Chem. Eng. Comm., 193:1038–1063, 2006 Copyright # Taylor & Francis Group, LLC ISSN: 0098-6445 print/1563-5201 online DOI: 10.1080/00986440500354275
Simulation of Bubble Breakup Dynamics in 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 breakup of bubbles in homogeneous turbulence under zero gravity conditions. The lattice Boltzmann method was used in the simulations. Homogeneous turbulence was generated by a random stirring force that acted on the fluid in a three-dimensional periodic box. The grid size was sufficiently small that the smallest scales of motion could be simulated for the underlying bubble-free flow. The minimum Weber number for bubble breakup was found to be about 3. Bubble breakup was stochastic, and the average time needed for breakup was much larger for small Weber numbers than for larger Weber numbers. For small Weber numbers, breakup was preceded by a long period of oscillatory behavior during which the largest linear dimension of the bubble gradually increased. For all Weber numbers, breakup was preceded by a sudden increase in the largest linear dimension of the bubble. When the Weber number exceeded 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 biochemical reactors and bubble columns. In these flows, the deformation and breakup of bubbles 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 conditions that lead to bubble deformation and breakup. Kolmogorov (1949) and Hinze (1955) developed a theory for bubble or drop breakup in turbulent flows. They suggested that a bubble breaks as a result of interactions 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 length scales 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]
1038
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strength of eddies having sizes comparable to the bubble. Hinze formulated a criterion for breakup based on a force balance. He pointed out that, in sufficiently strong turbulence, a bubble would deform and break when the surface tension was unable to balance the random pressure fluctuations that cause deformation. He defined a Weber number, We ¼ ql hdu2 ðde Þide =c, where ql is the liquid density, de is the equivalent spherical diameter of the bubble, c is the surface tension, and hdu2(d)i is the mean-square longitudinal velocity difference of the undisturbed flow over a distance d. He proposed that, when the Weber number exceeded a critical value, Wecr , the bubble would break. Based on the experimental results of Clay (1940a, b) for emulsions of drops, Hinze estimated that the critical Weber number for drop breakup was Wecr ¼ 1:18. Levich (1962) developed a criterion for bubble breakup that is similar to that of Kolmogorov and Hinze except that the density of the bubble as well as the liquid appears in the criterion. Shinnar (1961) used Taylor’s (1932, 1934) analysis of breakup due to viscous stresses to develop a criterion for bubble breakup based on the assumption that the bubble sizes are on the order of the Kolmogorov scale or smaller. Finally, Baldyga and Bourne (1995) generalized the above results to account for turbulent intermittency using a multifractal approach. The multifractal method accounts for the (often large) deviations of the local energy dissipation rate from the mean value. Following Kolmogorov and Hinze, many investigators have studied bubble or 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-Hinze theory, many formulas were proposed to predict the maximum stable bubble or drop size, and a wide range of critical Weber numbers was obtained based on different assumptions and experiments. Senhaji (1993) suggested that the critical Weber number was about 0.25 based on experimental studies on air bubbles in a uniform turbulent downflow under normal gravity conditions. The experiments of Sevik and Park (1973) and Risso and Fabre (1998) are particularly relevant to the present study. Sevik and Park (1973) predicted a critical Weber number equal to 2.6 by observing the splitting of air bubbles penetrating a water jet. They performed experiments with bubbles in the size range 4.0 to 5.8 mm. Although there are some apparent typographical errors in the article, it appears that the Taylor microscale Reynolds number of their turbulent flow was O(103), which is an order of magnitude larger than that to be considered in the present study. They postulated a resonance mechanism involving a bubble dynamics and turbulent fluctuations in addition to the force balance. According to their criterion, the threshold Weber number for breakup is determined by the condition that two characteristic frequencies are equal. One of these frequencies is that of the n ¼ 2 mode of bubble oscillation (Lamb, 1932). The other frequency is the characteristic frequency of turbulence fluctuations for eddies that are of the same size as the bubble. Since the oscillation frequency of a bubble or drop depends on the density ratio of the two phases, one might expect to observe a significant difference between the critical 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 is approximately 1.18. However, Sevik and Park obtained a critical Weber equal to 2.6 from their experiments. Their resonance criterion is consistent with this discrepancy in the critical Weber numbers for the two sets of experiments.
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Risso and Fabre (1998) obtained values of the critical Weber number between 2.7 and 7.8 from experimental data obtained under microgravity conditions. They performed experiments with two sets of bubbles: type A bubbles ranged in size from 2 to 6 mm, and type B bubbles ranged in size from 12.4 to 21.4 mm. Breakup was observed only for type B bubbles, but only about 50% of type B bubbles broke; this points to a stochastic mechanism of breakup. Risso and Fabre did not provide a Taylor microscale Reynolds numbers for their turbulence, which was weakly inhomogeneous. They identified two bubble breakup mechanisms: force imbalance and resonance oscillation. In weak turbulence, a bubble breaks through a resonance phenomenon in which the n ¼ 2 bubble oscillation mode is dominant. The n ¼ 2 mode of oscillation is a degenerate mode that consists of an axisymmetric mode and two non-axisymmetric modes (see, for example, Risso (2000) or LonguetHiggins (1989)) in which the bubble volume is conserved. The frequencies and damping constants of the oscillation modes may be found in Lamb (1932). A theoretical treatment of resonant bubble oscillations in time-periodic straining flows may be found in Kang and Leal (1990); their article incorporates nonlinear effects. However, Risso and Fabre found that. When the turbulence is sufficiently strong, the resonance mode is bypassed and the bubble breaks up abruptly. In the present article, the results of simulations of deformation and breakup of bubbles in homogeneous turbulence under zero gravity conditions are discussed. One goal of the work was to determine the feasibility of using the lattice Boltzmann method (LBM) to simulate bubble breakup in turbulence. The other goal of the simulations was to understand the breakup mechanism. The Reynolds numbers of the simulations, based on the spatial period and the turbulent intensity, were too small for the existence of an inertial sub-range. However, the Reynolds number based on the equivalent spherical bubble diameter and the turbulent intensity was typically O(102), so inertial effects were important. Thus, Shinnar’s theory was 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 of the subject than can be attempted here. It is useful to briefly discuss studies dealing with the numerical simulation of bubble or drop deformation or breakup in high Reynolds number flows. Ryskin and Leal (1984a, b) used an adaptive grid finite difference method to simulate the deformation of steadily rising axisymmetric bubbles. Dandy and Leal (1989) extended the Ryskin-Leal method to study the motion and deformation of axisymmetric droplets. Kang and Leal (1987) used the Ryskin-Leal method 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 to the present study since they demonstrated a Reynolds number dependence of the critical Weber number, and they also showed that the critical Weber number depended on the history of the bubble. For example, in some simulations, they subjected a bubble to a supercritical strain rate for a short period of time. When the strain rate was reduced to a ‘‘subcritical’’ value (as determined in simulations for which the initial bubble shape was spherical and the bubble was subjected to a single strain rate), they found that bubble broke if the strain rate was sufficiently close to the critical value. Similar behavior will be discussed in the present article in the context of turbulent flow. Han and Tryggvason (1999) presented simulations of the secondary breakup of axisymmetric drops that are accelerated by a constant body force. They used the front tracking finite difference method (Unverdi and Tryggvason, 1992) to perform
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the simulations. Their article also includes references to several studies on the breakup of drops using the volume of fluid method. Han and Tryggvason’s work differs from that and reported in the present article in several respects: the unsteadiness near their drops arose from the droplet motion rather than an external stirring force; they considered axisymmetric motion, while the present study considers non-axisymmetric deformation; and the driving force for deformation in their case was buoyancy, while the present study considers gravity-free conditions. Sankaranarayanan et al. (1999) used the LBM to study the velocity and deformation 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 or LBM method since zero gravity conditions were considered and the conventional method was found to be adequate. A recent article by Sankaranarayanan et al. (2003) provides validation tests of the LBM against the front tracking finite difference method. A primary difference between this work and that discussed in the above articles is that the present study deals with the effects of externally forced homogeneous turbulence on bubble breakup.
Numerical Methods The simulations in this article were performed with the LBM. The LBM is discussed in the books by Rothman and Zaleski (1997) and Succi (2001) and in the article by Chen and Doolen (1998). In this approach, one obtains approximate solutions of the Navier-Stokes equation by solving a kinetic equation for the probability distribution functions of an artificial lattice gas. The Chapman-Enskog procedure (Chapman and Cowling, 1961) may be used to show that the velocity and pressure fields obtained from the LBM are approximate solutions of the Navier-Stokes equation, provided they vary slowly in space and time. The LBM has the advantage that it is relatively easy to develop programs for multiphase flows and flows in complex geometries. The LBM is also well suited to parallel computations since the information transfer is local in time and space. Perhaps the greatest advantage of the method is that, for a given computational domain, the computational work is independent of the number of bubbles. In the LBM, it is convenient to work with quantities that are made dimensionless in terms of the time step and grid spacing. Thus, the dimensionless time step is unity and the speed of sound for the lattice gas is O(1). To avoid significant compressibility effects, the simulations to be reported were performed in parameter regimes for which the typical fluid velocities were small compared to unity. Simulations were performed for both single component fluids and twocomponent fluids. In what follows, the LBM will be described for each of these situations. In both cases, an exhaustive discussion of the methodology will not be attempted. Instead, the main features of the techniques that were used in the simulations will be presented. More detailed discussions may be found in the above references 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 by Frisch et al. (1986). Rothman and Zaleski (1997) discussed the LGA and its limitations.
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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 viewed as a discrete version of the continuum Boltzmann equation. In the LBM, the computational domain is represented by a lattice of nodes on which a set of particle probability distribution functions, fi, are computed. Each particle distribution function is associated with a specific lattice velocity, ei. The simulations reported were performed on a velocity cubic lattice. One of the lattice velocities was zero. The other velocities were chosen such that, in one time step, a particle traveled to one of its nearest lattice nodes. In the LBM a time step involves two sub-steps: Collision: fic ðx; t þ 1Þ ¼ fi ðx; tÞ þ Xi ðx; tÞ
ð1Þ
Streaming: fi ðx þ ei ; t þ 1Þ ¼ fic ðx; t þ 1Þ
ð2Þ
where fi is the particle distribution function, Xi is the collision term, x and t are the position vector and time, and the index c denotes collision. The index i varies from 0 to 14. The collision term satisfies conservation of mass and momentum: X Xi ðx; tÞ ¼ 0 ð3Þ X
i
ei Xi ðx; tÞ ¼ Fðx; tÞ
ð4Þ
i
where Fðx; tÞ is the force acting on the lattice site x at time t. Macroscopic fluid properties, such as the number density, q, and velocity, u, may be computed from the following equations: X X q¼ fi ; qu ¼ ei fi ð5Þ i
i
The simulations were performed with the Bhatnagar-Gross-Krook (BGK) form for the collision term: Xi ¼
fi ðx; tÞ fieq ðx; tÞ s
ð6Þ
where fieq is the equilibrium particle distribution function and s is a relaxation time. The 15-velocity cubic lattice eliminates the velocity dependence of the pressure term that is encountered for one-speed lattices. This lattice was proposed by Chen et 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.
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In the 15-velocity lattice, the equilibrium particle distribution function may be expressed as (Sankaranarayanan et al., 1999, 2002; Sankaranarayanan and Sundaresan, 2000). fieq
9 3 ¼ wi q 1 þ 3ei ðu þ saÞ þ ðei ðu þ saÞÞ2 ðu þ saÞ ðu þ saÞ 2 2
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: 8 2 > > ; i¼0 > > > 9 > > < 1 ð8Þ wi ¼ ; i ¼ 1; . . . ; 6 > > >9 > > > 1 > : ; i ¼ 7; . . . ; 14 72 The values for the weighting factors in Equation (8) are chosen so that when the expression 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) differs from that used by Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan (2000) because they used pffiffiffi a different unit of length; in their article, the length of the lattice spacing was 3, while, in the present study, the length of the lattice spacing is unity. The version of the LBM used in the present study is an ‘‘explicit’’ form. Sankaranarayanan et al. (2002) developed an implicit formulation of the LBM, where the relationship between the kinematic viscosity and the relaxation time is different than for the explicit formulation. To derive the equation of state, one uses the first and second moment of the LBM equation. The resulting expression for the pressure is: p ¼ q=3
ð10Þ
The expression for the pressure in Equation (10) differs from that in Sankaranarayanan et al. (2002) by a factor of three. This difference is caused by the difference in the lattice spacing. Isothermal conditions are assumed in the simulations reported by Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan (2000) and the present simulations. Using Equations (1–10), one may compute the distribution function at each lattice site at any time step and obtain the corresponding density, velocity, and pressure fields. The BGK form of the lattice-Boltzmann equation (LBE) may be obtained by combining Equations (1), (2), and (6): fi ðx þ ei ; t þ 1Þ fi ðx; tÞ ¼
fi ðx; tÞ fieq ðx; tÞ s
ð11Þ
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Homogeneous Turbulence Generation Stationary turbulence was generated in a three-dimensional periodic box using a method that was suggested by Eswaran and Pope (1988) and further developed by Ruetsch and Maxey (1991, 1992) and Wang and Maxey (1993). The fluid was stirred by a random external force field throughout the domain, including the interior of the bubbles. This force (per unit mass) appeared as a body force in the Navier-Stokes equation. As pointed out by Wang and Maxey (1993), it is desirable that the force field should vary randomly in time so that the resulting turbulence can be spatially homogeneous. Earlier forcing schemes (e.g., Siggia, 1981; Squires and Eaton, 1991) used a stationary forcing function so that the turbulence statistics were spatially inhomogeneous. Wang and Maxey used a stochastic forcing scheme that was suggested by Eswaran and Pope (1988). The force was created by exciting the low-order Fourier modes using a Uhlenbeck-Ornstein (UO) stochastic process. The reader is referred to Wang 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 spatially averaged root-mean-square value of any component of the fluctuating turbulent velocity. Typically, the initial 1000 time steps were needed to produce stationary turbulence intensities. For the simulations to be discussed below, this period of time corresponded to more than one eddy turnover time. Table I lists parameter values for the single-phase homogeneous turbulence simulations. In Table I, L is the box size; n is the kinematic viscosity; u0 is the turbulence intensity in one direction; Lf, k, and g are the turbulence integral, Taylor, and Kolmogorov length scales, respectively; e is the energy dissipation rate; Te is the eddy turnover time (Te ¼ u02 =e); kmax is the maximum wave number; and Rek is the Reynolds number based on the Taylor microscale. The box size, the kinematic viscosity, and the turbulent intensity were specified for each run. The latter quantity was determined by the strength of the stirring force. The other parameters may be regarded as ‘‘outputs’’ of the simulations. Turbulence characteristics, such as the intensity, 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.
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Table I. Parameter values for the single-phase turbulence simulations No.
L
n
u0
Lf
k
g
e 106
Te
kmax g
Rek
1 2 3 4 5 6 7 8
96 96 96 96 96 96 64 64
0.015 0.015 0.015 0.02 0.02 0.02 0.02 0.02
0.0217 0.0367 0.0506 0.0202 0.0348 0.0499 0.0348 0.0467
18.76 19.76 20.22 19.66 21.55 20.90 14.84 14.50
13.28 11.44 10.60 14.47 12.70 11.81 9.62 8.72
1.54 1.10 0.90 1.92 1.37 1.11 1.20 0.98
0.60 2.31 5.12 0.59 2.25 5.36 3.92 8.60
784 582 500 698 538 465 308 253
4.83 3.46 2.83 6.03 4.30 3.49 3.77 3.08
19.23 28.00 35.79 14.64 22.10 29.49 16.71 20.35
was established. In the simulations to be reported, all the values of kmax g were larger than unity. Eswaran and Pope (1988) found that when kmax g 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 discussed below. The spectra are scaled with Kolmogorov parameters (Wang and Maxey, 1993). Figure 3 also shows experimental results obtained by Comte-Bellot and Corrsin (1971) for Rek ¼ 60:7. It may be seen that when scaled with Kolmogorov variables, the results nearly collapse on a single curve. Therefore, in the simulations to be discussed, most of the turbulence energy was in the dissipation range; there was no 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 presented results for phase equilibria and surface tension as well as for rising bubbles that may distort significantly from a spherical shape.
Figure 3. One-dimensional energy spectra.
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In a multicomponent simulation, the particles of each component satisfy a set of equations similar to those described above. If it is desired to simulate a two-phase gas-liquid system, attractive interactions may be introduced between the particles of the condensable component. The solubility of the ideal gas component in the liquid may be reduced by introducing an interaction between the components. Macroscopic properties such as density, velocity, and pressure may be computed from 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 the following expressions: X X q¼ qr ; qu ¼ qr ur ð12Þ r
r
The equilibrium particle distribution function may be expressed as 9 rðeqÞ r fi ¼ wi q 1 þ 3ei ðu þ sr ar Þ þ ðei ðu þ sr ar ÞÞ2 2 3 r r r r ðu þ s a Þ ðu þ s a Þ 2
ð13Þ
Shan and Chen (1993) proposed the use of a particle interaction force to create gas-liquid equilibria. They introduced the following inter particle potential to describe the microscopic interactions: Vrr ðx; x0 Þ ¼ Grr ðx; x0 Þwr ðxÞwr ðx0 Þ
ð14Þ
where Grr ðx; x0 Þ is a Green’s function that describes the intensity of the interactions between components r and r at lattice positions x and x0 , and w is an ‘‘effective mass.’’ Shan and Chen recommended a nearest-neighbor interaction: ( 0 jx x0 j > lc 0 Grr ðx x Þ ¼ ð15Þ grr jx x0 j lc where grr is a constant and lc is the distance from a lattice site to its nearest neighbors. From this interparticle potential, the total interparticle force Frint ðxÞ acting on the component r at lattice site x can be expressed as: Frint ðxÞ ¼ wr ðxÞ
S X
grr
r¼1
b X
wr ðx þ ea Þea
ð16Þ
a¼0
where S is the total number of components and b is the total number of non zero lattice velocities. In the simulations to be reported, the ‘‘nearest neighbors’’ were interpreted more broadly than in Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan (2000) to include sites along diagonals in addition to the directions parallel to the coordinate axes. Thus, each lattice site had 14 nearest neighbors. For the 15-velocity lattice, Equation (16) may be approximated as follows: Frint ðxÞ 10wr ðxÞ
S X r¼1
grr rwr ðxÞ
ð17Þ
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The effect of the interparticle interaction force is accounted for through Equation r r r (13), where a ¼ F int=q . The Chapman-Enskog procedure yields the following result for the pressure: p¼
S X S X q þ5 grr wr wr 3 r¼1 r¼1
ð18Þ
In the two-phase flow runs, component 1 was condensable and component 2 was an ideal gas. For the pure components, the equations of state take the following form: p1 ¼
q1 þ 5g11 w21 ; 3
p2 ¼
q2 3
ð19Þ
The following expression for w, which was proposed by Shan and Chen (1993), was used in the simulations to be discussed below: q w ¼ q0 1 exp ð20Þ q0 Sankaranarayanan et al. (1999, 2002) and Sankaranarayanan and Sundaresan (2002) used alternative expressions for w. The constant q0 was chosen to be 10. A negative value of g11 , which represents an attractive interaction between the particles comprising component 1, was introduced to induce a phase transition. In the simulations, g11 was 0.016. Figure 4 shows the equation of state relation for each component. 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 of condensable vapor inside the gas phase; it was necessary to choose negative values to avoid unphysical behavior in the gas phase. Under these conditions, the density in the bulk liquid phase was about 18 and the density in the bulk gas phase was about 1.4. Thus, the density ratio was about 13; such a ratio would correspond to a high-pressure system. For simplicity, the kinematic viscosities of the components were assigned the same value.
Figure 4. Equations of state for pure components 1 and 2.
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As pointed out by Shan and Chen (1993), the interparticle interaction creates surface tension between the two phases. Provided that a drop or bubble is sufficiently large, 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 the turbulence characteristics were obtained from the single-phase flow simulations. Two steps were involved in a two-phase flow simulation: (1) a stationary bubble with a desired diameter was created and (2) the stirring force for the turbulence was imposed to generate turbulence. Typically, several thousand time steps were needed to approach stationary conditions. The initial density field prescribed to create a stationary bubble strongly affected the stability of the program and the time needed to reach the equilibrium state. In the simulations to be discussed, a final density field from a simulation in a small computational box was used to create an initial density field for a simulation in a larger box. This was done by assuming that the liquid density was uniform outside the region 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 the computed velocity field near the interface, a high-order expansion of the gradient formula was adopted (Qian, 2003). The velocity of the fluid was computed by averaging the values before and after the collision step (Shan and Chen, 1993) because there was a large density gradient in the interface region. Figure 5 shows a density contour plot and the density profile as a function of position along a line passing through the center of a stationary bubble. From the pressure field of the stationary bubble, the surface tension was computed by using the Young-Laplace law as c ¼ Dpr=2, where Dp is the computed difference in average pressure between the interior and the exterior of the bubble and r is the bubble radius. Typically, the bubble interface is about 3 4 lattice units thick. Over the range of conditions examined in the present simulations, it was observed that the interface thickness did not change significantly with the bubble size. Therefore, if one wishes to minimize the effect of the finite interfacial thickness, one needs to perform computations for a large bubble. After a stationary bubble was created, a stirring force was used to generate turbulence. The simulations provided velocity, pressure, and density fields on each
Figure 5. Cross-sectional view of density contours and the density profile for a stationary bubble used as the initial condition for Run 6 of Table II.
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time step. The density field was used to determine the location, size, and shape of the bubbles. The bubble boundary was identified as the set of locations where the value of the fluid density was equal to the average value of density in the bulk liquid and bulk gas phase. The bubble volume oscillated as a result of the turbulence fluctuations. It is known (see Feng and Leal, 1994) that shape and volume oscillations are coupled. Sankaranarayanan (2002) benchmarked the bubble oscillations obtained from LBM simulations against theory and experiment. For the simulations to be reported, the bubble volume oscillation was less than 20%; the corresponding variation in the equivalent spherical diameter was roughly 7%. The physical situation corresponding to the simulations was that of a liquid-gas mixture in which the liquid was close to the boiling point so that pressure variations around its surface could cause significant phase changes. Increasing the bubble size reduced the pressure difference and, therefore, decreased the volume oscillations.
Results Table II lists parameter values for the two-phase simulations in lattice units. By combining variables, one can create dimensionless groups that can be compared with experimental 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 were stopped after the bubble broke because of subsequent numerical instabilities that lead 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
96 96 96a 96 96a 96 96 96 96 96 96 96 96 96 96 96 64 64
0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02
44.98 44.98 44.98 44.98 44.98 34.16 34.16 34.16 44.26 44.26 44.26 34.42 34.42 34.42 27.26 27.26 27.75 27.75
19.23 28.00 28.00 35.79 35.79 19.23 28.00 35.79 14.64 22.10 29.49 14.64 22.10 29.49 22.10 29.49 16.71 20.35
1.025 2.900 2.900 5.368 5.368 0.930 2.470 4.600 0.827 2.184 4.551 0.641 1.722 3.590 1.364 2.867 1.607 2.848
0.0318 0.544 0.211 0.752 0.665 0.0125 0.0875 0.277 0.0213 0.125 0.615 0.0183 0.0757 0.449 0.0457 0.199 0.0615 0.170
0.0291 0.257 0.134 0.413 0.192 0.0033 0.106 0.215 0.0355 0.0840 0.3290 0.0094 0.0547 0.319 0.0368 0.172 0.0482 0.107
— 11.00 — 2.80 4.00 — — 4.40 — — 3.01 — — 6.02 — 12.04 — 37.94
0.686 0.686 0.686 0.686 0.686 0.573 0.573 0.573 0.787 0.787 0.787 0.724 0.724 0.724 0.636 0.636 0.684 0.684
a
Different turbulent series with the same strength.
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edge; Rek is based on the single-phase flow simulations; We is based on the singlephase averaged (over space and time) longitudinal velocity difference over distances equal to the bubble equivalent spherical diameter; S ¼ S=S0 1 is the fractional deviation (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 deviation of the area projected by the bubble on the x-y plane (computed to allow comparisons with the experiments of Risso and Fabre) relative to its initial value; h idenotes time averaging over the entire duration of a run; and Tb is the time of the first bubble breakup divided by the eddy turnover time from Table I. The surface tension (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 the bubble size. When performing a LBM simulation, it is not necessary to perform ‘‘numerical surgery’’ to permit a bubble or drop to break. In this respect, the LBM differs from some alternative numerical simulation methods, such as the finite element or boundary element methods, since it does not explicitly track an interface and simulates a fluid with density variations. However, one must still establish criteria to determine when 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 Weber number for the runs in Table II. For We < 3 the distortion of the bubble is small and the bubble does not break. For We > 3, bubble distortion is significant, the bubble breaks, and the data points become more scattered. Therefore, the simulations suggest a critical Weber number Wecr 3:0. When the critical Weber number was exceeded, 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 Risso and 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 the force balance mechanism were compared to the simulation results. Table III shows predictions 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 for the runs in Table II. The run numbers for selected runs appear in brackets [ ].
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Table III. Predicted maximum stable bubble sizes for Runs 2, 3, and 7 in Table II Model
dmax
Hinze
Levich
Baldyga and Bourne
Shinnar
18.4
42.4
28.9
206
and Bourne (1995), and Shinnar (1961): 8 0:725ðc=ql Þ0:6 e0:4 ðHinzeÞ > > > > > > c0:6 > > ðLevichÞ > 0:4 0:2 0:4 > ql qb e > > > 2 < 30:926 dmax ¼ c > 5 ðBaldyga-BourneÞ Lf 4 > > 5=3 > > e2=3 ql Lf > > > > 1=2 > > > 16ðlb =ll Þ þ 16 > cn l : ðShinnarÞ ll e1=2 19ðlb =ll Þ þ 16
ð21Þ
In the above formulae, c, qb, ql, e, lb, ll, and Lf denote the surface tension, density of the bubble, density of the liquid, average turbulent energy dissipation rate in the single-phase simulation, viscosity of the gas, viscosity of the liquid, and the integral length scale of the single-phase simulation, respectively. In calculating dmax, the values of the surface tension, liquid density, gas density, and dissipation rate were taken to be (in dimensionless LBM units) 0.7, 18, 1.4, and 2.31 106, 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 mean that the predictions of the models are incorrect since, by simply changing the initial seed used to compute the random stirring force, no breakup was observed in Run 3 even though all of the physical parameters were identical to those in Run 2. This is consistent with the idea that the fluctuations leading to breakup are intermittent and one cannot be sure that breakup would not occur if one performed a simulation (or zero gravity experiment) over a longer period of time. Indeed, it may be seen that, in Run 18, breakup occurred for a bubble having a diameter equal to 27.8. The singlephase dissipation rate in this case was 8.60 10 6. The predictions of the different models for Run 18 are given in Table IV. Table IV. Predicted maximum stable bubble sizes for Run 18 in Table II Author(s)
dmax
Hinze
Levich
Baldyga and Bourne
Shinnar
10.5
24.1
14.8
87.0
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Figure 7. Predictions of the Kolmogorov theory for hdu2(d)i in the inertial sub-range compared with the computed values in a single-phase run.
The only model that is clearly inconsistent with the simulations is that of Shinnar. This is not surprising since Shinnar’s theory is based on a small bubblescale Reynolds number assumption, and the bubble-scale Reynolds number was large compared to unity in all simulations. On the other hand, one could also not expect perfect agreement with the models of Hinze, Levich, or Baldyga and Bourne since those models were based on the assumption that the turbulence Reynolds number was sufficiently large to permit the existence of an inertial sub-range. Figure 7 shows the quantity hdu2(d)i for one of the single-phase simulations (Run 3). The prediction of the Kolmogorov theory for this quantity in the inertial sub-range is also shown in Figure 7. It may be seen that there is a significant difference between the Kolmogorov theory and the computed values. The difference becomes larger for smaller values of the separation, d, which is consistent with the notion that the energy spectrum in the simulations decreases more rapidly in the simulated turbulence 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 were comparable to or larger than the integral scale of the turbulence. Therefore, the maximum size of a bubble based on Hinze’s model should be given by 2 dmax ¼ 0:59c=ðql u0 Þ. 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 Risso and Fabre, the n ¼ 2 oscillation appears as a maximum in the frequency spectrum of the projected area. The frequency and damping constant of the oscillations of a bubble at large Reynolds number are given by Lamb (1932). For the conditions of Run 3 in Table II, the period of the oscillation is 989 LBM units or 1.26 eddy turnover times. It may be seen from Figure 8 that there is no clear evidence of an unusually strong oscillation at this period. On the other hand, the raw power spectra shown by Risso and Fabre also typically exhibited weak maxima at the n ¼ 2 mode frequency. It is also likely that the relatively large gas volume fraction and the interaction of a bubble with its periodic neighbors influenced the bubble dynamics. Welldefined maxima appeared only after Fougere’s method had been applied to the raw time data; this approach was not attempted in the present work. Although it is not
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Figure 8. Frequency power spectrum and time dependence of the deviation of the projected area.
possible to make a definitive statement regarding the n ¼ 2 mode based on the present simulations, complex oscillations followed by an abrupt breakup were observed in the simulations; this may be consistent with the notion of stochastic resonance suggested by Risso and Fabre. In the experimental observations reported by Risso and Fabre, it was not feasible to determine the variations in the total surface area of a bubble. An advantage of numerical simulations is that this information is accessible and may be related to other items of interest such as the times at which a bubble breaks up. Figure 9 shows a measure of the bubble deformation based on its surface area, S , as a function of the time measured in eddy turnover times for Runs 1 and 13. As may be seen, the bubble does not break. The Reynolds numbers of the single-phase flows based on the Taylor microscale were 19.23 and 22.1. The largest change in the bubble surface area 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 9. Variation of bubble surface area with time for a case in which the bubble did not break.
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Figure 10. Variation of bubble surface area with time for two cases (Run 2, 14) in which the bubble breaks.
time when the largest value of S is reached. In fact, for Run 2, S is decreasing at the point of breakup, and it continues to decrease for some time after breakup. This may occur when the bubble has been deformed to a shape like that shown in the first panel of Figure 12 in which a satellite bubble has nearly broken away from the rest of the bubble. When breakup finally occurs, the surface area actually decreases slightly. However, as may be seen in Figure 10, the surface area eventually increases again and S eventually reaches values close to unity. In some cases, the child bubbles formed by a breakup eventually coalesced into a larger bubble. This behavior may be due to the periodic boundary conditions, which prevent the child bubbles from moving far away from one another. Figure 11 shows the mode of breakup for most of the bubbles that broke in the simulations. No more than three child bubbles were ever observed in any of the simulations. However, as may be seen in Figure 4 of their article, Risso and Fabre found examples in which as many as 10 child bubbles formed. It seems possible that this difference is caused by the difference in the turbulent energy spectra in the simulations and the experiments. The energy spectrum in the simulations decreased more rapidly with wavenumber for wavelengths on the order of the bubble size than the corresponding spectrum in the experiments. Therefore, one would expect that there was less energy available to create the small-scale disturbances on the bubble surface that would be likely to create large numbers of child bubbles. Figure 12 shows images of the bubble just prior to and just after breakup for Run 4. The images in Figure 11 are not at equally spaced times; they were selected to show the typical behaviors of the bubbles. The bubble shapes are somewhat smoother than those shown by Risso and Fabre, which may be consistent with the notion that the energy spectrum for the simulated turbulence decreases more rapidly with wavenumber than the spectrum in the experiments. Figure 13 shows a projection onto the plane of the paper of the velocity vectors at the surface of the bubble at the same instants in time as those shown in Figure 12. It appears that the bridge between the two child bubbles is broken by a shearing rather than an extensional motion. This is consistent with the fact that, in many cases, the bubble surface area achieves a maximum value before breaking. In such cases, the extensional motion
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Figure 11. Bubble breakup patterns.
deforms the bubble into a highly elongated shape, which then breaks up at a later time when the surface area is somewhat smaller. However, once a neck forms in a bubble, capillarity should play an important role in causing the neck’s diameter to decrease. It is conceivable that capillarity, rather than shear, causes the breakup in Figure 12.
Figure 12. Bubble breakup for Run 4.
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Figure 13. Projections of the velocity vectors at the surfaces of the bubbles in Figure 11.
An effort was made to identify simple criteria for breakup based on the stretching 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 dimension of a bubble or the maximum magnitude of the difference between values of the fluid velocity at points on the bubble surface. In both cases, examples were found in which a large value was observed followed by a decrease to much smaller values, and, possibly after many more oscillations, breakup occurred at a value of the parameter that was often substantially smaller than the maximum value that had been observed. Table V summarizes results for the maximum difference between velocities at different 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 pffiffithe ffi surface of the bubble, dvmax, is compared with the root-mean-square value, 3u0 . The difference is at least three times the root-mean-square value. Much larger differences were, however, observed long before breakup in several cases. Table V also gives the values of the largest linear dimension of the bubble, lmax, divided by the spherical diameter of the bubble. The value of lmax is determined at the point of breakup. 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 in another case it is 133% larger. A related parameter is a Weber number based on the largest linear dimension and the maximum velocity difference across the bubble at the point of breakup. This quantity also varies widely, with values between 31 and
Table V. Velocity field and bubble characteristics at or near the point of breakup pffiffiffi lmax =de We0max Smax Amax No. dvmax=ð 3u0 Þ 2 4 5 8 11 14 16 18
4.343 3.539 5.473 4.228 3.132 3.823 3.569 4.639
2.078 1.973 2.456 1.329 1.809 2.147 2.333 1.590
43.51 48.04 122.27 31.20 33.82 56.03 47.95 38.10
1.022 0.578 0.871 0.395 0.368 0.638 0.375 1.081
0.547 0.352 0.495 0.535 0.251 0.504 0.487 0.590
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Figure 14. Fraction of bubbles with a given maximum amount of deformation, A , at four different times normalized by the eddy turnover time.
122. However, in all cases, this Weber number is an order of magnitude larger than the Weber number based on the average flow characteristics of the single-phase flow and the equivalent spherical diameter of the bubble. These results underscore the stochastic 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 experiments that experienced a given maximum deformation. Deformation was measured by A . They showed results for different elapsed times from the beginning of an experiment. A vertical line on their graph indicated the onset of break up. The corresponding maximum amount of deformation experienced by a bubble was approximately 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 the experiments than the fraction of bubbles that breakup or the maximum stable bubble
Figure 15. Fraction of bubbles with a given maximum amount of deformation, S , at four different times normalized by the eddy turnover time.
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size. For comparison, Figure 14 shows the computed results. It may be seen that the minimum amount of deformation needed for breakup in the simulations is about half of the experimental value. Figure 14 also shows a vertical line corresponding to the minimum amount of deformation needed to guarantee breakup. The value of 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 this measure, is approximately 0.37. The minimum amount of deformation needed to guarantee breakup is 1.0.
Conclusion In many respects, the results for bubble breakup in this paper agree well with the low-gravity bubble breakup experiments reported by Risso and Fabre (1998). In both 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 would exist in the absence of the bubble. In the simulations, this Weber number was approximately 3.0. The value of the Weber number below which breakup is not observed probably depends on the duration of the simulation. For a longer simulation, a lower value would probably have been obtained. The simulations indicate that the minimum amount of deformation, as measured by the fractional change in the bubble surface area, prior to breakup is about 0.37. The corresponding deformation, as measured by the fractional change in the area projected by the bubble on a plane is about 0.25. The latter value is smaller than the value suggested by Risso and Fabre. The mean deformation plots in Figure 6 of the present article are qualitatively similar to the experimental plot of deformation as measured by the projected area. In both the simulations and the experiments, there is a fairly abrupt transition to much larger values of the deformation near the critical Weber number. No simple criteria could be found that could be associated with breakup. In several cases, bubbles became highly extended and then returned to much more compact shapes before breaking. The maximum magnitude of the velocity difference between any two points on a bubble’s surface is also not a good indicator of whether or not a bubble will break. As with the linear dimension of the bubble, extremely large velocity differences were sometimes observed after which the velocity differences became substantially smaller before the bubble finally broke. Similar behavior was obtained for the surface area of a bubble. These observations appear to be consistent 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 process. 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 than simple criteria such as linear extension or surface area in determining when a bubble will break. As a consequence of the computational demands of the simulations, it was 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 the present 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
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that the latter difference may be due to the fact that, in the simulations, the energy spectrum of the turbulence decreases more quickly with wavenumber than in the experiments, 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 bubble with its periodic neighbors are also likely sources of discrepancies between the simulations and the experiments. Finally, small bubbles may simply dissolve due to Ostwald ripening. An advantage of the simulations is that they provide information about the surface area of the bubbles. For the largest bubble and Reynolds number, the average surface area was 80% larger than the area of a sphere with the same volume. However, the results also indicate a sensitivity to initial conditions that suggests a much longer simulation time would be needed to obtain accurate statistical results. The stochastic nature of the breakup process is an important point of agreement between the simulations and the experiments of Risso and Fabre.
Acknowledgments This work was supported by the U.S. Department of Energy under Grant DE-FG0288ER13919 and by a grant from DuPont. We acknowledge the support and facilities of the National Center for Supercomputer Applications at the University of Illinois at Urbana, Illinois. The authors would also like to express their appreciation to Dr. X. Shan for helpful discussions about the lattice Boltzmann method.
Nomenclature a A A0 A b de dmax ei E11 fi f fic fieq fmag F Fint Ft e F e0 F g G(x x0 ) k kmax l, m, n
force per unit mass bubble total projected area at x-y plane projected area of a spherical bubble at x-y plane bubble projected area variation relative to a spherical bubble total number of nonzero velocity states on a lattice site equivalent spherical diameter of a bubble maximum stable bubble size lattice velocity 1-D turbulence energy spectrum particle distribution function dimensionless frequency particle distribution function at collision step equilibrium distribution function parameter that controls turbulence intensity external force per unit mass in physical space interphase force per unit volume fraction of bubbles that experience a given maximum deformation external force per unit mass in spectral space modified external force per unit mass in spectral space Green’s function parameter Green’s function wave vector maximum wavenumber coordinates in spectral space
1060 lc L Lf N p p.s.d. r Rek S S0 S t t Tb Tf Te u u0 Vrr ðx; x0 Þ wi We Wecr x; x0 x, y, z ^; ^ x y; ^z x1 ; x2 y 1 ; y2
D. Qian et al. distance from a lattice site to its nearest neighbors. box size longitudinal scale of turbulence number of grid points along one of the box edges pressure power spectrum density initial radius of bubble Taylor microscale Reynolds number bubble total surface area surface area of a spherical bubble fractional change of bubble surface area relative to a spherical bubble time time measured in eddy turnover times dimensionless time of the first bubble breakup measured in eddy turnover times time period over which the turbulence is modulated eddy turnover time velocity turbulence intensity in one direction interparticle potential weighting factor Weber number critical Weber number for bubble breakup lattice site coordinates in physical space unit vectors random numbers between 0 and 1 Gaussian random number
Greek Letters a, b c hdu2(d)i dvmax Dp Ds e g k l n q q0 qb ql s
stochastic process constants surface tension mean square of the difference in the turbulent velocities over a distance equal to d maximum difference in the fluid velocity vectors between different points on the bubble surface pressure difference between the interior and the exterior of the bubble time interval in stochastic process energy dissipation rate Kolmogorov scale of turbulence Taylor microscale of turbulence dynamic viscosity of the liquid kinematic viscosity of the liquid density arbitrary constant gas density liquid density relaxation time
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effective mass collision term
Subscripts and Superscripts 1 2 b c cr eq l mag max r; r
liquid component gas component bubble phase collision critical equilibrium liquid magnitude maximum component label
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