Numerical simulation of risisng bubble

8/12/2019 Numerical simulation of risisng bubble

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NUMERICAL SIMULATION OF A SINGLE RISING BUBBLE BY VOF WITH SURFACE COMPRESSION 961

There are numerous papers available which demonstrate that marker and cell , VOF and LS areindeed fast and robust tools. However, as recently noticed by Hysing et al. [9], the accuracy ofthe numerical models is only seldom checked. Validation of two-phase flow simulations is oftenperformed by qualitative comparison of interface locations in CFD simulations with pictures fromcorresponding experiments, see, for example, Ferziger and Peri [10] or Yeoh and Barber [11].

Therefore, Hysing et al.[9] have proposed a quantitative benchmark case, the two-dimensional

rising bubble problem for two different set-ups. The benchmark is deduced from the numericalresults of independent CFD simulations with three different codes. Two of them are based on the LSapproach on fixed grids. The third one resolves the two-phase flow with an arbitrary LagrangianEulerian approach where the computing mesh follows the phase interface. This is often referred toas interface tracking approach. The results of the three different codes are very close to each other,which suggests that the modelling errors should be very small. Hysing et al.[9] deduced measuresfrom their results, which can be used to allow a quantitative comparison with results from othertwo-phase flow models.

The benchmark of Hysing et al.[9] has been already employed by [12] to check the accuracy ofa two-fluid approach. However, a quantitative comparison of the VOF model data with the LS andarbitrary LagrangianEulerian results in Hysing et al.[9] is missing up to now.

In the VOF approach, a phase transport equation is used to capture the interphase. There arebasically two methods for evaluating phase transport equation:

direct methods: which discretise the phase transport equations directly by using high res-olution discretisation schemes for the convection parts. Schemes specifically developed forinterface phenomena such as, for example, compressive interface capturing scheme for arbi-trary meshes [13] or high-resolution interface-capturing scheme [14]. There are no limitationsfor the underlying computing mesh for the direct methods.

reconstruction methods: where the phase transport equation is approximated typically in twosteps, a geometric interface reconstruction step and an interface propagation step. Methodsfollowing this approach include donoracceptor method [3], simple line interface calculation,piecewise linear interface construction [15] or efficient least-squares volume of fluid inter-face reconstruction algorithm [16]. All these methods are restricted to hexahedral (or quadraticin two dimensions) shaped mesh cells. The recently presented moments of fluids method byDyadechko and Shashkov [17,18] overcomes this limit and extends the reconstruction methodsto arbitrary cell shapes. However, these methods are more computational demanding.

zkan et al. [19] evaluated the two different VOF methods for a three-dimensional bubble-train flow.Their study reveals that there are some deficiencies for the evaluated direct methods (implementedin the commercial CFD-solvers STAR-CD, CFX and FLUENT).

In this paper, we evaluate the VOF approach implemented in the CFD library OpenFOAMversion1.5.1 through version 2.1.0 [20,21]. In OpenFOAM, the phase transport equation is solved directly,which is described in detail in Section 2.

The rest of the paper is outlined as follows: A short summary of the test cases and the numericalset-up of the present investigation is given in Section 3. The results of the VOF model applied to thetwo test cases are presented, compared with the findings of Hysing and co-workers [9] and discussedin detail in Section 4.

2. MATHEMATICAL MODEL AND COMPUTATIONAL METHODS

2.1. Governing equations

The rising bubble problem investigates the ascending motion of a bubble of fluid 2 that is surroundedby fluid 1. To track the two-phase flow of the rising bubble, the VOF method of Hirt and Nichols [3]is adopted. Here, a volume fraction 1is used to mark fluid 1 with the definition

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962 J. KLOSTERMANN, K. SCHAAKE AND R. SCHWARZE

The governing equations of the unsteady, laminar and incompressible flow are the equation ofcontinuity (Equation (2)) and the NavierStokes equation (Equation (3))

r Eu D 0 (2)

@

@t

.

Eu/

C r .

Eu

Eu/

D rp

C2

r h!D iC Eg C r1 (3)

Here,Euis the velocity, p is the pressure andEgis the gravitational acceleration. The mixed fluidproperties densityand viscosityare weighted by the volume fractions1and2 D 1 1of thetwo fluids

D 11 C .1 1/2, D 11C .1 1/2 (4)The rate of strain tensor is defined by

!DD 1

2

rEu C rEuT (5)

Because of the continuous surface force (CSF) approach of Brackbill et al. [22], the surface tensionforce is included as an additional source term

r1in the NavierStokes equation. The curvature

is defined as

Dr .En/ (6)the divergence of the normal vectorEnof the interface. Its definition is given in Section 2.2.2.

Solving the phase transport equation by the surface compression approach. The transport equationof each volume fraction1and 2in a incompressible two-fluid system is given by

@i

@tC r .Euii/ D 0, iD 1, 2 (7)

For the derivation of the surface compression approach, it is sufficient to consider the transportequation of the volume fraction1only

@1

@tC r .Eu11/ D 0 (8)

To solve this transport equation, the velocity Eu1 of fluid 1 is needed. In the widely used origi-nal VOF method by Hirt and Nichols [3], the velocity Eu1 is assumed to be equal to the mixedvelocityEu D Eu1

@1

@t C r .Eu1/ D 0 (9)

This is only valid if 1 is maintained as a step function throughout the domain, for example,numerical diffusion at the interface is not allowed. Furthermore, the upper boundedness 1 6 1for Equation (9) is not guaranteed because this formulation is not conservative [23].

According to Rusche [23], it was Weller who first developed and implemented a conservativeform in the CFD library OpenFOAM [20, 21]. He defined the mixedEuand the relative velocitiesEur between phases1and 2,

Eu D 1 Eu1 C 2 Eu2 D 1 Eu1 C .1 1/Eu2 (10)

EurD Eu1 Eu2 (11)Then, the addition of Equations (10) (multiplied by1) and (11) (multiplied by1 1) yields

1 Eu1 D 1 Eu C 1.1 1/Eur (12)

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids2013; 71:960982

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Finally, the relative velocity formulation of the transport of1in the surface compression approachis obtained by inserting Equation (12) in Equation (8)

@1

@tC r .Eu1/ C r Eur1.1 1/ D 0 (13)

with the explicitly fixed relative velocityEur , which is defined in Section 2.2.2. The surface compres-sion term

r

Eur1.1

1/contributes only in the region of the interface (0 < 1< 1) and limits the

smearing of the interface because of the compensation of the diffusive fluxes. The convective term inthe surface compression approach (Equation (13)) differ from the ones used in the other direct VOFmethods (e.g. compressive interface capturing scheme for arbitrary meshes [13] and [14]) where thethird term on the left-hand side (LHS) of Equation (13) is neglected.

2.2. Discretised model equations

Equations (2), (3) and (13) are integrated with the CFD library OpenFOAM [20,21]. OpenFOAM

uses the finite volume method in cell-centred formulation to solve systems of partial differentialequations on three-dimensional block-structured or unstructured meshes consisting of arbitrarilyshaped convex cells. With Gausss theorem applied to the convective and diffusive terms andusing the Euler implicit time scheme of Equation (3), the following semi-discretised system ofequations is derivedZ

VP

t

.Eun Euo/

dVCXf

f EunfDXf

fES r?fEun C0@ 1

Vp

Xf

Eun1 ES1A X

f

n1f ES

ZVP

.rpm Eg Exr C r/dV (14)

where D EufESrepresents the volumetric flux through the cell face S. The subscript./fdenotesface values, which are interpolated from the cell values indicated by ./p. The superscript ./o indi-cates previous (old) time step. The superscript ./n denotes the value of the actual iteration withinthe current time step, thus indicating implicit treatment. Respectively, the superscript ./n1 gives thevalue of the previous iteration, thus indicating explicit treatment. Note thatis always a face value,thus the subscript is omitted here. For simplification, the subscript of the volume fraction is alsoomitted, thusD 1. The operator r?fdenotes a surface normal gradient, which is the componentnormal to the cell face of the gradient. This term is part of the Laplacian term and is discretised as

followsr?fEun D EunNEunPjdj . The indexPdenotes the cell of interest, the index Nthe neighbouring

cell and jdj the distance between both neighbouring cells. The pressure p has been replaced by themodified pressurepmD p Eg ExwithExas the position vector. Its derivation is given by Rusche[23]. He states that this treatment enables an efficient numerical treatment of the steep density jumpat the interface by including the hydrostatic termEg Exrinto the RhieChow correction.

2.2.1. Segregated pressure-based solver (velocitypressure coupling). For the description of thevelocitypressure coupling, we follow the procedure of Rusche [23] and Medina [24].

Equations (2) and (3) are solved by the segregated pressure correction method, which will beoutlined in this section. If we take only the terms containing the velocities

Eufrom Equation (14), the

following system of linear algebraic equations can be defined.

A WDZVP

t.Un/dVC

Xf

fUnf

r.EuEu/

CXf

f r?fUnfCXf

Un1f

Xf

n1f j ESj r.rEuC.rEu/T/

(15)

By using the product rule, the diffusive term of the momentum equation (Equation (3) with Equation (5)) can beexpanded to:r .

rEuC

rEu/T

Dr .rEu/Cr .rEu/TCr .rEu/T Dr .rEu/CrEu r.

This time scheme is used for the demonstration of the discretisation process; for the simulations later on, theCrankNicolson time scheme is used.


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This (A) represents the full system of equations AU D b, with Abeing the momentum coefficientmatrix,Uthe vector of the unknown velocity vectorsEuandbthe vector containing the source terms.The following operators acting on A can be defined and are implemented in OpenFOAM: theD-operator AD gives the diagonal part ofA, the N-operator AN representing the lower andupper triangle (off-diagonal) part ofA, the S-operator AS extracting the source vector b andthe H-operator AHdefined by AHWD AS ANUD ADU. With the operators given earlier,the equation system A can be rewritten as .ADC AN/UD AS, which allows us to rewriteEquation (14) as

ADUn D AH C

ZVP

rpm Eg Exr C rC

tUo

d V (16)

Assembling a block matrix system of Equations (16) and (2) AD rr 0

Un

pm

D" AH

RVP

rpm Eg !r C rC tUo

d V

0

# (17)

and taking the Schur complement of it gives the pressure equation:

r .AD1/f

rpmD r

(18)

with flux predictor

D.AD1AH/fESC .AD1/fh.Eg Ex/fj ESjr?f C ./fj ESjr?f

i

source term

(19)

C .AD1/f

tUof

j ESj extra source from time derivative

The fluxis corrected by

D .AD1/fESr?fpm (20)

after the calculation of the pressure with Equation (18).This method gives an oscillation-free velocity field in line with the RhieChow correction [25],even though there is no explicit RhieChow correction.

To obey the conservation of the fluxes on the right-hand side of equations ((18), (19) and (20)),one has to assure consistent treatment of all terms because they are assembled explicit.

2.2.2. Surface compression. In OpenFOAM [20, 21], the transport equation for the volume frac-tionis given by Equation (13). By default, a relative velocityEurcannot be determined in the VOFmethod because only a single velocityEufor both fluids is considered in the whole domain. Thus,ur or rather the cell face flux r velocity has to be approximated as given in Equation (25). Thesemi-discretised form of Equation (13) is

nC1i D ni tr .Eu/ C Eur.1 / (21)Gausss theorem and integration over a control volumeVpgivesZ

VP

r .Eu/ C Eur.1 /dVDXf

./fC .r.1 //f (22)

Introducingas

WD .Eu/ C Eur.1 /fESD ./fC .r.1 //f (23)

This is the same as taking the divergence of Equation (15) with the continuity equation Equation (2)r EUn D 0.


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which results in the discretised transport equation for the phase indicatorZVP

nC1i dVDZVP

nid V tXf

(24)

The compression flux, r in Equation (23), is defined by the magnitude of the face flux velocityin the transition region .uc/fmultiplied with the normal face vector of the interface .bni/f, whichyields only the contribution normal to the interface.

rWD .uc/f .bni /f (25)The face flux velocity in the transition region .uc/fis then evaluated by

.uc/fD min

c j ESj

, j ESj

max

! (26)

which is bounded to the maximum face flux velocity

j ES j

max

in the flow field. The coefficient ccontrols the weight of the compression flux rand should be in the range of unity [24]. By choosingcD 0, the compression flux can be forced to rD 0. The contribution to the interface, which isnormal at the cell face, is derived by the scalar projection.bni/f of the normal vector of the interfacepointing from fluid 2 (D 0) to fluid 1 (D 1) onto the cell face vectorES. This term is determinedby the following inner product:

.bni /fD En ESD r?fjr?f

C bjES with b D 10

8

3

rPN

VP

N

(27)

Algorithm 1. Explicit two-phase numerical solution procedure.


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to prevent division by zero with VPdenoting the cell volume andNthe number of cells. The scalar.bni /fis also used for the determination of the curvature in Equation (6); here, the divergence termis rewritten with Gausss theorem:

Dr .En/ D Xf

.

bni /f (28)

2.2.3. Solution algorithm. The overall solution cycle is given in Algorithm 1. Here, the distribu-tion of the volume fraction field ( -subcyle at step 4) is calculated ahead of the well-known PISOalgorithm [26] (step 8) that updates the velocity and pressure fields.

The key element of the -subcyle is the solution of Equation (24). To guarantee a sharp andbounded solution, a numerical scheme designed for the multi-dimensional advection equationshould be employed (e.g. [27, 28]). For this study, the explicit multidimensional universal limiterwith explicit solution (MULES) implemented in OpenFOAM [20, 21] is used to integrate the-equation Equation (24) explicitly. Therefore, Equations (23) and (24) are solved together withthe mass fluxfi

fi

WDf

Euf

ES

D.1/f

C.1

/2f

D.1

2/f

C.2/f (29)

for a number ofNcorrcorrector steps. During this correction,r stays constant, see Algorithm 1.Finally, the mass flux is calculated according to Equation (24) as

f DNsubXiD1

t i

tfi (30)

with t i D t=Nsub. Equations (4), (6), (25) and (30) are iterated inside the so-called-subcycle. This sub cycle consists of solving the equations Nsub-times (sub time-stepping; here, alsor is updated, see in Algorithm 1) and then averaging the mass flux.

During the PISO loop, the mass flux f in the advection term is maintained constant. Once

the pressurevelocity equations are solved, the volumetric flux is updated. The overall numericalprocedure is sketched in Algorithm 1.

2.2.4. Discretisation schemes and used parameters. The applied discretisation schemes and theparameters of the numerical model are summarised in Table I. For convenience, the correspondingterminology of OpenFOAM is also given. The transient term in the momentum equation,Equation (14), is discretised by the CrankNicolson scheme [29]. Different interpolation schemesare applied to determine cell face values in the convective terms. For the convective part in themomentum equation, a total variation diminishing scheme with the flux limiter function .r/Dmax0,min.2 r , 1/is used. The smoothness parameter r is defined by the ratio of consecutive gra-dients. The van Leer [30] scheme with the flux limiter function.r/ D .rCjr j/=.1Cjr j/ is used forthe discretisation of the

r.

Eu/ term in Equation (13). Here, the flux is bounded between 0 and 1. For

the second convective partr .Eur.1 //in Equation (13), the so-called interfaceCompressionscheme is used. Here, the limiter function reads

.P, N/ D minmax

1 max

q1 4 P .1 p/,

p1 4 N .1 N/

, 0

, 1

(31)

which is bounded between 0 and 1, with the flux Pat the evaluated cell and the flux Nat theneighbour cell. In our study, we included the optional momentum predictor step that brought smallbut noticeable improvements in the results. This will be discussed later.


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Table I. Used discretisation and interpolation schemes, parameters of the numerical model.

Discretisation schemeTerm (OpenFOAM terminology) Method

@@t

.Eu/ CrankNicolson 1 CrankNicolson scheme [29]r .EuEu/ Limited linearV 1 TVD scheme, limiter function see text

r .

Eu/ vanLeer 01 TVD scheme, van Leer limiter see text

r .Eur .1 // InterfaceCompression Bounded limited scheme, limiter see textrEu, r Linear CDSr?f

Linear corrected Surface normal gradient: CDS from the neighbouringcell values with explicit correction on nonorthogonalmeshes [31]

r .rr/ Linear corrected Face values () approximated by CDS, and the resultingsurface normal gradient is evaluated as given earlierby CDS with nonorthogonal correction [31]

Term Interpolation scheme Method

./f Linear Default interpolation scheme for getting face valuesfrom cell values, for example,,and CDS fromthe neighbouring cell values

Parameter Value Notesc 1 Interface compression valid values between 0 and 1Ncorr 1 Number of corrector stepsNsub 2 Number of -sub cyclesNnonortho-corr 0 Only used for nonorthogonal meshesNcorr 15 Max. no. of PISO loopsmomPred true Momentum prediction step

TVD, total variation diminishing; CDS, central differencing scheme.The symbolstands for an arbitrary scalar or vector.

3. NUMERICAL SET-UP OF THE SIMULATIONS

3.1. Computational domain

On the basis of the benchmark definition of Hysing et al. [9], a two-dimensional computationaldomain with an aspect ratioxW yD 1 W 2is employed, see Figure 1. The bubble is initially centredat .x, y/D .0.5, 0.5/with rb0D 0.25as the initial radius. The viscosity and density of fluid 2(bubble2) are smaller than those of the surrounding fluid 1 (1). The domain is fully enclosed byno-slip walls (EuD.0, 0/) at the top and the bottom and free slip walls (uxD 0 and r?fEuD0) onthe left and the right. The gravity vectorEgpoints towards the bottom of the domain.

Hysinget al.[9] distinguished two different set-ups of the numerical experiment: ellipsoidal bub-ble (TC1) and skirted bubble (TC2). They differ by the density ratio 1=2 and the ratio of theviscosity 1=2 between both phases (index 1 for heavy and index 2 for light fluid). Physicalparameters of TC1 and TC2 are given in Table II.

All parameters in Table II are made dimensionless with characteristic scales for the lengthL D 2rb0 , for the timetD L=Ug and for the rising velocityUgD pg2rb0 . The Reynolds numberRe, the Etvs numberE oand the capillary numberC aare defined as

Re D 1UgL1

, Eo D 1U2gL

, C a D 1Ug

D E o

Re (32)

Thus, the surface tension force is more prominent for TC1 in comparison with TC2, see Table II.The simulations of TC1 and TC2 are performed with four different grid spacings h D

1=40, 80, 160, 320, respectively. The simulation time istfinalD 3with a grid size-dependent timestep oftD h=2.


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Figure 1. Simulation domain, boundary conditions and initial configuration of the rising bubble problemdue to Hysing et al. [9], length ratio x:y=1:2, gravity Eg, 1 and 2 regions of heavy and light phase,respectively, in green free surface (D 0.5), bubble centre at.x , y/ D .0.5, 0.5/and initial bubble diameter

2 rb0 D 0.5. The boundary conditions are slip (blue) and no-slip walls (red).

Table II. TC1 and TC2: Physical properties and similarity parameters.

Case 1 2 1 2 g Re Eo C a 1=2 1=2

TC1 1000 100 10 1 0.98 24.5 35 10 0.286 10 10TC2 1000 1 10 0.1 0.98 1.96 35 125 3.571 1000 100

Table III. TC1 and TC2: Simulation statistics, grid refinement 1= h, number of elementsnelements, total number of time steps nt and computation time CPU on one processor

(dual-core AMD Opteron 2.4 GHz).

1= h nelements nt CPU TC1 CPU TC2

40 3 200 240 1 min 1 min80 12 800 480 10 min 10 min

160 51 200 960 2.5 h 2.5 h320 204 800 1 920 35 h 37 h

The resulting number of elementsnelementsand time steps nt along with computing times CPUare given in Table III. The computing times are the measured wall clock times for serial simu-lations on a single 2.4 GHz dual-core AMD Opteron processor (AMD Corporation, Sunnyvale,California, USA). The computing times for both test cases differ only marginally and are compa-rable with the ones given by Hysing and co-workers [9] indicating that OpenFOAM has a similarcomputing performance.


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3.2. Benchmark quantities

The following specific values of the benchmark quantities are used for the comparison of oursimulation results with the reference data of Hysing et al.[9]:

3.2.1. Final vertical position of the centre of massYc.tD 3/. The final position of the centre of

mass is obtained from the centre of mass position EXcD .Xc , Yc/

EXcD

R1\2

ExcdAR1\2

dA (33)

Here,ExcD .xc , yc/is the centre of mass position of an individual cell in the computational mesh.

3.2.2. Minimum circularityCmin with corresponding incidence time. The minimum circularity isdeduced from the circularity

CD 2 rb0Pb

(34)

where 2 rb0 is the perimeter of the initial round bubble and Pb is the actual perimeter of thedeformed bubble. The actual perimeter depends strongly on the position of the interface. Never-theless, within the VOF method, the interface is defined somewhere in the 0 < < 1region.Theoretically, this region should tend to zero; in practice, the region is spread over a few cells sothat the surface position can vary within this region. To pinpoint the surface, we chose a valueofD 0.5.

3.2.3. Maximum rise velocityVmaxwith corresponding incidence time. The maximum rise velocityis deduced from the rise velocity V, which is evaluated for each time step as follows.

V

D R1\2

v dA

R1\2

dA

(35)

Here,v is the velocity iny-direction in an individual cell in the computational mesh.

(a)domain after patching (b)domain after relaxation

Figure 2. Distribution of both phases in the computational domain of the coarsest grid (h D 1=40), yellowcorresponds to D 1, blue to D 0 and green to 0 < < 1, situation after patching and relaxation.

(a) Domain after patching and (b) relaxation.


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3.3. Initialisation

At the beginning of the simulations, the cell values for are patched to eitherD 1for the bubbleor D 0for the surrounding fluid. An example of the staircase profile resulting from the patch-ing process can be seen in Figure 2(a); no cells with 0 < < 1 have been defined. A transientsimulation with surface tension but without gravity is employed to relax the unphysical staircaseprofile of the bubble surface. The results offrom the transient simulation with the zero gravitycondition attD 3serve as initial values for the rising bubble simulation, see Figure 2(b). However,after the relaxation, the interface is smeared across four cells no matter which grid spacing was used.

(a) (b)

(c) (d)

Figure 3. The nondimensional velocity field during initialisation and zero gravity condition indicating thepatterns of the parasitic currents for the conditions of TC1. The fields are nearly symmetric. For visualisationpurposes, the pictures have been divided: the left sides show the flow patterns indicated by vector arrows,

and the right sides show the coloured magnitude of the velocity.


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4. RESULTS OF THE BENCHMARK TEST CASES

4.1. Zero gravity condition

After the initialisation with the zero gravity condition, some first errors and uncertainties of thenumerical model can be estimated. Because of the action of the viscous forces, the momentum ofthe flow field should be damped out, and a final velocity of

Eu

D0is expected in1and 2. Thus,

the bubble should keep its initial position EXicD .0.5, 0.5/. However, small deviations from theseassumed values are found in the zero gravity simulations at tD 3. Spurious velocities are foundon both sides of the interface, which we interpret as parasitic currents. These observations are inagreement with the findings in, for example, [12, 3234] for a static viscous drop in equilibrium.The shape and the magnitude of the parasitic currents are shown in Figure 3 (for the parameters ofTC1 at t D 3). For all mesh resolutions, 16 counterrotating vortices can be identified in the flowfield of these parasitic currents. These parasitic currents arise from small deviations in the curvaturebecause of the implementation of the CSF model, see, for example, Lafaurie et al.[32].

In Figure 4, the influence ofcon the development of maxjupj (local parasitic current maximum)over time is shown. Obviously, the parasitic currents are damped for both cases with and without sur-face compression until a small but finite value maxjupj > 0. In detail, the development of maxjupjshows small dependence on the grid spacing and onc. In the case ofr

0(c

D1), a convergent

behaviour for a resolution finer than h 6 1=40is noticeable. Note that the parasitic current is pro-portional to the kinetic energy in the system. To obey energy conservation, the kinetic energy andthus the parasitic currents should approach zero because of the inclusion of viscous forces.

A snapshot of the pressure field for TC1 under zero gravity condition at tD 3is depicted inFigure 5. The pressure is normalised by the pressure jump across the surface given by the YoungLaplace equation pD 2 =rb0 resulting in p D p=pD p rb0=.2/. In the case of a staticbubble, the normalised pressure should read in 1:p D 0and 2:p D 1with a sharp pressurejump at the surface. This is not the case because the values range from p D0.83to p D 0.70(in2) for the mesh resolution ranging from h D 1=40toh D 1=320, respectively. Furthermore, atthe interface, some unphysical spikes (overshots and undershots) in the pressure field can be found.To reduce these spikes, we varied the interpolation scheme (least squares and cell-limited linearschemes) [20,21] for the rterm, which gave no improvements.

The resulting errors of the zero gravity simulations for TC1 and TC2 on the coarsest and thefinest grids are given in Table IV. The order of magnitude ofC ap remains nearly constant in bothtest cases. No grid convergence was found. Therefore, C ap 104 is determined as a measure toestimate the influence of the parasitic currents in our simulations.

However, the parasitic currents only have a marginal influence on the centre of mass positionbecause this error is much smaller than the nominal displacement length of the parasitic currents,

(a) (b)

Figure 4. Influence of compression fluxr on the parasitic currents. (a) cD 0and (b)cD 1.


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(a) (b)

(c) (d)

Figure 5. Pressure field after relaxation for initialisation of TC1. (a)h D 1=40, (b)h D 1=80, (c)h D 1=160and (d)h D 1=320.


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Table IV. The spatial errors in the zero gravity simulation attD 3for TC1 and TC2: centre of mass positionXc,Ycand difference jEXcD EXc EXic j with respect to the initial position EXic , maximum of the parasitic

current velocities maxjup.tD 3/j and their capillary numberC apD jupjmax=.

1= h X c Yc jEXc j maxjup.tD 3/j C aTC1 40 0.500011 0.499637 3.6 104 0.0112 4.6 104

320 0.500002 0.499948 5.2 104

0.00522 2.1 104

TC2 40 0.5 0.499776 2.2 104 0.00252 1.3 104320 0.500007 0.499908 9.0 105 0.00575 2.9 104

(a) t= 0 (b) t= 1.5 (c) t= 3

Figure 6. TC1: shapes of the rising bubble at different times. The bubble surface is indicated by a value ofD 0.5and fine gridh D 1=320. (a)tD 0, (b)tD 1.5and (c)tD 3.

(a)overall situation (b)detail

Figure 7. TC1: final shape of the bubble attD 3, volume of fluid simulations on the fineh D 1=320(red)and the coarse grids hD 1=40(blue) and reference solutions by Hysing et al.[9] black (TP2D) and grey

(FreeLIFE). (a) Overall situation and (b) detail.

jEXc

j max

jup.t

D3/

jt . The effects of

jEXc

jand max

jup.t

D3/

jon the results of the rising

bubble simulations are discussed later.

4.2. TC1

4.2.1. Simulation results. In Figure 6, typical results of the computed interface of the rising bub-ble for TC1 are shown. As illustrated for the stages (tD 0, tD 1.5and tD 3), the bubble getsdeformed, while staying compact as one connected region throughout the simulation. This is due tothe low viscosity and density ratios and low E oand C anumbers indicating the high influence ofthe surface tension force.

Figure 7 compares the bubble shape at the final time ( t D 3) in our simulations for thecoarsest (hD1=40) and the finest grids (hD1=320) with the corresponding results for the codes


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(a)complete simulation period (b)detail with benchmark point

Figure 8. TC1: temporal evolution of centre of mass position Yc, volume of fluid simulations on gridshD 1=40, 80, 160, 320 (coloured) and reference solutions from Hysing et al.[9] in black (TP2D) and

grey (FreeLIFE). (a) Complete simulation period and (b) detail with benchmark pointYc.tD 3/.


Figure 9. TC1: temporal evolution of circularity C, volume of fluid simulations on grids h D1=40, 80, 160, 320(coloured) and reference solutions in black (TP2D) and grey (FreeLIFE). (a) Completesimulation period and (b) detail with benchmark pointCmin.tmin).

TP2D/MooNMD and FreeLIFE|| given by Hysing et al.[9]. Note that the results of the TP2D andthe MooNMD codes are indistinguishable for TC1; thus, we use only TP2D to refer both of themin this test case. Obviously, the VOF solutions on the coarsest and the finest grids are very close toeach other, but there is a small but noticeable difference to the final bubble shape of the referenceresults, see Figure 7 (left). A detailed view at a typical section reveals that congruence betweenthe VOF simulations and the reference results seems to be unreachable, not even with much finergrid resolutions.

Figures 810 illustrate the temporal development of the quantities centre of mass position Yc ,circularityCand rising velocity Vfrom which the benchmark quantities are deduced. Again, the

results of the VOF simulations for the four-grid resolutions h D 1=40, 80, 160, 320are comparedwith the corresponding results of the codes TP2D and FreeLIFE given by Hysing et al.[9].

The temporal evolution ofYc in our simulations is nearly identical to the reference results, seeFigure 8 (left), but the close-up showing the range tD 2.53 in Figure 8 (right) shows that completeconvergence towards the benchmark results is again not achieved. On the contrary, the temporal evo-lution ofC in Figure 9 (left) andV in Figure 10 (left) shows more apparent differences between ourfindings and the corresponding benchmark data. Here, the close-ups near the minimum circularityCminin Figure 9 (right) and the maximum rising velocity Vmaxin Figure 10 (right) reveal that the

||The reference results have the following grid resolutionshD 1=640(TP2D) andhD 1=160(FreeLIFE) respectively.


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Figure 10. TC1: temporal evolution of rising velocity V, volume of fluid simulations on grids h D1=40, 80, 160, 320(coloured) and reference solutions in black (TP2D) and grey (FreeLIFE). (a) Complete

simulation period and (b) detail with benchmark pointVmax.tmax/.

Table V. TC1: the benchmark quantities minimum circularity Cmin with corresponding incidence timest .Cmin/, maximum rising velocity Vmax with t .Vmax/ and final position Yc of centre of mass at t D 3.

1= h 40 80 160 320 Benchmark Hysinget al.[9] trubeljet al.[12]

Cmin 0.8985 0.8966 0.8999 0.9044 0.9012 0.0001 0.8876t.Cmin/ 1.8500 1.9625 1.9375 1.9625 1.9 1.8915Vmax 0.2364 0.2373 0.2365 0.2348 0.2419 0.0002 0.2457t.Vmax/ 0.9250 0.9250 0.9219 0.9516 0.927 0.011 0.9235Yc.tD 3/ 1.0616 1.0655 1.0668 1.0696 1.081 0.001 1.0679

results of our simulations do not show any convergence towards the reference. Instead of this result,our simulations scatter slightly around the results of the benchmark.

Table V summarises the benchmark quantities for TC1. Minimum circularityCminand maximum

rising velocityVmax, both with the corresponding incidence timest .Cmin/andt .Vmax/, respectively,and the final position Yc of the centre of mass at t D 3for TC1 are given. For convenience, theresults of trubeljet al.[12] for TC1 are also included. The values ofh D 1=320serve as the basis.The maximum relative errors in our simulations for the grids hD 1=40, 80, 160are about 1%forCmin, Vmaxand Yc.tD 3/, 3% for t .Vmax/and 6% for t .Cmin/. Comparing our results of gridhD1=320with the benchmark data of Hysing et al.[9], the relative errors are in the order of 1%forCminand Yc.tD 3/and 3% forVmax,t .Vmax/andt .Cmin/.

As explained in the previous section, the parasitic currents induce some uncertainties in the VOFsimulations. From the findings of the zero gravity simulations for TC1, we estimate the magnitudeof the parasitic currents to be in the order ofuP 0.01. Therefore, we expect a relative uncertaintyin the flow field near the bubble interface in the order ofuP=jVj 0.01=0.25D4%. Indeed, thisis an upper limit of all the relative errors between our data and the benchmark results. So, we con-clude that the performance of our VOF model for TC1 is mainly dominated by the existence of theparasitic currents.

4.2.2. Influence of the model parameters.

Compression flux. For comparison, some simulations with zero compression fluxrD 0have beencarried out to reveal its influence on the sharpness of the surface and the temporal evolution ofthe circularity.

In Figure 11, the influence of r on the surface is depicted. Two grid resolutions (h D1=320and hD 1=40) are investigated. For both grid resolutions, a clear smearing of the surface


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(a) = 1/40

(b) = 1/320

Figure 11. Influence of compression flux r on the surface, rD 0 (left) and rD 1 (right), for twodifferent grid instances: (a)h D 1=40and (b)h D 1=320.

Figure 12. Influence of compression term onC.

can be identified in the case ofrD 0. In the case ofrD 1, the smearing is clearly reduced for thecoarse (h D 1=40), and for the fine grid (h D 1=320), it is invisible.

The influence on the evolution ofC shows exemplarily the influence of the compression flux,see Figure 12. In the case of the coarse mesh (hD1=40), the case ofrD 0 leads to a divergingbehaviour attD 3. For the fine mesh case (h D 1=320), a larger discrepancy to the reference solu-tions betweentD 1andtD 2.5can be distinguished, whereas fort > 2.5, the circularity convergestowards the reference solutions.


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(a)Crank-Nicolson scheme (b)backward scheme

Figure 13. Influence of time step and time discretisation scheme on the evolution of C for the gridh D 1=160. (a) CrankNicolson and (b) backward schemes.

Figure 14. Influence of the momentum predictor on the evolution ofC.

Time schemes and time step. The time step (dtD 1=160, 1=320, 1=640) and the time discretisa-tion scheme (CrankNicolson and backward) have been varied for TC1, with hD1=160to checkfor their influence. The temporal evolution ofC was chosen exemplarily to show the negligibleinfluence of these variations, see Figure 13. The choice of the time discretisation scheme had noinfluence on the evolution ofC.

Momentum predictor. In Figure 14, the influence of the momentum predictor step (seeSection 2.2.3) on the temporal evolution ofCfor the case TC1,hD1=40and 1=320is depicted.In the low-resolution case (h D 1=40), influence of an additional momentum predictor step(momPred

D 1) on the temporal evolution is negligible. For the finer resolution (h

D 1=320),

we see that omitting the momentum predictor leads to larger deviations; hence, all simulations arecarried out with the additional momentum predictor step.

4.3. TC2

Figure 15 shows typical results for the rising bubble in TC2 at three different stages (tD 0, tD1.5andtD 3). The distortion of the interface in TC2 is considerably larger than in TC1. The bubbleis deformed to a skirted bubble and eventually forms thin filaments with smaller satellite bubblesthat are detached. Compared with TC1, lower surface tension forces indicated by the higher Eoand C a numbers as well as higher viscosity and density ratios are the reasons for the formationof a skirted bubble. The correct simulation of these physics is especially challenging as there is


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(a) = 0 (b) = 1.5 (c) = 3

Figure 15. TC2: shapes of the rising bubble at different times, bubble surface indicated by a value ofD 0.5and simulation on gridh D 1=320. (a)tD 0, (b)tD 1.5andtD 3.

(a)overall situation (b)in detail

Figure 16. TC2: final shape of the bubble at tD 3, volume of fluid simulations on fine hD 1=320(red)and coarse grids hD1=40(blue) and reference solutions from Hysing et al.[9] in black (TP2D) and grey

(FreeLIFE). (a) Overall situation and (b) in detail.


Figure 17. TC2: temporal evolution of centre of mass position Yc , volume of fluid simulations on gridshD 1=40, 80, 160, 320 (coloured) and reference solutions from Hysing et al.[9] in black (TP2D) and

grey (FreeLIFE). (a) Complete simulation period and (b) detail with benchmark pointYc.t3/.

no unique numerical result for t > 2available. For YC and V , the difference is less pronouncedas forC.

In Figure 16, the bubble shapes at the final time (t D 3) of our simulations for the coarsest(hD 1=40) and the finest grids (hD 1=320) with the corresponding results of the codes TP2Dand FreeLIFE are compared. Note that we omit the results of MooNMD code because the applied


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(a)complete simulation period (b) detail with benchmark point

Figure 18. TC2: temporal evolution of circularity C, volume of fluid simulations on grids h D1=40, 80, 160, 320(coloured) and reference solutions in black (TP2D) and grey (FreeLIFE). (a) Complete

simulation period and (b) detail with benchmark pointCmin.tmin/.

(a)complete simulation period (b) with benchmark points

Figure 19. TC2: temporal evolution of rising velocity V, volume of fluid simulations on grids h D1=40, 80, 160, 320(coloured) and reference solutions in black (TP2D) and grey (FreeLIFE). (a) Completesimulation period and (b) with benchmark pointsVmax1.tmax1/, Vmax2.tmax2/.

arbitrary LagrangianEulerian method does not allow for bubble breakup [9]. However, our solu-tions on the finest grid are very close to the final bubble shape of the reference calculations, seeFigure 16 (left). The close-up at a range between xD 0.65 and xD 0.75 reveals that congru-ence between our VOF simulations and the reference result of FreeLIFE is achieved at this gridresolution, see Figure 16 (right).

The temporal development of the quantitiesYc ,CandVis shown in Figures 1719.Similar to TC1, the evolution ofYc in our simulations is nearly identical to the reference results,

see Figure 17 (left). The close-up of Figure 17 (right) confirms the good agreement between ourresults on gridh D 1=320and the reference results; even so, convergence, that is a grid-independentsolution, is again not achieved.

Contrary to the findings for TC1, the evolution ofCin Figure 18 (left) and V in Figure 19 (left) ofour simulations are also closer to the corresponding results of FreeLIFE. Please note the differencesin the benchmark results: in TP2D, there is a separation of satellite bubbles, whereas in FreeLIFE,the bubble remains skirted with elongated filaments. This difference between TP2D and FreeLIFEis documented by the very different temporal evolution ofC.

In our simulations, the rising bubble also remains skirted, although the separation of the satellitebubbles is found close to the final situation at tD 3. The close-ups near the minimum circularityCmin in Figure 18 (right) and the maximum rising velocities Vmax1and Vmax2in Figure 19 (right)confirm that the results of our simulations approximate the results of FreeLIFE, although completeconvergence is again not achieved.


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Table VI. TC2: the benchmark quantities minimum circularity Cmin with corresponding incidence timest .Cmin/, maximum rising velocitiesVmax1/2witht .Vmax1/2/and final positionYcof centre of mass attD 3.

Benchmark1= h 40 80 160 320 (FreeLIFE) Hysinget al.[9]

Cmin 0.5000 0.4999 0.5051 0.4945 0.4647t .Cmin/ 3.0000 2.8125 2.9875 2.9500 3.0000

Vmax1 0.2388 0.2441 0.2431 0.2474 0.2514t .Vmax1/ 0.6875 0.7000 0.7250 0.7156 0.7281Vmax2 0.2134 0.2240 0.2302 0.2353 0.2440t .Vmax2/ 1.7000 1.8688 1.9594 2.0047 1.9844yc.tD 3/ 1.0862 1.1014 1.1089 1.1218 1.1249

Table VI summarises the benchmark quantities minimum circularity Cmin and maximum ris-ing velocitiesVmax1,Vmax2, with the corresponding incidence times t .Cmin/,t .Vmax1/and t .Vmax2/,respectively, and the final positionYc of the centre of mass at tD 3for TC2. The relative errors ofour simulations decrease for an increasing grid resolution** up to the grid resolution ofh D 1=160,ifh D 1=320is taken as a basis. Nevertheless, complete convergence has not been achieved with thefinest grid. Comparing our results of gridh

D1=320with the benchmark data of FreeLIFE in [9],

the relative errors are below 1% forYc.tD 3/, around 2% fort .Cmin/,Vmax1,t .Vmax1/andt .Vmax2/,around 4% forVmax2and around 6% forCmin.

From the findings of the zero gravity simulations for TC2, we estimate the magnitude of the para-sitic currents to be in the order ofuP 0.005. Therefore, we expect a relative uncertainty in the flowfield near the bubble interface in the order ofuP=jVj 0.005=0.25D2%. We see that this valueis again an upper limit of most of the relative errors between our data and the benchmark results.Similar to TC1, we conclude that the performance of our VOF model for TC2 is also dominated bythe existence of the parasitic currents.

5. CONCLUSIONS

We present a numerical model for the rising bubble problem. The numerical model is based on thevolume of fluid method. The CSF approach is employed to describe the influence of the surfacetension. Surface compression is used to sharpen the resolution of the bubble interface. The modelequations are solved with the open-source CFD library OpenFOAM. The implementation of thesurface compression algorithm into OpenFOAM is described in detail.

Two different realisations of the rising bubble problem are investigated with the numerical modelwith four different grid resolutions. Both test cases differ in the Reynolds number Re, the EtvsnumberE o, the density and the viscosity ratios. Therefore, the final shape of the rising bubbles isellipsoidal in one and skirted in the other test case. These two test cases have been also investigatedby several other groups in the past. Their data for the circularity, the centre of mass and the risingvelocity of the rising bubbles, serve as benchmark values for the VOF simulations.

The comparison of our results with the benchmark data shows that the proposed model canresolve the bubble behaviour in reasonable agreement with the benchmark. In the case of the ellip-

soidal bubble (TC1), our results deviate from the reference results of the benchmark data. It is veryimportant to note that TC1 results do not show any grid convergence. This is especially criticalbecause a grid-converged solution is commonly used if there is no benchmark (or experimental)result available.

In the case of the skirted bubble (TC2), physics and the associated bubble shape are more com-plex. Here, correct simulation of the formation of the shape and detachment of satellite bubbles is

**Nevertheless, the relative error is not decreasing monotonously, see, for example,Vmax1(h D 1=80and h D 1=160).A reason for this might be the nonlinear temporal evolution of the parasitic currents. Here, much more researchis necessary.


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especially critical and is still a challenge. As seen in the paper by Hysing et al. [9], a unique bench-mark solution does not exist for TC2. For this test case, we were able to show grid convergencecompeting with the results given by [9].

In both cases, parasitic currents are observed in the flow field. These currents result from thenumerical realisation of the CSF approach. We show that the magnitude of the parasitic currentsdefines an upper measure for the uncertainty of the VOF simulations. The presented VOF approach

shows deficiencies with cases that are surface force dominated (Ca < 1/as was the case for TC1.The study of zkan et al.[19] indicates that such problems occur especially for VOF methods thatsolve the phase transport equations directly (direct methods). However, we did not experience theseproblems, such as the mentioned solver divergence problems. Although in our study, the capillarynumber is one order of magnitude higher, the deviations are much smaller compared with theirstudy. We conclude that the surface compression approach is a more improved method comparedwith high-resolution interface-capturing scheme approach investigated by zkanet al. [19].

The advantages of the presented VOF approachare are

1. boundedness of the volume fraction 0 6 6 1,2. mass conservation without any reinitialisation step (as in LSs),3. maintenance of sharp interfaces through the surface compression approach,4. the surface compression approach gives better results compared to the ones without

surface compression.Nevertheless, on the basis of the results presented here, we propose that the used surface com-

pression approach should be further improved in future versions of the VOF solver. It remains anopen question whether the source of these deviations for surface-tension-dominated flows comesfrom the direct VOF method, from the CSF method or from the interaction of both methods. Forfurther improvements, we suggest the following:

1. The implementation of the CSF approach should be refined to minimise the parasitic currents.Or alternative methods, such as the continuous surface stress method, should be tested.

2. The improvement of the determination of the curvature (approximation of the interfaceposition) and the correct determination of the pressure jump needs to be addressed.Here, a reconstruction method, for example, moments of fluid method [17, 18] could givesome improvements.

The results presented in this paper should raise the awareness to use the direct VOF method,currently implemented in OpenFOAMversion 1.5.1 through version 2.1.0 [20, 21], cautiouslyfor surface-tension-dominated flows (e.g. free surface liquid metal flows). Here, we recommendcareful validation with appropriate reference solutions (results from experiments and publishedbenchmark results).

ACKNOWLEDGEMENTS

This work presents a part of the results obtained under contract number SFB 799 C1 sponsored by theDeutsche Forschungsgemeinschaft (DFG), Germany.

REFERENCES1. Maiwald A, Scheller P, Brcker C, Schwarze R. Flow-induced emulsification of mold powder slag into liquid steel.

STEELSIM2009 3rd International Conference Simulation and Modelling of Metallurgical Processes in Steelmaking ,Leoben, 8.-10. September, 2009; 162167.

2. Harlow F, Welch J. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.Physics of Fluids1965; 8(12):21822189.

3. Hirt CW, Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries.Journal of ComputationalPhysics1981; 39:201225.

4. Osher S, Sethian JA. Fronts propagating with curvature-dependent speedalgorithms based on HamiltonJacobiformulations.Journal of Computational Physics1988; 79(1):1249.

5. Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow.Journal of Computational Physics1994; 114(1):146159.


DOI: 10.1002/fld


23/23


6. Rider WJ, tracking K, DB. Reconstructing volume.Journal of Computational Physics1998; 141(2):112152.7. Scardovelli R, Zaleski S. Direct numerical simulation of free-surface and interfacial flow.Annual Review of Fluid

MechanicsJan 1999; 31(1):567603.8. Sethian J, Smereka P. Level set methods for fluid interfaces.Annual Review of Fluid Mechanics 2003; 35:341372.9. Hysing S, Turek S, Kuzmin D, Parolini N, Burman E, Ganesan S, Tobiska L. Quantitative benchmark computations

of two-dimensional bubble dynamics.International Journal for Numerical Methods in Fluids 2008; 60:12591288.10. Ferziger JH, Peric M. Computational Methods for Fluid Dynamics. Springer: Berlin, Germany, 2002.

11. Yeoh GH, Barber T. Assessment of interface capturing methods in computational fluid dynamics (CFD) codesacase study.The Journal of Computational Multiphase Flows Jun 2009; 1(2):201215.12. trubelj L, Tiselj I, Mavko B. Simulations of free surface flows with implementation of surface tension and interface

sharpening in the two-fluid model.International Journal of Heat and Fluid Flow 2009; 30:741750.13. Ubbink O. Numerical prediction of two fluid systems with sharp interfaces.PhD Thesis, Imperial College, University

of London, 1997.14. Muzaferija S, Peric M. Computation of free-surface flows using the finite-volume method and moving grids.

Numerical Heat Transfer Part B-FundamentalsDecember 1997;32(4):369384.15. Youngs D. Time-dependent Multi-material flow with Large Fluid Distortion. Academic Press Inc:

(London) Ltd, 1982. http://www.scopus.com/inward/record.url?eid=2-s2.0-0019936463&partnerID=40&md5=f1dbae4707c3493b06e31cc4da893bb7, cited By (since 1996) 19.

16. Popinet S. An accurate adaptive solver for surface-tension-driven interfacial flows.Journal of Computational Physics2009; 228(16):58385866. http://www.scopus.com/inward/record.url?eid=2-s2.0-67649472402&partnerID=40&md5=6fcd38f15413e444be2e8a6c13465c45, cited By (since 1996) 35.

17. Dyadechko V, Shashkov M. Moment-of-fluid interface reconstruction. Technical Report, Los Alamos National

Laboratoty, P.O. Box 1663, Los Alamos, NM 87545, 2005.18. Dyadechko V, Shashkov M. Reconstruction of multi-material interfaces from moment data. Journal of Computa-tional Physics2008; 227(11):53615384. http://www.sciencedirect.com/science/article/pii/S0021999107005748.

19. zkan F, Wrner M, Wenka A, Soyhan H. Critical evaluation of CFD codes for interfacial simulation of bubble-train flow in a narrow channel. International Journal for Numerical Methods in Fluids 2007; 55(6):537564.DOI: 10.1002/fld.1468. http://www.scopus.com/inward/record.url?eid=2-s2.0-35248859711&partnerID=40&md5=b1e4b01a78a0402c575caf2d0f551eae, cited By (since 1996) 10.

20. Jasak H, Gschaider B, Nilson H, Beaudoin M, et al. Openfoam-1.5-dev extend project on sourceforge. http://openfoam-extend.svn.sourceforge.net/viewvc/openfoam-extend/trunk/Core/OpenFOAM-1.5-dev/;2009.

21. OpenCFD Limited. User guide openfoam 1.6.Technical Report. http://www.opencfd.co.uk/openfoam;2009.22. Brackbill JU, Kothe DB, Zemach C. A continuum method for modeling surface tension.Journal of Computational

Physics1992; 100(2):335354.23. Rusche H. Computational fluid dynamics of dispersed two-phase flows at high phase fractions.PhD Thesis, Imperial

College of Science, Technology and Medicine, London, England, 2002.24. de M, P B R. Study and numerical simulation of sediment transport in free-surface flow.PhD Thesis, University of

Mlaga, July 2008.25. Rhie CM, Chow LW. Numerical study of the turbulent flow past an airfoil with trailing edge separation.AIAA1983;21:15251532. http://ci.nii.ac.jp/naid/80001467460/en/.

26. Issa RI. Solution of the implicitly discretised fluid flow equations by operator-splitting. Journal of ComputationalPhysics1986; 62(1):4065.

27. LeVeque RL.Finite Volume Methodes for Hyperbolic Problems. Cambridge University Press: Cambridge, UK, 2002.28. Hirsch C. Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid

Dynamics, 2nd ed. Butterworth-Heinemann: Oxford, UK, 2007.29. Crank J, Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the

heat-conduction type.Advances in Computational MathematicsDecember 1996;6(1):207226. http://dx.doi.org/10.1007/BF02127704.

30. van L, B. Towards the ultimate conservative difference scheme. IV. monotonicity andconservation combined in asecond order scheme.Journal of Computational Physics1974; 14:361370.

31. Jasak H. Error analysis and estimation for the finite volume method with applications to fluid flows. PhD Thesis,Imperial College, University of London, 1996.

32. Lafaurie B, Nardone C, Scardovelli R, Zaleski S, Zanetti G. Modelling merging and fragmentation in multiphaseflows with surfer.Journal of Computational Physics1994; 113(113):134147.33. Harvie D, Davidson M, Rudman M. An analysis of parasitic current generation in volume of fluid simula-

tions.Applied Mathematics ModelOctober 2006; 30(10):10561066. http://www.sciencedirect.com/science/article/B6TYC-4HC6M40-2/2/aba2563fc39451c85a3adf331f9c1816.

34. Sussman M, Smith KM, Hussaini MY, Ohta M, Zhi-Wei R. A sharp interface method for incompressible two-phaseflows.Journal of Computational Physics2007; 221(2):469505.

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