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A coupled volume-of-ﬂuid and level set (VOSET) method for computing

incompressible two-phase ﬂows

D.L. Sun, W.Q. Tao

State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China

article info

Article history:

Received 24 February 2009

Received in revised form 28 August 2009

Accepted 23 September 2009

Available online 10 November 2009

Keywords:

VOF method

LS method

VOSET method

Volume fraction function

Level set function

abstract

A coupled volume-of-ﬂuid and level set (VOSET) method, which combines the advantages and overcomes

the disadvantages of VOF and LS methods, is presented for computing incompressible two-phase ﬂows. In

this method VOF method is used to capture interfaces, which can conserve the mass and overcome the

disadvantage of nonconservation of mass in LS method. An iterative geometric operation proposed by

author is used to calculate the level set function / near interfaces, which can be applied to compute

the accurate curvature

and smooth the discontinuous physical quantities near interfaces. By using

the level set function / the disadvantages of VOF method, inaccuracy of curvature and bad smoothness

of discontinuous physical quantities near interfaces, can be overcome. Finally the computing results

made with VOSET method are compared with those made with VOF and LS methods.

Ó 2009 Published by Elsevier Ltd.

1. Introduction

Flows with a spatial variation of ﬂuid properties, such as gas–

liquid interfaces due to density and viscosity differences, can be

found in many natural and industrial processes such as chemical

reactor, power plant, copper reﬁning and internal combustion

engine. The generation of vorticity by the discontinuous ﬂuid prop-

erties produces a complex ﬂow structure, which presents a compu-

tational challenge.

In the past several decades a number of different methods have

been developed to simulate complex two-phase ﬂow problems.

The most important methods include the front tracking method

[1–4], the marker particle method [5,6], the volume of ﬂuid (VOF)

method [7–17] and the level set (LS) method [18–24]. Since the

development of VOF method by Hirt and Nichols [7] in 1981, the

method has become very popular and is widely used in the free-sur-

face modeling. The LS method was developed in 1988 by Osher and

Sethian [18]. It has become popular in many disciplines, such as im-

age processing, computer graphics, computational geometry and

computational physics. Thus, among the four major methods men-

tioned above, the VOF method and LS method are probably the most

widely usedmethods in the literatures. Needless to say, each method

has its own advantages and drawbacks. In the present work based on

a comprehensive analysis to VOF and LS methods, a coupled volume-

of-ﬂuid and level set method (hereafter VOSET), which combines the

advantages and overcomes the disadvantages of VOF and LS

methods, is presented for computing the incompressible two-phase

ﬂows. In the following a brief review on the VOF and LS methods and

their advantages/disadvantages are presented.

In the VOF method, a volume fraction function C, whose value

lies between 0 and 1, is deﬁned to denote whether a space is occu-

pied by the dispersed phase or continuous phase. When the value

of C is unity, the space is occupied by the dispersed phase; when

the value of C is zero, the space is occupied by the continuous

phase; when the value of C is between 0 and 1, the space contains

both the dispersed and continuous phases, where by the deﬁnition

a free surface exists.

For a given ﬂow ﬁeld, the standard advection equation governs

the evolution of C:

u 

C ¼ 0 ð1Þ

If the ﬂow is incompressible, the C advection equation can be recast

in the conservative form:

ð

uCÞ¼0 ð2Þ

For the C advection equation, the standard ﬁnite-difference approx-

imations would lead to a smearing of the C function and the inter-

faces would lose their deﬁnition. Therefore, the volume tracking

algorithms are applied to capture the interfaces. Most volume track-

ing algorithms published to date fall into one of these two interface

reconstruction categories: one is the Piecewise Constant Volume

0017-9310/$ - see front matter Ó 2009 Published by Elsevier Ltd.

doi:10.1016/j.ijheatmasstransfer.2009.10.030

* Corresponding author. Tel./fax: +86 29 82669106.

E-mail address: wqtao@mail.xjtu.edu.cn (W.Q. Tao).

International Journal of Heat and Mass Transfer 53 (2010) 645–655

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

Tracking Methods, which include Hirt and Nichols’s donor-acceptor

algorithm [7]; the other is the Piecewise Linear Volume Tracking

Methods, which include the PLIC (Piecewise Linear Interface Con-

struction) algorithm due to Youngs [8]. In our study the PLIC algo-

rithm is adopted to solve Eq. (2) and reconstruct the interfaces,

because this algorithm is more accurate than other algorithms.

Usually, in the VOF method the smoothed volume fraction func-

tion

C is used to compute the curvature

. The smoothed volume

fraction function

C can be obtained by

i;j

m;n

Kðj

i;j



m;m

Þdxdy ð3Þ

where the smoothening function K is given by the cubic B-spline

proposed by Monaghan [25]:

Kðr;

Þ¼

1  6



þ 6





1 





< 1

0 otherwise

ð4Þ

where

denotes the width of transition region used for smoothen-

ing. In the present paper typically

is used in the VOF method

where

represents the grid size.

The curvature

can be obtained by the following equation:

¼



ð5Þ

The curvature can also be written as



Cj





ð6Þ

The curvature value numerically calculated by Eq. (5) is quite inac-

curate, because the principal contributions to Eq. (5) come from the

edges of the transition region rather than center. However, numer-

ical approximations to Eq. (6), for which the principal contributions

come from the center of the transition region, give better results in

practice [26]. So Eq. (6) is adopted to calculate the curvature in the

VOF method.

The discontinuous physical quantities (density and viscosity)

near interfaces can be smoothed by

ðCÞ¼

C þ

ð1  CÞð7Þ

ðCÞ¼

C þ

ð1  CÞð8Þ

where the subscripts d and c denote, respectively, the dispersed and

continuous phases.

An advantage of VOF method is the fact that accurate algo-

rithms can be used to advect the volume fraction function so that

the mass is conserved while still maintaining a sharp representa-

tion of the interfaces [27]. However, because the volume fraction

function C is a step function, it is difﬁcult to obtain the accurate

curvature and smooth the discontinuous physical quantities near

the interfaces. Even though the smoothed volume fraction function

C is used to compute the curvature, the accuracy of the curvature is

still not good. This is the disadvantage of VOF method [27].

In the LS method, a smooth function /, called level set function,

is used to represent the interfaces. Generally, a signed distance

function, on which an extra condition of |5/| = 1 is imposed, is de-

ﬁned as the level set function. The signed distance function can be

expressed as

/ð

r; tÞ¼

dð

ðtÞÞ < 0if

r 2 the dispersed phase;

¼ 0; if

r 2 the interface;

dð

ðtÞÞ > 0if

r 2 the continuous phase:

ð9Þ

where

denotes the interfaces, d is the distance function which is

the shortest distance from some point to the interfaces. In the LS

method, the location is indicated by /: / < 0 in the dispersed- phase

region, / > 0 in the continuous-phase region, and / = 0 on the

interfaces.

Since the interfaces move with ﬂuid ﬂow, the evolution of / is

given by

u 

/ ¼ 0 ð10Þ

In order to keep the solution accurate, the level set function / needs

to be reinitialized after every time step. This is achieved by solving

the following transient problem to the steady state:

¼ Sð/

Þð1 j

 /jÞ

/ð

r; 0Þ¼/

rÞ



ð11Þ

where /

is the time derivative and S is the sign function. Eq. (11)

has the property that / remains unchanged at the interfaces, there-

fore, /

and / at the zero level set are the same. In Eq. (11) the level

set function / will converge to |5/|=1 with the evolution of time.

Although this approach improves conservation of mass consider-

ably, it fails to achieve mass conservation exactly.

From the level set function the curvature

can be obtained by

the following equation:





ð12Þ

For smoothing the discontinuous physical quantities near inter-

faces, the smoothed Heaviside function is used which is given by

ð/Þ¼

0if/ < 

1 þ



sinð



if j/j

1if/ >

ð13Þ

Typically

= 1.5

is used where

represents the grid size. Then the

smoothed density and viscosity can be calculated by

ð/Þ¼

ð1  H

ð/ÞÞ þ

ð/Þð14Þ

ð/Þ¼

ð1  H

ð/ÞÞ þ

ð/Þð15Þ

The advantages of LS method are the fact that the curvature can be

computed accurately and the smoothness of discontinuous physical

quantities near interfaces is very good by using the level set func-

tion / – a smooth function.

However, the LS method produces more numerical error than

VOF method, especially when the interfaces experience severe

stretching or tearing. A common symptom of such a disadvantage

is the loss/gain of mass. To alleviate the problem of mass being not

conserved, Sussman et al. [20] have developed a constraint that

signiﬁcantly improves the accuracy of solving Eq. (11). Nourgaliev

et al. [28] have showed that reducing the spatial discretization er-

rors in Eq. (10) can improve mass conservation. The fast marching

method [29] is another technique for calculating distance func-

tions, which can improve mass conservation property. However,

it must be noted that none of the above techniques can exactly

conserve mass. This is the disadvantage of LS method [27].

On the basis of the above analysis to VOF and LS methods, it can

be found that they have the complementary advantages and disad-

vantages, so it is an inevitable trend to develop a method combin-

ing the VOF and LS methods.

In 2000 a CLSVOF method, which combines the VOF and LS

methods, was put forward by Sussman [27]. Although this method

extracts the advantages of VOF and LS methods, it is more compli-

cated than VOF or LS method. Because in this method both the C

and / advection equations need to be solved, and the level set

function needs to be coupled to the volume fraction function by

assigning the level set function to be the exact singed normal dis-

646 D.L. Sun, W.Q. Tao / International Journal of Heat and Mass Transfer 53 (2010) 645–655

tance to the reconstructed interfaces. Subsequently, improved

CLSVOF [30], MCLS [31] and ACLSVOF [32] methods were put for-

ward. Just as the original CLSVOF, all of these methods need to

solve both the C and / advection equations.

The present coupled volume-of-ﬂuid and level set (VOSET)

method is simpler than CLSVOF method and the others mentioned

above. This is because in the present method only the C advection

equation needs to be solved and the level set function is calculated

by a simple iterative geometric operation. In the following the VO-

SET method will ﬁrst be introduced in detail. Then four examples of

the incompressible two-phase ﬂow will be illustrated to compare

the VOF, LS and VOSET methods. For the simplicity of presentation,

all the derivations and computations are conducted on the uniform

grids in two-dimensional Cartesian coordinates.

2. Iterative geometric operation to calculate the level set

function

Before calculating the level set function by iterative geometric

operation, the interface shapes and locations should be ﬁrstly con-

ﬁrmed by reconstructing the interfaces. In PLIC algorithm the

interface reconstruction is based on the idea that a normal vector

together with the volume fraction function C determines a unique

linear interface cutting the cell. The normal vector to an interface is

estimated by a ﬁnite-difference formula of the following equation:

n ¼

C j

Cj= ð16Þ

There are 16 possible cases for the interface shape in the PLIC algo-

rithm. For n

> 0 and n

> 0 there exist 4 cases as shown in Fig. 1.

After the interface shape is reconstructed based on the PLIC

algorithm, the level set function /, i.e. the signed distance function,

can be calculated by an iterative geometric operation. The detailed

procedure of the operation is presented as follows:

Step 1: Set the initial value for the signed distance function / in

the whole computational domain.

i;j

M if C

i;j

 0:5

M if C

i;j

< 0:5



ð17Þ

where M is the max geometrical size of physical model as shown in

Fig. 2. This can guarantee that the accurate signed distance function

near the interfaces is included in the initial value of /, i.e. M < /

< M.

Step 2: Flag the cells near the interfaces. We ﬂag the cells in the

region of 3-

width on each side of the interfaces where

repre-

sents the grid size, as shown in Fig. 3. Such ﬂagged region is wide

enough to compute the curvature

and smooth the discontinuous

physical quantities near the interfaces. This ﬂagging step can avoid

calculating the signed distance function in the whole computa-

tional domain and thus can save computational time.

Step 3: Calculate the signed distance function / in the ﬂagged

region near the interfaces. Before calculating the signed distance

function /, we should ﬁrst compute the shortest distance d to

the interfaces for each grid point within the ﬂagged region. The

shortest distance from grid point (i, j) to the interfaces can be ob-

tained by comparing all of the minimum distances from grid point

(i, j) to any interface in cells within a 7  7 stencil around the cell (i,

j). The reason of selecting 7  7 stencil to compute the shortest dis-

tance is that the signed distance function is calculated only in the

region of 3-

width on each side of the interfaces.

Fig. 4 shows the method of calculating the minimum distance to

an interface in a cell. In the ﬁgure, the thick line segment BC de-

notes the interface. When h

>90°, AB is the minimum distance;

when h

<90°, AC is the minimum distance; when h

<90° and

<90°, the line AD vertical to the interface is the minimum

distance.

The shortest distance from grid point (i, j) to interfaces can be

obtained by comparing all of the minimum distances to any inter-

Case4Case3Case2Case1

Fig. 1. The interface shape in computational cell for n

> 0 and n

<0.

interface

-M

Fig. 2. Initial value /

of the signed distance function.

interface

flagged region

Fig. 3. The ﬂagged region near interfaces.

D.L. Sun, W.Q. Tao / International Journal of Heat and Mass Transfer 53 (2010) 645–655

647

face in cells within the 7  7 stencil around the cell (i, j), as shown

in Fig. 5. The signed distance function can be obtained by

i;j

d if C

i;j

> 0:5

0ifC

i;j

¼ 0:5

d if C

i;j

< 0:5

ð18Þ

Then the normal vector to the interface is calculated again with the

signed distance function by the following equations:

n ¼

/=j

/jð19Þ

Based on the more accurate normal vector determined by Eq.

(19), the interface is reconstructed again. Then return to Step 1. Re-

peat the geometric operation until the iteration times is equal to

the pre-speciﬁed times N. By the iteration, we can acquire more

accurate signed distance function. Usually, the iteration times N

is set as 1 for coarse mesh and 3 for ﬁne mesh.

Fig. 6 shows the signed distance function near the interfaces

with different shapes, which is calculated by the iterative geomet-

ric operation. In Fig. 6 the thick lines denote the interfaces and the

thin lines represent the contour lines of the signed distance func-

tion. From Fig. 6 we can ﬁnd that accurate signed distance func-

tions can be calculated by the proposed iterative geometrical

operation.

It is worth noting that the above-mentioned method for deter-

mining the level-set function is the major novelty of the present

paper. Because the simplicity of the proposed method the present

minimum

distance

minimum

distance

minimum

distance

θ <90

θ <90 and

θ >90 θ >90

(c)(b)(a)

(i, j )

Fig. 4. The minimum distance to an interface in a cell.

interface

shortest distance

minimum distances to any interface

in cells within a 7

× 7 stencil around

the cell (i, j )

i, j

Fig. 5. The shortest distance determined by comparing all of the minimum distances.

(a)

(b)

(c)

(d)

Fig. 6. The signed distance function near the interfaces with different shapes

calculated by the iterative geometrical operation.

648 D.L. Sun, W.Q. Tao / International Journal of Heat and Mass Transfer 53 (2010) 645–655

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