Modeling of bubble coalescence and break-up in

turbulent bubbly flow

R.V. Mukin∗

Paul Scherrer Institute, 5232, Villigen PSI, Switzerland

Abstract

This paper deals with the issue of modeling isothermal mono- and polydis-perse bubbly flows in the framework of Eulerian approach, based on theDiffusion Inertia Model (DIM) coupled with method of δ-function approx-imation. The model takes into account the interphase momentum transferdue to drag and non-drag forces, bubble break-up and coalescence processesas well as dependency of the relative velocity, drag and lift coefficients on thebubble diameter.

In order to take into account polydispersity effect, a new model of bubblebreak-up and coalescence is presented in comparison to a well-known modelby Yao and Morel (2004). The considered models are based on the mecha-nism of exchanging energy between turbulent eddies and interfacial area ofa bubble.

The study starts with validation of the expressions for interfacial forcesbased on simulation of monodisperse bubbly flows. This has allowed sep-arating polydispersity effect and carefully analysing the role of interfacialforces, acting on bubbles. The presented model for the polidisperse bubblyflow has been validated by means of comparison to experiments MT-Loop(where the tube diameter is 50.8 mm) and TOPFLOW (with the tube diam-eter of 195 mm), providing an adequate agreement with experimental data.All numerical calculations were performed in an open-source CFD packageOpenFOAM.

Keywords: Polydisperse bubbly flow; Coalescence; Break-up; MT-Loop;TOPFLOW; Diffusion Inertia Model.

∗Corresponding author. Tel.: +41 5631 044 29Email address: roman.mukin@psi.ch (R.V. Mukin)

Preprint submitted to Int. J. Multiphase Flow February 26, 2014

1. Introduction

Turbulent bubbly flows occur in many technical processes and industrialfacilities. Particularly wide application they have in nuclear industry. Mod-eling of bubbly turbulent flows is of a great practical interest as well and,therefore quite a number of publications is devoted to it. Predictions of bub-ble distribution (void fraction) and other parameters of the two-phase flowin channels of a nuclear reactor can be crucial to the safety analysis.

The complexity of bubbly turbulent flows modeling is associated with alarge number of various physical phenomena involved, such as heat transfer,phase transitions, bubble coalescence and break-up, etc. In general, all ap-proaches in modeling of gas-liquid two-phase flows can be divided as follows.First one is based on the mixture formulation, in which mixture momentumequation are used to describe the two-phase flow. This approach is relativelysimple and can decrease computational cost, however, it is difficult to obtainthe relative velocity correlations, which can appropriately and simultane-ously represent the dynamic behaviors of both phases. Second, Euler-Eulerapproach (for instance, two-fluid and volume of fluid models) is based on theconcept of inter-penetrating continua. In this approach the two phases areconsidered separately, such that dynamic interactions between the phases canbe taken into account, using a separate momentum balance for each phasewith interfacial transfer terms appearing in each of the equations. And thelast approach, named Euler-Lagrange, is based on treating the liquid phaseas a continuum and tracking each individual bubble using the Newtonianequations of motion containing correlations for the force terms acting oneach bubble.

Even though in Euler-Lagrange methods treating of the dispersed phaseis more natural, they are still computationally too expensive. This explainsEuler-Euler models (e.g. Ishii and Mishima (1984)) being widely used forcomplex or industrial applications. Modeling of an isothermal monodispersebubbly turbulent flow in pipes based on the two-fluid approach is very welldeveloped in the papers by Lopez de Bertodano et al. (1994); Hill et al.(1995); Chahed et al. (2003); Troshko and Hassan (2001a); Kashinsky et al.(2006). However, the gas phase, as a rule, is a polydisperse system consist-ing of bubbles of different sizes. The most common way to calculate sucha polydisperse medium is based on the decomposition of the entire system

2

of bubbles into groups (classes), called the class method (CM), followed bymodeling of the transport of mass, momentum and heat for each group. Inthis case, the two-fluid approach becomes multi-fluid. Obviously, the compu-tational resources and time required should be substantially increased duringthe transition from modeling monodisperse bubbly flow to simulation of flowwith polydisperse bubbles. Therefore, in order to reduce the number of solvedequations usually some simplifying assumptions are applied. For example, toreduce the number of balance equations, several groups by size are combinedinto a single velocity group (Krepper et al., 2008), and the authors of Yeohand Tu (2006) even assumed that the bubbles of all groups have the samevelocity. Evidently, such assumptions lead to a significant accuracy reduc-tion of the calculated results not only in terms of bubble velocity, but alsoof their concentration.

In the papers by Piskunov and Golubev (1999); Avetisyan et al. (2002);Marchisio and Fox (2005); Passalacqua and Fox (2011); Vikas et al. (2011) theequivalent methods of modeling the spectrum evolution of particle size, basedon an approximation of a continuous distribution as a sum of δ-functions ormoments, as it is called in Marchisio and Fox (2005); Passalacqua and Fox(2011); Vikas et al. (2011), are presented. Extensive comparison of thismethod to CM has been done in Selma et al. (2010), where a rectangularbubble column is studied. It should be noted that the approximation ofthe continuous spectrum by δ-functions, i.e. polydispersed particle systemrepresentation, as a sum of moments, is used very much like in the frameworkof the Lagrangian approach, where the equations of motion and heat andmass transfer to the dispersed phase are integrated along individual particletrajectories. The methods in the previously mentioned papers are adapted tothe Eulerian continuum approach for two-phase flows. As it is shown there,the method of δ-approximation is very effective in application to variousphysical problems with coalescence, break-up, nucleation, condensation orevaporation of the dispersed phase and requires a small number of groupsin size compared to other known methods of polydisperse flow modeling, seeAvetisyan et al. (2002); Selma et al. (2010); Morel et al. (2010).

There are several physical phenomena responsible for changes in size andshape of bubbles in two-phase liquid flow: break-up and coalescence, gascompressibility, phase transitions and deformation of bubbles as a result ofinterfacial forces. Only processes of coalescence and break-up are taken intoaccount here, which are the main mechanisms leading to a polydisperse dis-tribution of bubble sizes. Obviously, as a result of deformation, coalescence

3

and other processes, the spherical shape of bubbles is not preserved. But forsimplicity, assuming in the present paper that the bubbles remain sphericalin shape, allows considering only the change in size of spherical bubbles as aresult of the processes of coalescence and break-up.

In Zaichik et al. (1997a,b) DIM is proposed to describe the dispersionand deposition of low-inertia particles in turbulent flows. The model is basedon the kinetic equation for the probability density function of the velocitydistribution of particles (Zaichik and Alipchenkov, 2007) in turbulent flowand is valid for two-phase flows with particle density much greater than thedensity of the carrier flow. The idea of the model is to express the particlevelocity through the load characteristics of a continuum and thus to reducethe problem of simulating transport of the dispersed phase to the solutionof the diffusion type equation for the concentration of particles. In Zaichiket al. (2004) a generalization of DIM on turbulent flows to calculate thedistribution of monodisperse bubbles in a vertical tube is made.

The use of DIM allows, instead of solving the momentum balance equa-tions, to express the particle velocity in terms of characteristics of each group,bearing a continuum, and thus to reduce the problem of modeling dispersedphase transport to the solution of diffusion equation for the particle concen-tration. As a result, it leads to reduction in computing costs. Note thatin the case of bubbly flows using DIM does not lead to loss of accuracy indescribing the void fraction distribution. This is explained by the fact, thatthe dispersed phase (bubbles) is low-inertial and quick to react to the fluctu-ations in the turbulent flow. This forms the conditions of DIM applicability,see Zaichik et al. (2004, 2010a). Another distinctive feature of DIM is theexpression for the relative velocity between dispersed phase and current flow,which takes into account the polydispersity of the first phase and dependson local parameters of the flow.

An experimental study of bubbly turbulent flows in vertical channels isdescribed in a large number of papers (e.g. Kashinsky et al. (2006); Serizawaet al. (1975); Wang et al. (1987); Liu and Bankoff (1993)), providing themeasured distributions of the volume concentration of gas, liquid and gas ve-locities and turbulent characteristics of the liquid. However, the informationabout the structure of the flow would be incomplete if no experimental dataon the distribution of the bubble diameter or interface density is presented.Such information has been obtained by the authors of Hibiki and Ishii (1999);Tay-Jian and Liu (1997); Shawkat et al. (2008). In the papers Prasser et al.(1998, 2001) a method is developed for measuring the location and size of

4

bubbles in a section of the channel using a grid of sensors, based on whichthe MT-Loop and TOPFLOW facilities have been set up to study two-phaseflows. The experimental data, obtained from these plants, allowed construct-ing a detailed database on the characteristics of gas-liquid flows in bubblyand slug regimes (Lucas et al., 2005, 2010a,b). These experimental data areused to adjust (calibrate) and validate various multi-fluid models (Krepperet al. (2008, 2005); Frank et al. (2008); Liao et al. (2011)). Our validation ofDIM for the calculation of polydisperse bubbly flows is also based on theseexperimental data: Lucas et al. (2005, 2010a,b).

The paper starts from presenting governing equations of DIM for the poly-dispersed system of particles (here and after meaning, the term ”particles”for aerosols, droplets or bubbles) under an arbitrary ratio of the densities ofthe dispersed and continuous phases in Sec. 2.1. Analysis of expressions forinterfacial forces is performed in Sec. 2.4. The models for bubble coalescenceand break-up are presented and analyzed in Sec. 3.

Validation analysis is divided into two successive parts. In the first partvalidation of the expressions for interfacial forces, based on simulation ofmonodisperse bubbly flows, is presented. Monodisperse flow is a simplifiedapproach meaning that all the bubbles have the same diameter, in otherwords single size approach. It has allowed us to separate polydispersity effectand carefully analyse the role of interfacial forces acting on bubbles in theabsence of phase transitions.

The second part is validation of coalescence and break-up models for poly-dispersed flow by means of comparison with experimental data for bubblyflow in vertical tubes. Validation in tubes with diameter 50 mm has beenseparated in two subsections. First we studied experiments performed byHibiki et al. (2001), where a size distribution of small bubbles was comparedin a pipe cross-section. Typical size of bubbles here were not more than 3.6mm. Relatively larger sizes of bubbles have been studied in the next subsec-tion (Sec. 5.2), where comparison with MT-Loop experiments is presented.Behaviour of turbulent bubbly flow in large diameter tube, i.e. simulationof TOPFLOW experiments with inner tube diameter 195 mm, is presentedin Sec. 5.3. Note that, in pipes starting with certain diameter, slug bubblescannot exist because of Taylor instability of bubble interface. The charac-teristics of the two-phase flow in large pipes are much different from thosein small pipes. Therefore validation of presented models for bubbly flow in alarge tube is required also. Considering pipes with sufficiently large diame-ter is a topic of interest in Boiling Water Reactors, where natural circulation

5

serves to increase the reactor safety. This design includes a large chimneysection above the reactor, which provides natural circulation through the re-actor core. And in case of accidental event it could help reactor to cool itself.A summary of the study can be found in Sec. 6.

2. Model development

2.1. Diffusion Inertia Model

In the method of δ-approximation a polydisperse ensemble of particles ismodeled by a system of monodisperse groups, where continuous probabilitydensity function of mass distribution of the dispersed phase is approximatedby the sum of δ-functions:

P(m) =A∑α=1

Nαδ(m−mα), (1)

where mα ≡ Mα/Nα is an average weight of α-group of particles, definedas the ratio of the mass concentration Mα of particles in α-group to theirnumerical concentration Nα; A is the number of groups of the particle spec-trum. The number and mass of all particles per unit volume according toEq.(1) are defined by:

M =A∑α=1

Mα, N =A∑α=1

Nα.

The equations for Nα and Mα in DIM have the form:

∂Nα

∂t+∂NαWi

∂xi+

∂

∂xi

{τpαNα

1 + `

[(1− A)

(gi −

DWi

Dt

)+ FLαi + FWαi

]}=

=∂

∂xi

[1

1 + `

(DT

∂Nα

∂xi+Nα

∂qαDT

∂xi

)]+ SNα,

(2)∂Mα

∂t+∂MαWi

∂xi+

∂

∂xi

{τpαMα

1 + `

[(1− A)

(g i −

DWi

Dt

)+ FLαi + FWαi

]}=

=∂

∂xi

[1

1 + `

(DT

∂Mα

∂xi+Mα

∂qαDT

∂xi

)],

(3)

6

DWi

Dt=∂Wi

∂t+Wj

∂Wi

∂xj, qα =

τpα (fpα − A)

TL, A =

(1 + CA) ρfρp + CAρf

,

` =M

(1− Φ) ρf, Φ =

A∑α=1

Φα =M

ρp, Φα =

Mα

ρp,

here Wi is the average velocity of two-phase flow; τpα, qα, Φα are the parti-cle relaxation time, eddy migration coefficient and volume concentration ofparticles of α-group; gi is the acceleration due to gravity; TL is Lagrangianintegral time scale of turbulence; fpα is a coefficient of involving particles intoturbulent flow; DT ≡ νT/ScT is a coefficient of turbulent diffusion of inertia-less mixture; νT is an eddy viscosity; ScT is a turbulent Schmidt number; mis a ratio of mass concentration of the dispersed and continuous phases; CAis an added mass coefficient; ρf , ρp are densities of the continuous and thedispersed phase; Φ is a volume concentration of the dispersed phase. Thequantities FLαi and FWαi denote respectively the transverse accelerations offorces due to the gradient of the averaged velocity (the so-called lift force)and the effect of flow limiting by the wall (wall lubrication force). Left-handside terms in Eqs. (2) and (3) describe the change in numerical and massconcentrations of particles as a result of convective transport with mean flowvelocity and action of various forces (gravity, inertia, and lift forces).

The first terms in the right-hand sides of Eqs. (2) and (3) characterize theprocesses of turbulent transport: turbulent diffusion and turbulent migration(turbophoresis). The last terms in Eqs. (2) and (3) indicate the source ofnumerical concentration of particles due to polydispersity, coalescence andbreak-up in the absence of phase transitions.

2.2. Governing equations for two-phase flow

In the considering model, instead of equations for a continuous medium,usually used in the two-fluid (or multi-fluid) approach, we use the equationsfor a mixture of two-phase flow characteristics. The mass balance equationhas the following form for two-phase flow:

∂ρ

∂t+∂ρWi

∂xi= 0, (4)

ρ = (1− Φ) ρf +M, Wi =1

ρ

[(1− Φ) ρfUi +

A∑α=1

MαVαi

],

7

where ρ is the density of the mixture.The continuous phase velocity is expressed by the following relation:

Ui = Wi −

A∑α=1

MαVrαi

ρ.

Obviously, in the gas-liquid bubbly flow away from the critical point (i.e.when ρp � ρf and ρ ' (1− Φ) ρf ) averaged velocities of the two-phasemedium and continuous phase are almost the same (Wi ' Ui).

The equation of momentum conservation of two-phase flow can be writtenas:

∂ρWi

∂t+∂ρWiWj

∂xj= −∂P

∂xi− ∂

∂xj

{[(1− Φ) ρf +

A∑α=1

Mαfpα

]〈u′iu′j〉

}+

+∂

∂xj

[(1− Φ)µf

(∂Ui∂xj

+∂Uj∂xi− 2

3

∂Uk∂xk

δij

)]+ ρgi,

(5)where P is pressure; 〈u′iu′j〉, µf are the turbulent stresses and the coefficientof dynamic viscosity of the continuous phase. Equation (5) is written underthe assumption that the dispersed phase does not contribute to the laminartwo-phase stresses, and the contribution of the turbulent kinetic stress ischaracterized by the entrainment of particles in turbulent motion fpα.

Turbulence characteristics are calculated based on k−ε turbulence model,modified for two-phase flow. In this model the turbulent stresses of continu-ous phase are presented as follows:

〈u′iu′j〉 =2

3kδij − νT

(∂Ui∂xj

+∂Uj∂xi− 2

3

∂Uk∂xk

δij

), νT = Cµ

k2

ε,

where the eddy viscosity νT is defined by a turbulent two-phase flow of energyk and its dissipation rate ε.

The turbulent energy balance equation is the following:

∂ρk

∂t+∂ρkWi

∂xi=

∂

∂xi

{[(1− Φ)µf +

ρνTσk

]∂k

∂xi

}−

−[(1− Φ) ρf +

A∑α=1

Mαfpα

]〈u′iu′j〉

∂Wi

∂xj− ρε+ Sk1 − Sk2,

(6)

where

Sk1 =3Ckρf

4

A∑α=1

ΦαCDα|Vrα|3

dα,

8

Sk2 =3ρf (1− A)2 k

2

A∑α=1

Φα (1− fuα)CDα|Vrα|

dα,

here Sk1 is a source of turbulent energy due to interfacial hydrodynamicresistance (causing pseudoturbulence in the wake of the particles); Sk2 is anextra dissipation due to involvement of particles in turbulent motion.

The equation of the turbulent energy dissipation rate can be representedas follows:

∂ρε

∂t+∂ρεWi

∂xi=

∂

∂xi

{[(1− Φ)µf +

ρνTσε

]∂ε

∂xi

}−

−Cε1εk

[(1− Φ) ρf +

A∑α=1

Mαfpα

]〈u′iu′j〉

∂Wi

∂xj− Cε2ρε

2

k+

+ε

k(Cε3Sk1 − Cε4Sk2) .

(7)

The constants in the turbulence model are assumed to be standard: Cµ =0.09, σk = 1.0, σε = 1.3, Cε1 = Cε3 = 1.44, Cε2 = Cε4 = 1.92, ScT = 0.9.

The rate of dispersed phase contribution in the turbulent stresses of two-phase flow is characterized by the coefficient of particles entrainment into theturbulent motion:

fpα = A2 +(1− A2

)fuα,

where the response rate to the particle velocity fluctuations of a continu-ous phase fuα corresponds to the two-scale bi-exponential autocorrelationfunction (Zaichik and Alipchenkov, 2007):

fuα =2τpαTL + τ 2

T

2τpαTL + 2τ 2pα + τ 2

T

.

Response coefficients fpα and fuα are a measure for the response of dis-persed phase to velocity fluctuations of the fluid and thereby they control adegree of coupling between the fluid and dispersed phases. The response co-efficients are functions of the dispersed phase related to the fluid fluctuatingvelocity correlation timescales seen by particles (the so-called eddy-particleinteraction time scales).

Lagrangian integral time scale of turbulence is defined as:

TL =C

1/2µ k

ε.

9

Differential Taylor time scale is calculated on the basis of the relations(Zaichik and Alipchenkov, 2007):

τT =

(2Reλνf151/2a0ε

)1/2

, Reλ =

(20k2

3ενf

)1/2

,

a0 =a01 + a0∞Reλa02 +Reλ

, a01 = 11, a02 = 205, a0∞ = 7.

2.3. Law of the wall for two-phase flow

Correct modeling of bulk turbulence needs to have proper boundary con-ditions on the walls. In experimental studies Sato et al. (1981); Nakoryakovet al. (1996) the discrepancy was shown between flow velocity profile for sin-gle phase flow and two phase flow, therefore single-phase logarithmic law isnot valid for the two-phase turbulent flow. Here simplified approach similarto single-phase flows, presented in Wilcox (1998) and adopted for two phaseflows in Politano et al. (2003), has been chosen. More detailed analysis ofthe wall law can be found in Troshko and Hassan (2001b).

In accordance to Politano et al. (2003) we did similar simplifications forthe governing equations. In addition to usual simplification for single-phaseflows (extensive description can be found in Wilcox (1998)), here was assumedthat void fraction and the turbulent kinetic energy are essentially constant inthe wall region. Also terms Sk1 and Sk2 in Eqs. (6) and (7) were neglected,assuming that other terms are greater. Taking into account this assumptions,boundary conditions for Eqs. (5), (6) and (7) are the following:

k =U2?

(1− Φ)√Cµ, ε =

U3?

(1− Φ)1.5Kyw,

Ui =U?

(1− Φ)0.5

(1

Kln y+ + b

),

where the constants of the logarithmic law and distance from the wall areassumed the same as in Politano et al. (2003). These boundary conditionswere applied for the wall distance in the range 30 ≤ y+ ≤ 100. The proposedwall law was implemented into OpenFOAM.

10

2.4. Interface forces

Dynamic relaxation time of the particle, which characterizes the effect ofhydrodynamic resistance is given by:

τpα =4 (ρp + CAρf ) dα

3ρfCDα|Vrα|

,

where values dα, CDα are the diameter and drag coefficient of the particles ofα-group respectively; V r

αi ≡ Vαi−Ui is an averaged relative velocity betweenthe α-group particle velocity Vαi and the one of the continuous phase Ui.

The drag coefficient of the deformable particles (drops or bubbles) isdetermined by the interpolation formula (Loth, 2008):

CD = CWep→0D + ∆CD

(CWep→∞D − CWep→0

D

), (8)

where CWep→0D is a drag coefficient of non-deformable spherical particles in

the limit of zero Weber number; CWep→∞D is a drag coefficient of the particle

at a large Weber numbers, and ∆CD is an interpolation function describingtransition from zero to a large Weber number. In accordance to Loth (2008)the following relations can be written:

CWep→0D =

24

Rep

(1 + 0.15Re0.687

p

)if Rep ≤ 103,

0.44 if Rep > 103,

CWep→∞D =

8

3+

24

Rep, ∆CD = tanh

[0.0038

(WepRe

0.2p

)1.6],

Rep =Vrd

νf, Wep =

ρfV2rd

σ,

where σ is surface tension.In the transverse direction the lift and wall lubrication forces, turbulent

diffusion and turbophoresis act, determining together the radial distributionof the void fraction in the vertical tube cross-section. However, the majorinfluence on the radial distribution is given by the lift and wall lubricationforces. Turbulent diffusion and turbulent migration do not contain any em-pirical coefficients and are determined by the right-hand side of Eqs. (2) and(3).

11

The lift force is written as

FLi =CLρfV

rj

ρp + CAρf

(∂Ui∂xj− ∂Uj∂xi

).

The lift coefficient, following Hibiki and Ishii (2007), can be written inthe form:

CL = max(−0.27, fL(Wep)C

Wep→0L

), (9)

where CWep→0L is a lift coefficient of a non-deformable spherical bubble, and

fL(Wep) is a correction function, which takes into account the bubble defor-mation, the value of −0.27 is taken with accordance to the experimental datafrom Tomiyama et al. (2002). In Legendre and Magnaudet (1998) by gener-alizing numerical and analytical results for a single non-deformable sphericalbubble moving in a viscous shear layer, the following expression for the liftcoefficient was obtained:

CWep→0L =

[1.88

RepSrp (1 + 0.2Rep/Srp)3 +

(1 + 16Re−1

p

)2

4(1 + 29Re−1

p

)2

]1/2

,

Srp =dp|rotU||Vr|

.

Comparison of the lift coefficient expressions used in Eq. (9) is presented inFig. 1.

The direction of the lift force acting on bubbles in vertical channels mayvary not only depending on the direction of the flow (upward or downward),but also on the bubble size. Apparently, the first effect of the bubble size ontheir distribution in the upward flow was found in Sekoguchi et al. (1974).Thus, it was found that small bubbles move towards the wall, and largebubbles (approximately greater than 5 mm) move close to the flow axis.Further experimental studies (for example, Liu and Bankoff (1993); Hibikiand Ishii (1999); Tomiyama et al. (2002)) have confirmed the influence of thebubble size on the void fraction distribution. In Tomiyama et al. (2002) itwas found that the bubble deformation may lead to a change in the directionof the lift. On the basis of generalization of experimental data of Hibiki andIshii (2007) a correction function was proposed to account for the bubbledeformation on the lift coefficient. This correction in terms of Wep functioncan be written as:

fL(Wep) = 2− exp(0.0295 ·We2.21

p

).

12

Near the wall of the tube the velocity difference between the bubble andthe wall is less than the one between the bubble and the external flow. Thisleads to difference in pressure, and as consequence does not allow bubbles toslip on the wall. In order to handle this behaviour of bubbles near the wallan additional wall lubrication force was introduced, the expression for whichwas proposed in Antal et al. (1991):

FWi =CWρf |Vr|2ni(ρp + CAρf ) d

, CW = max

(CW1 + CW2

dpy, 0

),

where ni is a unit vector normal to the wall; y is a distance from the wall,and the constants were assumed to be CW1 = −1.0, CW2 = 1.0. Similarformulas for the wall lubrication force have also been proposed in Franket al. (2008); Tomiyama (1998). They have been compared in recent paperby Rzehak et al. (2012). One of the conclusions is that despite of its ratherweak physical basis, the model of Antal et al. (1991) is still widely used inmultiphase CFD simulations and gives sufficiently good results compared toother models in literature.

The lift and wall lubrication forces play a major role in the distributionsof the flow velocity and void fraction, mainly in the transverse cross-section ofthe tube flow. However, the drag force contributes mainly in the longitudinaldirection and plays a significant role in forming the relative velocity. Therelative velocity itself can be determined by the following expression fromZaichik et al. (2010a):

V rαi = τpα (1− A)

(gi −

DWi

Dt

), (10)

which does not take into account the contribution of the lift and wall lubri-cation forces, as well as turbophoresis.

3. Break-up and coalescence models

In the case of low void fraction for describing the processes of bubblecoalescence and break-up in turbulent flows, the following assumptions areused: 1) a bubble breaks only into two bubbles of different sizes, and coales-cence process arises only between two bubbles; 2) bubbles move in isotropichomogeneous turbulence (it means that statistics of small scale motion of tur-bulent eddies is universal); 3) bubble diameter lies in the inertial turbulent

13

range, and relative velocity of bubble in turbulent flow may be expressed asa function of the turbulence dissipation rate and bubble diameter as (εd)1/3.

The last term in the balance equation of dispersed phase numerical con-centration (2) describes the processes of bubble coalescence and break-up inturbulent flow:

SNα = Scoα + Sbrα . (11)

3.1. Bubble coalescence

According to Prince and Blanch (1990), coalescence mechanism of twobubbles in turbulent flow can be decomposed in three steps. First, collidingof two bubbles with trapping a small amount of liquid between them. Sec-ond, draining of the liquid film, separating two bubbles, until it reaches acritical film thickness. Otherwise, the bubbles bounce back without coalesc-ing. Finally, rupturing of the liquid film, which leads to coalescence of thetwo bubbles.

There is a spherical symmetry of the relative motion of two collidingparticles in isotropic turbulence. Therefore, the number of collisions of par-ticles, belonging to different groups, per unit volume per unit time is definedas (Zaichik et al., 2010b)

Scoα = −Nα

2

A∑α1=1

βαα1Nα1 , (12)

where βαα1 is a coalescence kernel function of α- and α1-groups of particles.To determine βαα1 two models of coalescence, proposed by Zaichik et al.

(2010b) and by Yao and Morel (2004), are used. These two models are basedon the mechanism of particle collisions due to their involvement into theturbulent motion of the flow. In addition, Zaichik model takes into accountthe impact, caused by the presence of the averaged relative velocity betweenparticles of different groups. Moreover, this model allows to describe thecoalescence kernel single dependence in the whole range of the density ratioof the dispersed and continuous phases, ρp/ρf , i.e. is valid both for bubblesin liquid, and drops in gaseous environment.

3.1.1. Zaichik coalescence model

Following Prince and Blanch (1990), the coalescence kernel can be ex-pressed as the product of the collision cross-sectional area π/4d2

αα1, the char-

acteristic velocity Vt, and coalescence efficiency ηco. In Zaichik model the

14

coalescence kernel function is written in the following form:

βαα1 = 4π1/2d2αα1

Vtφ(Σ)Γηco, (13)

Vt =(V 2in + V 2

sp

)1/2,

Vin = |1− A|(fuα + fuα1

2− fuαfuα1

)1/2(2k

3

)1/2

,

Vsp = [(1− A) fuα + A]1/2 [(1− A) fuα1 + A]1/2(Sll(dαα1)

2

)1/2

,

here Vt denotes the total turbulent collision rate, Vin is caused by a lack ofthe velocity correlation of two particles due to their inertia, and Vsp allowsfor a lack of the velocity correlation of two particles, which are separated bydistance dαα1 . Thus, in accordance with (13), there are two mechanisms ofturbulent collisions. The first (inertial) mechanism arises due to the pres-ence of a relative velocity between inertial particles, located at the same spacepoint; this exhibits a local one-point pattern and may be modeled within thepoint description of particles. The second (spatial) mechanism of the emer-gence of a relative velocity between particles, which can lead to collisions, isconnected with the finite particle size, and hence is non-local (two-point).

The longitudinal structure function Sll(dαα1) describes the velocity dif-ference at two points of the continuum at a distance of the collision radiusdαα1 ≡ (dα + dα1)/2. In accordance with Zaichik et al. (2010b), to determineit over entire spatial range, here the following approximation proposed byBorgas and Yeung (2004) is used:

Sll(d) = 2 (εd)2/3

[1− exp

(− ε1/4d

(30νf )3/4

)]4/3 [1 +

(3

2k

)6

(εd)4

]−1/6

.

The modification factor Γ takes into account the accumulation effect ofthe dispersed phase in flow and is given by Fede and Simonin (2005):

Γ =

(1− Φ

Φm

)2.5Φm

, (14)

where Φm = 0.64 is maximum packing value, which describes the case of acertain void fraction, when the bubbles touch each other and the average freetravelling time becomes zero.

15

Small void fractions, i.e. Γ = 1, correspond to the case of a dilute two-phase flow, where no correction is needed. If the void fraction reaches itslimiting value Φm, Γ = 0, and the collision radius becomes null.

The function φ(Σ) in Eq. (13) takes into account the contribution of theaveraged relative velocity between two particles (particle-particle drift) inthe coalescence kernel function (Zaichik and Alipchenkov, 2007):

φ(Σ) =exp(−Σ2)

2+π1/2

2

(Σ +

1

2Σ

)erf(Σ), Σ =

V rαα1

2Vt,

where parameter Σ is the ratio of inter-particle effects of drift and turbulenceto the coalescence kernel function. Similarly, in Eq. (10) inter-particle drift isdetermined by taking into account only the gravitational and inertial forces:

V rαα1

=

∣∣∣∣(τpα − τpα1) (1− A)

(g − DW

Dt

)∣∣∣∣ .Collision efficiency ηco in Eq. (13) describes the coalescence probability

of colliding particles, here the same expression as in model Yao and Morel(2004) is used:

ηco = exp

[−Kco

(We∗αα1

We∗cr

)1/2], (15)

where constant Kco is assumed to be equal to unity, and the Weber numberof the colliding particles is defined as:

We∗αα1=ρfSll(dαα1)dαα1

σ, (16)

The critical Weber number in Eq. (15) is described by the following depen-dence according to Sevik and Park (1973):

We∗cr =24

π2

ρf3ρp + 2ρf

. (17)

3.1.2. Yao-Morel coalescence model

Assuming that the coalescence time consists of the free travelling time andthe interaction time, and based on Prince and Blanch (1990) assumptionsthat two bubbles will coalesce, if the contact time between them, dependingon the surrounding turbulent eddies, is larger than the liquid film drainage

16

time, the following relation for the coalescence kernel function is suggestedin Yao and Morel (2004):

βαα1 =πd

7/3αα1ε

1/3ηco

6[Γ +KcΦ

(We∗αα1

/We∗cr)1/2] , (18)

where Weber numbers instead of (16) and (17) are defined as:

We∗αα1=

2ρfd5/3αα1ε

2/3

σ, We∗cr = 1.24. (19)

The radial distribution function in (18) instead of (14) is given by:

Γ =

[1−

(Φ

Φm

)1/3], Φm =

π

6= 0.52.

The collision efficiency in Eq. (18) is described according to Eq. (15). Theconstants in Eqs. (18) and (15) are chosen in accordance to critical Webernumber in Eq. (19) Kco = 1.017, Kc = 1.922.

3.2. Bubble break-up

First theory for break-up in turbulent flows was independently proposedby Kolmogorov (1949) and Hinze (1955). This theory is based on two funda-mental assumptions: (i) inertia dominates for eddies larger than Kolmogorovmicroscale, (ii) only velocity fluctuations over a distance close to the bubblediameter are capable of causing large deformations. The Kolmogorov-Hinzetheory then postulates the existence of a critical value beyond which break-upoccurs.

Following Luo and Svendsen (1996) the rate of bubble break-up is deter-mined by examining the interaction of bubbles with turbulent eddies. Whenturbulent eddies are hitting the bubbles, only the eddies of length scale equalto the bubble size can induce bubble break-up, since the small eddies havenot enough energy to overcome the interfacial forces. Larger eddies merelytransport the bubble. For the same reason only daughter bubbles with adiameter smaller than the length scale of the eddy can be formed. Largerdaughter bubbles cannot be grasped by the eddy. The rate of bubble break-up is determined by examining the interaction of bubbles with turbulenteddies. To describe the bubble break-up process two models, correspondingto two coalescence models presented above, are used.

17

3.2.1. Zaichik break-up model

Zaichik break-up model describes the process of bubbles splitting in theLuo and Svendsen assumption that the main mechanism of break-up is the in-teraction of particles with turbulent eddies (Alipchenkov et al., 2004). There-fore the term, describing the process of break-up in Eq. (11), can be writtenthrough of critical threshold function as follows:

Sbrα =Nα

τ brαH(We∗α −We∗cr)ηbr, (20)

where Heaviside function H(x) (H(x) = 0 at x < 0, H(x) = 1 at x > 0) isused to describe a critical value, beyond which break-up occurs; ηbr is a break-up efficiency; Weber number and critical Weber number are determined inaccordance with Eqs. (16) and (17) correspondingly.

Following Alipchenkov et al. (2004), break-up time in Eq. (20) is thesum of deformation time to the critical size τ defα , caused by interaction withturbulent eddies, and dissociation time τ disα :

τ brα = τ defα + τ disα ,

where

τ defα = Kb

[(1 + 2ρp/ρf ) We∗cr

1−We∗cr/We∗α

]1/2dα

S1/2ll (dα)

, Kb = 1.1,

τ disα =πd

3/2α

4

(3ρp + 2ρf

3σ

)1/2

.

According to Alipchenkov et al. (2004), break-up efficiency can be writtenin the form:

ηbr = exp

(−We∗cr

We∗α

), (21)

3.2.2. Yao-Morel break-up model

Bubble break-up mechanism in the model by Yao and Morel (2004) isbased on bubble-eddy resonance mechanism. Turbulence acts on each bubbleas a succession of colliding eddies, with random arrival times and intensities,instead of a single isolated eddy. If the frequencies of these successive eddiesare close to the resonance frequency of the bubble, the bubble may startoscillating, and these oscillations can break the bubble, even if We < Wecr,

18

as opposed to Zaichik model, where bubbles can break-up only if We > Wecr.Break-up term in Yao-Morel model is written in the following form:

Sbrα =Kb1Φ (1− Φ) ε1/3ηbr

3d11/3α

[1 +Kb2 (1− Φ) (We∗α/We∗cr)

1/2] . (22)

Weber number in Eq. (22) is determined by the relation:

We∗α = 2ρfd5/3α ε2/3/σ,

where critical Weber number We∗cr, as in Eq. (19), equals to 1.24. Constantsin Eq. (22) have the following values: Kb1 = 1.6, Kb2 = 0.42. Break-upefficiency ηbr is determined according to Eq. (21).

Here and after in the text the coalescence and break-up models in Eqs. (13)and (20) will be called Zaichik model, and the models by Yao and Morel inEqs. (18) and (22) will be refereed as Yao-Morel model. In Sec. 5 validationof the these models will be presented.

4. Model validation for monodisperse bubbly flow

Validation of the presented model was performed by comparing the re-sults of calculations with experimental data. The following experimentaldata have been selected: Wang et al. (1987); Serizawa et al. (1975); Liu andBankoff (1993) for the monodisperse bubbly flow in vertical pipes. Theybecame a kind of benchmark data due to popularity in many other similarinvestigations, like Troshko and Hassan (2001a); Frank et al. (2008); Usti-nenko et al. (2008). All experiments were carried out with water-air mixtureat atmospheric pressure and constant room temperature. Void fraction mea-surements were done by using the resistive probe technique, and hot filmanemometry was used to measure the liquid velocity profiles. Table 1 showsthe main parameters of the selected experiments. Detailed information aboutthe distributions of inlet parameters is missing, so flat profiles were set in thecalculations for all the known variables in Eqs. (2) – (7). The values of theinitial concentrations were determined by integrating the experimental data,given the velocity values, and calculated initial velocity of the carrier medium.At the outlet of the computational grid axial gradients of all variables, exceptpressure, were taken to be zero.

The numerical simulations were performed using the open-source compu-tational fluid dynamics CFD package OpenFOAM. Eqs. (2)–(7), (11) have

19

been implemented into a new solver in accordance with OpenFOAM method-ology. Presented calculation results are grid independent and were achievedwith several types of nonuniform axisymmentric computational meshes, keep-ing y+ = 30. In particular, the following numbers of grid elements in pipecross-section were chosen: 20, 25, 30, 35 and 40 cells by radius.

Figs. 2 and 3 show simulation results compared with experimental dataof Wang et al. (1987) for hydrodynamically developed upward (W1-W3) anddownward (W4-W6) flows in a vertical tube. In Fig. 2(a), the calculatedand experimental radial distributions of void fraction for the upward floware presented. DIM shows a good agreement for the void fraction near thewall, where in maximum of void fraction the lift and wall lubrication forcesare compensating each other in radial direction.

Fig. 2(b) presents the radial distribution of the longitudinal velocity ofthe developed turbulent carrier flow. Since the bubbles move towards thewalls of the tube, an additional flow turbulence is generated in the wakeof the bubble; it leads to diminishing velocity gradient near the wall. Thiseffect of bubble induced liquid velocity flattening is explained in details inLopez de Bertodano et al. (1994). In this case, numerical results show agood agreement with experimental data except near the wall region wherethe measured velocities are higher than those predicted by the model. Theexperimental liquid velocity at the wall is not zero, which is not the case inthe model. Also, the drag coefficient in Eq. (8) does not take into account awall since it was obtained in approach of unbounded bubbles, which leads toan additional inaccuracy in calculating the relative velocity near the walls.

In the case of downward flow lift force is directed towards the flow axis, sothat the calculated void fraction distribution has no peak near the wall (Fig.3a). The appearance of small peaks in the experimental distributions of voidfraction, could be explained probably by some inaccuracy in measurementsnear the wall (Wang et al., 1987). The liquid velocity profile in Fig. 3b isflattened by the presence of bubbles and has a peak near the wall; calculatedoff center velocity peak agrees with experiment. Bubbles near the wall arealmost absent, they are tended to the flow axis, reducing the flow rate there,see Figs. 3a and 3b. Thus, the flow velocity of downward flow will be greaterin region, where void fraction is minimum, and on the contrary, where it ismaximum in the case of upward flow. The same behaviour appears, whensufficiently large bubbles tend to the core region of the pipe in upward flow.It easily can be observed in calculations of experiments L1, L2 and L3, wheremeasurements for the various size of bubbles were performed, with keeping

20

the void fraction for the chosen experiments almost the same, see Fig. 5. Ingeneral, simulation results for presented void fractions and liquid velocitiesare in a good agreement with the experimental data.

As the next step, a comparison to experiments S1, S2 and S3, wherethe bubble size was assumed to be kept the same with varied void fraction,was performed. The void fraction and flow velocity distribution in the tubecross-section are presented in Fig. 4. Distribution of void fraction near thewall is similar to the experiments W1 – W3 for the upward flow, see Fig. 2a.However, in experiments of Serizawa et al. (1975) the bubbles were slightlylarger in size, see Tab. 1, therefore the width of the peak in Fig. 4a is largerthan in W1 – W3 experiments. The influence of polydispersity of bubblesbecomes more pronounced in S3 experiment, where besides the peak nearthe wall an increased void fraction on the tube axis is observed as a result ofa higher void fraction in comparison to S1 and S2.

A monodispersed upward bubbly flow in a rather broad range of bubblesizes was studied experimentally in Liu and Bankoff (1993). It was foundthat for the upward flow, if the bubble diameter is larger than 5 mm, gasvoid fraction in the axial zone is greater than near the wall of the tube.This has been taken into account by changing a sign of the lift coefficientCL in Eq. 9, see Fig. 1. Fig. 5a shows a comparison of calculated radial voidfraction profiles with the experimental data of L1 – L3. The model correctlypredicts the void fraction increasing towards the tube axis for L1 experimentand its transfer to the wall for experiments L2 and L3. In the case of L1, aswell as S3, due to the relatively high values of void fraction and bubbles size,effect of polydispersity can not be neglected. The high rate of fluid in thecore region, obtained in the simulation of the experiment L1 compared to L2and L3, is due to high values of void fraction and therefore greater impact ofthe bubbles on the flow (Fig. 5b).

5. Model validation for polydisperse bubbly flow

In the previous section validation of interfacial forces expressions basedon simulation of monodisperse bubbly flows was done. In order to separatepolydispersity effect from the influencing of uncertainty of interfacial forcescoefficients. Then, keeping the same coefficients, the study of polydispersityeffect was performed; the results will be presented in the following section.Yao-Morel and Zaichik models for bubble coalescence and break-up, pre-sented in Sec. 3, will be compared in the next two subsections. And the

21

last subsection is devoted to simulation of bubbly flow in tubes with largediameter; only Zaichik model was used there.

5.1. Validation on the experiments by Hibiki et al. (2001)

Hibiki et al. performed local measurements of the radial profiles of gasvolume fraction, liquid and bubble velocities and mean diameter of the bub-bles using hot film anemometer probes and a dual-sensor resistivity probe.The parameters of selected experiments are presented in Table 2.

The model results for distributions of void fraction, liquid and gas veloc-ities, mean bubble diameter with different coalescence and break-up modelsare compared to the experimental data in Figs. 6, 7 and 8. In void fractionand velocities of gas and carrier flow the numerical results showed a goodagreement to the experimental data. Since the size of bubbles is sufficientlysmall, peak of void fraction is near the wall due to lift and wall lubricationforces. This case is similar to the cases which have been described in previoussection for the W1 – W3 experiments. Comparison of the flow velocities forsingle phase flow and liquid flow with bubbles shows increasing of velocitynear the wall. With increasing gas void fraction average size of bubbles is alsoincreasing. This is explained by the fact that with increasing void fractionthe balance between coalescence and break-up of bubbles is shifted towardthe prevalence of coalescence, because coalescence rate is proportional to thesquare of bubble concentration, see Eq. (12), whereas the break-up rate – tothe first power, see Eq. (20).

Calculated gas velocity is in a good agreement with the measured data,and near the wall Yao-Morel and Zaichik models perform sufficiently wellfor both velocities. However, the flow velocity in the core region for theexperiments H2 and H3 with higher void fraction (see Figs. 7b, c) is under-predicted, compared to the experiment. This inconsistency could be relatedto measurement difficulties of the used technique in the core region.

From the bubble size measurements, see Fig. 8, in the beginning of thetest tube (z/D=6) we defined the initial size of bubbles for one δ-functionin Eqs. (3) and (2), since bubble variation is in between 2.35 mm and 2.85mm. Experimental measurements showed that larger bubbles are closer tothe walls, where void fraction is larger, and smaller bubbles are distributed inthe core region of the pipe. However the results based on Zaichik model areoverpredicted in the bulk region and underestimated near the wall. We canexplain this: the smaller bubbles can pack better near the wall, where voidfraction is higher, compared to larger bubbles. At the same time due to the

22

facts the main mechanism of break-up in Zaichik model is interaction withflow turbulence, and turbulence is higher near the wall, the resulting bubblesize is smaller. However, average size of bubbles at z/D = 53.5 cross-sectionis closer to the experimental value. Bubble coalescence and break-up modelsby Yao-Morel give a good agreement in the bulk region, but the average sizeof bubbles at the outlet cross-section is underestimated.

5.2. Validation on MT-Loop experiments

Series of MT-Loop experiments became a very well known benchmark andit had been used for validation in many studies (for instance, Frank et al.(2008); Rzehak et al. (2012); Morel et al. (2010)). The measurements wereperformed at the MT-Loop facility; a detailed description is given in Krepperet al. (2008); Lucas et al. (2005, 2010a); Prasser et al. (2003). The workingregion is a vertical pipe 4 m high with an internal diameter of 51.2 mm. Theair bubbles were injected by means of 19 capillaries uniformly distributed overthe section of the pipe. A grid sensor was used in the facility, which makes itpossible to measure the radial distribution of the bubble concentration andbubble size. Measurements were performed at 10 different heights for 100different combinations of the reduced liquid and gas velocities.

Depending on the injected gas and velocity of the current flow, the authorsof Lucas et al. (2005) classified bubbly flows into several types: (1) Bubblyflow with wall peak (MT-Loop 074); (2) Bubbly flow in the transition region;(3) Bubbly flow with centre (core) peak (MT-Loop 107); (4) Bubbly flow witha bimodal bubble size distribution. Based on this classification experimentsMT-Loop 074 for bubbly flow with wall peak and MT-Loop 107 for bubblyflow with centre peak have been chosen. Parameters of these experimentsare presented in Table 2. According to the definition of the bubble sizedistribution:

h =dΦ

ddb,

the integral over the distribution yields the total gas void fraction:

Φ =

∫ ∞0

h(db) ddb.

Numerical concentration can be obtained from the following relation:

N =6Φ

πd3b

.

23

Measured size distribution of bubbles was divided into two or four parts ac-cording to number of δ-functions and then integrated inside each divided partin order to obtain gas void fraction for the certain δ-function. In Figs. 10 and12 the examples of approximation by two and four δ-functions for measuredbubble size distribution in experiments MT-Loop 074 and 107 are shown.Black solid lines are initial sizes of bubbles, and their heights represent voidfraction for a certain δ-function, to be more illustrative. Red and blue linesare the calculated δ-functions for Zaichik and Yao-Morel models respectively.The numbers above denote the δ-function number.

Fig. 9 shows comparison of the gas void fraction profile for MT-Loop074 experiment in the approximation of two and four δ-functions for bubblesize distribution, presented in Fig. 10. This experiment is characterized bylow concentration of the injected gas. Qualitatively similar concentrationdistribution of bubbles for different models of coalescence and break-up areobtained. For low superficial gas velocity the balance between coalescenceand break-up is reached for small bubbles. For Yao-Morel model the maxi-mum void fraction is overpredicted near the wall, in the contrast we obtaineda good comparison for Zaichik model. Such a difference in gas void fractiondistributions can be explained by the observation in Fig. 10. It is clearly seenthat Yao-Morel model generates smaller bubbles compared to Zaichik model.As a result the lift force moves smaller bubbles towards the wall comparedto larger bubbles, which are moving into the core region of the pipe.

Since the coalescence rate increases with the square of the bubble den-sity, while the break-up rate is only proportional to the bubble density, theequilibrium between coalescence and break-up moves toward a larger bub-ble diameter. If void fraction leads to the formation of bubbles larger than5.5 mm, then bubbles start to migrate towards the core region of the tube.Therefore, we can observe a case, when some bubbles are near the wall andanother ones – in the core region of the tube. In order to simulate this,MT-Loop 107 case has been chosen.

Fig. 11 shows the distribution of gas volume fraction in MT-Loop 107experiment. Approximation by two δ-functions gives qualitatively differentresults for the used models. For Zaichik model the bubble size is more than5.5 mm, therefore they move to the center. In the opposite, Yao-Morelbubbles are sufficiently small (see Fig. 12a), and they stay near the wall. Itcauses increasing of the gas void fraction near the wall. However none of theconsidered models reproduces bubbles segregation in the flow. For four δ-functions results of Zaichik model are better, see Fig. 11b. In this case major

24

part of bubbles have sufficiently small bubble size, however there are bubbleslarger than 5.5 mm which are tending towards the core region. δ-functionnumber two (red line) in Fig. 12b has the largest size of bubbles, howeverother δ-functions together have larger void fraction, and the center peak isnot seen. The simulations with four δ-functions based on Yao-Morel modelgive small diameter of bubbles, and they are collected even more near thewall region of the pipe, see blue lines in Fig. 12b.

5.3. Validation on TOPFLOW experiments

The experimental data at the thermal fluid dynamic test facility TOPFLOW(Prasser et al., 2007) provide an opportunity to test the model against twophase flow characteristics measured in a large tube with inner diameter of195.3 mm and height 9000 mm. By using the wire-mesh sensor measuringtechnique the authors of Prasser et al. (2007) were able to visualize andquantify individual gas bubbles in the measuring cross-section. Analysinginformation of the measured bubble size distributions, one can get bubblecoalescence and break-up frequencies. This is important for understandingof the dominating process, which defines the size of bubbles. As it wasmentioned in the Introduction, bubbly flow behaviour in tubes with large di-ameter differs from the one in tubes with smaller diameter, so that in a largepipe slug bubbles can no longer exist due to Taylor instability of the bubbleforming. From the comparison to MT-Loop experimental data, presented inthe previous subsection, it follows that Yao-Morel model predicts too smallbubble sizes for the reason mentioned above. Therefore, only Zaichik modelwas used to simulate TOPFLOW experiments, where initial bubble sizeswere sufficiently larger.

The data of TOPFLOW 074 and 107 experiments were chosen. Theparameters of these experiments are presented in Table 2. Based on themeasured bubble size distribution (see Figs. 14 and 16) at the level z/D =1.1 of the tested tube, initial δ-functions were defined. The bubble size at thebeginning of the tube is large enough for the most of the bubbles not to tendto the wall. Since void fraction is sufficiently small, bubble size decreases.At the outlet section bubble size is still sufficiently large, and they are dis-tributing uniformly in the core region of the tube. Zaichik model reproducesthis behaviour and has a good agreement with experimental results for gasvoid fraction and gas velocity, see Figs. 13, and also reproduces decreasingof the bubble size, see Fig. 14. In TOPFLOW 107 gas void fraction is larger,which leads to increasing of the bubble size. In the experiments measured

25

size distribution of the bubbles at the outlet section shows that majority ofthe bubbles became six times larger than at the inlet tube section. It ex-plains the peak at the center of the tube for gas void fraction and gas velocity,(see Figs. 15), since all large bubbles are tending to this region of the tube.Zaichik model shows increase in bubble size, reproducing the major size ofbubbles, however, a little less than in the experiment, see Fig. 16.

6. Conclusions and outlook

The method of modeling two-phase turbulent polydisperse bubbly flows,which is based on coupling DIM for two-phase turbulence flows and themethod of δ-function approximation, is presented. Such approach allowssimulating turbulent bubbly flows with low computational cost compared tothe class methods, represented by the MUltiple SIze Group (MUSIG) model,where continuous range of bubble sizes is discretized into many series offinite size classes (Frank et al., 2008). Another distinctive feature of thepresented approach is using an algebraic expression for the relative velocity,see Eq. (10), depending on the local bubble diameter, which also gives com-putational benefits without loosing accuracy. In addition, the relationshipsEqs. (8) and (9) are novel in the sense that they are formulated in termsof the local Weber and Reynolds numbers, as opposed to static parametersonly valid at terminal velocity conditions (e.g. Bond number and Mortonnumbers).

Two different coalescence and break-up models were used in the calcula-tions: a newly developed Zaichik model (Eqs. (13) and (20)) was comparedto a well-known Yao-Morel one (Eqs. (18) and (22)). Since coalescence andbreak-up models are based on the mechanism of energy exchange betweenbubbles and turbulent eddies, they depend strongly on turbulent characteris-tics of the flow. Turbulent kinetic energy and its dissipation were calculatedby modified k−ε turbulence model, where additional source terms have beenadded for taking into account the turbulence induced by the bubbles. Thewall law adapted for two-phase flows was used as boundary conditions. Itcould however be improved in future, since the constants were taken for thesingle-phase flow. We focused here on the modeling of bulk turbulence andleft the detailed analysis of turbulent bubbly flow near the wall for furtherinvestigations.

DIM together with coalescence and break-up models was validated againstexperimental data of MT-Loop and TOPFLOW experiments. As it was

26

shown, the structure of the bubbly flow in tube, presented in TOPFLOW ex-periment, is dramatically changed, compared to the flow in MT-Loop experi-ment. However, calculated results are in a good agreement with TOPFLOWdata, which means that the proposed model can be used for simulating flowin large pipes. It might be quite helpful to refine empiric relations or predictnew ones for analysis of such systems, which include flow in pipes of largediameter, since using the same relations as for small pipes is questionable.

From the comparison of the obtained results with the experimental datait follows that Zaichik model works well in the whole range of considered gasvoid fractions. It reproduces the situation, when two major sizes of bubbleappear in the flow. Yao-Morel model gives reliable results only for low voidfractions. Bubble sizes are underestimated in this model for all the consideredcases, which results in the overestimated void fraction near the wall. Itcan be explained by the differences in the assumptions of break-up models.Yao and Morel suggested that bubble can break even if We < Wecr dueto resonance interaction of bubble shape and small-energy turbulent eddy.On the contrary, in Zaichik model break-up is possible only at We > Wecr.Moreover, Zaichik critical Weber number in Eq. (17) is higher than Yao-Morelone in Eq. (19).

Considering isothermal flow allowed us to concentrate our attention oninfluencing of interfacial forces and processes of bubble coalescence and break-up in the absence of thermal effects and phase transition. Continuationof the following study will be further development of proposed models forsimulations of gas-liquid nonisothermal polydisperse bubbly flows with heattransfer and phase transitions: boiling and condensation.

Acknowledgements

This work was started in a close collaboration and under the supervisionof Professor Leonid Zaichik, and was finished in his light memory.

27

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Tables

Experiment J inC , m/s Φ dbubble, mm Direction Dtube,mm

Wang et al. (1987)W1 0.43 0.132 2.8 Up 57.15W2 0.43 0.310 3.0 Up 57.15W3 0.43 0.383 3.2 Up 57.15W4 0.71 0.145 2.8 Down 57.15W5 0.71 0.288 3.0 Down 57.15W6 0.71 0.371 3.2 Down 57.15

Serizawa et al. (1975)S1 1.03 0.0397 4.0 Up 60S2 1.03 0.1023 4.0 Up 60S3 1.03 0.1627 4.0 Up 60

Liu and Bankoff (1993)L4 1.0 0.087 6.6 Up 57.2L5 1.0 0.095 3.7 Up 57.2L6 1.0 0.106 2.81 Up 57.2

Table 1: Global flow conditions of experiments used in validation for monodisperse flow.J inC is a superficial velocity, Φ is a gas void fraction.

Experiment J inC , m/s Φ Ref Dtube,mm

Hibiki et al. (2001)H1 0.986 0.203 4.3× 104 50.8H2 0.986 0.108 4.3× 104 50.8H3 0.986 0.0512 4.3× 104 50.8

Lucas et al. (2005)MTLOOP 074 1.017 0.04 4.5× 104 51.2MTLOOP 107 1.017 0.13 4.5× 104 51.2

Prasser et al. (2007)TOPFLOW 074 1.017 0.04 2.6× 105 195.3TOPFLOW 107 1.017 0.13 2.6× 105 195.3

Table 2: Global flow conditions of experiments used in validation for polydisperse flow.

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Figure Captions

Figure 1. Comparison of lift force coefficients as a function of bubble size.

Figure 2. Comparison of numerical (lines) and experimental (points) Wanget al. (1987) results for upward flow. Void fraction (a) and liquid velocity (b)as functions of dimensionless radial distance, where r is the distance fromthe tube axis, R is the tube radius. Blue, red and green are for W1, W2 andW3 experiments respectively.

Figure 3. Comparison of numerical (lines) and experimental (points) Wanget al. (1987) results for downward flow. Void fraction (a) and liquid velocity(b) as functions of dimensionless radial distance. Blue, red and green are forW4, W5 and W6 experiments respectively.

Figure 4. The same as in Fig. 2. Blue, red and green are for S1, S2 and S3experiments respectively.

Figure 5. The same as in Fig. 2. Blue, red and green are for L4, L5 and L6experiments respectively.

Figure 6. Comparison of calculated void fraction profiles with experimentsH1 (a), H2 (b) and H3 (c) at z/D=53.5: model results (all curves) and exper-imental data of Hibiki et al. (2001) (all symbols). Red lines are calculationresults based on Zaichik model, blue lines are Yao-Morel model. Legends tothe figures are the same as in figure (a).

Figure 7. Comparison of calculated liquid (solid lines) and gas (dash lines)velocity profiles with experiments H1 (a), H2 (b) and H3 (c) at z/D=53.5.Circles are measured gas velocity and squares are measured liquid velocity.Red lines are calculation results based on Zaichik model, blue lines are Yao-Morel model. Black line in figure (a) represents calculated liquid velocity in

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comparison to measurements (cross symbol) of liquid velocity in the case ofsingle flow. Legends to the figures are the same as in figure (c).

Figure 8. Comparison of calculated bubble size profiles with experiments H1(a), H2 (b) and H3 (c): model results (all curves) and experimental dataof Hibiki et al. (2001) (all symbols). Black dots are measured bubble sizedistribution at z/D=0.15. Dashed line is initial size of bubble for all models.Red lines and blue lines represent the same models as in Fig. 7. Legends tothe figures are presented in figures (a) and (b).

Figure 9. Void fraction profile distribution. Red and blue lines are calcu-lated size of δ-functions by Zaichik and Yao-Morel models respectively. Redcross symbol is MTLoop 074 experimental results: (a) Comparison for 2δ-functions; (b) Comparison for 4 δ-functions.

Figure 10. Bubble size distribution. Grey area is a measured distribution atthe level z/D = 39.9 in MTLoop 074. Black vertical lines are representinginitial sizes of δ-functions; red and blue lines are calculated sizes of δ-functionsby Zaichik and Yao-Morel models respectively: (a) Comparison by using 2δ-functions and (b) by using 4 δ-functions.

Figure 11. Void fraction profile distribution for MTLoop 107 (red cross sym-bols). Notations are the same as in Fig. 9

Figure 12. Bubble size distribution for experiment MTLoop 107 (red crosssymbols). Notations are the same as in Fig. 10

Figure 13. Comparison with experiment TOPFLOW 074 (red cross symbol):(a)Void fraction profile. Dash line is representing calculations based on 1δ-function, solid line calculation based on 2 δ-functions approximation; (b)Gas (solid blue line) and liquid velocity (dash blue line) profiles.

Figure 14. Comparison with experiment TOPFLOW 074 (black solid lineat the level z/D=1.33, red solid line at the top level: (a) black dash line is

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an initial position of 1 δ-function and red dash line is bubble sizes calculatedby Zaichik model with 1 δ-function approximation; (b) black solid verticallines are initial positions of 2 δ-functions and red vertical lines are bubblesizes calculated by Zaichik model with 2 δ-function approximation.

Figure 15. Comparison with experiment TOPFLOW 107 (red cross symbol):(a) Void fraction profile. Dash line is representing calculations based on 1δ-function, solid line – calculation based on 2 δ-functions approximation; (b)Gas (solid blue line) and liquid velocity (dash blue line) profiles.

Figure 16. Comparison with experiment TOPFLOW 107 (black cross sym-bols at the level z/D=1.33 and red thin line at the top level): (a) black dashline is an initial position of 1 δ-function and red dash line is bubble sizescalculated by Zaichik model with 1 δ-function approximation; (b) black solidvertical lines are initial positions of 2 δ-functions and red vertical lines arebubble sizes calculated by Zaichik model with 2 δ-functions approximation.

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