cfd simulation of polydispersed bubbly two-phase flow

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2009, Article ID 320738, 12 pagesdoi:10.1155/2009/320738

Research Article

CFD Simulation of Polydispersed Bubbly Two-PhaseFlow around an Obstacle

E. Krepper,1 P. Ruyer,2 M. Beyer,1 D. Lucas,1 H.-M. Prasser,3 and N. Seiler2

1 Institute of Safety Research, Forschungszentrum Dresden-Rossendorf (FZD), P.O. Box 510119, 01314 Dresden, Germany2 Institut de Radioprotection et de Surete Nucleaire, CE Cadarache, Bat. 700, BP 3, 13 115 Saint Paul lez Durance Cedex, France3 Institute of Energy Technology, ETH-Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland

Correspondence should be addressed to E. Krepper, [email protected]

Received 10 January 2008; Accepted 18 February 2008

Recommended by Iztok Tiselj

This paper concerns the model of a polydispersed bubble population in the frame of an ensemble averaged two-phase flowformulation. The ability of the moment density approach to represent bubble population size distribution within a multi-dimensional CFD code based on the two-fluid model is studied. Two different methods describing the polydispersion are presented:(i) a moment density method, developed at IRSN, to model the bubble size distribution function and (ii) a population balancemethod considering several different velocity fields of the gaseous phase. The first method is implemented in the Neptune CFDcode, whereas the second method is implemented in the CFD code ANSYS/CFX. Both methods consider coalescence and breakupphenomena and momentum interphase transfers related to drag and lift forces. Air-water bubbly flows in a vertical pipe withobstacle of the TOPFLOW experiments series performed at FZD are then used as simulations test cases. The numerical results,obtained with Neptune CFD and with ANSYS/CFX, allow attesting the validity of the approaches. Perspectives concerning theimprovement of the models, their validation, as well as the extension of their applicability range are discussed.

Copyright © 2009 E. Krepper et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Many flow regimes in Nuclear Reactor Safety Research arecharacterized by multiphase flows, with one phase being aliquid and the other phase consisting of gas or vapor ofthe liquid phase. The flow regimes found in vertical pipesare dependent on the void fraction of the gaseous phase,which varies, as void fraction increases, from bubbly flowto slug flow, churn turbulent flow, annular flow, and finallyto droplet flow at highest void fractions. In the regimes ofbubbly and slug flows, a spectrum of different bubble sizes isobserved. While dispersed bubbly flows with low gas volumefraction are mostly monodispersed, an increase of the gasvolume fraction leads to a broader bubble size distributiondue to breakup and coalescence of bubbles. The exchangearea for mass, momentum, or heat between continuous anddispersed phases thus cannot be simply modeled based onthe knowledge of just the void fraction and a mean diameter.Moreover, the forces acting on the bubbles may depend ontheir individual size which is the case not only for drag

but also for nondrag forces. Among the forces leading tolateral migration of the bubbles, that is, acting in normaldirection with respect to the main drag force, bubble liftforce was found to change the sign as the bubble size varies.Consequently, in the context of pipe flows, this leads to aradial separation between small and large bubbles and tofurther coalescence of large bubbles migrating toward thepipe center into even larger Taylor bubbles or slugs.

An adequate modeling approach must consider all thesephenomena. The paper presents two different approachesboth being based on the Eulerian modeling framework.On one hand, a generalized inhomogeneous multiple sizegroup (MUSIG) model was applied, for which the dispersedgaseous phase is divided into N inhomogeneous velocitygroups (phases), each of these groups being subdivided intoMj bubble size classes. On the other hand, a moment densitymethod is used to model the evolution of the bubble sizedistribution function along the flow, where main statisticsof the distribution being described with a reduced set oftransport equations. For both models, bubble breakup and

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2 Science and Technology of Nuclear Installations

Gas

Sizefractions

V

d1 dk dM

Bubblebreak-up

Bubblecoalescence

Figure 1: Scheme of the standard MUSIG model: all size fractionsrepresenting different bubble sizes move with the same velocityfield.

Velocity groupsJ = 1 · · ·N

Size fractionsK = 1 · · ·∑MJ

V1 V2 V· · ·

d1 dM1 dM1+1 dM1+M2 d∑MJ

Bubblebreak-up

Bubblecoalescence

Figure 2: Improvement of the MUSIG approach: the size fractionsMj are assigned to the velocity field Vj .

coalescence processes are taken into account by appropriatemodels.

2. Model for the Local Size Distribution ofa Bubble Population

2.1. Closure Models for Momentum Exchange and for BubbleCoalescence and Breakup. While modeling a two-phase flowusing the Euler/Eulerian approach, the momentum exchangebetween the phases has to be considered. Apart from thedrag acting in flow direction, the so-called nondrag forcesacting mainly perpendicularly to the flow direction must beconsidered. Namely, the lift force, the turbulence dispersionforce, the virtual mass force, and the wall force play animportant role.

The turbulent dispersion force acts on smoothing thegas volume fraction distribution and can be evaluatedeither from a single expression related to drag turbulentcontribution, [1] or deduced from a model for dispersedphase agitation contribution to the momentum balance, forexample, the Tchen model [2]. To avoid the maximum gasfraction at the wall, Tomiyama et al. [3, 4] propose a wallforce.

The lift force �FL considers the interaction of the bubblewith the shear field of the liquid. For a single bubble, it reads

�FL = −CLρl πd3

6( �wg − �wl)× rot(�wl), (1)

where ρl is the liquid density, d the bubble diameter, and�wg and �wl are the bubble and liquid velocities, respectively.

The classical lift force, which has a positive coefficient CL,acts in the direction of decreasing liquid velocity. In case ofcocurrent upwards pipe flow, this is the direction towards thepipe wall. Numerical [5] and experimental [3] investigationsshowed that the direction of the lift force changes its sign if asubstantial deformation of the bubble occurs. Tomiyama [4]investigated single bubble motion and derived the followingcorrelation for the coefficient of the lift force from theseexperiments:

CL =

⎧⎪⎪⎨⎪⎪⎩

min[0.288 tanh(0.121 Re ), f

(Eod

)], Eod < 4

f(Eod

), for 4 < Eod < 10

−0.27 Eod > 10

with

f(Eod

) = 0.00105Eo3d − 0.0159Eo2

d − 0.0204Eod + 0.474.(2)

This coefficient depends on the modified Eotvos numberEod, which is formulated for the maximum horizontalbubble size in flow direction [6] and on the Reynolds numberRe based on bubble size. The sign of the lift coefficient, and,consequently, the direction of the lift force, depends, thus,on the bubble size diameter. This behavior, originally foundfor single bubbles of air in glycerol, was also established bydifferent experiments for gas-water polydispersed flows (e.g.,[7]) and for different fluids too.

For several flow configurations, this bubble size depen-dency of the lift force direction can lead to the separationbetween small and large bubbles. This effect has been shownto be a key phenomenon for the development of the flowregime.

The bubble coalescence model takes into account the ran-dom collision processes between two bubbles. The appliedmodel is based on the work of Prince and Blanch [8]. Thebubble breakup model considers the collision between aliquid turbulent eddy of a certain characteristic size anda bubble (see [9]). Coalescence and breakup mechanisticmodels are common to both methods for taking intoaccount the polydispersion in size of the bubble population.We introduce corresponding “breakup” and “coalescence”numerical coefficients FB and FC used to scale the originalcorrelations.

2.2. Population Balance Approach

2.2.1. The MUSIG Model by Lo. In principle, the Euleriantwo-fluid approach can be extended to simulate a continuousliquid phase and several gaseous dispersed phases solvingthe complete set of balance equations for each phase.The investigations, however, showed that for an adequatedescription of the gas volume fraction profile including apopulation balance model decades of bubble size classeswould be necessary. In a CFD code, such a procedure islimited by the increased computational effort to obtainconverged flow solutions. To solve this problem, the multiplesize group model first implemented by the code developers inCFX-4 solves only one common momentum equation for allbubble size classes (homogeneous MUSIG model, see [10],

Science and Technology of Nuclear Installations 3

Figure 1). Mathematically, the multiple size group model(MUSIG) is based on the population balance method and thetwo-fluid modeling approach. The dispersed phase is dividedinto M size fractions. The population balance equation isapplied to describe the mass conservation of the size fractionstaking into account the interfraction mass transfer resultingfrom bubble coalescence and breakup. This model approachallows a sufficient number of size fraction groups requiredfor the coalescence and breakup calculation to be used andhas found a number of successful applications to large-scaleindustrial multiphase flow problems.

Nevertheless, the assumption also restricts its applicabil-ity to homogeneous dispersed flows, where the slip velocitiesof particles are almost independent of particle size and theparticle relaxation time is sufficiently small with respect toinertial time scales. Thus, the asymptotic slip velocity canbe considered to be attained almost instantaneously. Thehomogeneous MUSIG model described above fails to predictthe correct phase distribution when heterogeneous particlemotion becomes important. One example is the bubbly flowin vertical pipes, where the nondrag forces play an essentialrole on the bubble motion. The lift force was found to changeits sign, when applied for large deformed bubbles, which aredominated by the asymmetrical wake ([3], see Section 2.1).The lift force in this case has a direction opposite to theshear-induced lift force on a small bubble. For this reason,large bubbles tend to move to the pipe core region resultingin a core void maximum, whereas a near-wall void peak ismeasured for small bubbles. The radial separation of smalland large bubbles cannot be predicted by the homogeneousMUSIG model. This has been shown to be a key mechanismfor the establishment of a certain flow regime.

2.2.2. New Strategy: the Inhomogeneous MUSIG Model. Acombination of the consideration of different dispersedphases and the algebraic multiple size group model wasproposed to combine both the adequate number of bubblesize classes for the simulation of coalescence and breakupand a limited number of dispersed gaseous phases to limitthe computational effort [11]. The inhomogeneous MUSIGmodel was developed in cooperation with ANSYS CFX and isimplemented in CFX since the version CFX-10 ([12–14] seeFigure 2).

In the inhomogeneous MUSIG model, the gaseousdispersed phase is divided into a number N of the so-calledvelocity groups (or phases), where each of the velocity groupsis characterized by its own velocity field. Further, the overallbubble size distribution is represented by dividing the bubblediameter range within each of the velocity groups j in anumber Mj , j = 1, . . . ,N , bubble subsize fractions. Thepopulation balance model, considering bubble coalescenceor bubble breakup is applied to the subsize groups. Hencethe mass exchange between the subsize groups can exceedthe size ranges assigned to the velocity groups resulting inmass transfer terms between the different phases or velocitygroups.

The lower and upper boundaries of bubble diameterintervals for the bubble size fractions can be controlled byeither an equal bubble diameter distribution, an equal bubble

mass distribution, or can be based on user definition of thebubble diameter ranges for each distinct bubble diameterfraction. The subdivision should be based on the physicsof bubble motion for bubbles of different size, for example,different behavior of differently sized bubbles with respect tolift force or turbulent dispersion. Extensive model validationcalculations have shown that in most cases, N = 2 or 3,velocity groups are sufficient in order to capture the mainphenomena in bubbly or slug flows [15, 16].

2.3. The Moment Density Method Approach. The momentdensity method proposes an alternative way to modelpolydispersion. This method allows to model the time andspace evolution of a realistic distribution with the help of avery reduced number of transport equations, for example,Kamp et al. [17], Hill [18]. This is an interesting property,especially from a numerical point of view, with regard toalternative methods like population balance methods. In thepresent study, we restrict our attention to adiabatic bubblyflows of an incompressible gas inside a continuous liquidphase, and only consider a dispersion in bubbles size. We herepresent the basic formalism of the moment density method.

2.3.1. A Distribution Function Parameterized by Its StatisticalMoments. The moment density method requires the useof an approximate representation of the bubble populationthanks to a presumed shape continuous function for thebubble size distribution function f , the one being thus totallydetermined by a finite number of parameters. The momentdensities are intensive physical quantities characterizingmean properties of the population, like the density of bubblearea or volume, the so-called volumetric interfacial area ai,and the volumetric fraction α of the dispersed phase. Bubblesbeing assumed as spherical and of diameter d, it yields

ai =∫πd2 f ∂d,

α =∫πd3

6f ∂d.

(3)

In this work, we introduce f as being simply quadratic inthe bubble size d,

f =

⎧⎪⎨⎪⎩

3 n(�x, t) d4 d3

1(�x, t)(2d1(�x, t)− d) if d ≤ 2d1,

0 elsewhere,(4)

where n is the local density number of bubbles (m−3) and d1

is the mean diameter of the distribution. It is easy to showthat n, d1, ai, and α are related through:

2d1ai = 9α,

243π n α2 = 10 a3i .

(5)

The knowledge of the two moment densities ai and αis thus sufficient to reconstruct the whole distributionfunction f . The choice for these two parameters is dictatedby their importance in the description of the two-phase flow,especially concerning the transfers between the phases.


2.3.2. Transport Equations for the Size Distribution. Thetransport of the distribution function f obeys a Liouville-Boltzmann equation from which one can derive transportequations for the main moment densities as well as fortheir transport velocities [19]. The main idea of the methodconsists in solving a finite number of moment densitiestransport equations in the framework of the two-fluidmodel. This solving allows to determine the distributionparameters and thus the whole population evolution. Thuswe consider the solving of the following transport equations:

∂ai∂t

+∇(ai〈�w〉ai) =∫

2πd(dd

dt

)f ∂d +

∫πd2Fc,b∂d,

∂α

∂t+∇·(α〈�w〉α) =

∫πd2

2

(dd

dt

)f ∂d,

(6)

where Fc,b is the source term of the Liouville-Boltzmannequation. The transport velocities 〈�w〉ai and 〈�w〉α of aiand α, respectively, are assumed to be equal to the meantransport velocity of the bubbles �wg (no correlation betweendispersions in velocity and in size is taken into account at thecurrent stage of development of the model). 〈�w〉α is definedby

α〈�w〉α =∫

πd3

6�w f ∂d, (7)

where �w is the velocity of a bubble of size d. It satisfies thetransport equation:

∂α〈�w〉α∂t

+∇·(α〈�w〉α〈�w〉α)

= −∇·(α 〈�w′ �w′〉α) +∫

πd3

6

(d�wdt

)f ∂d.

(8)

2.3.3. Closure Relations. The closure of the system of (3)–(8) concerns models (i) for coalescence and breakup eventsthrough Fc,b, (ii) for the bubble size evolution along itstrajectory dd/dt, (iii) for the hydrodynamic forces actingon individual bubbles, through d�w/dt, as well as (iv) forthe kinetic stress tensor ∇·(α 〈�w′ �w′〉α). The closure issue isbased on the mechanistic model of all these phenomena atthe scale of a single bubble. Source terms of the equations arethen evaluated from the integral of individual contributionsover the whole bubble population, whose size distributionfunction is known. The simplicity of the analytical expression(4) for f allows to easily derive expressions for the integralsfrom the classical models described in Section 2.1. In thisstudy, since the dispersed phase is considered as incompress-ible and incondensable, the bubble size evolution along itstrajectory is zero. Detailed expressions for the source termscan be found in [20].

It is worth pointing out that solving the system of (3)–(8), coupled with the system of balance equations for thecontinuous phase, is sufficient to describe the evolutionof a broad spectrum of bubble sizes. This simplicity ofthe formulation is advantageous especially in terms ofcomputational cost. Further extension of the model, takinginto account the dispersion in velocity, can be considered

Table 1: Water and gas superficial velocities JL and JG of the testsinvestigated in this paper.

JG [m/s]

— 0.0368 0.0898

JL [m/s]1.611 — Run n◦ 097

1.017 Run n◦ 074 Run n◦ 096

using a similar formalism. For example, size-related velocitydistribution can be introduced to model size-dependentbubble migration due to lift force, like it is done for theinhomogeneous MUSIG model. Moreover, this method hasalready been successful to consider more general particlesvelocity dispersion, for example, the work of Fox [21]concerning crossing trajectories.

3. The Experiment–Bubbly FlowAround an Obstacle

In the presented experiment performed at the TOPFLOWfacility of Research center Dresden-Rossendorf (FZD), thelarge test section with a nominal diameter of DN200 wasused to study the flow field around an asymmetric obstacle(see Figure 3). This is an ideal test case for CFD codevalidation, since the obstacle creates a pronounced three-dimensional two-phase flow field. Curved stream lines,which form significant angles with the gravity vector, arecirculation zone in the wake and a flow separation at theedge of the obstacle are common in industrial componentsand installations.

The wire-mesh technology was applied to measure thegas volume fraction and the gas velocity in different distancesup- and downstream the obstacle [22]. The sensor providesdetailed data on the instantaneous flow structure with ahigh resolution in space and time. In particular, they allowvisualizing the structure of the gas-liquid interface [23].

The tests were performed both for air/water and forsteam/water. In the current paper, only adiabatic air/watertests were considered. The parameters are summarized inTable 1.

4. Numerical Settings

4.1. CFX and Population Balance Method. Pretest calcu-lations using ANSYS/CFX and applying a monodispersedbubble size approach were performed for the conditionsof test run 074 (JL = 1.017 m/s, JG = 0.0368 m/s) (see[24–26]). In the calculation, a fluid domain was modeled1.5 m upstream and downstream the obstacle. Half of thetube including a symmetry boundary condition set at thexz-plane of the geometry was simulated. In the presentpaper, the inhomogeneous MUSIG model approach wasapplied to air/water obstacle experiments run 096 (JL =1.017 m/s, JG = 0.0898 m/s) and run 097 (JL = 1.611 m/s,JG = 0.0898 m/s). In the presented calculations for run 096and run 097, 25, and 20, respectively, subsize gas fractionsrepresenting equidistant bubble sizes up to 25 mm and20 mm, respectively, were simulated, assigned to 2 dispersed


Wire mesh sensor

Movable obstacle

Variation of the distanceobstacle-sensor:−520 · · · + 520 mm

Gas injection

φ67

φ25

φ193.7

1500

1000

500

500

2000

2990

Wire-meshsensor

Movableobstacle

Verticaltest section

DN200

Air orsteam

injection

Waterinjection

Figure 3: Sketch of the movable obstacle with driving mechanism—a half-moon shaped horizontal plate mounted on top of a toothed rod.

gaseous phases. The first 6 size groups were assigned to thefirst gaseous phase (or velocity group) and the remainingsize groups were assigned to the second gaseous phase. Thebubble size distribution measured at the largest upstreamposition was set as an inlet boundary condition for thecalculation.

4.2. Neptune CFD and Moment Density Method.Neptune CFD code [27], which is based on the two-fluid model formulation, has been used to performnumerical simulations using the moment density method.The test run 074 has been simulated using a calculationdomain corresponding to a 1 m long half pipe centeredon the obstacle of around 150 000 cells that takes benefitof the symmetry of the experimental setup. Neptune CFDcalculations of test 074 are based on a k-ε model for theliquid flow, the Tchen correlation being used to model thefluctuations related to bubbles turbulent motion. Bubblehydrodynamics model include both laminar and turbulentevaluations of drag, lift, and virtual mass forces, thanksto a drift model for the relative velocity. Inlet boundaryconditions correspond to flat profiles for α, ai, and thevelocities, allowing to recover the experimental surfaceaveraged values for superficial velocities and mean bubblediameter. The surface averaged volumetric fraction andmean diameter are used to reconstruct the inlet bubble sizedistribution function. The numerical method is based ona pressure-based method. Mass, momentum, and energyequations are coupled by an iterative procedure within atime step. Spatial discretization is based on finite volumeframework on unstructured meshes, all the variables beingcomputed at the center of cells.

5. Comparison of Measured andCalculated Results

5.1. The Main Observed Phenomena. Both the steady-state ANSYS CFX calculations applying the inhomogeneousMUSIG model, and the Neptune CFD calculations applyingthe moment density method, could reproduce all qualitativedetails of the flow structure of the two-phase flow fieldaround the diaphragm. The structure of the flow for the hereconsidered test cases 074, 096, and 097 are essentially similar.These different tests have been selected for the purpose ofinvestigating certain phenomena which are more or lesspronounced.

The numerical results have been compared to three-dimensional wire-mesh sensor data in Figure 4 (ANSYS CFXfor the run 096) and Figure 5 (Neptune CFD for the run074). The water velocity and the total gaseous void fractionare represented. All qualitative details of the structure ofthe two-phase flow field around the obstacle could bereproduced.

Shortly, behind the obstacle a strong vortex of theliquid combined with the accumulation of gas is observed.The measured and calculated shape and extension of therecirculation area agree very well. Upstream the obstacle, astagnation point with lower gas content is seen in experimentand calculation. Details, like the velocity and void fractionmaxima above the gap between the circular edge of theobstacle and the inner wall of the pipe, are also found in agood agreement between experiments and calculations. Inthe unobstructed cross sectional part of the tube a strongjet is established. Main discrepancies between experimentaland Neptune CFD results concern the volumetric fraction


Calculation Measurement

Volume fraction (-)

0.18

0.135

0.09

0.045

0

0.3

0

(a)


Water velocity (m/s)2.4

1.8

1.2

0.6

0

4

3.2

2.4

1.6

0.8

0

(m/s)

(b)

Figure 4: Comparison of time averaged values calculated by CFX (left) and measured (right) up- and downstream of the obstacle in theair-water test run 096, JL = 1.017 m/s, JG = 0.0898 m/s, (FB = FC = 0.05).

0.06

0.045

0.03

0.015

0

21.5

10.5

0

Figure 5: Neptune CFD numerical (left-hand side of each pair)versus TOPFLOW experimental (right-hand side of each pair)results for run 074. The two left-hand side pictures represent thedispersed phase volume fraction (scale from 0 to 6·10−2) across thesymmetry plane of the pipe and the two right-hand side pictures theliquid velocity norm (scale from 0 to 2 ms−1).

upward (below) the obstacle and can be associated to the flatprofiles used as inlet bottom boundary conditions.

The structure of the flow is studied in more detail in thefollowing sections.

5.2. Phenomena in the Wake of the Obstacle

5.2.1. Size Distribution of the Bubble Population. Moredetailed understanding of the flow situation can be gained,considering the bubble size distribution. According to theapplied bubble breakup model of Luo and Svendsen [9],

Water turbulenceeddy dissipation (m2s−3)

5

3.75

2.5

1.25

0

X Y

Z

Figure 6: Turbulence eddy dissipation (run 096) (CFX).

bubble breakup can be expected in regions showing high-turbulent eddy dissipation. Figure 6 presents maximum val-ues of the numerically evaluated turbulent eddy dissipationat the edges of the obstacle. At the same time, the appliedbubble coalescence model of Prince and Blanch [8] indicatesstrong importance of coalescence in regions of bubbleaccumulation, that is, in the wake behind the obstacle. Bothbubble coalescence (see gas accumulation shown in Figures4 and 5) and bubble breakup (see distribution of turbulencedissipation Figure 6), which might partially compensate eachother, are expected shortly behind the obstacle.

Figure 7 presents measured cross-sectional averaged bub-ble size distributions upstream (z = −0.52 m), shortlybehind (z = 0.08 m) and downstream the obstacle (z =0.52 m) for the run 096. The measurements show in the


2520151050

DB (mm)

0

0.4

0.8

1.2

1.6

2

dVF/dDB

(%m

m−1

)

Run 096

z = −0.52 mz = 0.08 mz = 0.52 m

Figure 7: Measured bubble size distribution for run 096.

bubble accumulation zone at z = 0.08 m the cross-sectional average shows a shift toward larger bubbles.The experimentally measured bubble size distribution ofrun 074 is essentially similar to the run 096. It can beseen on the volumetric fraction maps, Figure 8. Above theobstacle, where turbulent intensity is strong, breakup is notexperimentally attested, contrarily to the Luo and Svedsen[9] model prediction, but coalescence occurs, leading tothe formation of big bubbles (over 7 mm) that concentratethereafter in the central part of the pipe.

Both ANSYS CFX and Neptune CFD calculated bubblesize distributions, however, show a shift of the mean bubblediameter toward smaller bubbles shortly behind the obstaclewhen both coalescence and breakup are taken into account.In the calculations, the bubble breakup is overestimated.The corresponding results are provided by Figure 9 for therun 096 and on Figure 10 for run 097. This disagreementwas found not solvable by simple changes of breakup orcoalescence coefficients, which were set here to FB = FC =0.05. Similar deviations would arise at other locations ofthe flow domain. Neptune CFD results for the run 074 withstandard coefficients FB = FC = 1 are shown in Figure 11(the details can be found in [20]).

This suggests to perform computations for whichbreakup is neglected, but that still consider the Prince andBlanch [8] coalescence model with standard coefficient. Thecorresponding Neptune CFD results show that both bubblesize repartition and order of magnitude are consistent withexperimental results: big bubbles of more than 7 mm arecreated and accumulate in the recirculation zone just abovethe obstacle (see Figure 14). Big bubbles stay in the LHS ofthe pipe, the unobstructed cross-sectional part of the pipe isfree of big bubbles, that is consistent with the experimentalobservation.

As partial conclusions, (i) the Luo and Svendsen breakupmodel tends to overestimate the breakup of TOPFLOW

experimental tests, whereas (ii) the use of the Prince andBlanch model for coalescence within the framework ofthe moment density method allows to predict satisfyingevolution of bubble size across the flow.

5.2.2. Bubbles Streamlines. More detailed effects of lateralmotion of small and large bubbles can be revealed bystudying bubble streamlines and by analyzing lift forcesacting on bubbles of different size. On one hand, the liquidvelocity flow carries the small bubbles into the region behindthe obstacle (see Figures 12 and 14, right-hand side for thebubble streamlines). Lateral deviation due to lift force isillustrated on Figure 13 for the lift force arrows with thepopulation balance method and Figure 14, right-hand sidefor the lift coefficient with the moment density method. Onthe other hand, the air accumulation in the wake region leadsto bubble coalescence and the generation of large bubblesas revealed by the analysis of experimental results. Thisphenomenon is underestimated in the calculations takingbreakup into account.

Caused by the lift force, large bubbles are redirected intothe downstream jet (see Figure 13) once they can be formedin the wake by coalescence. The streamline representation(see Figure 12) clearly shows this phenomenon for largebubbles already present in the upstream flow.

In the actual version of the moment density methodused in Neptune CFD, dynamics of the bubble populationis estimated using a single transport velocity. The averagedlift contribution takes into account both the bubble sizedistribution and the Tomiyama correlation. When a majorityof bubbles are locally above the critical Eotvos number, thelift coefficient changes its sign. In this case, the directionof the lift force is changed, as it can be seen on Figure 14.As a consequence, the bubble streamlines above the obstacledeviate to the center of the pipe. This explains the lowvalue of the dispersed phase volumetric fraction near theleft-hand side part of the pipe wall above the obstacle forNeptune CFD numerical results (see Figure 5). This result isfully consistent with experimental results.

As a partial conclusion concerning the bubble stream-lines calculations, both methods showed their ability toconsider the effect of bubble size on the lateral deviation ofbubble streamlines due to lift force. This provides a moreprecise understanding and a more accurate prediction ofbubbles repartition across the flow.

5.3. Phenomena in the Jet. In the cross-sectional area besidethe obstacle, a strong jet is established creating strong shearflow. The resulting phenomena are more pronounced withincreasing water velocity, like in run 097, where the liquidvelocity was increased to JL = 1.611 m/s. Figure 16 representsmeasured and ANSYS CFX calculated cross-sectional gasfraction distributions for this run. In the most downstreamcross section of the measurements an almost gas bubble freeregion in the centre of the jet is found. Bubbles are collectedat the edges of the jet in regions of largest water velocitygradient.


0

−0.5

−1

−1.5

−1.9

−2.4

−2.9

−3.4

−3.9

eps (%)

0 < dB < 4.8

(a)

0

−0.3

−0.6

−0.9

−1.2

−1.4

−1.7

−2

−2.3

eps (%)

4.8 < dB < 5.8

(b)

0

−0.3

−0.5

−0.8

−1.1

−1.4

−1.6

−1.9

−2.2

eps (%)

5.8 < dB < 7

(c)

0

−0.8

−1.7

−2.5

−3.3

−4.1

−5

−5.8

−6.6

eps (%)

7 < dB < 1000

(d)

Figure 8: TOPFLOW experimental volumetric fractions of air bubbles according to different size classes, extracted from [24].

2520151050

DB (mm)

0

1

2

3

4

dVF/dDB

(%m

m−1

)

Run 096

−0.52 m0.08 m0.52 m

Figure 9: Bubble size distributions for run 096 (JL = 1.017 m/s,JG = 0.0898 m/s) (CFX) (FB = FC = 0.05).

The streamline representation of the ANSYS CFX cal-culations, however, (Figure 12 for run 096, which is fairlysimilar to run 097) indicates large bubbles being directedinto the jet caused by the lift force. This discrepancy betweenexperiment and ANSYS CFX calculations can possibly beexplained by the strong water velocity gradient near the jet.This strong shear flow induces bubble breakup which is not

20151050

DB (mm)

0

1

2

3

4

5

dVF/dDB

(%m

m−1

)

Run 097

−0.52 m0.08 m0.52 m

Figure 10: Bubble size distributions for run 097 (JL = 1.611 m/s,JG = 0.0898 m/s) (CFX) (FB = FC = 0.05).

yet considered in the model of Luo and Svendsen [9]. In thetests, the big bubbles could migrate toward the jet, but befragmented at the relatively sharp boarder of this jet. Onlya small fraction of the small bubbles created by this breakupprocess can enter the jet by action of the turbulent dispersionforce.


6e − 3

4.5e − 3

3e − 3

1.5e − 3

0

Figure 11: Neptune CFD result for mean Sauter bubble diameter(m), both breakup and coalescence being taken into account (FB =FC = 1) run 074.

Air velocity (ms−1)

2.5

1.875

1.25

0.625

0

dB < 6 mm dB > 6 mm

Figure 12: Streamlines for small (left) and large (right) bubbles(run 096) (CFX).

The gas distribution resolved by bubble size classes (seeFigure 15) shows clearly the deviations between measure-ments and calculations. In the experiment, the gas accumula-tion behind the obstacle leads to a strong coalescence and thecreation of large bubbles. In the CFX-calculations, this effectis underestimated.

In Neptune CFD calculations based on the momentdensity method the bubble breakup was neglected. Asattested by Figure 14, left-hand side, in this case the bubblespresent in the jet are small bubbles. Big bubbles created

(kg m−2 s−2)

5000

3750

2500

1250

0

F1lift

dB < 6 mmF2lift

dB > 6 mm

Figure 13: Bubble lift force vectors for the different gas velocitygroups (run 096) (CFX).

are concentrated in the center of the pipe where the largestvalue of volumetric fraction can be found (Figure 5). Thestructure of the gas repartition in the jet for both calculationand experiments is represented on Figure 17. In the upperpart, the structure of the flow of the numerical calculationis consistent with experimental results. The most importantdeviation is observed in the region just above the obstacle(z = 0.08m). A gain in accuracy of these results could beachieved by considering size-dependent velocity dispersionwithin the moment density method, in particular for bubblesize-dependent lateral migration in the regions where thereexists a strong mixing of different bubble sizes (mainlyjust above the obstacle as far as run 074 is concerned, seeFigure 8).

Nevertheless, the main trends of the bubble dynamicsdownward the obstacle are in agreement with experi-mental results (see Figure 16 for CFX and Figure 17 forNeptune CFD).

6. Summary and Perspectives

In this study, we focused on the model of the bubble sizedistribution in the numerical simulation of bubbly flows.More deep understanding of the flow structure is possiblewhen considering a more accurate characterization of thepolydispersion. For upward two-phase flow in vertical pipesthe core peak in the cross-sectional gas fraction distributioncould be reproduced very well both by the moment densityand by the MUSIG population balance methods. For com-plex flows, the general three-dimensional structure of theflow could be well reproduced in the simulations.

These test cases of pipe flow with internal obstacledemonstrate the complicated relationship and interferencebetween size-dependent bubble migration, bubble coales-cence, and breakup effects for real flows. With an appropriategiven distribution function, the numerical effort of themoment density method is lower compared to the multiple


9.61e − 3

8.21e − 3

6.81e − 3

5.41e − 3

4.01e − 3

Diameter (m)

X

Y

Z

(a)

0.288

0.146

0.005

−0.137

−0.278

CL

X

Y

Z

(b)

Figure 14: Neptune CFD numerical results for local mean bubble diameter and corresponding streamlines colored by the equivalent meanlift coefficient (run 074) (FB = 0;FC = 1).

(-)0.1

0.075

0.05

0.025

0

dB < 6 mm dB > 6 mm

(a)

Obstacle

0 < dB < 5.8max = 7%

5.8 < dB < 200max = 28.3%

(b)

Figure 15: Calculated by CFX (left) and measured (right) gas distributions up- and downstream of the obstacle resolved to bubble sizeclasses (run 096 JL = 1.017 m/s, JG = 0.0898 m/s, FB = 0,FC = 0.05).

bubble size group method (MUSIG). On the other hand,applying the MUSIG method the simulation of a flowsituation allows to deal with more general shapes for thedistribution function. Both methods enable to consider theeffect of polydispersion in size on the bubble populationdynamics, in particular on the evaluation of interphase

momentum transfers associated with lift and drag. The inho-mogeneous population balance model, using several velocityfields for the bubbly phase, is able to deal with size separationof a locally polydispersed in size population, whereas themoment density method accounts for local diversity in bub-ble hydrodynamics thanks to a single-averaged contribution


Volume fraction (-)0.22

0.165

0.11

0.055

0


Figure 16: Calculated by CFX (left) and measured (right) gas cross-fractional distributions downstream the obstacle (run 097 JL =1.611 m/s, JG = 0.0898 m/s, FB = FC = 0.05). Calculations (obstacleshown), distances at z = 0.08 m, 0.16 m, 0.25 m, 0.37 m, and 0.52 m.Measurements (obstacle in the upper left area). distances at z =0.01 m, 0.015 m, 0.02 m, 0.04 m, 0.08 m, 0.16 m, 0.25 m, and 0.52 m.

Volume fraction (-)0.06

0.04

0.03

0.01

0

z = 0.5 m

z = 0.25 m

z = 0.16 m

z = 0.08 m

Figure 17: Calculated by Neptune CFD (left) and measured (right)gas volumetric fraction run 074 for different elevations (obstacle,symbolized in gray, is at z = 0 m and covers the “upper” part of thehalf-pipe cross-section representation).

of interphase transfers. These two promising refinements ofthe two-fluid model for bubbly flows have shown their abilityto recover consistent description of the lift force in the upperpart of the flow, where complex flow structures are observed.

While the closure models on bubble forces, which areresponsible for the simulation of bubble migration, allow

to explain the bubbles hydrodynamic behavior observedexperimentally, clear deviations occur for bubble coalescenceand breakup. Both methods based on similar coalescenceand breakup models lead to the same conclusion: in thesimulations of the TOPFLOW experiments, the Luo andSvendsen model leads to an overestimation of the breakupthat appears as negligible in the experiments. On the otherhand, the coalescence model of Prince and Blanch seems ableto recover the correct bubble size, if used solely, as attested bycorresponding Neptune CFD calculations. For both meth-ods, the presently applied models describing bubble breakupand coalescence could be proven as weak points in numerousCFD analyses. These bubble breakup and coalescence modelsdepend to a large extent on the turbulence properties ofthe two-phase flow, which were not measured and couldnot be validated in the pipe flow test cases. Therefore,further investigations are necessary to determine whether thecurrently used multiphase flow turbulence models deliverappropriate and verifiable quantities that can be used for thedescription of bubble dynamics processes.

Extensions of both the moment density method andthe MUSIG method to nucleate boiling regime numericalsimulation are in progress. This includes the phenomenaof compressibility, phase-change, and wall nucleation. Tomodel the bubble size-dependent lateral migration phe-nomenon, the moment density method should also include amodel for the bubble velocity distribution. This can be doneusing a similar formalism to the present model for bubblesize distribution function.

Nomenclature

ai: Specific interfacial area [m−1]CL: Lift force coefficient [-]d: Bubble diameter [m]d: Lagrangian derivativeEo: Eotvos number [-]f : Size distribution function [m−4]Fc,b: Breakup and coalescence related variations of f

[m−4s−1]FB/C : Breakup, coalescence coefficients [-]FL: Lift force [kg m s−2]J : Superficial velocity [m s−1]n: Bubbles number density [m−3]N : Number of velocity groups [-]M: Number of sub-size groups [-]Re: Reynolds number [-]V : Velocity [m s−1]w: Bubble velocity [m s−1]z: Axial coordinate [m]l: Liquidg: Gasα: Volumetric fraction [-]ρ: Density [kg m−3].

Acknowledgments

Part of this study is carried out at the Institute of SafetyResearch of the FZD as a part of current research projects


funded by the German Federal Ministry of Economics andLabour, Project nos. 150 1265 and 150 1271. The other partof this study carried out at the Institut de Radioprotectionet de Surete Nucleaire (IRSN) has been achieved in theframework of the Neptune project, financially supported byCEA (Commissariat a l’Energie Atomique), EDF (Electricitede France), IRSN, and AREVA-NP. The authors express theirgratitude to the technical TOPFLOW (FZD) team.

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