Accepted Manuscript
Numerical simulation of melt filling process in complex mold cavity with insets using IB-CLSVOF method Qiang Li PII: DOI: Reference:
S0045-7930(16)30099-8 10.1016/j.compfluid.2016.04.005 CAF 3144
To appear in:
Computers and Fluids
Received date: Revised date: Accepted date:
2 November 2015 24 March 2016 4 April 2016
Please cite this article as: Qiang Li , Numerical simulation of melt filling process in complex mold cavity with insets using IB-CLSVOF method, Computers and Fluids (2016), doi: 10.1016/j.compfluid.2016.04.005
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Highlights
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A new coupled level-set and volume-of-fluid (CLSVOF) method is proposed. A local mass-correction method is used to conserve the liquid mass. The shape-LS function is employed to describe the immersed boundary of mold cavity. The IB-CLSVOF method is applied to 3D melt filling process in mold with insets. The IB-CLSVOF method can depict some important physical phenomena successfully.
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Numerical simulation of melt filling process in complex mold cavity with insets using IB-CLSVOF method Qiang Li* School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
ABSTRACT
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A coupled level-set and volume-of-fluid (CLSVOF) method is proposed for tracking interface of incompressible two-phase flows, where the VOF advection is carried out only, and a zero level-set (LS) formula and LS reinitialization equation are employed to reestablish the signed distanced field near the interface, which can program easily. Meanwhile, a mass-correction scheme based on the interface curvature is used to conserve the liquid mass correctly. Firstly, the performance of the CLSVOF method is evaluated through two- and three-dimensional
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deformation field tests. Then, the CLSVOF method is adopted to simulate the rising bubble and droplet falling problems. Finally, a shape LS function is used to describe and treat the irregular surface (immersed boundary) of mold cavity. With the aid of the shape function, the coupled immersed boundary (IB) and CLSVOF (IB-CLSVOF) method is applied to the melt filling process in the complex mold cavity with insets. The numerical results show that the proposed IB-CLSVOF method has the capability to cope with complex interface variations accurately. And also,
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the numerical results demonstrate that IB-CLSVOF method is able to successfully depict the phenomena of race-tracking, air entrapment and weldline development, which are very helpful for determining processing
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conditions and designing mold cavities.
KEY WORDS: VOF; Level set; CLSVOF; Interfaces; Immersed boundary; Injection molding
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1. Introduction
Numerical simulations of interfacial flows are relevant to problems found in various industries
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such as chemical, oil and food industry. If the fluid flows involve both gas and liquid, the fluid is known as two-phase flows. A lot of numerical methods have been developed for simulating
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two-phase free-surface flows, in which the volume-of-fluid (VOF) method and the level-set (LS) method are the most common numerical strategies used to predict interface motion based on Eulerian description. The VOF method has been widely used for solving the two-phase free-surface flows, in which the interface is captured by the VOF function F representing the volume fraction occupied by the liquid phase in each computational cell [1]. The LS method represents the interface as an isosurface of a scalar function, and can handle topology changes easily, which has been applied in various areas in recent years [2]. Needless to say, each method has its own advantages and disadvantages. Herein, a brief review on the VOF and LS methods and their *
Corresponding author. Tel.: +86 391 3987791; fax:+86 391 3987001 E-mail addresses:
[email protected]. 2
ACCEPTED MANUSCRIPT advantages/disadvantages are presented. In the VOF method, a volume fraction function F, is defined as the liquid volume fraction in a cell, whose value lies between zero and one. An advantage of the VOF method is its good mass conservation [3]. However, the big disadvantage of VOF method is that it is difficult to calculate the interface normal and curvature due to the fact that the volume function F is a step function. The VOF function is advected to get new values according to the fluid velocity per time step. The VOF advection schemes can be generally classified as the split and unsplit schemes [4]. The split advection schemes contain the dimensional advection and reconstruction steps in each coordinate
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direction, which are straightforward to implement compared to unsplit schemes. Although the unsplit schemes can offer discrete conservation, the unsplit advection schemes are computationally intensive and complicated [5]. The LS function is defined as a distance function, whose major virtue is that it works in any number of space dimensions, handles topological merging and breaking naturally [6]. However, the LS method suffers from the flaw of liquid mass loss/gain,
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especially when the interfaces experience severe stretching or tearing.
To solve the above problems, many improved VOF and LS methods are proposed, such as mass-conserving VOF method [4], mass-conserving LS method [7], particle LS method [8] and the coupled LS and VOF (CLSVOF or VOSET) method [9-18]. The CLSVOF or VOSET method can combine the advantages and overcome the disadvantages of both VOF and LS methods, which is a good tradeoff between the computational complexity and performance. In the CLSVOF method, the
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VOF and LS functions are advected concurrently, and then the coupled step is implemented to obtain the signed distance field near the interface to obtained the signed distance field and conserve
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the mass of the reference fluid. The method of reestablishing the signed distance field used by Son [10, 11] and Wang et al. [15] is based on geometric algorithm, in which the shortest distances from the cell nodes to the interfaces are obtained by geometric means. Son and Tao [16], Ling et al.[17]
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propose a VOSET method, in which the VOF advection is only carried out, and then based on the interfaces the signed distance function is also obtained by using geometric operations. Following
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the geometric algorithm of Son [10,11] for constructing distance field, Li et al. [18] adopt the CLSVOF method with a global mass-correction formula to simulate the gas assisted injection molding process. But the geometric algorithm of creating distance field function near the interface
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is complex to implement especially in three-dimensional case. Meanwhile, the global mass-correction formula does not work perfectly. In many practical flows the complex boundaries should usually be dealt with. Here, this work
focuses on a key issue of the plastic injection molding, especially in the complex molds with insets. The simulation of the filling processes are of most importance to understand and design the mold cavities, many researches have been done to simulate the complex flows in injection molding with the finite element (FE) and finite volume (FV) methods [19-21]. To deal with complex boundaries, for FVM, unstructured grids or body-fitting grids are usually employed to conform to complex boundaries of the mold cavities [21]. But the mesh generation is time consuming. In recent years, 3
ACCEPTED MANUSCRIPT the immersed boundary method (IBM) has achieved great progress. One of its advantages is that the governing equations can be solved on Cartesian grid easily with a body force prescribed on boundaries [22]. In addition, there is another class of methods, usually referred to as “Cartesian grid methods”, which is originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids [23]. These methods have been extended to simulate unsteady viscous flows and thus have capabilities similar to those of IBM. Another branch of Cartesian grid method, cut-cell method, has succeeded in simulating two-phase flows with embedded solid boundaries, which has been successfully applied to the Euler equations in two and three dimensions,
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and to flows involving both moving bodies and moving material interfaces [24,25]. In this review, the term IBM is used to contain all such methods that simulate viscous flows with immersed (or embedded) boundaries on Cartesian grid system.
In this work, a new CLSVOF method is proposed, in which the LS advection is not implemented and the VOF function is advected only, then a relational expression between VOF and
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LS functions is proposed to get the zero contour. Based on the zero contour and LS reinitialization, a signed distance function is obtained using a fast algorithm near the interface [26]. Since the whole calculation process may cause numerical errors, the mass conservation can not be ensured exactly. Thus, following the method of Luo et al. [7], a mass-correction formula based on local interface curvature is adopted near the interface cells, which can ensure the mass conservation. The performances of the LS, VOF and CLSVOF methods are evaluated through two- and
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three-dimensional deformation field tests. Then the CLSVOF method is adopted to simulate two-dimensional rising bubble and three-dimensional droplet falling problems. Finally, with the aid
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of the shape LS function to represent the irregular surface of the mold cavity, the coupled IB and CLSVOF (IB-CLSVOF) method is applied to the melt filling process in the complex mold cavity.
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2. CLSVOF method
2.1. The VOF advection algorithm
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The VOF function F is advanced by the fluid velocity u using the following equations
F u F 0 t
(1)
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In order to take into account the divergence error present in the velocity field, instead of Eq.
(1), solve
F uF F u t
(2)
To solve Eq. (2), the operator-split method used by Son [11] is adopted
F * F n uF n u F* t x x
(3a)
F ** F * vF* v F* t y y
(3b)
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ACCEPTED MANUSCRIPT F n 1 F ** wF * w F* t z z
(3c)
Integration of Eqs. (3a)-(3c) over a computational cell (i, j, k) yields
Fi*, j , k dxi dy j dzk Fi ,nj , k dxi dy j dzk utF n dydz
i 1 2, j , k
utF n dydz
Fi*, j , k utdydz i 1 2, j , k utdydz i 1 2, j , k
i 1 2, j , k
(4a)
Fi*, *j , k dxi dy j dzk Fi*, j , k dxi dy j dzk vtF *dzdx i , j 1 2, k vtF *dzdx i , j 1 2, k
Fi ,nj,1k dxi dy j dzk Fi*, *j , k dxi dy j dzk wtF **dxdy
i , j , k 1 2
(4b)
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Fi*, j , k vtdzdx i , j 1 2, k utdydz i , j 1 2, k
wtF **dxdy
Fi*, j , k wtdxdy i , j , k 1 2 wtdxdy i , j , k 1 2
i , j , k 1 2
(4c)
where only Fi 1 2, j , k , Fi , j 1 2, k and Fi , j , k 1 2 are not defined. Son [11] adopted a kind of
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geometric calculation procedure to evaluate them. For example, Fi 1 2, j , k is defined as the liquid volume fraction in the region ui 1 2, j , k tyz which is assumed to be advected in the x direction during time step t
Fi 1 2, j , k
VF i 1 2, j , k
u tyz
(5)
i 1 2, j , k
VF i 1 2, j ,k
denotes the advected liquid volume as shown in Fig. 1. Similarly, the liquid
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where
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volumes advected in the y and z directions can be obtained in the same way. Son [11] had described the VOF advection algorithm very clearly, and the interface is reconstructed via a PLIC (Piecewise Linear Interface Calculation) scheme. Even the VOF method
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could ensure mass conservation, undershoot/overshoot values with small magnitude will still happen when solving the VOF advection function. Also, the truncation error will cause the liquid
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mass fluctuation. In order to conserve the liquid mass exactly, we adopt a mass-correction scheme in the present study, which is provided in [7]. The following is the summary for the mass-correction method.
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(i) Filter the undershoot/overshoot values,
1, if Fk 1 106 Fk 0, if Fk 10 6 F , else k
(6)
(ii) Compute the difference Gt for the liquid mass between the theoretical volume M t ,init and the computational volume M t at time t, Gt M t ,init M t , where M t ,init
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K
F k 1
0 k
UAt ,
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M t Fkt , K is the total number of grid. k 1
n
F k 1
0 k
is the liquid volume at initial time t=0, U is
the liquid velocity at the inlet boundary, A is the area of the inlet boundary, and t is the time duration up to n time step. (iii) Distribute Gt at interface cells (i.e., 0
(7a)
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dGt , p
p max Gt m p1 p max
In [18], we used a simple approach to distribute the lost volume equally to the interface, which is similar to the following,
Gt N
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dGt , p
(7b)
where p and m are the pth cell and the total cells of the interface within a narrow band around the interface, respectively. N is the number of the interface cells. p and
max are the local
interface curvature and the maximum value of interface curvature within a narrow band,
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respectively. p is calculated with the LS function, which is given in Section 2.2. (iv) For every interface cell, the accurate liquid volume fraction is recalculated by (8)
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Fpn Fpn dGt , p
Using Eq.(7a), the total liquid mass can be conserved. By the way, Luo et al. [7] use a series of formula to calculate the volume fraction from the LS
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function, and then employ a simple Newton iteration to realize the volume fraction remedy. However, in this study, the volume fraction remedy is quite easy since the volume fraction is given
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explicitly at each time step.
2.2. LS function
The LS function is a signed distance function which has many good properties that can
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compute the interface normal vector n accurately and smooth the discontinuous physical quantities. It is known that the signed distance function can be obtained by the traditional reinitialization algorithm [4], which is mainly related to the zero-contour of the LS function, and the traditional redistancing algorithm is
sign( 0 )(1 ) (x,0) 0 (x) where
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(9)
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sign0
(10)
min( x, y ) 2 2 0
The information is spread from the zero-contour to both sides [26], so the signed distance function can be obtained from the zero-contour by solving Eq. (9). The following is an example to show how the LS reinitialization function works. The initial LS function is
1 (r, ) (r 0.5
0.15r sin(6 ))3 , whose zero contour is shown in Fig. 2. In order to test the reinitialization
and
2 (r, ) 4(r 0.5 0.15r sin(6 ))3
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algorithm, the following two LS functions are chosen:
3 (r, ) 0.4(r 0.5 0.15r sin(6 ))3 , whose zero contours are the same as 1 (r , ) .
Although 1 (r , ) ,
2 (r , ) and 3 (r , ) are different at the initial time, the final LS functions,
shown in Fig. 3(d), are in the same shapes (contour taken over [-0.5:0.05:0.5]).
In fact, the zero contour of the LS function can be obtained from the VOF function F. In the
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interface reconstruction via the PLIC scheme, a parameter s is used which is the shortest distance from the local origin to the interface (the origin is included in the reference fluid, and see Fig.1 for example) [10, 11]. When the normal vector n is the unit normal vector, s is just the shortest distance from the local origin to the interface. There are only four and five possible interface configurations in two-dimensional and three-dimensional cases, respectively. Take an interface cell in two-dimensions as an example, the liquid volume given by Fdxdy can be expressed as
s dy1
2
(11)
a max( a,0) , dx1 nx dx and dy1 ny dy . Similarly, the air volume can be
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where
2
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2Fdx1dy1 s s dx1
written as
2
sm s dy1
2
(12)
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2(1 F )dx1dy1 (sm s) 2 sm s dx1
where sm dx1 dy1 . Defining Fc min( F ,1 F ) , sc min( s, sm s) , dxc max( dx1 ,
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dy1 ) and dyc min( dx1 , dy1 ) . Using the above definitions, Eqs.(11) and (12) can be expressed as in a unified form,
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2Fc dxc dyc sc2 sc dyc
2
(13)
From Eq.(13), s c and s can be obtained as
sc 2 Fc dxc dyc
if sc dyc (or Fc dyc 2dxc )
(14a)
sc Fc dxc 0.5dyc
if sc dyc (or Fc dyc 2dxc )
(14b)
s sc
if F 0.5
(15a)
s sm sc
if F 0.5
(15b)
And it is much more complicated that the relation between volume function F and s in three
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i , j si , j 0.5 nx x ny y i , j Similarly, in the three-dimensions the formula is,
(16a)
i , j ,k si , j ,k 0.5 nx x ny y nz z i , j ,k
(16b)
Thus, the zero contour of the LS function can be obtained with Eq.(16a) or Eq.(16b). And then
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by solving reinitialation function Eq.(9) within a narrow band, the signed distance function is achieved near the interface. Meanwhile, the reinitialation function is discretized by the fifth-order WENO scheme in space and third-order TVD Runge-Kutta scheme in time, respectively [20]. Herein the fast algorithm proposed by Peng et al. [26] for LS reinitialization is employed, which could save CPU time.
In the mass-correction formula of Eq.(7a), the interface curvature is calculated using LS
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function, i.e.,
2 2 2 x ( yy zz ) y ( xx zz ) z ( xx yy ) 2 x y xy 2 x z xz 2 y z yz 1.5 x2 y2 z2
(17)
2.3. Numerical implementation for CLSVOF method
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The Navier-Stokes equation is solved and velocity field u is obtained to move interfaces from
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F n to F n1 . The LS function is obtained by using Eq.(16a) (or Eq.(16b)) and the reinitialization algorithm based on the PLIC reconstruction. And then the normal n and curvature of the interface can be calculated using n and n from the LS function, which are
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employed during VOF advection and mass correction processes at each time step, respectively. For convenience, the flow chart of CLSVOF method is summarized as shown in Fig. 5, and the
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coupling of the LS and VOF functions is illustrated in the dashdotted box.
3. Numerical methods
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3.1. Governing equations and their discretizations To fix ideas we shall call one of the fluids a liquid and the other a gas. All the fluids (including
liquid and gas) are assumed to be incompressible, so the velocity field is divergence free, that is
( u) 0 t
(18)
When taking into account the interfacial tension forces and the fact that the density and the viscosity are variable over the whole region, the nondimensional momentum and energy equations are given by
u 1 1 uu p 2D H 2 f t Re We Fr 8
(19)
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(20)
where u u(x, t ) is the velocity field, p p(x, t ) is the pressure field, and are the density and viscosity of the fluid, respectively. D 1 2 (u u ) . is the LS function, T
H ( ) is the smoothed Heaviside function. Furthermore, Re lUL l , Fr U / gL , We l LU 2 , Pe l ClUL l and Br lU 2 lT0 denote the Reynolds number, the
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Froude number, the Weber number, Peclet number and Brinkman number, respectively. Here, L and U are the length and velocity scales, l and l are the liquid values of density and viscosity, where the subscript l represents the liquid.
is surface tension, and g denotes the gravitational
acceleration. f is the body force(virtual force), which is prescribed on the Cartesian mesh in IBM. is the solid volume fraction in a computational cell, whose value is 0 1 for the
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interface cells, zero in fluid regions, and one in solid regions, respectively.
In each individual phase the properties of the density and viscosity are constant, and across the interface the fluid properties are discontinuous. In order to calculate conveniently, the density and viscosity can be defined with the smoothed Heaviside function
(21)
a l a H
(22)
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a l a H
where the subscript a represent the air. The smoothed Heaviside function is
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0 1 1 H 1 sin 2 1
if if
(23)
if
where is a parameter related to the space step size, and 1.5x in this study.
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The governing equations are discretized by the finite volume SIMPLE (semi-implicit method for pressure linked equations) methods on a non-staggered mesh [18,27], which stores all the
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variables at the same physical location and employs only one set of control volumes, and nodes for the physical quantities lie in the center of the control volume. The finite volume SIMPLE methods are described in [27] in details.
3.2. Body Force Solvers For IBM, the body force f , containing f x , f y and f z , should be calculated. For simplicity, here take the calculation of f x as an example. In each computational cell the fluid velocity u is firstly calculated without body force f x , and is recomputed with (1 )u , then the body force f x 9
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can be obtained using Eqs. (24-25) [28]
u * u f xn t where u
*
(24)
is the fluid velocity recomputed by 1 u . So the following formula can be obtained,
0, u * u , f x 0, u * 0, u 1 u , f x , t
in f (25)
in s or on
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where f , s and represent the fluid region, solid region and fluid/solid interface, respectively. is the solid volume fraction which can be calculated using following formulas from the shape LS function
A 6 D D D
kf 0
1 kf , kf
(26a)
kf 0
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f , which is provided in Section 5 [29].
(26b)
where A max A ,0 max B ,0 max C ,0 max d ,0 max E ,0 3
3
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
3
3
1 2
1 2
1 2
1 2
1 2
1 2
3
A kf D D D , B kf D D D
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C kf D D D , D kf D D D
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E kf D D D , D max Dx , Dy , Dz
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D min Dx , Dy , Dz , D Dx Dy Dz D D
f f f , Dy y , Dz z y k z k x k
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kf f x k , Dx x
Similarly, f y and f z can be obtained with the above formulas.
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4. Numerical test 4.1. Two- and three-dimensional deformation To demonstrate the ability of the CLSVOF method to capture two-and three-dimensional
deformations, the CLSVOF method is compared with the pure LS and VOF methods. For LS method the LS advection and reinitialization equations are discretized by the fifth-order WENO scheme in space and third-order TVD Runge-Kutta scheme in time, respectively. In VOF method the VOF function is solved using the operator-split scheme, and then the interface is reconstructed based on the PLIC scheme [10, 11]. Except for the mass-correction steps in CLSVOF method, the main difference between CLSVOF and VOF methods is the calculation of the normal vector n
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u sinx sin2y cost T 2
v siny sin2x cost T 2
(27)
For three-dimensional case the following velocities are used
u 2 sin 2 x sin2y sin2z cost / T
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v sin2x sin 2 y sin2z cost / T
(28)
w s i n2x s i n2y s i n2 z c o st / T
where t is time, and T is the time at which the flow returns back to its initial shape. T 8 for two-dimensional case, and T 3 for three-dimensional case.
In two dimensions, the computations are performed on a 1282 mesh. The results obtained by
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using different methods at t 4 are shown in Fig. 6(a) where the maximal deformation is reached. It is obvious that the VOF and CLSVOF methods well maintain the thin strips with small droplets at the tails, whereas serious mass loss occurs at the tail of the filament for the LS method. Fig. 6(b) shows the results at t 8 when the flow field returns back to its initial shape. The recovered shape is approximately a disk by using the CLSVOF method with mass-correction based on local
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interface curvature. While for the LS method, the recovered shape is far from a disk with serous mass loss. The interface obtained by the VOF method is slightly rough compared with that obtained
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by the CLSVOF method. The comparisons are also given in Fig. 7 between the results by the mass conserving LS [7], unsplit VOF [5] and present CLSVOF methods. Although the mass conserving LS method is relatively easy to implement, it could cause relatively larger spatial position error (Fig.
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7(a)), while the present CLSVOF method is comparable to the unsplit VOF method (Fig. 7(b)). Fig.8 shows the interfaces on different meshes at t=8 without mass-correction, with
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mass-correction based on the local interface curvature and with mass-correction based on the equational volume. For a 642 mesh, the result of the CLSVOF method without mass-correction loses about 7.74% of the volume and the result with mass-correction can conserve the mass.
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Because of the low-resolution mesh, it is not very obvious that the superiority of the mass-correction based on the local curvature over that based on the equational volume. For a 1282 mesh, to better illustrate the interface location at the final time t=8, the upper feature of the interfaces are shown in Fig. 9. It is noted that the result with mass-correction based on the local curvature is closer to the exact location than that with the mass correction based on the equational volume. Table 1 provides the relative mass loss at t=8 for two dimensional deformation on different grids and with different methods. It is shown that the mass loss is greatly reduced with the mass correction. In the three-dimensions, a sphere of radius 0.15 is placed within a unit computational domain at
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ACCEPTED MANUSCRIPT (0.35,0.35,0.35), and a 1003 mesh is used. At t=1.5 the deformed shape reaches the maximum stretch and the velocity field is reversed. Fig. 10 shows the deformed shapes at t=0.75,1.5,1.5,2.25 and 3.0. Fig. 10(a) illustrates that the LS, VOF and CLSVOF methods give the almost the same shapes, and Figs. 10(b) and 10(c) show that LS method can cause a lot of mass loss than the other methods, which can not resolve the thin interface film produced at the middle section of the stretched shape. A better recovered shape is obtained by the present CLSVOF method than the LS method as shown in Fig. 10(d). And it is also observed that the interfaces obtained by the VOF method are not smooth, and have some small scars. The relative mass losses for different methods
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are listed in Table 2. From Fig.10 and Table 2, it can be easily found that, among these three different approaches, the present CLSVOF method with mass-correction based on the interface curvature can provide the most satisfactory interfaces.
According to above benchmark problems, the LS method can not conserve the liquid mass, as indicated by the fact that the relative mass ratio deviates largely from the exact value of 1 in both
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two- and three-dimensional cases. In the CLSVOF method with mass-correction based on the interface curvature the mass ratio is nearly equal to the exact value of 1. Hereafter, the present CLSVOF method will be employed to simulate the incompressible two-phase flows.
4.2. Rising bubble problem
Rising bubble problems are classical examples for testing the accuracy and the efficiency of the numerical schemes. Here, a rising bubble is simulated in a quiescent liquid. The physical
l g 1000 .The predicted rising bubble interfaces, obtained on a 1002 mesh with
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density ratio
Re 100 and We 200 , are illustrated in Fig. 11. Good qualitative agreement is found with the
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results obtained by using conservative LS method in [30].
4.3. Droplet falling problem
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Droplet falling problem is a good example to illustrate the robustness of the free surface flow algorithms since it involves complicated flow deformations such as coalescence and jet formation.
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In this case, a water drop falls into a quiescent water pool resulting in a splash. The whole process of the droplet free falling and the impact of droplet on liquid surface are simulated. A water drop of 2.9 mm diameter, formed using a hypodermic needle, fall from height of
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about 36 cm on a water pool. Fig. 12 shows the break-up of the Rayleigh jet and droplet formation. A visual comparison between the numerical and experimental results in [31] shows good agreement.
5. Numerical Simulation of Complex Filling Process In this section, the injection molding processes of Cross fluid are simulated using the coupled IB and CLSVOF (IB-CLSVOF) method. The immersed solid boundary, i.e., mold boundary, is represented using a shape LS function , and then the irregular boundary of the mold cavity f
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can be implicitly defined as the zero-contour of the shape-LS function , which is negative in the f
fluid region and positive in the solid region . Meanwhile, the boundary corresponds to f
s
the zero contour of the shape-LS function, i.e.,
dist , x 0, dist ,
x f x
(29)
x s
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where dist denotes the minimum distance from the mesh point x to the mold boundary.
In order to build the shape-LS function , the Constructive Solid Geometry (CSG) algorithm f
[6] is used, which can construct complex domains out of basic geometries such as spheres, cylinder.
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For example, in order to construct a cylindrical-cubic cavity, let 1 and 2 be the cylinder and cubic boundaries 1 and 2 , respectively. And their LS functions are,
1 R
x x0 2 y y0 2
(30b)
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2 min y y1 , y2 y
(30a)
where R is the cylindrical radius, and ( x0 , y0 ) is the center. y1 and y 2 are two boundaries of
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the cubic cavity along y-axis. Thus, the shape-LS function of the ring-shaped mold cavity (cylindrical-cubic cavity) is expressed as
f max( 2 , 1 ) (see Fig.13). Other complex
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geometries could also be created by Boolean CSG operations [6]. And the solid volume fraction in Eq. (19) is calculated from the shape-LS function by using Eq. (26) [29].
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f
Furthermore, in order to release the air out of the mold cavity, the exhaust holes are needed. In
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this simulation, the outflow boundary condition u n | 0 and no-slip wall boundary condition
u | 0 are used for the air region ( 0 ) and the polymer melt region ( 0 ), respectively,
where is the boundary of mold cavity.
5.1 Filling process in ring-shaped mold cavity without insets The numerical simulation of melt filling process is performed for a polymer, namely Taiwan PP6733, where the Cross model has been chosen to assess the total viscosity of the polymer melt [32]. The expression of the Cross model is of the form
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0 T , p
1 0 *
,
1n
(31)
The zero shear rate viscosity of the melt is presented by
.
A1 T T * 0 D1 exp * A2 T T
(32)
~
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where T * D2 D3 P , A2 A2 D3 P , P is the pressure. A1 , A2 , D1 , D2 and D3 are parameters related to the material.
As a test example, the filling process is simulated in a cylindrical-cubic mold, and Fig. 13 shows the geometry and size of the mold cavity. The material parametric constants corresponding to n, , D1, D2, D3, A1 and A2 of the viscosity model are 0.219, 56, 1.3×1014, 263.15, 0, 28.44,
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*
51.6, respectively. The melt and mold temperatures are 533K and 323 K, respectively. The injection velocity is 30, and the space and time intervals are 0.075 and 0.012, respectively. Fig. 14 demonstrates the melt interface in the ring-shaped mold cavity at different times. The geometric size of the mold cavity is shown in Fig. 13 and the thickness (along z-axis) is 1. Without
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insets the polymer melt goes well in the mold cavity, and the fountain phenomenon can be seen clearly from Fig. 15, which is a phenomenon that the middle velocity is larger than the mean velocity of the melt front and the melt flows from the middle to both side of the mold cavity.
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5.2 Filling process in ring-shaped mold cavity with two cylindrical insets Here three types of cylindrical insets are considered in the ring-shaped mold cavity, whose
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thicknesses are 1, 0.75, 0.5 and 0.25, respectively. Fig. 16 shows a mold cavity with two cylindrical insets of one thickness along z-axis. Fig. 17 demonstrates the melt interface in the mold cavity at different times. After the melt turns around the circular inserts two melt branches encounter and air
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is entrained among melt and inserts (Fig. 17(b)). And then the entrained air escapes from mold cavity gradually (Figs. 17(c) and 17(d)) since there are the exhaust holes in the fusion area.
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Meanwhile, the weldline appears within the injection molding plastic components, which forms when separate melt fronts meet at the meeting point and disappear when the meeting angle is more than 150° [33]. Figs. 17(b) and 17(c) reveal the weldline formation progress, which are undesirable in the injection molding. Fig. 18 demonstrates the melt interface in ring-shaped mold cavity with two cylindrical insets of 0.75-thickness at different times. In a short time, the difference between two melt branches is unobservable. Meanwhile, the fountain phenomenon could be seen clearly, as shown in Fig. 18(a). At t=2.2, the three branches emerge (Fig. 18(b)). Then the difference enlarges gradually due to race-tracking effect. After flowing around and over the insets, the three branches meet each other and the weldlines begin to form (Fig. 18(c)). Fig. 18(d) reveals the phenomenon of air traps over 14
ACCEPTED MANUSCRIPT the thin inset at t=4.05. The above phenomena could only be simulated successfully in the three-dimensions. In order to better study the weldline development, the temperatures of three different points on x–y plane (z=0.1) have been considered, i.e., A(4.1,1.2), B(4.7, 1.0) and C(4.9,0.8), where the melt would pass through. Fig. 19 shows the temperatures for the three points in the molds with insets of 1- and 0.75-thickness, which decrease with time. And the low temperatures will facilitate the formation of the weldline. Figs. 20 and 21 demonstrate the melt interface at different times in the ring-shaped mold
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cavity with two cylindrical insets of 0.5- and 0.25-thickness, respectively. Since the insets have thinner thickness, the flow resistance is smaller. Thus, the race-tracking effect is not significant. From Figs. 20 and 21 it can been concluded that the thin insets can make melt filling smoothly without any air entrapped in the region where the polymer melt flows over the insets. Comparing Figs. 20 and 21 with Figs. 17 and 18, it could be found that the weldline disappear with thinner
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insets.
6. Conclusions
A new CLSVOF method, in which the VOF function is solved only, is proposed and implemented for the interface simulations of the incompressible two-phase flows. To conserve the liquid mass, following the method of Luo et al. [7], a local mass-correct remedy is employed
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according to the local curvature near interface. Based on the constructed interfaces, the signed distance function near the interface cells can be obtained using the LS reinitialization which is calculated by a fast algorithm. The performance of this method is first evaluated through the
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benchmark problems, including two- and three-dimensional deformation field tests, which give better interface than the LS and VOF methods. Then the CLSVOF method is adopted to simulate
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rising bubble and droplet falling problems. At last, a sharp representation of the immersed mold boundary is represented by a shape-LS function, and the IB-CLSVOF method is proposed to simulate the melt filling processes in the ring-shaped molds without insets, and with two cylindrical
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insets of the same/different thickness, respectively. And the conclusions can be drawn as follows. The race-tracking effect is a phenomenon that polymer melt front along the thicker edge moves
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faster than that in the thinner area, so the melt flow is unbalanced in the thick and thin segments in the mold, which could cause weldline to form or air to be trapped in the polymer melt. And the polymer temperatures decrease with time in the fusion zone. Thus in the injection molding process the injection port should be added in the thin sections or the each part thickness of the mold should be consistent. The proposed IB-CLSVOF model can be applied to mold filling processes in the arbitrary mold cavities, such as the melt filling and the gas-assisted injection molding process, which is our next work.
Acknowledgements
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ACCEPTED MANUSCRIPT This work was partically supported by National Natural Science Foundation of China (No.11301157), NSFC Tianyuan Fund for Mathematics (No.11326232), Natural Science Foundation of Education Department of Henan Province (No. 15A110001) and the Research Fund for the Doctoral Program of Henan Polytechnic University (B2013-057, B2012-030).
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13. T. Ménard, S. Tanguy, A. Berlemont, Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet, Int. J. Multiphas. Flow. 33(5) (2007) 510–524. 14. X.Yang, A.J. James, J. Lowengrub, X. Zheng, V.Cristini, An adaptive coupled level-set/volume-of-fluid
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interface capturing method for unstructured triangular grids, J. Comput. Phys. 217(2) (2006) 364–394. 15. Z. Wang, J. Yang, B. Koo, F.Stern, A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves, Int. J. Multiphas. Flow. 35(3) (2009) 227–246.
16. D.L.Son, W.Q.Tao, A coupled volume-of-fluid and level set (CLSVOF) method for computing incompressible two-phase flows, Int. J.Heat. Mass. Tran. 53(4)(2010) 645–655. 17. L. Kong, H.L.Zhao, L.S.Dong, L.H.Ya, W.Q.Tao, A three-dimensional volume of fluid & level set (VOSET) method for incompressible two-phase flow, Coput. Fluids. 118(2) (2015) 293–304. 18. Q. Li, J. Ouyang, B.Yang, X.Li, Numerical simulation of gas-assisted injection molding using CLSVOF method, Appl.Math. Model. 36(5)(2012) 2262–2274. 19. A. Polynkin, J.F.T. Pittman, J. Sienz, Gas assisted injection molding of a handle: Three-dimensional simulation and experimental verification, Polym. Eng. Sci. 45(6)(2005) 1049–1058. 16
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20. Q. Li, J. Ouyang, G. Wu, X. Xu, Numerical Simulation of Melt Filling and Gas Penetration in Gas Assisted Injection Molding. CMES-Comp. Model. Eng. 82(3-4)(2011) 215–232. 21. B.J. Araujo, J.C.F. Teixeira, A.M. Cunha, C.P.T. Groth, Parallel three-dimensional simulation of the injection molding process, Int. J.Numer. Meth Fl. 59(7)(2009) 801–815. 22. J.I.Choi, R.C.OPeroi, J.R.Edwards, J.A.Rosati, An Immersed Incompressible
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23. M.J. Aftosmis, M.J. Berger, J.E. Melton, Robust and efficient Cartesian mesh generation for component-based geometry, AIAA. J. 36(6)(1998) 952–960.
Cartesian Grids, Coput. Fluids. 85(2013) 135–142.
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24. M. Meinke, L. Schneiders, C. Günther, W. Schröder, A Cut-Cell Method for Sharp Moving Boundaries in
25. C. Günther, M. Meinke, W. Schröder, A flexible level-set approach for tracking multiple interacting interfaces in embedded boundary methods, Coput. Fluids. 102(2014) 182–202.
26. D. Peng, B.Merriman, S. Osher, H. Zhao, M. Kang, A PDE-based fast local level set method, J. Comput. Phys. 155(2)(1999) 410–438.
27. Q. Li, J. Ouyang, B.Yang, T.Jiang, Modelling and simulation of moving interfaces in gas-assisted injection
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molding process, Appl.Math. Model. 35 (1)(2011) 257–275.
28. E.A. Fadlun, R. Verzicco, P. Orlandi, J.Mohd-Yusof, Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations, J. Comput. Phys. 161(1) (2000) 35–60. 29. S.P. van der Pijl, A. Segal, C. Vuik, P. Wesseling, Computing three-dimensional two-phase flows with a mass-conserving level set method, Comput. Vis. Sci. 11 (2008) 221–235.
30. T.W.H. Sheu, C.H. Yu, P.H. Chiu, Development of a dispersively accurate conservative level set scheme for
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capturing interface in two-phase flows, J. Comput. Phys. 228(3)(2009) 661–686.
31. J.Hernández, J. López, P.Gómez, C. Zanzi, F. Faura. A new volume of fluid method in three dimensions-Part I: Multidimensional advection method with face-matched flux polyhedra, Int. J. Numer. Meth. Fl. 2008,
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58(8):897–921.
32. T. Boronat, V.J. Segui, M.A. Peydro, M.J. Reig, Influence of temperature and shear rate on the rheology and
2735–2745.
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processability of reprocessed ABS in injection molding process, J. Mater. Process. Tech. 5(1)(2009)
33. S. Zheng, J. Ouyang, Z. Zhao, L. Zhang, An adaptive method to capture weldlines during the injection mold
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filling, Comput. Math. Appl. 64(9)(2012) 2860–2870.
Table captions:
Table 1. Relative mass loss at t=8 for two-dimensional deformation on different meshes.
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Table 2. Relative mass loss at t=3 for three-dimensional deformation.
Figure captions: Fig. 1. Schematic for evaluation of the liquid volume advected in the x direction,
(VF )i1 2, j ,k , for (a) u>0
and (b) u<0. Fig. 2. Zero contour of
1 (r , ) .
Fig. 3. The initial and finial LS functions. Fig. 4. The relationship between zero LS function
i, j
and
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si , j in each control cell in two-dimensions.
ACCEPTED MANUSCRIPT Fig. 5. Flow chart of CLSVOF method. Fig. 6. Two-dimensional deformation test at t=4 (a) and t=8 (b) (red line for computational interface; black line for initial interface). From left to right: LS; VOF; CLSVOF. Fig. 7. Two-dimensional deformation test at t=8 with different methods: (a) conserved LS method in [17] (Red line for mass remedy; Blue line for no mass remedy); (b) VOF method in [5]; (c) the present CLSVOF method (red line for computational interface; black line for initial interface). Fig. 8. The interface comparisons at t=8 (a) on a 642 mesh (b) on a 1282 mesh without mass-correction (green line), with mass-correction based on local curvature (blue line) and with mass-correction based on equational volume(red line).
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Fig. 9. The upper interface comparisons on a 1282 mesh at t=8 without mass-correction (green line), with mass-correction based on local curvature (blue line) and with mass-correction based on equational volume(red line). Fig. 10. Deformation of a sphere at t=0.75 (a); t=1.5 (b); t=2.25 (c); t=3 (d). From left to right: LS method; VOF method; CLSVOF method.
Fig. 11. Shapes of a bubble rising in a quiescent liquid. (a)t=2.8; (b)t=3.6; (c)t=4.8; (d)t=5.6.Left: present result; Right: results of Tony et al. (red dotted line) and Sussman et al. (black solid line)[25].
Fig. 12. Evolution of the interface for water droplet impacting on water pool at (a)t=15.9; (b)t=30.6; (c)t=35.8;
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(d)t=40.1; Above: present results; Below: results in [31].
Fig. 13. Geometry and computational domain of a concentric cylindrical: (a) three-dimensional perspective; (b) front view show (from z-axis).
Fig. 14. Melt front in the ring-shaped mold cavity without insets at different times: (a) t=1.0; (b) t=2.0.
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Fig. 15. Velocity vectors of polymer melt on (a) x-y plane and (b) y-z plane at t=1.5. Fig. 16. Mold cavity with two cylindrical insets of one-thickness: (a) front view; (b) side view. Fig. 17. Melt front in the ring-shaped mold cavity with two cylindrical insets of one-thickness at different times: (a) t=2.2; (b) t=3.85; (c) t=4.05; (d) t=4.15.
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Fig. 18. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.75-thickness at different times: (a) t=1.2; (b) t=2.2; (c) t=3.85; (d) t=4.05.
Fig. 19. Temperatures at three points in the ring-shaped mold cavity with two cylindrical insets of (a) one-thickness
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and (b) 0.75-thickness at different times.
Fig. 20. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.5-thickness at different times: (a) t=2.2; (b) t=3.3; (c) t=3.85; (d) t=3.95.
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Fig. 21. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.25-thickness at different times: (a)
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t=2.2; (b) t=3.3; (c) t=3.85; (d) t=3.90.
(a)
(b)
Fig. 1. Schematic for evaluation of the liquid volume advected in the x direction, (VF )i1 2, j ,k , for (a) u>0
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1 (r , ) .
(a)
1 (r , )
(b)
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Fig. 2. Zero contour of
2 (r , )
(c)
3 (r , )
(d) finial LS function
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Fig. 3. The initial and finial LS functions.
o s i, j
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Fig. 4. The relationship between zero LS function
Interface
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and
si , j in each control cell in two-dimensions.
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Start
0 Initialize LS
0 Initialize VOF F
Advect VOF
Obtain LS
n
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n Get F and reconstruct interface
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Normal n and curvature
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Normal n and curvature
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Advect VOF and correct mass
Fig. 5. Flow chart of CLSVOF method.
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n 1 Get F and reconstruct interface
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1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
Y
1 0.9
Y
Y
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0.5
0.5
0.4
0.4
0.4
0.3
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0
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(a)
(b)
Fig. 6. Two-dimensional deformation test at t=4 (a) and t=8 (b) (red line for computational interface; black line for
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initial interface). From left to right: LS; VOF; CLSVOF.
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(b)
(c)
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(a)
Fig. 7. Two-dimensional deformation test at t=8 with different methods: (a) conserved LS method in [17] (Red line for mass remedy; Blue line for no mass remedy); (b) VOF method in [5]; (c) the present CLSVOF method (red
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line for computational interface; black line for initial interface).
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(a)
(b) 2
2
Fig. 8. The interface comparisons at t=8 (a) on a 64 mesh (b) on a 128 mesh without mass-correction (green
volume(red line).
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line), with mass-correction based on local curvature (blue line) and with mass-correction based on equational
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mass-correction based on local curvature (blue line) and with mass-correction based on equational volume(red line).
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(b)
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(c)
(d)
Fig. 10. Deformation of a sphere at t=0.75 (a); t=1.5 (b); t=2.25 (c); t=3 (d). From left to right: LS method; VOF
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method; CLSVOF method.
(a)
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(b)
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(c)
(d)
Fig. 11. Shapes of a bubble rising in a quiescent liquid. (a)t=2.8; (b)t=3.6; (c)t=4.8; (d)t=5.6.Left: present result;
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Right: results of Tony et al. (red dotted line) and Sussman et al. (black solid line)[25].
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ACCEPTED MANUSCRIPT (a)
(b)
(c)
(d)
Fig. 12. Evolution of the interface for water droplet impacting on water pool at (a)t=15.9; (b)t=30.6; (c)t=35.8;
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(d)t=40.1; Above: present results; Below: results in [31].
(a)
(b)
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Fig. 13. Geometry and computational domain of a concentric cylindrical: (a) three-dimensional perspective; (b)
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front view show (from z-axis).
(a)
(b)
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Fig. 14. Melt front in the ring-shaped mold cavity without insets at different times: (a) t=1.0; (b) t=2.0.
(a)
(b)
Fig. 15. Velocity vectors of polymer melt on (a) x-y plane and (b) y-z plane at t=1.5.
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(a)
(b)
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Fig. 16. Mold cavity with two cylindrical insets of one-thickness: (a) front view; (b) side view.
(a)
(b)
(c)
(d)
(a) t=2.2; (b) t=3.85; (c) t=4.05; (d) t=4.15.
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Fig. 17. Melt front in the ring-shaped mold cavity with two cylindrical insets of one-thickness at different times:
(a)
(b)
(c)
(d)
Fig. 18. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.75-thickness at different times: (a) t=1.2; (b) t=2.2; (c) t=3.85; (d) t=4.05.
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Fig. 19. Temperatures at three points in the ring-shaped mold cavity with two cylindrical insets of (a) one-thickness
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and (b) 0.75-thickness at different times.
(a)
(b)
(c)
(d)
Fig. 20. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.5-thickness at different times:
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(a) t=2.2; (b) t=3.3; (c) t=3.85; (d) t=3.95.
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(a)
(b)
(c)
(d)
Fig. 21. Melt front in the ring-shaped mold cavity with two cylindrical insets of 0.25-thickness at different times: (a)
Mesh
64
2
128
2
t=2.2; (b) t=3.3; (c) t=3.85; (d) t=3.90.
Table 1. Relative mass loss at t=8 for two-dimensional deformation on different meshes.
No mass-correction
Mass-correction error based on local
Mass-correction error based on
error (%)
curvature (%)
equational volume (%)
7.741
0.672
0.518
0.923
0.056
0.043
Table 2. Relative mass loss at t=3 for three-dimensional deformation.
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VOF
CLSVOF with mass correction
Volume loss(%)
32.109
0.423
0.018
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Method
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