic mass balance requires that for eachtime step�t the interface moves a length�x such that�t D�x, the same proof was carried out byUsmani et al. [27]. The transient position of the interface according to the F and G values is shownin Figure 5. It can be noticed how the dispersion of the interface is practically nullified; therefore,the interface is located in a narrow region where 0 < F < 1.
4. VERIFICATION: FILLING OF A STEP CAVITY
In the previous section the proposed scheme for tracking the position of an interface was verified fortwo physical situations. In the first case, the obtained results are exactly the same as those obtainedby Swaminathan and Voller [20], and in the second case, it is demonstrated that the scheme obeysthe mass conservation law.
To verify the proposed scheme with more complex problems, the filling process proposed byDhatt et al. [28] is replicated. In this process, a fluid subjected to gravity enters a step cavity with aconstant velocity (0.1 m/s). Initially the front of the filling fluid (fluid 1) is located at x D 0.02 mand the rest of the cavity is occupied by the displaced fluid (fluid 2). Both fluids are consideredNewtonian with constant physical properties. The physical properties of the fluids and details of thephysical domain are depicted in Figure 6(a).
Free slip condition at the walls of the cavity and fully developed flow at the outlet are consideredas boundary conditions for the velocities. Meanwhile, a null traction condition at the interface is theadopted boundary condition for the pressure.
Previously, a numerical consistency study with respect to the time step was performed observingthe convergence for the solution of the interface position for three different time steps (�t D 0.05;0.01I 0.001s/. Nonrelevant differences between results were found (Figure 7), accordingly a timestep of 0.01 s was chosen. The discretization of the domain was performed through the imple-mentation of an open code program EASYMESH, which generates unstructured two-dimensionalmeshes based on Delaunay triangulation [29]. A uniform mesh of 1566 nodes shown in Figure 6(b)is generated to verify the present numerical results.
This physical situation was chosen as well to evaluate the ETILT algorithm developed byCruchaga et al. [19]. In their work, the finite element method was implemented and the domainwas discretized by an uniform mesh composed of 700 isoparametric four-noded elements and theuse of the time step of 0.01 s was reported. The Navier–Stokes equations were solved through ageneralized streamline operator technique.
Figure 6. Physical domain for the filling process of a two-dimensional step cavity (a) and the mesh usedwith 1566 nodes and 2888 elements (b).
Figure 7. Time step study for the filling process of the two-dimensional step cavity. Interface position att D 0.8 s.
The predicted transient position of the interface is in good agreement with those reported in theliterature (Figure 8). In this study both approaches previously described for volume fraction distri-bution were implemented. Noticeable differences between results are not observed; in both casesthe predicted particular characteristics of the interface are preserved. Despite being a filling processinfluenced by gravity, the way in which the distribution stage is performed does not affect the shapeof the interface; therefore, the position of the interface is mainly dependent on the a priori way thatthe velocity field is calculated.
5. APPLICATION: NONISOTHERMAL INJECTION MOULDING OF ANON-NEWTONIAN FLUID
Low density polyethylene (LPDE) is injected with a constant velocity (0.1 m/s) through a duct of0.1 m in length (w D0.02 m) to a rectangular cavity (Figure 9(a)). The non-Newtonian behavior ofLPDE is modeled by the power-law model modified with an Arrhenius-type term to describe thevariation of viscosity with temperature (26)
� . P� ,T /D �0 P�n�1e�b.T�Tref/I P� > 0
�.T /D 0e�b.T�Tref/I P� D 0. (26)
The parameters of the power-law model and the thermophysical properties of the fluid are givenin Table III. The constant inlet temperature of the fluid is 200 °C and the cavity walls are kept at aconstant temperature (60 °C). The heat transfer is modeled by a pure diffusive differential equation,which is a good approximation for low Reynolds numbers .� 10�3/. At the interface a constantvalue equal to the inlet temperature is considered.
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
Figure 8. Transient front positions during the filling process of a two-dimensional step cavity with aNewtonian fluid. Results obtained for front positions in the present study (columns 1 and 2) and in [19]
(column 3).
Figure 9. Physical domain (a) for the filling process of a two-dimensional mold with LPDE and the meshused (b) with 1733 nodes and 3264 elements.
Table III. Power-law parameters of LPDE and thermophysical properties of the fluids during the injectionmolding of a two-dimensional cavity with LPDE.
Power-law parameters Thermophysical properties LPDE Air
The position of the interface at two different times for five meshes is shown in Figure 10. It canbe seen that the influence of the mesh size in the predicted shape of the interface is more importantduring the early times of the filling process when the fluid expansion takes place. In Table IV themean position of the interface along the x-axis with the respective standard deviations is shown.According to these results, the conservativeness in all the cases is assured; nevertheless, the shape ofthe interface is not the same. Meanwhile the results with coarser meshes show curvilinear shapes forthe interfaces, those obtained with finer meshes predict flatter interface shapes. All the used meshesare uniform except for the mesh of 1448 nodes, which has a higher concentration of nodes near thefluid expansion region. That is why the results of the interface position at 8.7 s are deviated withrespect to those obtained with the finer mesh. According to this mesh study, a mesh of 1733 nodes ischosen (Figure 9(b)); meanwhile, a constant time step is used during the calculations (�t D 0.005s/.
In Figure 11 the transient position of the interface and the isobars observed during the polymerinjection are shown. As can be seen in the left column of Figure 11, after passing the duct thepolymer expands filling up the available volume of the mold and avoiding the formation of voidspaces or bubbles. To keep the quality of the final product, this kind of filling behavior is desirablein the polymer injection molding process.
It can be noticed in the right column of Figure 11 how a uniform distribution of the pressure alongthe duct region of the mold is built up. Once the expansion of fluid takes place, a broad isobaricregion can be noticed. Regarding the transient behavior of temperature during the injection molding(Figure 12), only noticeable changes are observed near the mold walls, which could representa potential development of a phase change layer. Meanwhile, under the established conditionsunimportant variations of temperature are noticed in the bulk of the filled mold. This behavior istypical during the process because phase change must be avoided and fluidity held.
Figure 10. Interface position (represented by F D 0.5) at two different times (t D 4.8 and 8.7 s) and meshesduring the nonisothermal filling of the cavity represented in Figure 9.
Table IV. Mean position of the interface at two differenttimes during the injection molding of a two-dimensional
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
Figure 11. Transient (t D 0.5, 2, 4, 8, and 10 s) behavior of the interface location (left column) and pressure(right column) during mold filling of LPDE.
Figure 12. Unsteady temperature distribution for LPDE non-Newtonian fluid injection molding (t D 2, 4,8, and 10 s).
6. CONCLUSIONS
A new conservative technique without numerical smearing for tracking a mobile interface basedon the VOF method has been implemented. The convective term of the differential equation thatdescribes the position of the interface was modified by a parameter that restricts the fluid flowonly through the interfaces of control volumes that are completely full. The underlying algorithmconsists of three stages: (i) accumulation stage in which the volume of fluid that enters during agiven time is accumulated in the control alongside the interface; (ii) distribution stage where the
excess accumulated volume of fluid of the previous stage is distributed among the neighbor controlvolumes with FP < 1.0; and (iii) a correction stage during which the excess of volume of fluidoccasionally confined in a finite volume is redistributed in the interface and an overall correctionmass balance is performed.
The computational cost of the method is not considerably increased in comparison with thesolution of the classic VOF equation, because the parameter G defined by the fraction of volumecalculated is restricted to only two possible values (G D 1 or G D 0).
The described technique along with the Pressure Implicit Momentum Explicit algorithm hasbeen successfully verified for one-dimensional and two-dimensional classical injection cases, andthen implemented to describe the transient interface location, pressure, velocity, and temperaturedistributions of a power law non-Newtonian fluid injection process in a two-dimensional mold.
ACKNOWLEDGEMENTS
The authors acknowledge CONICYT-Chile for support received in the FONDECYT project 1111067.Diego A. Vasco acknowledges the financial support given by the Doctoral Fellowship of the AdvancedHuman Capital Program CONICYT-Chile.
REFERENCES
1. Khor CY, Ariff ZM, Ani FC, Mujeebu MA, Abdullah MK, Abdullah MZ, Joseph MA. Three-dimensionalnumerical and experimental investigations on polymer rheology in meso-scale injection molding. InternationalCommunications in Heat and Mass Transfer 2010; 37:131–139.
2. Maier RS, Rohaly TF, Advani SG, Fickie KD. A fast numerical method for isothermal resin transfer mold filling.International Journal for Numerical Methods in Engineering 1996; 39:1405–1417.
3. Hieber CA, Shen SF. A finite element/finite difference simulation of the injection molding filling process. Journal ofNon-Newtonian Fluid Mechanics 1980; 7:1–32.
4. Gao DM. A three-dimensional hybrid finite element-volume tracking model for mould filling in casting processes.International Journal for Numerical Methods in Fluids 1999; 29:877–895.
5. Lewis RW, Ravindran K. Finite element simulation of metal casting. International Journal for Numerical Methodsin Engineering 2000; 47:29–59.
6. Lewis RW, Usmani AS, Cross JT. Efficient mould filling simulation in castings by an explicit finite element method.International Journal for Numerical Methods in Fluids 1995; 20:493–506.
7. Cruchaga M, Celentano D, Breitkopf P, Villon P, Rassineux A. A front remeshing technique for a Lagrangian descrip-tion of moving interfaces in two-fluid flows. International Journal for Numerical Methods in Engineering 2006;66:2035–2063.
8. Hirt CW, Cook JL, Butler TD. A Lagrangian method for calculating the dynamics of an incompressible fluid withfree surface. Journal of Computational Physics 1970; 5:103–124.
9. Lakehai D, Meier M, Fulgosi M. Interface tracking towards the direct simulation of heat and mass transfer inmultiphase flows. International Journal of Heat and Fluid Flow 2002; 23:242–257.
10. Van Wachem BGM, Almstedt AE. Methods for multiphase computational fluid dynamics. Chemical EngineeringJournal 2003; 96:81–98.
11. Hirt CW, Nichols BD. Volume of Fluid (VOF) method for the dynamics of free boundaries. Journal of ComputationalPhysics 1981; 39:201–225.
12. Voller VR, Peng S. An algorithm for analysis of polymer filling of molds. Polymer Engineering and Science 1995;22:1758–1765.
13. Mashayek F, Ashgriz N. A hybrid finite-element-volume-of-fluid method for simulating free surface flows andinterfaces. International Journal for Numerical Methods in Fluids 1995; 20:1363–1380.
14. Rudman M. Volume-tracking methods for interfacial flow calculations. International Journal for Numerical Methodsin Fluids 1997; 24:671–691.
15. Ramshaw JD, Trapp JA. A numerical technique for low-speed homogeneous two-phase flow with sharp interface.Journal of Computational Physics 1976; 21:438–453.
16. Van Leer B. Towards the ultimate conservative difference scheme IV: A new approach to numerical convection.Journal of Computational Physics 1977; 23:276–299.
17. Estacio KC, Mangiavacchi N. Simplified model for mould filling using CVFEM and unstructured meshes.Communications in Numerical Methods in Engineering 2007; 23:345–361.
18. Kim MS, Lee WI. A new VOF-based numerical scheme for the simulation of fluid flow with free surface. Part I:New free surface-tracking algorithm and its verification. International Journal for Numerical Methods in Fluids2003; 42:765–790.
19. Cruchaga M, Celentano D, Tezduyar TE. Moving-interface computations with the edge-tracked interface locatortechnique (ETILT). International Journal for Numerical Methods in Fluids 2005; 47:451–469.
NON-NEWTONIAN INJECTION MOLDING WITH A NEW CVFEM/VOF METHOD
20. Swaminathan CR, Voller VR. A time-implicit filling algorithm. Applied Mathematical Modelling 1994; 18:101–108.21. Maliska CR, de Vasconcellos JFV. An unstructured finite volume procedure for simulating flows with moving fronts.
Computer Methods in Applied Mechanics and Engineering 2000; 182:401–420.22. Baliga BR, Patankar SV. A new finite element formulation for convection diffusion problems. Numerical Heat
Transfer Part B 1980; 3:393–409.23. Liyong Y, Lee LJ, Koelling KW. Flow and heat transfer simulation of injection molding with microstructures.
Polymer Engineering and Science 2004; 44:1866–1876.24. Estacio KC, Nonato LG, Mangiavacchi N, Carey GF. Combining CVFEM and meshless front tracking in Hele–Shaw
mold filling simulation. International Journal for Numerical Methods in Fluids 2008; 56:1217–1223.25. Salinas CH, Chavez CA, Gatica Y, Ananias RA. Simulation of wood drying stresses using CVFEM. Latin American
Applied Research 2011; 41:23–30.26. Maliska CR, Raithby GD. Calculating 3D fluid flows using orthogonal grids. In Proceedings of the Third
International Conference on Numerical Methods in Laminar and Turbulent Flows, Seattle, WA, 1986; 656–666.27. Usmani AS, Cross JT, Lewis RW. A finite element model for the simulation of mould filling in metal casting and the
associated heat transfer. International Journal for Numerical Methods in Engineering 1992; 35:787–806.28. Dhatt G, Gao DM, Cheick AB. A Finite element simulation of metal flow in moulds. International Journal for
Numerical Methods in Engineering 1990; 30:821–831.29. Cavendish JC, Field DA, Frey WH. An approach to automatic three-dimensional finite element mesh generation.
International Journal for Numerical Methods in Engineering 1985; 21:329–347.
