Panahi 29B6

Iranian Journal of Science & Technology, Transaction B, Engineering, Vol. 29, No. B6 Printed in The Islamic Republic of Iran, 2005 © Shiraz University

COMPARISON OF INTERFACE CAPTURING METHODS IN TWO PHASE FLOW *

R. PANAHI, E. JAHANBAKHSH AND M. S. SEIF** Dept. of Mech. Eng., Sharif University of Technology, P. O. Box 11365-9567 Tehran, I. R. of Iran

Email: [email protected]

Abstract– In two phase flow investigation, there is a need for robust methods capable of predicting interfaces, in addition to treating the traditional governing equations of fluid mechanics (Navier-Stokes Eqs.). Such methods in a finite volume approach can be classified into two typical categories called interface tracking and interface capturing methods. According to their abilities, interface capturing methods are of more interest in free surface modeling, especially when complex interface topologies such as wave breaking are included. These methods solve a scalar transport equation in order to find the distribution of two phases all over the computational domain. That is, all properties of the effective fluid will be calculated. In this paper, some existing schemes will be reviewed and two high order composite schemes will be applied on a discretised form of the volume fraction convection equation. A discussion on the performance of methods will be presented as a result of different scalar distribution convection in various velocity fields according to the program that has been developed in this manner. It will be presented to show that CICSAM (Compressive Interface Capturing Scheme for Arbitrary Mesh) can be used as a good choice for free surface simulation in practice.

Keywords– Interface modeling, two phase flow, volume methods, high resolution schemes

1. INTRODUCTION

In many engineering applications, from open channel flow [1] to near interface hydrofoil [2], there is a need to simulate the free surface. In general, the flow of two immiscible fluids can be classified into three categories based on the structure of the interface between them [3]:

• Two fluids separated by a sharp interface, e.g. flows in open channels without wave breaking. • Transitional flows, in which parts of the interface break into regions filled by both fluids, e.g. air

bubbles trapped in liquid. • Dispersed flows, in which two fluids make a suspension without a clearly defined interface, e.g. in

a rigorously shaken tank. Numerical methods, which are nearly applicable in the first two problems, can be categorized into two

main groups [3]: • Interface tracking or surface methods (Fig. 1) • Interface capturing or volume methods (Fig. 2) Interface tracking or moving mesh methods mark and track the interface with calculations only on one

phase based on the satisfaction of two conditions [4]: • Free surface is a sharp interface between two fluids and there is no flow across it (interface

kinematics condition). • Forces acting on fluid in a free surface are in equilibrium (interface dynamic condition).

∗Received by the editors May 8, 2005; final revised form November 16, 2005. ∗∗Corresponding author

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Surface methods maintain sharp interfaces for which the exact position is known throughout the calculation, but there is a great disadvantage when treating the large deformation of the free surface [5]. However this can be overcome with some modification in mesh movement algorithms [6].

In interface capturing methods, different fluids are marked with massless particles (MAC (Marker And Cell)) [7] or by volume fraction (Eq. (1)) (Fig. 2). The advantage of such methods is their ability to cope with the arbitrary shaped interfaces and large deformations, as well as rupture and coalescence in a natural way [5].

volumecontrolofvolumetotalfluidofVolume

FractionVolume1

== α (1)

Fig.1. Interface tracking method Fig.2. Interface capturing method Considering the value of the volume fraction, all properties of the effective fluid will be calculated

)1()1(

21

21

αµαµµαραρρ

−+=−+= (2)

The SLIC (Simple Line Interface Calculation) is one of the first methods that falls in the volume

methods category. The fluid distribution of a cell that contains part of the interface is obtained by using the volume fraction distribution of neighboring cells. It approximates the interface in each cell as a parallel line to one of the coordinate axes [8], and can not represent the exact interface [9].

The idea of a donor-acceptor scheme, in which the volume fraction value of a downwind cell (acceptor cell) of a cell face is used to predict the level of volume fraction transported through it during a time step [9], was the basic idea of the VoF (Volume of Fluid) scheme. The problem associated with the donor-acceptor formulation is that it causes incorrect steeping on interfaces which are aligned with flow direction. The VoF Scheme improved that formulation by including some information on the slope of the interface into the fluxing algorithm in the discretisation of the scalar transport equation (Eq. (3)) [10]. It has also been extended to three dimensions [11].

0)( =∇+∂∂ u

tαα (3)

In the same methods, this equation is used to calculate the distance of fluid to the interface (level set

methods) [12, 13]. Although the VoF scheme ensures physical volume fraction values (overall boundedness between zero and one), it doesn’t preserve local boundedness; i.e., a volume fraction value which initially lies between the values of its neighbors, does not necessarily preserve this property when advected in the absence of source or sink [5]. In addition to this problem, working with a transport equation includes numerical diffusion. Composite schemes have been introduced to solve local boundedness, in addition to less numerical diffusion and keeping a transitional area through one cell, in which all were the deficiencies of simple methods. The key issue in composite schemes is not just when to switch, but how to switch [5]. This is done by introducing a weighting factor based on the angle between the interface and the direction of the motion. The HRIC (High Resolution Interface Capturing) scheme, which is a popular composite scheme, uses an upwind differencing scheme and a special higher order

Fluid

Free Surface Fluid 1

Fluid 2

Comparison of interface capturing methods in…

December 2005 Iranian Journal of Science & Technology, Volume 29, Number B6

541

interpolation. This scheme is widely used in many applications [3, 14, 15] despite its numerical diffusion. CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) [5], which is another composite scheme, uses CBC (Convection Boundedness Criteria) [16] interpolation as the most compressive method and UQ (ULTIMATE-QUICKEST) [17], with a novel weighting factor [5].This results in less numerical diffusion.

In this paper, the two above mentioned high resolution composite schemes will be presented and differences between their switching (when and where) and their effect on convection results in analyzing various bench marks will be discussed.

2. TRANSPORT EQUATION The finite volume discretisation of the volume fraction transport equation, (Eq. (3)) is based on integration over control volume and the time step

∫ ∫∫ ∫++

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂∂ tt

t V

tt

t V

dtdVudtdVt

δδ

αα 0. (4)

If P is the centroid of the constant-volume cell, then the first term of Eq. (4) reduces to

∫∫ ∫+

++

−=⎟⎠

⎞⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂ tt

tP

tP

ttPP

Ptt

t V

VdtVt

dtdVt

δδ

δ

αααα )( (5)

where VP is the volume of the cell. Gauss theorem can be applied to the volume integration of the convection term (second term in Eq. (4)) to give

∫ ∑ ∑∂ = =

==V

n

f

n

ffffff FuAudS

1 1

.. ααα (6)

where fF is the volumetric flux at the face

fff uAF .= (7)

In order to approximate the time integral of the convection term, an assumption about the variation of variables with time should be taken, although it is possible to use any value in this interval. A general approach is to introduce a weighting parameter η between zero and one

[ ] [ ]( )∫ ∑∑+

=

+

=+−=⎟

⎟⎠

⎞⎜⎜⎝

⎛tt

t

n

f

ttff

tff

n

fff tFFdtF

δδ δαηαηα

11)1( (8)

Common choices of η are 0=η for the explicit, 1=η for the implicit and 5.0=η for the Crank-Nicholson schemes.

The Crank-Nicholson scheme, which uses a linear variation in face values over time and is second order accurate in time, introduces less numerical diffusion. Taking this into account and choosing a small enough time step, one can use the most recent value of F (variation of F can be neglected in comparison withα ) to get

PSF

tV

f

n

f

ttf

PttP α

δδ αδ

α =+∑=

++

1 21 (9)

where the source term is

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f

n

f

tf

PtP F

tV

SP ∑

=

−=1 2

1αδ

αα (10)

Eq. (10) uses the volume fraction values in cell centers as well as in face centers. That is, an interpolation must be applied. As mentioned earlier, popular interpolations such as the central differencing scheme introduce some problems, and high resolution composite schemes can resolve them in good manner.

3. HIGH RESOLUTION DIFFERENCING SCHEMES High resolution differencing schemes can be represented with different means such as TVD (Total Variation Diminishing) [18], FCT (Flux Corrected Transport) [19] and NVD (Normalized Variable Diagram) [17]. However, NVD is the most popular concept for applying boundedness criteria (face control), in a discretised form of convection equations [9].

NVD can be used to give a normal expression for Dα (volume fraction of the donor cell) and fα (volume fraction of the cell face) (Fig. 3)

UA

UDD αα

ααα

−−

=~ (11)

UA

Uff αα

ααα

−

−=~ (12)

where subscripts D, U and A represent donor, acceptor and upwind cells according to the flow direction as demonstrated in Fig.3.

Fig. 3. One dimensional control volume

Most composite differencing methods take some information about the interface slope into account with different switching functions and properly switch between high order and low order interpolation schemes, in which they use all their benefits to yield a bounded scalar field with less numerical diffusion.

Two high order differencing methods which are constructed on an NVD basis will be discussed in the next sections. a) CICSAM

All composite differencing methods switch between different schemes. The main distinction is on how they switch between these schemes. For example, original VoF switches upwind from downwind, when the angle between the interface and direction of motion is 450.

In order to reach less numerical diffusion, CICSAM uses CBC as the most compressive scheme stipulating robust local bounds on fα~ (Fig. 4), which in explicit flow calculations is represented as

⎪⎩

⎪⎨

⎧

>≤

≤≤⎭⎬⎫

⎩⎨⎧

=1~,0~~

1~0~

,1min~

DDD

DD

D

f

when

whenCCBC

ααα

αα

α (13)

Flow direction

fα

Dα AαUα



543

where ∑= ⎭

⎬⎫

⎩⎨⎧ −

=n

j D

fD V

tFC

10,max

δis the face donor cell Courant number.

Fig. 4. Explicit representation of CBC CBC does not actually preserve the shape of the interface [5], thus it is necessary to switch to another scheme which will preserve the interface shape better. CICSAM uses the UQ scheme to switch when it is needed to reach its aim, which in explicit form can be represented as:

⎪⎪⎩

⎪⎪⎨

⎧

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

><

≤≤+−+

=

1~,0~~

1~0~,8

)3~6)(1(~8min~

DDD

DCBCf

DfDf

UQf

when

whenCC

ααα

αααα

α (14)

A key issue in composite schemes is how to switch. CICSAM introduces a weighting factor

fγ (Fig.5), based on the angle between the interface and the direction of the motion to predict the normalized face value as

UQffCBCfff αγαγα ~)1(~~ −+= (15) Finally, the face volume fraction value can be calculated

ttAf

ttDff

δδ αβαβα ++ +−= )1( (16)

where

D

Dff α

ααβ ~1

~~

−

−= (17)

Fig. 5. CICSAM switching between two schemes The weighting factor fβ , which implicitly contains the upwind value Uα , carries all the information regarding the fluid distribution in the donor, acceptor and upwind cells, as well as the interface orientation relative to flow direction.

Upwind differencingInterpolation local boundedness

1

1

Dα~

fα~

CD

Angle (deg)

0 20 40 60 80 100 120 140 160 180

Switc

hing

func

tion

1 0.5 0

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In this manner, discretisation of the scalar transport equation (Eq. (3)) has been completed by substituting face values with such an interpolation. But as mentioned earlier, all basics of the CICSAM method are one dimensional in their original form, and extracting such a three dimensional scheme leads to non physical values in some cases. To overcome this problem, a correction step is added to the volume fraction calculation procedure, which changes the weighting factor fβ implicitly in order to reach physical distribution, according to mass conservation [5]. b) HRIC

This method also switches between two schemes which are upwind differencing and an especial scheme as well as

⎪⎪⎩

⎪⎪⎨

⎧

<<<

<<<

=

D

D

D

D

D

D

D

f

αα

αα

α

αα

α

~11~5.05.0~0

0~

~1

~2

~

~ ** (18)

D

D

D

D

DDfD

f

f

CCC

C

<<<<

⎪⎪⎩

⎪⎪⎨

⎧−

−+=7.0

7.03.03.0

~ 4.07.0

)~~(~

~

~ **

**

*

α

ααα

α

α (19)

where CD is donor cell Courant Number. Here, the switching function is θcos (Fig. 6) which includes the effect of two schemes in the calculation of normal face value according to the angle between the interface and the direction of the motion:

)cos1(~cos~~ * θαθαα −+= Dff (20)

Fig. 6. HRIC switching between two schemes

4. RESULTS AND DISCUSSION The test cases in this section focus on the interface advection of different shapes exposed to translation and shear flow according to CICSAM, HRIC, Hyper-C and UQ interpolations in a developed program. Both Hyper-C and UQ are simple methods, besides Hyper-C uses the CBC as its interpolation.

For the purpose of comparing, the solution error is defined as

∑

∑ −= cellsall

ii

oi

cellsall

ii

eii

ni

Vol

VolVolE

α

αα (21)

0 20 40 60 80 100 120 140 160 180

Switc

hing

func

tion

1 0.5 0

Angle (deg)



545

where nα is the calculated solution after n time steps, eα the exact solution, and oα the initial condition [5]. Two meshes are employed for these benchmarks:

• 100×100 uniformly spaced cells for ],0[, π∈yx • 200×200 uniformly spaced cells for ]2,0[∈x and ]1,0[∈y

The time step in all problems has been adjusted to get the maximum Courant number of 0.25. a) Constant oblique velocity field

The first test case is the convection of different scalar distributions with a one dimensional velocity field (u,v) = (2,1) which is constant in the whole computational domain (Fig.7).

Close-up views of the final advected shapes of circle and hollow square are given in addition to the exact scalar distribution, which is constant during advection (Figs.8 and 9). CICSAM and Hyper-C methods are of better performance in preserving the shape of the interface besides less numerical diffusion relative to HRIC. This could be predicted because HRIC uses the upwind value as one of its switching schemes that is the source of diffusion (Fig. 6).

Fig.7. Advection pattern and scalar distribution in computational domain

Fig.8. Circle distribution after 500 time steps: (1) exact

solution, (2) advection according to CICSAM, (3) advection according to Hyper-C,

(4) advection according to HRIC

Fig.9. Hollow square distribution after 500 time steps:

(1) exact solution, (2) advection according to CICSAM, (3) advection according to Hyper-C,

(4) advection according to HRIC The Hyper-C scheme represents less numerical diffusion in comparison with other schemes (Table. 1). However, in one dimensional flow, it renders wrinkles in the interface of complex flows [20]. The UQ scheme has the most numerical diffusion among interpolations. This results in serious disability to preserve the shape of the scalar field.

u

Scalar field distribution

v

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Table.1. Errors of different methods in scalar translation

Hollow square advection Circle advection Method

0.1068 0.2009 CICSAM 0.6078 0.5147 HRIC 0.0957 0.0994 Hyper-C 1.1787 1.4152 UQ

b) Shear flow

Most real two phase flows have shear velocity field. That is, it is important to evaluate the performance of such methods in the presence of shear flow, when the interface deforms considerably rather than keeping its shape.

In this section a shear velocity field is enforced onto a scalar distribution (as in Fig. 10). The capability of schemes illustrated before is tested by showing a comparison on the shape deformation and on the diffusion error measured by Eq. (21). The test is performed as follows: the initial volume fraction field is subjected to an assigned velocity field for certain time intervals; then the velocity field is changed in sign, and computation is repeated for the same time interval, in order to force the volume fraction to return to the initial condition.

The x and y components of velocity for forward rotation are

)(sin)(cos),()(cos)(sin),(

yxyxvyxyxu

ππππ

−=+= (22)

The sign of the expressions are flipped for the reverse-rotation part of the computation. The initial

position of the scalar circular scalar field is )12.0,5.0(),(ππ+

=yx with a radius of 0.2 The results just before velocities reversation and at the end of the calculation for 2000 time steps are

given in Fig. 11 for Hyper-C, HRIC and CICSAM methods. It seems that CICSAM is the most robust method in preserving actual interface deformation with

minimum numerical diffusion, the case in which HRIC has a serious disadvantage. From the numerical diffusion view point, the Hyper-C scheme is more applicable than HRIC, but it has less accuracy in interface shape preservation, which comes from its limitation to one dimensional flow rather than HRIC.

Fig.10. Scalar distribution in shear velocity field Errors have been calculated according to convected and exact scalar distributions after applying one forward and backward velocity field for 1000 and 2000 time steps according to Eq. (21) and are given in Table 2. However, the UQ method represents high numerical diffusion and is not in the order of the other methods in shear flow advection.



547

Table 2. Errors of different methods in shear flow advection

Error for 2000 time step forward followed by its

backward

Error for 1000 time step forward followed by its

backward Method

0.0589 0.0316 CICSAM 0.1377 0.0763 HRIC 0.1327 0.0467 Hyper-C

Fig.11. Results for the shear flow case (a) Hyper-C, (b) HRIC, (c) CICSAM

5. CONCLUSION

In order to treat the two phase flow, finite volume methods use different interface schemes that can be categorized as interface tracking and interface capturing methods. Interface capturing methods have some advantages in complex interface deformations like wave breaking that hold them as robust and interesting choices, although they must solve an additional scalar convection equation to give volume fraction distribution all over the computational domain that exert more computational effort. One needs a proper interpolation scheme that results in less numerical diffusion and local boundedness preservation, in addition to taking a real, well defined interface shape to apply on a discretised form of the volume fraction transport equation. CICSAM as a high order composite method sets an appropriate equilibrium between these challenges, and with its correction step, implicitly modifies nonphysical volume fraction values in a fully conservative procedure, although Hyper-C represents fewer errors in one dimensional flows.

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