simulation of bubble breakup dynamics in homogeneous turbulence

Simulation of Bubble Breakup Dynamicsin Homogeneous Turbulence

D. QIAN AND J. B. MCLAUGHLIN

Department of Chemical Engineering, Clarkson University,Potsdam, New York

K. SANKARANARAYANAN AND S. SUNDARESAN

Department of Chemical Engineering, Princeton University,Princeton, New Jersey

K. KONTOMARIS

DuPont Central Research and Development, Wilmington, Delawore

This article presents numerical simulation results for the deformation and breakupof bubbles in homogeneous turbulence under zero gravity conditions. The latticeBoltzmann method was used in the simulations. Homogeneous turbulence wasgenerated by a random stirring force that acted on the fluid in a three-dimensionalperiodic box. The grid size was sufficiently small that the smallest scales of motioncould be simulated for the underlying bubble-free flow. The minimum Weber numberfor bubble breakup was found to be about 3. Bubble breakup was stochastic, and theaverage time needed for breakup was much larger for small Weber numbers than forlarger Weber numbers. For small Weber numbers, breakup was preceded by a longperiod of oscillatory behavior during which the largest linear dimension of the bubblegradually increased. For all Weber numbers, breakup was preceded by a suddenincrease in the largest linear dimension of the bubble. When the Weber numberexceeded the minimum value, the average surface area increased by as much as 80%.

Keywords Bubble breakup; Multiphase flow; Numerical simulation; Turbulence

Introduction

Gas-liquid turbulent flows occur in industrial systems such as stirred tank biochemi-cal reactors and bubble columns. In these flows, the deformation and breakup ofbubbles strongly affect the interfacial area, which, in turn, affects the rates of heat,mass, and momentum transfer. It is, therefore, of interest to determine the conditionsthat lead to bubble deformation and breakup.

Kolmogorov (1949) and Hinze (1955) developed a theory for bubble or dropbreakup in turbulent flows. They suggested that a bubble breaks as a result of inter-actions with turbulent eddies that are of approximately the same size as the bubble.They assumed that the bubble size was in the inertial sub-range of turbulence lengthscales so that Kolmogorov’s universal energy spectrum could be used to estimate the

Address correspondence to J. B. McLaughlin, Department of Chemical Engineering,Clarkson University, Potsdam, NY 13699. E-mail: [email protected]

Chem. Eng. Comm., 193:1038–1063, 2006Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440500354275

1038

strength of eddies having sizes comparable to the bubble. Hinze formulated acriterion for breakup based on a force balance. He pointed out that, in sufficientlystrong turbulence, a bubble would deform and break when the surface tensionwas unable to balance the random pressure fluctuations that cause deformation.He defined a Weber number, We ¼ qlhdu2ðdeÞide=c, where ql is the liquid density,de is the equivalent spherical diameter of the bubble, c is the surface tension, andhdu2(d)i is the mean-square longitudinal velocity difference of the undisturbed flowover a distance d. He proposed that, when the Weber number exceeded a criticalvalue, Wecr, the bubble would break. Based on the experimental results of Clay(1940a, b) for emulsions of drops, Hinze estimated that the critical Weber numberfor drop breakup was Wecr ¼ 1:18.

Levich (1962) developed a criterion for bubble breakup that is similar to that ofKolmogorov and Hinze except that the density of the bubble as well as the liquidappears in the criterion. Shinnar (1961) used Taylor’s (1932, 1934) analysis of break-up due to viscous stresses to develop a criterion for bubble breakup based on theassumption that the bubble sizes are on the order of the Kolmogorov scale or smal-ler. Finally, Baldyga and Bourne (1995) generalized the above results to account forturbulent intermittency using a multifractal approach. The multifractal methodaccounts for the (often large) deviations of the local energy dissipation rate fromthe mean value.

Following Kolmogorov and Hinze, many investigators have studied bubbleor drop size distributions in turbulent flows theoretically and experimentally(Coulaloglou and Tavlarides, 1977; Walter and Blanch, 1986; Prince and Blanch, 1990;Bouaifi and Roustan, 1998). Although most researchers used the Kolmogorov-Hinzetheory, many formulas were proposed to predict the maximum stable bubble or dropsize, and a wide range of critical Weber numbers was obtained based on differentassumptions and experiments. Senhaji (1993) suggested that the critical Weber num-ber was about 0.25 based on experimental studies on air bubbles in a uniform turbu-lent downflow under normal gravity conditions.

The experiments of Sevik and Park (1973) and Risso and Fabre (1998) are parti-cularly relevant to the present study. Sevik and Park (1973) predicted a criticalWeber number equal to 2.6 by observing the splitting of air bubbles penetrating awater jet. They performed experiments with bubbles in the size range 4.0 to5.8 mm. Although there are some apparent typographical errors in the article, itappears that the Taylor microscale Reynolds number of their turbulent flow wasO(103), which is an order of magnitude larger than that to be considered in thepresent study. They postulated a resonance mechanism involving a bubble dynamicsand turbulent fluctuations in addition to the force balance. According to their cri-terion, the threshold Weber number for breakup is determined by the condition thattwo characteristic frequencies are equal. One of these frequencies is that of the n ¼ 2mode of bubble oscillation (Lamb, 1932). The other frequency is the characteristicfrequency of turbulence fluctuations for eddies that are of the same size as the bub-ble. Since the oscillation frequency of a bubble or drop depends on the density ratioof the two phases, one might expect to observe a significant difference between thecritical Weber numbers for drops and bubbles. Hinze’s (1955) analysis of Clay’s(1940a, b) data for emulsions of drops indicates that the critical Weber number isapproximately 1.18. However, Sevik and Park obtained a critical Weber equal to2.6 from their experiments. Their resonance criterion is consistent with this discrep-ancy in the critical Weber numbers for the two sets of experiments.

Simulation of Bubble Breakup Dynamics 1039

Risso and Fabre (1998) obtained values of the critical Weber number between2.7 and 7.8 from experimental data obtained under microgravity conditions. Theyperformed experiments with two sets of bubbles: type A bubbles ranged in size from2 to 6 mm, and type B bubbles ranged in size from 12.4 to 21.4 mm. Breakup wasobserved only for type B bubbles, but only about 50% of type B bubbles broke; thispoints to a stochastic mechanism of breakup. Risso and Fabre did not provide aTaylor microscale Reynolds numbers for their turbulence, which was weakly inho-mogeneous. They identified two bubble breakup mechanisms: force imbalance andresonance oscillation. In weak turbulence, a bubble breaks through a resonancephenomenon in which the n ¼ 2 bubble oscillation mode is dominant. The n ¼ 2mode of oscillation is a degenerate mode that consists of an axisymmetric modeand two non-axisymmetric modes (see, for example, Risso (2000) or Longuet-Higgins (1989)) in which the bubble volume is conserved. The frequencies and damp-ing constants of the oscillation modes may be found in Lamb (1932). A theoreticaltreatment of resonant bubble oscillations in time-periodic straining flows may befound in Kang and Leal (1990); their article incorporates nonlinear effects. However,Risso and Fabre found that. When the turbulence is sufficiently strong, the reson-ance mode is bypassed and the bubble breaks up abruptly.

In the present article, the results of simulations of deformation and breakup ofbubbles in homogeneous turbulence under zero gravity conditions are discussed. Onegoal of the work was to determine the feasibility of using the lattice Boltzmannmethod (LBM) to simulate bubble breakup in turbulence. The other goal of thesimulations was to understand the breakup mechanism. The Reynolds numbers ofthe simulations, based on the spatial period and the turbulent intensity, were toosmall for the existence of an inertial sub-range. However, the Reynolds numberbased on the equivalent spherical bubble diameter and the turbulent intensitywas typically O(102), so inertial effects were important. Thus, Shinnar’s theorywas inapplicable since it is based on Stokes flow.

There is a large literature on the breakup of bubbles and drops in various flows.The reader is referred to the article by Risso (2000) for a more comprehensive review ofthe subject than can be attempted here. It is useful to briefly discuss studies dealingwith the numerical simulation of bubble or drop deformation or breakup in highReynolds number flows. Ryskin and Leal (1984a, b) used an adaptive grid finite dif-ference method to simulate the deformation of steadily rising axisymmetric bubbles.Dandy and Leal (1989) extended the Ryskin-Leal method to study the motion anddeformation of axisymmetric droplets. Kang and Leal (1987) used the Ryskin-Lealmethod to study the deformation and breakup of bubbles in an axisymmetric flow;they did not include buoyancy in their study. Their work is particularly relevant tothe present study since they demonstrated a Reynolds number dependence of the criti-cal Weber number, and they also showed that the critical Weber number depended onthe history of the bubble. For example, in some simulations, they subjected a bubble toa supercritical strain rate for a short period of time. When the strain rate was reducedto a ‘‘subcritical’’ value (as determined in simulations for which the initial bubbleshape was spherical and the bubble was subjected to a single strain rate), they foundthat bubble broke if the strain rate was sufficiently close to the critical value. Similarbehavior will be discussed in the present article in the context of turbulent flow.

Han and Tryggvason (1999) presented simulations of the secondary breakup ofaxisymmetric drops that are accelerated by a constant body force. They used thefront tracking finite difference method (Unverdi and Tryggvason, 1992) to perform

1040 D. Qian et al.

the simulations. Their article also includes references to several studies on the break-up of drops using the volume of fluid method. Han and Tryggvason’s work differsfrom that and reported in the present article in several respects: the unsteadinessnear their drops arose from the droplet motion rather than an external stirringforce; they considered axisymmetric motion, while the present study considersnon-axisymmetric deformation; and the driving force for deformation in their casewas buoyancy, while the present study considers gravity-free conditions.

Sankaranarayanan et al. (1999) used the LBM to study the velocity and defor-mation of freely rising bubbles in periodic arrays. Sankaranarayanan et al. (2002)extended the above work to smaller Morton numbers by developing an ‘‘implicit’’formulation of the LBM that is more stable at small viscosities than the conventional‘‘explicit’’ formulation. The work discussed in the present article uses the explicit orLBM method since zero gravity conditions were considered and the conventionalmethod was found to be adequate. A recent article by Sankaranarayanan et al.(2003) provides validation tests of the LBM against the front tracking finite differ-ence method. A primary difference between this work and that discussed in theabove articles is that the present study deals with the effects of externally forcedhomogeneous turbulence on bubble breakup.

Numerical Methods

The simulations in this article were performed with the LBM. The LBM is discussed inthe books by Rothman and Zaleski (1997) and Succi (2001) and in the article by Chenand Doolen (1998). In this approach, one obtains approximate solutions of theNavier-Stokes equation by solving a kinetic equation for the probability distributionfunctions of an artificial lattice gas. The Chapman-Enskog procedure (Chapman andCowling, 1961) may be used to show that the velocity and pressure fields obtainedfrom the LBM are approximate solutions of the Navier-Stokes equation, providedthey vary slowly in space and time. The LBM has the advantage that it is relatively easyto develop programs for multiphase flows and flows in complex geometries. The LBMis also well suited to parallel computations since the information transfer is local intime and space. Perhaps the greatest advantage of the method is that, for a given com-putational domain, the computational work is independent of the number of bubbles.

In the LBM, it is convenient to work with quantities that are made dimension-less in terms of the time step and grid spacing. Thus, the dimensionless time step isunity and the speed of sound for the lattice gas is O(1). To avoid significant com-pressibility effects, the simulations to be reported were performed in parameterregimes for which the typical fluid velocities were small compared to unity.

Simulations were performed for both single component fluids and two-component fluids. In what follows, the LBM will be described for each of these situa-tions. In both cases, an exhaustive discussion of the methodology will not beattempted. Instead, the main features of the techniques that were used in the simula-tions will be presented. More detailed discussions may be found in the above refer-ences and a dissertation by Qian (2003).

LBM for a Single-Component, Single-Phase Flow

The LBM originated from the method of lattice gas automata (LGA), proposed byFrisch et al. (1986). Rothman and Zaleski (1997) discussed the LGA and its limitations.


Qian et al. (1992) suggested the LBM as a more efficient way of performing simulations.He and Luo (1997) and Shan and He (1998) pointed out that the LBM may be viewedas a discrete version of the continuum Boltzmann equation.

In the LBM, the computational domain is represented by a lattice of nodes onwhich a set of particle probability distribution functions, fi, are computed. Each par-ticle distribution function is associated with a specific lattice velocity, ei. The simula-tions reported were performed on a velocity cubic lattice. One of the lattice velocitieswas zero. The other velocities were chosen such that, in one time step, a particletraveled to one of its nearest lattice nodes.

In the LBM a time step involves two sub-steps:

Collision: f ci ðx; tþ 1Þ ¼ fiðx; tÞ þ Xiðx; tÞ ð1Þ

Streaming: fiðxþ ei; tþ 1Þ ¼ f ci ðx; tþ 1Þ ð2Þ

where fi is the particle distribution function, Xi is the collision term, x and t are theposition vector and time, and the index c denotes collision. The index i varies from0 to 14. The collision term satisfies conservation of mass and momentum:X

i

Xiðx; tÞ ¼ 0 ð3ÞXi

eiXiðx; tÞ ¼ Fðx; tÞ ð4Þ

where Fðx; tÞ is the force acting on the lattice site x at time t. Macroscopic fluidproperties, such as the number density, q, and velocity, u, may be computed fromthe following equations:

q ¼X

i

fi; qu ¼X

i

ei fi ð5Þ

The simulations were performed with the Bhatnagar-Gross-Krook (BGK) form forthe collision term:

Xi ¼ �fiðx; tÞ � f eq

i ðx; tÞs

ð6Þ

where f eqi is the equilibrium particle distribution function and s is a relaxation time.

The 15-velocity cubic lattice eliminates the velocity dependence of the pressureterm that is encountered for one-speed lattices. This lattice was proposed by Chenet al. (1992). Figure 1 shows the lattice and the corresponding lattice velocity vectors.

Figure 1. The 15-velocity lattice and the corresponding lattice velocity vectors.

1042 D. Qian et al.

In the 15-velocity lattice, the equilibrium particle distribution function may beexpressed as (Sankaranarayanan et al., 1999, 2002; Sankaranarayanan andSundaresan, 2000).

f eqi ¼ wiq 1þ 3ei � ðuþ saÞ þ 9

2ðei � ðuþ saÞÞ2 � 3

2ðuþ saÞ � ðuþ saÞ

� �ð7Þ

where a denotes the sum of all forces per unit mass at a given lattice point, and wi

denotes a set of weighting factors that have the following values:

wi ¼

2

9; i ¼ 0

1

9; i ¼ 1; . . . ; 6

1

72; i ¼ 7; . . . ; 14

8>>>>>>><>>>>>>>:ð8Þ

The values for the weighting factors in Equation (8) are chosen so that when theexpression for the equilibrium distribution function is substituted into Equation(5), the correct values of the fluid density and velocity are obtained. He and Luo(1997) discussed the choice of weighting factors.

The relaxation time for this lattice may be expressed as

s ¼ 3n þ 0:5 ð9Þ

where n is the kinematic viscosity of the fluid. The expression in Equation (9) differsfrom that used by Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan andSundaresan (2000) because they used a different unit of length; in their article, thelength of the lattice spacing was

ffiffiffi3p

, while, in the present study, the length of thelattice spacing is unity. The version of the LBM used in the present study is an‘‘explicit’’ form. Sankaranarayanan et al. (2002) developed an implicit formulationof the LBM, where the relationship between the kinematic viscosity and the relaxationtime is different than for the explicit formulation.

To derive the equation of state, one uses the first and second moment of theLBM equation. The resulting expression for the pressure is:

p ¼ q=3 ð10Þ

The expression for the pressure in Equation (10) differs from that inSankaranarayanan et al. (2002) by a factor of three. This difference is caused bythe difference in the lattice spacing. Isothermal conditions are assumed in the simula-tions reported by Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan andSundaresan (2000) and the present simulations.

Using Equations (1–10), one may compute the distribution function at each lat-tice site at any time step and obtain the corresponding density, velocity, and pressurefields.

The BGK form of the lattice-Boltzmann equation (LBE) may be obtained bycombining Equations (1), (2), and (6):

fiðxþ ei; tþ 1Þ � fiðx; tÞ ¼ �fiðx; tÞ � f eq

i ðx; tÞs

ð11Þ


Homogeneous Turbulence Generation

Stationary turbulence was generated in a three-dimensional periodic box using amethod that was suggested by Eswaran and Pope (1988) and further developed byRuetsch and Maxey (1991, 1992) and Wang and Maxey (1993). The fluid was stirredby a random external force field throughout the domain, including the interior of thebubbles. This force (per unit mass) appeared as a body force in the Navier-Stokesequation.

As pointed out by Wang and Maxey (1993), it is desirable that the force fieldshould vary randomly in time so that the resulting turbulence can be spatially homo-geneous. Earlier forcing schemes (e.g., Siggia, 1981; Squires and Eaton, 1991) used astationary forcing function so that the turbulence statistics were spatially inhomoge-neous. Wang and Maxey used a stochastic forcing scheme that was suggested byEswaran and Pope (1988). The force was created by exciting the low-order Fouriermodes using a Uhlenbeck-Ornstein (UO) stochastic process. The reader is referred toWang and Maxey and Qian (2003) for the details.

In the single-phase flow runs, turbulence was developed from a motionless state.Figure 2 shows the time evolution of the turbulence intensity, u0, defined as the spa-tially averaged root-mean-square value of any component of the fluctuating turbu-lent velocity. Typically, the initial 1000 time steps were needed to produce stationaryturbulence intensities. For the simulations to be discussed below, this period of timecorresponded to more than one eddy turnover time.

Table I lists parameter values for the single-phase homogeneous turbulencesimulations. In Table I, L is the box size; n is the kinematic viscosity; u0 is the turbu-lence intensity in one direction; Lf, k, and g are the turbulence integral, Taylor, andKolmogorov length scales, respectively; e is the energy dissipation rate; Te is the eddyturnover time (Te ¼ u02=e); kmax is the maximum wave number; and Rek is theReynolds number based on the Taylor microscale. The box size, the kinematic vis-cosity, and the turbulent intensity were specified for each run. The latter quantitywas determined by the strength of the stirring force. The other parameters may beregarded as ‘‘outputs’’ of the simulations. Turbulence characteristics, such as theintensity, length and time scales, energy spectra, and two-point velocity statistics,were computed by averaging over both space and time after a stationary condition

Figure 2. Turbulence intensity versus time.

1044 D. Qian et al.

was established. In the simulations to be reported, all the values of kmaxg were largerthan unity. Eswaran and Pope (1988) found that when kmaxg is greater than unity,the smallest scales of turbulent motion are resolved.

Figure 3 shows the one-dimensional energy spectra for the simulations to be dis-cussed below. The spectra are scaled with Kolmogorov parameters (Wang andMaxey, 1993). Figure 3 also shows experimental results obtained by Comte-Bellotand Corrsin (1971) for Rek ¼ 60:7. It may be seen that when scaled with Kolmogorovvariables, the results nearly collapse on a single curve. Therefore, in the simulationsto be discussed, most of the turbulence energy was in the dissipation range; there wasno inertial sub-range.

LBM for Two-Component, Two-Phase Flow

The LBM may be applied to multi component and=or multi phase systems.Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan(2000) described the results of such simulations for bubbly flows. They presentedresults for phase equilibria and surface tension as well as for rising bubbles thatmay distort significantly from a spherical shape.

Table I. Parameter values for the single-phase turbulence simulations

No. L n u0 Lf k g e� 106 Te kmaxg Rek

1 96 0.015 0.0217 18.76 13.28 1.54 0.60 784 4.83 19.232 96 0.015 0.0367 19.76 11.44 1.10 2.31 582 3.46 28.003 96 0.015 0.0506 20.22 10.60 0.90 5.12 500 2.83 35.794 96 0.02 0.0202 19.66 14.47 1.92 0.59 698 6.03 14.645 96 0.02 0.0348 21.55 12.70 1.37 2.25 538 4.30 22.106 96 0.02 0.0499 20.90 11.81 1.11 5.36 465 3.49 29.497 64 0.02 0.0348 14.84 9.62 1.20 3.92 308 3.77 16.718 64 0.02 0.0467 14.50 8.72 0.98 8.60 253 3.08 20.35

Figure 3. One-dimensional energy spectra.


In a multicomponent simulation, the particles of each component satisfy a set ofequations similar to those described above. If it is desired to simulate a two-phasegas-liquid system, attractive interactions may be introduced between the particlesof the condensable component. The solubility of the ideal gas component in theliquid may be reduced by introducing an interaction between the components.Macroscopic properties such as density, velocity, and pressure may be computedfrom the distribution functions using mixture rules.

A superscript r that denotes the component may be introduced into Equations(1)–(6), (9), and (11), and the fluid density and velocity may be computed by thefollowing expressions:

q ¼X

r

qr; qu ¼X

r

qrur ð12Þ

The equilibrium particle distribution function may be expressed as

frðeqÞi ¼ wiq

r 1þ 3ei � ðuþ srarÞ þ 9

2ðei � ðuþ srarÞÞ2

�� 3

2ðuþ srarÞ � ðuþ srarÞ

�ð13Þ

Shan and Chen (1993) proposed the use of a particle interaction force to creategas-liquid equilibria. They introduced the following inter particle potential todescribe the microscopic interactions:

Vrrðx; x0Þ ¼ Grrðx; x0ÞwrðxÞwrðx0Þ ð14Þ

where Grrðx;x0Þ is a Green’s function that describes the intensity of the interactionsbetween components r and r at lattice positions x and x0 , and w is an ‘‘effectivemass.’’ Shan and Chen recommended a nearest-neighbor interaction:

Grrðx� x0Þ ¼0 jx� x0j > lc

grr jx� x0j � lc

(ð15Þ

where grr is a constant and lc is the distance from a lattice site to its nearest neigh-bors. From this interparticle potential, the total interparticle force Fr

intðxÞ acting onthe component r at lattice site x can be expressed as:

FrintðxÞ ¼ �wrðxÞ

XS

r¼1

grr

Xb

a¼0

wrðxþ eaÞea ð16Þ

where S is the total number of components and b is the total number of nonzero lattice velocities. In the simulations to be reported, the ‘‘nearest neighbors’’ wereinterpreted more broadly than in Sankaranarayanan et al. (1999, 2002) andSankaranarayanan and Sundaresan (2000) to include sites along diagonals inaddition to the directions parallel to the coordinate axes. Thus, each lattice sitehad 14 nearest neighbors.

For the 15-velocity lattice, Equation (16) may be approximated as follows:

FrintðxÞ � �10wrðxÞ

XS

r¼1

grrrwrðxÞ ð17Þ

1046 D. Qian et al.

The effect of the interparticle interaction force is accounted for through Equation(13), where a

r ¼ Frint=q

r.

The Chapman-Enskog procedure yields the following result for the pressure:

p ¼ q3þ 5

XS

r¼1

XS

r¼1

grrwrwr ð18Þ

In the two-phase flow runs, component 1 was condensable and component 2 wasan ideal gas. For the pure components, the equations of state take the following form:

p1 ¼q1

3þ 5g11w

21; p2 ¼

q2

3ð19Þ

The following expression for w, which was proposed by Shan and Chen (1993),was used in the simulations to be discussed below:

w ¼ q0 1� exp � qq0

� �� ð20Þ

Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan(2002) used alternative expressions for w. The constant q0 was chosen to be 10. Anegative value of g11, which represents an attractive interaction between the particlescomprising component 1, was introduced to induce a phase transition. In the simula-tions, g11 was �0.016. Figure 4 shows the equation of state relation for each compo-nent. The other Green’s functions were chosen as follows: �0:008 � g12 ¼ g21

� �0:006 and g22 ¼ 0. The values of g12 were adjusted to minimize the fraction ofcondensable vapor inside the gas phase; it was necessary to choose negative valuesto avoid unphysical behavior in the gas phase. Under these conditions, the densityin the bulk liquid phase was about 18 and the density in the bulk gas phase wasabout 1.4. Thus, the density ratio was about 13; such a ratio would correspond toa high-pressure system. For simplicity, the kinematic viscosities of the componentswere assigned the same value.

Figure 4. Equations of state for pure components 1 and 2.


As pointed out by Shan and Chen (1993), the interparticle interaction createssurface tension between the two phases. Provided that a drop or bubble is sufficientlylarge, they found that the surface tension, as determined by the Young-Laplace law,was independent of the radius of the drop or bubble. Sankaranarayanan (2002)documented the fact that the surface tension is constant during deformation.

Both single-phase and two-phase flow simulations were performed. Most of theturbulence characteristics were obtained from the single-phase flow simulations.Two steps were involved in a two-phase flow simulation: (1) a stationary bubble witha desired diameter was created and (2) the stirring force for the turbulence wasimposed to generate turbulence. Typically, several thousand time steps were neededto approach stationary conditions.

The initial density field prescribed to create a stationary bubble strongly affectedthe stability of the program and the time needed to reach the equilibrium state. In thesimulations to be discussed, a final density field from a simulation in a small compu-tational box was used to create an initial density field for a simulation in a largerbox. This was done by assuming that the liquid density was uniform outside theregion in which the density field from the small box was used.

In order to improve the isotropy of the flow, and improve the accuracy of thecomputed velocity field near the interface, a high-order expansion of the gradientformula was adopted (Qian, 2003). The velocity of the fluid was computed by aver-aging the values before and after the collision step (Shan and Chen, 1993) becausethere was a large density gradient in the interface region. Figure 5 shows a densitycontour plot and the density profile as a function of position along a line passingthrough the center of a stationary bubble. From the pressure field of the stationarybubble, the surface tension was computed by using the Young-Laplace law asc ¼ Dpr=2, where Dp is the computed difference in average pressure between theinterior and the exterior of the bubble and r is the bubble radius. Typically, the bub-ble interface is about 3�4 lattice units thick. Over the range of conditions examinedin the present simulations, it was observed that the interface thickness did not changesignificantly with the bubble size. Therefore, if one wishes to minimize the effect ofthe finite interfacial thickness, one needs to perform computations for a large bubble.

After a stationary bubble was created, a stirring force was used to generateturbulence. The simulations provided velocity, pressure, and density fields on each

Figure 5. Cross-sectional view of density contours and the density profile for a stationarybubble used as the initial condition for Run 6 of Table II.

1048 D. Qian et al.

time step. The density field was used to determine the location, size, and shape of thebubbles. The bubble boundary was identified as the set of locations where the valueof the fluid density was equal to the average value of density in the bulk liquid andbulk gas phase.

The bubble volume oscillated as a result of the turbulence fluctuations. It isknown (see Feng and Leal, 1994) that shape and volume oscillations are coupled.Sankaranarayanan (2002) benchmarked the bubble oscillations obtained fromLBM simulations against theory and experiment. For the simulations to be reported,the bubble volume oscillation was less than 20%; the corresponding variation in theequivalent spherical diameter was roughly 7%. The physical situation correspondingto the simulations was that of a liquid-gas mixture in which the liquid was close tothe boiling point so that pressure variations around its surface could cause signifi-cant phase changes. Increasing the bubble size reduced the pressure differenceand, therefore, decreased the volume oscillations.

Results

Table II lists parameter values for the two-phase simulations in lattice units. By com-bining variables, one can create dimensionless groups that can be compared withexperimental results. The initial condition for all simulations was a single bubble,of diameter de, in a three-dimensional periodic box with either 963 or 643 grid points.The duration of most runs was 10,000 time steps. However, Runs 4, 5, 8, and 14 werestopped after the bubble broke because of subsequent numerical instabilities thatlead to unphysical behavior. In Table II, L is the length of the computational box

Table II. Parameter values for the two-phase turbulence simulations

No. L n de Rek We hSi hA�i Tb� c

1 96 0.015 44.98 19.23 1.025 0.0318 0.0291 — 0.6862 96 0.015 44.98 28.00 2.900 0.544 0.257 11.00 0.6863 96a 0.015 44.98 28.00 2.900 0.211 0.134 — 0.6864 96 0.015 44.98 35.79 5.368 0.752 0.413 2.80 0.6865 96a 0.015 44.98 35.79 5.368 0.665 0.192 4.00 0.6866 96 0.015 34.16 19.23 0.930 0.0125 0.0033 — 0.5737 96 0.015 34.16 28.00 2.470 0.0875 0.106 — 0.5738 96 0.015 34.16 35.79 4.600 0.277 0.215 4.40 0.5739 96 0.02 44.26 14.64 0.827 0.0213 0.0355 — 0.787

10 96 0.02 44.26 22.10 2.184 0.125 0.0840 — 0.78711 96 0.02 44.26 29.49 4.551 0.615 0.3290 3.01 0.78712 96 0.02 34.42 14.64 0.641 0.0183 0.0094 — 0.72413 96 0.02 34.42 22.10 1.722 0.0757 0.0547 — 0.72414 96 0.02 34.42 29.49 3.590 0.449 0.319 6.02 0.72415 96 0.02 27.26 22.10 1.364 0.0457 0.0368 — 0.63616 96 0.02 27.26 29.49 2.867 0.199 0.172 12.04 0.63617 64 0.02 27.75 16.71 1.607 0.0615 0.0482 — 0.68418 64 0.02 27.75 20.35 2.848 0.170 0.107 37.94 0.684

a Different turbulent series with the same strength.


edge; Rek is based on the single-phase flow simulations; We is based on the single-phase averaged (over space and time) longitudinal velocity difference over distancesequal to the bubble equivalent spherical diameter; S� ¼ S=S0 � 1 is the fractionaldeviation (at any instant) of the bubble total surface area relative to its initial value,S0, for the spherical bubble released at t ¼ 0; A� ¼ A=A0 � 1 is the fractional devi-ation of the area projected by the bubble on the x-y plane (computed to allow com-parisons with the experiments of Risso and Fabre) relative to its initial value;h idenotes time averaging over the entire duration of a run; and T�b is the time ofthe first bubble breakup divided by the eddy turnover time from Table I. The surfacetension (computed from the initial stationary bubble using the Young-Laplace law)is given in the final column; it depends weakly on the kinematic viscosity and thebubble size.

When performing a LBM simulation, it is not necessary to perform ‘‘numericalsurgery’’ to permit a bubble or drop to break. In this respect, the LBM differs fromsome alternative numerical simulation methods, such as the finite element or bound-ary element methods, since it does not explicitly track an interface and simulates afluid with density variations. However, one must still establish criteria to determinewhen a bubble has broken. These criteria are based on the density of the fluid.

Figure 6 shows the relation between the mean deformation, hS�i, and the Webernumber for the runs in Table II. For We < 3 the distortion of the bubble is small andthe bubble does not break. For We > 3, bubble distortion is significant, the bubblebreaks, and the data points become more scattered. Therefore, the simulations sug-gest a critical Weber number Wecr � 3:0. When the critical Weber number wasexceeded, the time-averaged total bubble surface area increased by 20 to 80%.The hA�i plot is shown to facilitate comparison with the corresponding plot by Rissoand Fabre.

A goal of this study was to better understand the mechanism of bubble breakup.To that end, predictions for the maximum stable bubble diameter, dmax, based on theforce balance mechanism were compared to the simulation results. Table III showspredictions based on the formulae suggested by Hinze (1955), Levich (1962), Baldyga

Figure 6. Mean variation of bubble surface area and projected area versus Weber number forthe runs in Table II. The run numbers for selected runs appear in brackets [ ].

1050 D. Qian et al.

and Bourne (1995), and Shinnar (1961):

dmax ¼

0:725ðc=qlÞ0:6e�0:4 ðHinzeÞc0:6

q0:4l q0:2

b e0:4ðLevichÞ

Lfc

e2=3qlL5=3f

24 350:926

ðBaldyga-BourneÞ

cn1=2l

lle1=2

16ðlb=llÞ þ 16

19ðlb=llÞ þ 16

� �ðShinnarÞ

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð21Þ

In the above formulae, c, qb, ql, e, lb, ll, and Lf denote the surface tension, den-sity of the bubble, density of the liquid, average turbulent energy dissipation rate inthe single-phase simulation, viscosity of the gas, viscosity of the liquid, and the inte-gral length scale of the single-phase simulation, respectively. In calculating dmax, thevalues of the surface tension, liquid density, gas density, and dissipation rate weretaken to be (in dimensionless LBM units) 0.7, 18, 1.4, and 2.31� 10�6, respectively.The value of the dissipation rate is that for Run 2 in Table I. The predicted dmax

values may be compared with Runs 2, 3, and 7 in Table II.Breakup was observed for Run 2, for which the initial bubble diameter was 45.

The fact that bubble breakup was not observed in Run 7 does not necessarily meanthat the predictions of the models are incorrect since, by simply changing the initialseed used to compute the random stirring force, no breakup was observed in Run 3even though all of the physical parameters were identical to those in Run 2. This isconsistent with the idea that the fluctuations leading to breakup are intermittent andone cannot be sure that breakup would not occur if one performed a simulation (orzero gravity experiment) over a longer period of time. Indeed, it may be seen that, inRun 18, breakup occurred for a bubble having a diameter equal to 27.8. The single-phase dissipation rate in this case was 8.60� 10�6. The predictions of the differentmodels for Run 18 are given in Table IV.

Table III. Predicted maximum stable bubble sizes for Runs 2, 3,and 7 in Table II

Model

Hinze Levich Baldyga and Bourne Shinnar

dmax 18.4 42.4 28.9 206

Table IV. Predicted maximum stable bubble sizes for Run 18 inTable II

Author(s)

Hinze Levich Baldyga and Bourne Shinnar

dmax 10.5 24.1 14.8 87.0


The only model that is clearly inconsistent with the simulations is that ofShinnar. This is not surprising since Shinnar’s theory is based on a small bubble-scale Reynolds number assumption, and the bubble-scale Reynolds number waslarge compared to unity in all simulations. On the other hand, one could also notexpect perfect agreement with the models of Hinze, Levich, or Baldyga and Bournesince those models were based on the assumption that the turbulence Reynolds num-ber was sufficiently large to permit the existence of an inertial sub-range. Figure 7shows the quantity hdu2(d)i for one of the single-phase simulations (Run 3). The pre-diction of the Kolmogorov theory for this quantity in the inertial sub-range is alsoshown in Figure 7. It may be seen that there is a significant difference between theKolmogorov theory and the computed values. The difference becomes larger forsmaller values of the separation, d, which is consistent with the notion that theenergy spectrum in the simulations decreases more rapidly in the simulated turbu-lence than it would in the inertial sub-range of high Reynolds number turbulence.

It may also be seen from Figure 7 that the bubbles used in the simulations werecomparable to or larger than the integral scale of the turbulence. Therefore, themaximum size of a bubble based on Hinze’s model should be given bydmax ¼ 0:59c=ðqlu

02Þ.Figure 8 shows the frequency power spectrum of the variations in the bubble

projected area and the corresponding time series for Run 3. According to Rissoand Fabre, the n ¼ 2 oscillation appears as a maximum in the frequency spectrumof the projected area. The frequency and damping constant of the oscillations of abubble at large Reynolds number are given by Lamb (1932). For the conditions ofRun 3 in Table II, the period of the oscillation is 989 LBM units or 1.26 eddy turn-over times. It may be seen from Figure 8 that there is no clear evidence of an unu-sually strong oscillation at this period. On the other hand, the raw power spectrashown by Risso and Fabre also typically exhibited weak maxima at the n ¼ 2 modefrequency. It is also likely that the relatively large gas volume fraction and the inter-action of a bubble with its periodic neighbors influenced the bubble dynamics. Well-defined maxima appeared only after Fougere’s method had been applied to the rawtime data; this approach was not attempted in the present work. Although it is not

Figure 7. Predictions of the Kolmogorov theory for hdu2(d)i in the inertial sub-range com-pared with the computed values in a single-phase run.

1052 D. Qian et al.

possible to make a definitive statement regarding the n ¼ 2 mode based on thepresent simulations, complex oscillations followed by an abrupt breakup wereobserved in the simulations; this may be consistent with the notion of stochastic res-onance suggested by Risso and Fabre.

In the experimental observations reported by Risso and Fabre, it was not feas-ible to determine the variations in the total surface area of a bubble. An advantage ofnumerical simulations is that this information is accessible and may be related toother items of interest such as the times at which a bubble breaks up. Figure 9 showsa measure of the bubble deformation based on its surface area, S�, as a function ofthe time measured in eddy turnover times for Runs 1 and 13. As may be seen, thebubble does not break. The Reynolds numbers of the single-phase flows based onthe Taylor microscale were 19.23 and 22.1. The largest change in the bubble surfacearea is roughly 28%. Figure 10 shows S� for two cases (Runs 2 and 14 of Table II)for which the bubble breaks. It may be seen that the bubble does not break at the

Figure 8. Frequency power spectrum and time dependence of the deviation of the projectedarea.

Figure 9. Variation of bubble surface area with time for a case in which the bubble did notbreak.


time when the largest value of S� is reached. In fact, for Run 2, S� is decreasing at thepoint of breakup, and it continues to decrease for some time after breakup. This mayoccur when the bubble has been deformed to a shape like that shown in the firstpanel of Figure 12 in which a satellite bubble has nearly broken away from the restof the bubble. When breakup finally occurs, the surface area actually decreasesslightly. However, as may be seen in Figure 10, the surface area eventually increasesagain and S� eventually reaches values close to unity. In some cases, the child bub-bles formed by a breakup eventually coalesced into a larger bubble. This behaviormay be due to the periodic boundary conditions, which prevent the child bubblesfrom moving far away from one another.

Figure 11 shows the mode of breakup for most of the bubbles that broke in thesimulations. No more than three child bubbles were ever observed in any of thesimulations. However, as may be seen in Figure 4 of their article, Risso and Fabrefound examples in which as many as 10 child bubbles formed. It seems possible thatthis difference is caused by the difference in the turbulent energy spectra in the simu-lations and the experiments. The energy spectrum in the simulations decreased morerapidly with wavenumber for wavelengths on the order of the bubble size than thecorresponding spectrum in the experiments. Therefore, one would expect that therewas less energy available to create the small-scale disturbances on the bubble surfacethat would be likely to create large numbers of child bubbles.

Figure 12 shows images of the bubble just prior to and just after breakup forRun 4. The images in Figure 11 are not at equally spaced times; they were selectedto show the typical behaviors of the bubbles. The bubble shapes are somewhatsmoother than those shown by Risso and Fabre, which may be consistent with thenotion that the energy spectrum for the simulated turbulence decreases more rapidlywith wavenumber than the spectrum in the experiments. Figure 13 shows a projec-tion onto the plane of the paper of the velocity vectors at the surface of the bubbleat the same instants in time as those shown in Figure 12. It appears that the bridgebetween the two child bubbles is broken by a shearing rather than an extensionalmotion. This is consistent with the fact that, in many cases, the bubble surface areaachieves a maximum value before breaking. In such cases, the extensional motion

Figure 10. Variation of bubble surface area with time for two cases (Run 2, 14) in which thebubble breaks.

1054 D. Qian et al.

deforms the bubble into a highly elongated shape, which then breaks up at a latertime when the surface area is somewhat smaller. However, once a neck forms in abubble, capillarity should play an important role in causing the neck’s diameter todecrease. It is conceivable that capillarity, rather than shear, causes the breakup inFigure 12.

Figure 12. Bubble breakup for Run 4.

Figure 11. Bubble breakup patterns.


An effort was made to identify simple criteria for breakup based on the stretch-ing of a bubble or the difference in fluid velocities at different points on its surface.However, no critical values could be established for the maximum linear dimensionof a bubble or the maximum magnitude of the difference between values of the fluidvelocity at points on the bubble surface. In both cases, examples were found in whicha large value was observed followed by a decrease to much smaller values, and, poss-ibly after many more oscillations, breakup occurred at a value of the parameter thatwas often substantially smaller than the maximum value that had been observed.Table V summarizes results for the maximum difference between velocities at differ-ent points on the bubble and the largest linear dimension of the bubble at breakup.The maximum difference in the fluid velocity vectors between different points on thesurface of the bubble, dvmax, is compared with the root-mean-square value,

ffiffiffi3p

u0.The difference is at least three times the root-mean-square value. Much larger differ-ences were, however, observed long before breakup in several cases. Table V alsogives the values of the largest linear dimension of the bubble, lmax, divided by thespherical diameter of the bubble. The value of lmax is determined at the point ofbreakup. It may be seen that the largest linear dimension of the bubble varies widely;in one case it is only 33% larger than the equivalent spherical diameter, while inanother case it is 133% larger. A related parameter is a Weber number based onthe largest linear dimension and the maximum velocity difference across the bubbleat the point of breakup. This quantity also varies widely, with values between 31 and

Figure 13. Projections of the velocity vectors at the surfaces of the bubbles in Figure 11.

Table V. Velocity field and bubble characteristics at or near the point of breakup

No. dvmax=ðffiffiffi3p

u0Þ lmax=de We0max S�max A�max

2 4.343 2.078 43.51 1.022 0.5474 3.539 1.973 48.04 0.578 0.3525 5.473 2.456 122.27 0.871 0.4958 4.228 1.329 31.20 0.395 0.53511 3.132 1.809 33.82 0.368 0.25114 3.823 2.147 56.03 0.638 0.50416 3.569 2.333 47.95 0.375 0.48718 4.639 1.590 38.10 1.081 0.590

1056 D. Qian et al.

122. However, in all cases, this Weber number is an order of magnitude larger thanthe Weber number based on the average flow characteristics of the single-phase flowand the equivalent spherical diameter of the bubble. These results underscore thestochastic nature of breakup, which is consistent with the results of Risso and Fabre.

Finally, Risso and Fabre showed a plot of the fraction of bubbles in their experi-ments that experienced a given maximum deformation. Deformation was measuredby A�. They showed results for different elapsed times from the beginning of anexperiment. A vertical line on their graph indicated the onset of break up. The cor-responding maximum amount of deformation experienced by a bubble was approxi-mately 0.5. This means that A� reached this value at or before the time of breakup.Intuitively, it seems likely that this quantity is less sensitive to the duration of theexperiments than the fraction of bubbles that breakup or the maximum stable bubble

Figure 14. Fraction of bubbles with a given maximum amount of deformation, A�, at four dif-ferent times normalized by the eddy turnover time.

Figure 15. Fraction of bubbles with a given maximum amount of deformation, S�, at fourdifferent times normalized by the eddy turnover time.


size. For comparison, Figure 14 shows the computed results. It may be seen that theminimum amount of deformation needed for breakup in the simulations is abouthalf of the experimental value. Figure 14 also shows a vertical line correspondingto the minimum amount of deformation needed to guarantee breakup. The valueof this quantity is 0.6. The corresponding experimental value is 1.0.

Figure 15 is similar to Figure 14 except that the deformation is measured by S�

rather than A�. The minimum amount of deformation needed for breakup, by thismeasure, is approximately 0.37. The minimum amount of deformation needed toguarantee breakup is 1.0.

Conclusion

In many respects, the results for bubble breakup in this paper agree well with thelow-gravity bubble breakup experiments reported by Risso and Fabre (1998). Inboth cases, a Weber number can be identified below which breakup is not observed.This Weber number is based on the statistics of the single-phase flow that wouldexist in the absence of the bubble. In the simulations, this Weber number wasapproximately 3.0. The value of the Weber number below which breakup is notobserved probably depends on the duration of the simulation. For a longer simula-tion, a lower value would probably have been obtained. The simulations indicatethat the minimum amount of deformation, as measured by the fractional changein the bubble surface area, prior to breakup is about 0.37. The corresponding defor-mation, as measured by the fractional change in the area projected by the bubble ona plane is about 0.25. The latter value is smaller than the value suggested by Rissoand Fabre. The mean deformation plots in Figure 6 of the present article are quali-tatively similar to the experimental plot of deformation as measured by the projectedarea. In both the simulations and the experiments, there is a fairly abrupt transitionto much larger values of the deformation near the critical Weber number.

No simple criteria could be found that could be associated with breakup. In sev-eral cases, bubbles became highly extended and then returned to much more com-pact shapes before breaking. The maximum magnitude of the velocity differencebetween any two points on a bubble’s surface is also not a good indicator of whetheror not a bubble will break. As with the linear dimension of the bubble, extremelylarge velocity differences were sometimes observed after which the velocity differ-ences became substantially smaller before the bubble finally broke. Similar behaviorwas obtained for the surface area of a bubble. These observations appear to be con-sistent with the conclusion of Risso and Fabre that breakup is a stochastic process.They argued that stochastic resonance played an important role in the breakup pro-cess. The simulations provide some indirect support for this idea since, in most cases,breakup was preceded by a slow, secular growth of the maximum linear dimension.It appears that the spatial structure of the flow in the vicinity is more important thansimple criteria such as linear extension or surface area in determining when a bubblewill break. As a consequence of the computational demands of the simulations, itwas not possible to obtain statistical results for the characteristics of the flow field,but this would be a useful goal for a future work.

A point of disagreement between the experiments of Risso and Fabre and thepresent simulations lies in the number of child bubbles that result from breakup.In the simulations, no more than 3 child bubbles resulted from breakup. However,in the experiments, as many as 10 child bubbles were observed. It seems plausible

1058 D. Qian et al.

that the latter difference may be due to the fact that, in the simulations, the energyspectrum of the turbulence decreases more quickly with wavenumber than in theexperiments, in which the bubbles were in the inertial sub-range of length scales.The finite size of the computational domain and the associated interaction of a bub-ble with its periodic neighbors are also likely sources of discrepancies between thesimulations and the experiments. Finally, small bubbles may simply dissolve dueto Ostwald ripening.

An advantage of the simulations is that they provide information about the sur-face area of the bubbles. For the largest bubble and Reynolds number, the averagesurface area was 80% larger than the area of a sphere with the same volume. How-ever, the results also indicate a sensitivity to initial conditions that suggests a muchlonger simulation time would be needed to obtain accurate statistical results. Thestochastic nature of the breakup process is an important point of agreement betweenthe simulations and the experiments of Risso and Fabre.

Acknowledgments

This work was supported by the U.S. Department of Energy under Grant DE-FG02-88ER13919 and by a grant from DuPont. We acknowledge the support and facilitiesof the National Center for Supercomputer Applications at the University of Illinoisat Urbana, Illinois. The authors would also like to express their appreciation toDr. X. Shan for helpful discussions about the lattice Boltzmann method.

Nomenclature

a force per unit massA bubble total projected area at x-y planeA0 projected area of a spherical bubble at x-y planeA� bubble projected area variation relative to a spherical bubbleb total number of nonzero velocity states on a lattice sitede equivalent spherical diameter of a bubbledmax maximum stable bubble sizeei lattice velocityE11 1-D turbulence energy spectrumfi particle distribution functionf� dimensionless frequencyfi

c particle distribution function at collision stepfi

eq equilibrium distribution functionfmag parameter that controls turbulence intensityF external force per unit mass in physical spaceFint interphase force per unit volumeFt fraction of bubbles that experience a given maximum deformationeFF external force per unit mass in spectral spaceeFF0 modified external force per unit mass in spectral spaceg Green’s function parameterG(x� x0) Green’s functionk wave vectorkmax maximum wavenumberl, m, n coordinates in spectral space


lc distance from a lattice site to its nearest neighbors.L box sizeLf longitudinal scale of turbulenceN number of grid points along one of the box edgesp pressurep.s.d. power spectrum densityr initial radius of bubbleRek Taylor microscale Reynolds numberS bubble total surface areaS0 surface area of a spherical bubbleS� fractional change of bubble surface area relative to a spherical bubblet timet� time measured in eddy turnover timesTb� dimensionless time of the first bubble breakup measured in eddy turn-

over timesTf time period over which the turbulence is modulatedTe eddy turnover timeu velocityu0 turbulence intensity in one directionVrrðx; x0Þ interparticle potentialwi weighting factorWe Weber numberWecr critical Weber number for bubble breakupx; x0 lattice sitex, y, z coordinates in physical spacexx; yy; zz unit vectorsx1; x2 random numbers between 0 and 1y1; y2 Gaussian random number

Greek Letters

a, b stochastic process constantsc surface tensionhdu2(d)i mean square of the difference in the turbulent velocities over a distance

equal to ddvmax maximum difference in the fluid velocity vectors between different

points on the bubble surfaceDp pressure difference between the interior and the exterior of the bubbleDs time interval in stochastic processe energy dissipation rateg Kolmogorov scale of turbulencek Taylor microscale of turbulencel dynamic viscosity of the liquidn kinematic viscosity of the liquidq densityq0 arbitrary constantqb gas densityql liquid densitys relaxation time

1060 D. Qian et al.

wr effective massXi collision term

Subscripts and Superscripts

1 liquid component2 gas componentb bubble phasec collisioncr criticaleq equilibriuml liquidmag magnitudemax maximumr; r component label

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simulation of bubble breakup dynamics in homogeneous turbulence

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