Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.4.1 - A simple and conservative algebraic VOF method for resolving interface in one cell without geometrical calculations on structured and unstructured grids
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 Material Information
Title: 10.4.1 - A simple and conservative algebraic VOF method for resolving interface in one cell without geometrical calculations on structured and unstructured grids Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Sun, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: PLIC-VOF
PLIC/SN
algebraic VOF
conservation
unstructured
structured
adaptive grid
 Notes
Abstract: A simple and exactly conservative method is proposed for capturing sharp interfaces on arbitrary grids. The classic PLIC-VOF method resolves a sharp interface (e.g, Rider and Kothe 1998), but necessitates sophisticated geometrical reconstructions that can be prohibitive for general grids. The idea of this work is to define the volume flux by using linear algebraic formulas that approximate geometrical reconstructions of the PLIC-VOF. In order to control the appearance of flotsams, the approximated volume flux of a small volume is maintained not less than that of the geometric PLIC-VOF. It is not a method based on high-order finite-difference approximation to the advection equation, which resolves a sharp but still smeared interface. The present method resolves the interface in one grid cell as a typical PLIC-VOF without geometrical calculations. Since the method has approximated all geometrical reconstruction and volume flux evaluation algebraically, its extension to arbitrary grids is straightforward by using the finite volume method.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00255
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 1041-Sun-ICMF2010.pdf

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ICMF 2010, Tampa, FL, May 30-June 4, 2010


A simple and conservative algebraic VOF method for resolving interface in one
cell without geometrical calculations on structured and unstructured grids


Mingyu Sun

Center for Interdisciplinary Research, Tohoku University, Sendai 980-8578, Japan
sNtIGcil Iohokli ,1c ip
Keywords: PLIC-VOF, PLIC/SN, algebraic VOF, conservation, unstructured, structured, adaptive grid




Abstract

A simple and exactly conservative method is proposed for capturing sharp interfaces on arbitrary grids. The classic
PLIC-VOF method resolves a sharp interface (e.g, Rider and Kothe 1998), but necessitates sophisticated geometrical
reconstructions that can be prohibitive for general grids. The idea of this work is to define the volume flux by
using linear algebraic formulas that approximate geometrical reconstructions of the PLIC-VOF. In order to control
the appearance of flotsams, the approximated volume flux of a small volume is maintained not less than that of
the geometric PLIC-VOF. It is not a method based on high-order finite-difference approximation to the advection
equation, which resolves a sharp but still smeared interface. The present method resolves the interface in one grid
cell as a typical PLIC-VOF without geometrical calculations. Since the method has approximated all geometrical
reconstruction and volume flux evaluation algebraically, its extension to arbitrary grids is straightforward by using the
finite volume method.


Introduction

The volume-of-fluid (VOF) is one of the most widely
used methods for the numerical simulation of interfa-
cial phenomena (Hirt & Nicholas,1981). In the VOF
method, the evolution of an interface is predicted gen-
erally by solving the following advection equation

t + u V = 0, (1)

where u is the velocity. Function p is the volume frac-
tion or color function at the discrete level. It is unity in
a cell filled with one phase, and is zero if the cell lies in
the other phase. There are two approaches to solve (1).
One approach is based on the interface geometry in
a cell, and then to update the volume fluxes in a La-
grangian fashion. Excellent reviews on this approaches
have been given by Rider & Kothe (1998) and Scar-
dovelli & Zaleski (1999). The majority of these geomet-
ric VOF methods are based on two key procedures. One
is to reconstruct an interface in a cell, and the other to
advance the volume fraction in time and space. Both are
not trivial, because the volume fraction is discontinuous
across a sharply resolved interface. These procedures re-
quires sophisticated geometrical calculations, and their
extension to general grid structures is even more diffi-
cult. However, the accurate interface reconstruction with


a proper advection algorithm provides the best accuracy
in the VOF family.
An alternate approach is to discretize the advection
equation with a differencing scheme that guarantees
bounded volume fractions while preventing the smear-
ing of interfaces over several mesh intervals. The well-
known high-resolution schemes such as TVD methods
are too diffusive for this problem. A rather successful
strategy is to introduce artificial compression by down-
wind differencing (e.g., Ubbink and Issa 1999). This
approach relies much less on the cell geometry, and
can be fairly readily extended for general grid. How-
ever, an interface is often spread in two and more cells.
The interface is practically represented by isocontours of
volume fractions, instead of reconstructed interfaces as
commonly adopted in the geometric VOF approach.
The present work, following the methodology of the
geometric VOF, tries to reconstruct and advect interfaces
using linear algebraic formulas that approximate geo-
metrical reconstructions of the PLIC-VOF


Numerical methods

The advection equation (1) is solved by the finite vol-
ume method. The volume flux is evaluated in the direc-
tion normal to the grid line, and integrated in a unsplit




ICMF 2010, Tampa, FL, May 30-June 4, 2010


manner. Given a flow field u, the advection equation is
rewritten as


Ot + V (up) = pV u,


4t + V (up) 0, (3)
for incompressible velocity field. Equation (3) is dis-
cretized as,

2+' = k (un A kAt)j' (4)


where Pk is the volume fraction of phase k defined at the
grid interface, satisfying Ek =P 1. (2k is the volume
of phase k, 2k p = k2, satisfying

Q= 1: (5)


The normal velocity across grid interface u, equals to

Un = U S,

where s is the outward normal vector with a magnitude
of the length of grid line j. In short, (u, Pn At)j repre-
sents the volume flux crossing face j in time At.
Without losing generality, the phase with the smaller
volume is denoted as slave volume, Q2, and the other
phase master volume, 2,. In the present method, for
maintaining the positivity or the boundedness of volume
fractions, the slave volume is integrated by (4),

2+ 1 -= 2 5(u, ,pAt)j, (6)


where 4 is introduced to ensure the boundedness

2 1E [0, Q]. (7)

For the master volume,


2"+1 (2
"TO "T


S(uanpmAt)j,


where ,m = 1 (Ds. The limiter function 4) is defined
such that the outflow of the slave volume should be no
more than that inside, for cell i,


S for
1


fs > Q2
otherwise,


where fs is total outflow volume flux of the slave vol-
ume,
fs= (unAt)j.
j,un>O


The limiter function at grid line is taken from the up-
stream cell,
4i u, >0
Dj u, < 0'
The master volume is then updated using (5),

Q,91 Q-91. (10)

With this saturated relation, it seems that the equation
(8) is not necessary for volume update. In fact, the small
and master phases are in general not the same fluid in
two neighboring cells, so both volume fluxes have to be
calculated in order to update the volume of the small
phase. Furthermore, the volume flux calculation in (8) is
required to define the numerical fluxes for other conser-
vation laws.
It is obvious that the sum of (6) and (8) at the new
time step satisfies the saturated constraint (5), if

S(uaAt)j= 0, (11)

which is the finite-volume discretization of the
divergence-free velocity field. In other words,
divergence-free relation (11) is the sufficient condition
for maintaining the volume conservation of the present
method.
It is noted that the 4) limiter is different from the re-
distribution algorithm adopted in some VOF algorithms.
The redistribution algorithm is often triggered when an
abnormal volume that is beyond [0, Q2] is found in the
solution, and then redistribute the volume to somehow
empirically chosen cells nearby. The D limiter is to ad-
just the volume flux such that the abnormal volume will
not appear. This limiting method is simple and general
for any grid system, and it is also necessary for the fol-
lowing approximate volume flux formulas to preserve
the boundedness of the phase volumes.
Before introducing the present algebraic VOF method
to evaluate the volume flux (unAt)j, we first re-
view how a geometric VOF method works. The inter-
face in a cell is not tracked explicitly, but reconstructed
and approximated by a simple geometry. The simple
line (piecewise constant) interface calculation (SLIC)
assumes the geometry is a line parallel to one of the
grid lines. Currently most widely used method is based
on the piecewise linear interface calculation (PLIC). A
historic review of the piecewise constant and linear re-
constructions was given by Rider and Kothe (1998). In
PLIC, the surface normal vector, n, is required to con-
struct the linear interface


n r dl = 0,


where d1 is the line constant. The normal vector is in-
ferred from the spatial distribution of p. If p is a smooth




ICMF 2010, Tampa, FL, May 30-June 4, 2010


function, the normal vector satisfies

n = V. (13)

In the present work, the normal vector is treated as an
independent variable that are solved separately follow-
ing our previous work (Sun, 2010). Therefore in order
to uniquely reconstruct the line segment in the cell, one
needs to find the line constant di. Among many advec-
tion scheme, a simple way is to define the volume or area
cut by the grid line j shifted upstream by -u,At as the
volume flux (u,a At)j. These two procedures are geo-
metrically complicated, especially in axisymmetric and
3-D cases on unstructured grids. Therefore, in order to
simplify two procedures, instead of calculating the geo-
metric relations precisely, this work tries to approximate
them using simple formulas. We need
1. a simple formula to approximate line constant di,
and

2. a simple formula to evaluate the area cut by the grid
line shifted upstream.


The distance function in the cell is bounded, d e
[dmin, dma,], where din and dax are two extreme
values in those defined at all vertexes of the cell,

dmin = min(dj), d.x. = max(dj).

It is clear that the volume fraction of the dark phase in
the figure satisfies

(d) dmax
1, d = din

The basic idea of this work is to make use of this prop-
erty, and to construct a linear function to approximate
the line constant and the volume flux. The linear func-
tion is unique,


di dmax s(dmax dmin),


for interface reconstruction. ~, is the volume fraction of
the slave phase. The tilde over p, indicates the value will
be modified to prevent the appearance of flotsams, to be
defined later. The linear function can be reformulated as

dmaxi d (15)
Ts = (15)
dmaxl dminl

for the calculation of volume flux, where dmint and
dma, are taken as the maximum and the minimum value
at two vertexes of the grid line ', as shown in Fig. lb.
Formulas (14) and (15) are exact for grid lines par-
allel or perpendicular to the grid lines on a rectangular
grid. It is less accurate for other angles, and attains the
maximum error at 45. Denoting

sin20
1 + sin20'

where 0 is the angle between interface and grid line nor-
mal vectors. We introduce a modification of 0s,


Ps, P r < s
Vo4's4, 0' > 4s


b. d_, rn
Figure 1: Line constant calculation (a) and volume flux
calculation (b) using algebraic approximation

Consider a general quadrilateral as shown in Fig. 1.
Given the surface normal n, we introduce a function, d,
representing the relative distance to the interface,

d = n r.


such that the volume flux of the slave phase is not
less than that of the exact geometric reconstruction for
square cells. The proof is neglected here.

Numerical results

The initial phase volumes intersected with interfaces are
exactly calculated. For example, the sum of all phase
volumes inside a circle differs from the exact area by the
SThe definition for the maximum and the minimum values in the eval-
uation of volume fluxes is chosen, so that we may find a modifi-
cation (16) to prevent this approximate VOF method from gen-
erating flotsams. Other options may be possible, but still remain
unexploited, because of the difficulty in controlling flotsams.




ICMF 2010, Tampa, FL, May 30-June 4, 2010


order of 0(1016). For all tests, the velocities at cell
faces are also exactly specified, to exclude any possible
error in the treatment of velocity. The CFL number is
taken as 1/8 if not specified. For drawing interfaces, lin-
ear segments reconstructed in all interface cells are plot-
ted. They are not contours of equal volume fractions.
Surface normal vectors in the interface cells and the cells
filled with the dark phase are also plotted in most fig-
ures. The vectors in other cells are used in computation
as well, but not plotted for the sake of clarity. The ar-
row indicates the direction of the normal vector, starting
from the cell center, and its length is proportional to the
magnitude of the vector.
The geometric error measure, L1 norm, is defined as


1 N
Cy= N = QL \0a
>2 ZJa


Ii (17)


where Ocal and exact are the calculated and exact vol-
ume fractions in cell j with volume (2j. N is the total
number of cells used in the domain.
Translation of circular surfaces on square grid cell In
this test case, the computational domain is a square of
(0,1)2, with open or free boundary outside. Initial sur-
face normal vectors for circles are specified the same as
those define in Sun (2010). The initial circle of 0.3 in
diameter centered at (0.25, 0.5) is translated in the con-
stant velocity field of (u, v) = (1, 0) fort = 0.5. Fig.
2 shows the reconstructed interfaces at the last step, to-
gether with associated surface normal vectors located at
the cell center. It is emphasized that, even for a low res-
olution of d/Ax < 3, as shown in Figs. 2a, the particle
is reasonably advected and reconstructed.
Translation of hollow shapes The second test is the
advection of a hollow square and a hollow circle in an
oblique velocity field of (2, 1), which has been gener-
ally tested previously (Rudmann 1997, Ubbink and Issa
1999). The side lengths of outer and inner interfaces of
the square is 0.8 and 0.4 respectively, and for the circle
the outer and inner radii are 0.4 and 0.2 respectively. The
shapes are initially centered at (-1.2, -1.2) in a square
domain of (-2, 2)2 with a 2002 mesh. The solutions
are recorded at t = 1.25. The final shapes are plot-
ted in Fig. 3, and the corresponding error distributions
of volume fraction are shown in Fig. 4. The interfaces
are resolved sharply in one cell without smearing, which
has never been realized by the finite difference VOF ap-
proach. The solution errors together with a few calcu-
lated results in literature, are listed in Table 1. For the
consistency the literature, the error is slightly different
from (17),


Figure 2: A circular particle of d 0.3 after translating
from (0.25, 0.5) to (0.75, 0.5) in x-direction on different
meshes: a) d/Ax = 2.4, 8 x 8; b) d/Ax 4.8, 16 x 16.
c) d/Ax = 9.6, 32 x 32.


1 N

6 j=91 .exactJt j=l


i:I-




ICMF 2010, Tampa, FL, May 30-June 4, 2010


, '- - i- L I u ( . .1 u 1 n '




U u'I, ,' , i'
i~t
'4-124


/'


"` FIE


Figure 3: Advection of a hollow square (a) and a hollow
circle (b) with an oblique uniform velocity (2,1) using a
2002 square grid.


The small time step is found to improve the accuracy of
the present algebraic method, and the accuracy is com-
parable with other methods. A not-trivial improvement
is that all interfaces are resolved in one cell without pro-
ducing flotsams.


Table 1: Geometric error for oblique translation. a er-
rors obtained from Rudman (1997), and b obtained from
Ubbink and Issa (1999)
SLICa FCT-VOFy CICSAM-Ub Present Present
(CFL=1/8) (CFL=1/16)
Square 1 32E-1 1 63E-8 3 97E-2 6 83E-2 6 28E-2
Circle 9 18E-2 3 99E-2 2 84E-2 3 83E-2 2 14E-2

The last example is the deformation of a circular sur-
face of d 0.3 in a vortex velocity field. The circle is
initially centered at (0.5, 0.75). The time-resolved ve-
locity field is specified as,

u = -cos(7t/T)sin (7rz)sin(27y),


Figure 4: Contours for volume fraction error in advec-
tion of a hollow square (a) and a hollow circle (b) with
an oblique uniform velocity (2,1) using a 200 2 square
grid.


v = cos(wt/T)sin (2(y)sin(2wx).

The test was taken from Rider and Kothe (1998). This
is a vortical velocity field, which will deform the cir-
cle and promote topology changes. It is periodic with
a period of T. The circle will undergo the maximum
deformation at t T/2, and then return to its initial
state. After one period, the circle is supposed to re-
cover its original shape. The sequential snapshots to-
gether with the corresponding grids are plotted in Fig. 5.
A magnified view of the circle at the last step is shown
in Fig. 6. The solution-adaptive unstructured quadrilat-
eral grid (Sun and Takayama 1999) is used. The back-
ground coarse grid contains 620 cells, and the number
is increased to 1943 at the end by using a three-level re-
finement. The interface is shown by thick segments that
are exactly reconstructed from the volume fraction be-
tween zero and one. Neither interface diffusion and nor
flotsams are seen. This can be more clearly seen from


I s1





La- meI




ICMF 2010, Tampa, FL, May 30-June 4, 2010


~u~40//i
3


fttra mlq-t//tt


J


e. I I I I I I I I f. L++ + -

Figure 5: Results for the circular fluid body placed in
the time-reversed single-vortex flow field on a solution-
adaptive unstructured grid, T=2. (a) t=0.0; (b) t=2.0;
(c)t=0.4; (d)t=1.6; (e)t=0.8; (f)t=1.2. Figures in the right
column are supposed to be the same as those in the left.

the fact that fine grids are always clustered near the in-
terface.

Concluding remarks

An algebraic VOF method has been proposed. The
method is based on a linear approximation to the PLIC-
VOF method. Although its accuracy is not as good as ad-
vanced geometric VOF methods (e.g. L6pez et al. 2005,
Pilliod & Puckett 2004), it is still comparable with those
finite-difference based methods. The real advantages of
this method are its simplicity and its easy implementa-
tion on arbitrary grid structures. The method generates
no flotsams, without explicitly introducing debris sup-
pressing tricks, and exactly conserves the phase volumes
with a divergence-free velocity field.

References

C.W. Hirt and B.D. Nichols, Volume of Fluid (VOF)
method for the dynamics of free boundaries, J. Comput.


Figure 6: The zoomed view of Fig. 5b. The interface is
shown by thick segments that are reconstructed exactly
for all cells with volume fraction between zero and one.
Neither interface diffusion and nor flotsams are seen.
Phys. 39 (1981) 201-225.
J. L6pez, J. Hernmndez, P. G6mez and F. Faura, An im-
proved PLIC-VOF method for tracking thin fluid struc-
tures in incompressible two-phase flows, J. Comput.
Phys. 208 (2005) 51-74.
J.E. Pilliod Jr., E.G. Puckett, Second-order accurate
volume-of-fluid algorithms for tracking material inter-
faces, J. Comput. Phys. 199 (211 14 465-502.
W.J. Rider, D.B. Kothe, Reconstructing volume track-
ing, J. Comput. Phys. 141 (1998) 112-152.
M. Rudman, Volume-tracking methods for interfacial
flow calculations, Intl. J. for Numerical Methods in Flu-
ids, V24:671-691, (1997).
R. Scardovelli, S. Zaleski, Direct numerical simulation
of free-surface and interfacial flow, Annu. Rev. Fluid
Mech. 31 (1999) 567-603.
M. Sun, Volume-tracking of subgrid particles, to appear
in Intl. J. for Numerical Methods in Fluids, (2010).
M. Sun, K. Takayama, Conservative smoothing on an
adaptive quadrilateral grid, J. Comput. Phys. 150 (1999)
143-180.
O. Ubbink, R.I. Issa, A method for capturing sharp fluid
interfaces on arbitrary meshes, J. Comput. Phys. 153
(1999) 26-50.


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