Numerical Simulation of Buoyancy-driven Bubble Motion Using Level Set Method

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Numerical Simulation of Buoyancy-driven BubbleMotion Using Level Set MethodA. Salih a & S. Ghosh Moulic ba Department of Aerospace Engineering , Indian Institute of Space Science and Technology ,Thiruvananthapuram, Indiab Department of Mechanical Engineering , Indian Institute of Technology , Kharagpur, IndiaPublished online: 25 Jun 2010.

To cite this article: A. Salih & S. Ghosh Moulic (2010) Numerical Simulation of Buoyancy-driven Bubble Motion Using LevelSet Method, International Journal for Computational Methods in Engineering Science and Mechanics, 11:4, 211-229, DOI:10.1080/15502287.2010.483242

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International Journal for Computational Methods in Engineering Science and Mechanics, 11:211–229, 2010Copyright c© Taylor & Francis Group, LLCISSN: 1550–2287 print / 1550–2295 onlineDOI: 10.1080/15502287.2010.483242

Numerical Simulation of Buoyancy-driven Bubble MotionUsing Level Set Method

A. Salih1 and S. Ghosh Moulic2

1Department of Aerospace Engineering, Indian Institute of Space Science and Technology,Thiruvananthapuram, India2Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India

This paper discusses the numerical simulation of buoyancy-driven bubbles in various shape regimes using level set method.The effects of both inertia and viscous forces are considered inthe mathematical model. Surface tension force at the fluid inter-face is implemented through the CSF model of Brackbill et al. [2].The incompressible Navier-Stokes equations were solved on a finitevolume based staggered grid using an explicit projection method.Although the level set method offers an elegant way of computingmultiphase flows, it is often plagued by mass conservation prob-lems. In the present work, the level set method is augmented with avolume-reinitialization scheme, which helps preserve the individ-ual fluid masses within gas and liquid phases. This enabled us tocompute the motion of bubbles in all important shape regimes ona relatively coarse grid.

Keywords Multiphase Flows, Bubble Dynamics, Shape Regimes,Level Set Method, Mass Conservation, Volume-reinitialization

1. INTRODUCTIONOne of the fundamental examples of multiphase flows is an

isolated gas bubble rising through a viscous liquid. A thoroughknowledge of this problem is of major importance to under-stand more complex multiphase flow systems. However, despiteits apparent simplicity, the physics of buoyancy driven bubblemotion through a viscous liquid is very complicated. The ris-ing process is usually accompanied by significant deformationof the bubble, through complex interaction between the inertia,viscous, and surface tension forces. The diverse shapes of thebubble resulting from this deformation cause a large variety offlow patterns around the bubble. In the limit of low and highReynolds numbers, the flow description can be simplified byeither assuming that the inertia effects are negligible (Stokes’flow) or that the viscous effects are small and the flow is irro-

Address correspondence to A. Salih, Department of Aerospace En-gineering, Indian Institute of Space Science and Technology, Thiru-vananthapuram 695022, India. E-mail: [email protected]

tational. In both instances, analytical solution may be obtainedfor the bubble dynamics [17]. However, for the study of bubbledynamics at finite Reynolds numbers one has to resort to eitherexperimental or numerical techniques. Numerous experimentalstudies have been performed to understand dynamics of singlebubble rising in otherwise quiescent liquids, see for example,Hnat and Buckmaster [12], Clift et al. [4], Bhaga and Weber[1], and Wu and Gharib [26]. Some of the recent work on singlebubble rising in viscous liquid can be found in Hua et al. [13]and Hysing et al. [14].

Many numerical studies on bubble rising in viscous liquidhave been reported in the literature. There are three major cat-egories, viz., interface fitting methods, interface tracking meth-ods, and interface capturing methods, to resolve interface mo-tion when numerical methods are used to simulates the bubbledynamics. The interface fitting method introduced by Leal andco-workers (see for example, Ryskin and Leal [21]), employsthe boundary fitted coordinate system that maintains the sharp-ness of interface and helps to apply the interfacial conditionsat the exact locations. However, when the interface becomeshighly distorted, it is difficult to adequately resolve the geo-metrical complexities while maintaining the desired mesh con-trol. The front tracking method popularized by Tryggvason andco-workers is an interface tracking method that has achievedremarkable success in two- and three-dimensional bubble simu-lations (see for example, [6], [7], and [25]). The key idea behindtheir front tracking method is to use explicit marker points to fol-low the fluid interface while Navier-Stokes equations are solvedon a stationary Eulerian grid. The advantage of front trackingmethod is their ability to resolve features of the interface thatare smaller than the mesh spacing of the regular Eulerian gridon which the interface is overlaid. The major disadvantage ofthe front tracking method is that the topological changes in theinterface, like merging and breaking-up, require intensive logi-cal manipulation and awkward subjective methods must be usedto add or remove marker points as they get too far apart or tooclose together.

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212 A. SALIH AND S. GHOSH MOULIC

The difficulties associated with remeshing procedure in theinterface fitting and interface tracking methods are absent inthe interface mcapturing methods. The immensely popularVolume-of-Fluid (VOF) method (Hirt and Nichols [11]) andthe level set method belong to this category. The interface cap-turing methods are very robust with wide range of applicability;however, they require higher mesh resolution. The VOF methoduses a color function F (x, t) whose cell-averaged value indi-cates the instantaneous volume fraction of fluid inside each cell.The color function F is advected in the fluid and the interface isreconstructed based on the volume fraction of fluid. Althoughthe VOF method possesses excellent mass conservation proper-ties within each phase of the fluid, it is difficult to calculate thegeometric features of the interface from the computed volumefractions (see for example, Rider and Kothe [20]). In recentyears, the level set method introduced by Osher and Sethian[18] has become quite popular in multiphase flow computa-tions (Sussman et al. [24], Haario et al. [10], Yu and Fan [27]).It offers a simple and effective way to treat moving interfaceproblems involving complex topological changes. Since the in-terface is captured implicitly, the level set algorithm is capableof capturing the intrinsic geometric properties of highly com-plicated interfaces in a quite natural way. However, the majordrawback of the level set method is its inability to preserve thefluid masses within each phases of the fluid. This is a major issuewith the level set method and we will address this problem insection 2.5.

In the present study, we use level set method for the study ofan isolated gas bubble rising through viscous liquid in variousshape regimes. We consider the effects of both inertia and vis-cous forces as our simulations are for finite Reynolds numbers.The level set method is augmented with a volume-reinitializationscheme (developed in [22]) in order to preserve the individualfluid masses within gas and liquid phases.

2. GOVERNING EQUATIONS AND NUMERICALMETHOD

2.1. Dimensionless NumbersThe flow around the bubble is characterized by different di-

mensionless numbers. Among these, two important numbers arethe density ratio, λ(= ρb/ρo), the viscosity ratio, η = (µb/µo).Here the subscript b refers to the bubble (dispersed gas phase)and o refers to the ambient fluid (continuous liquid phase).The other important dimensionless numbers are the Reynoldsnumber, Froude number, and the Weber number, respectively,defined as

Re = ρodbU

µo

, F r = U 2

gdb

, We = ρodbU2

σ

where σ is the surface tension and U is the reference velocity.In this investigation, U = is taken to be

√gdb . In the study of

bubble motion, it is customary to regroup these numbers (seeClift et al. [4]) to yield the Eotvos number, Eo, and the Morton

number, Mo, defined as

Eo = We

Fr= ρogd2

b

σand Mo = We3

FrRe4= gµ4

o

ρoσ 3

where db is the effective diameter of the bubble. For 2-dimensional bubble it is the diameter of a circle with the samearea as the bubble, and for axisymmetric, 3-dimensional bub-ble it is the diameter of a sphere with the same volume as thebubble. For constant g the Morton number depends only on theproperties of fluid and is independent of the bubble dimension.Eotvos number relates the effect of surface tension to the effectof gravity in keeping the surface area a minimum.

2.2. Level Set FormulationThe outline of the level set method is briefly described here.

The basic idea behind the level set method is to define theinterface as the zero level set of a smooth scalar function, φ. Thelevel set function is initially defined as a signed normal distancefunction from the interface so that φ < 0 in liquid phase andφ > 0 in the dispersed gas phase. When the interface is advectedin the background velocity field, u, the evolution of φ is givenby

∂φ

∂t+ u · ∇φ = 0, φ(x, 0) = φ0(x) (1)

where φ0 embeds the initial location of the interface. Since φ isa smooth function, Eq. (1) can be solved by standard finite dif-ference schemes. In the level set formulation, the discontinuousdensity and viscosity across the interface are typically smoothedas follows

ρε(φ) = 1 + (λ − 1)Hε(φ) (2)

µε(φ) = 1 + (η − 1)Hε(φ) (3)

where ρε and µε are the dimensionless, smoothed density andviscosity, respectively. The Hε is the smeared-out Heavisidefunction defined as

Hε(φ) =

⎧⎪⎪⎨⎪⎪⎩

0, φ < −ε;

φ + ε

2ε+ 1

2πsin

(πφ

ε

), |φ| ≤ ε;

1, φ > ε.

(4)

where ε is a tunable computational parameter of the order of thesize of a mesh cell close to the interface.

When evolving the advection Eq. (1), the level set functionceases to be the normal distance function even if it is initial-ized as the normal distance function at t = 0. This leads thenumerical interface thickness to become too large or too small.Therefore, φ must be periodically reinitialized to recover normaldistance function property. For the purpose of reinitialization,we use the method described in Sussman et al. [24] where they

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NUMERICAL SIMULATION OF BUOYANCY-DRIVEN BUBBLES 213

solve the following equation

∂φ

∂τ+ Se(φ)

(|∇φ| − 1) = 0 (5)

to steady state in pseudo-time, τ . The term Se(φ) in Eq. (5) isthe smoothed sign function defined as

Se(φ) = φ√φ2 + |∇φ|2e2

where the parameter e is taken equal to the grid size.

2.3. Equation of MotionFor a two-phase flow system of immiscible, Newtonian flu-

ids, the Navier-Stokes equations govern the fluid motion in both

FIG. 1. The steady-state rising bubble in Stokes flow regime: Bubble shapeand corresponding streamline pattern at t = 1.5 for λ = 0.05, η = 0.05, andRe = 0.5.

phases of the fluid. As we are dealing with flows involving morethan one phase, the surface tension at the fluid interface has tobe accounted for. The dimensionless Navier Stokes equationsfor an incompressible flow with variable density and viscosityare

∇ · u = 0 (6)∂u∂t

+ (u · ∇)u = 1

ρε

[− ∇p + 1

Re∇· [

µε

(∇u + (∇u)T)]

− 1

Weκδε(φ)∇φ

]− 1

Frez (7)

where t denotes the time, u the velocity field, p the pressure,ρε and µε are the smoothed density and viscosity as defined byEqs. (2)-(3). In the present study, the surface tension is mod-eled as a body force concentrated at interface by employingthe Continuum Surface Force (CSF) model of Brackbill et al.[2]. The term involving the Weber number – in Eq. 7 – repre-sents the surface tension force, where κ and δε(φ) are the local

0 0.5 1 1.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time

Ris

e ve

loci

ty

16 x 4824 x 7236 x 108

0 0.5 1 1.50.997

0.998

0.999

1

1.001

1.002

1.003

Time

Nor

mal

ized

vol

ume

of b

ubbl

e 16 x 4824 x 7236 x 108

FIG. 2. Transient evolution of rise velocity (top) and normalized volume ofbubble (bottom).

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FIG. 3. The steady-state rising bubble for low Reynolds numbers: 2-D bubbleat t = 5.5 (left frame) and axisymmetric bubble at t = 5.0 (right frame) forλ = 0.05, η = 0.05, Eo = 1, Mo = 1 × 10−3, and Re = 5.623.

FIG. 4. The steady-state rising bubble for moderate Reynolds numbers: 2-Dbubble at t = 7.0 (left frame) and axisymmetric bubble at t = 7.0 (right frame)for λ = 0.1, η = 0.1, Eo = 2, Mo = 1 × 10−5, and Re = 29.91.

curvature of the interface and the regularized Dirac delta func-tion, respectively. The local curvature is expressed in terms ofφ and is given by

κ = ∇ · n = ∇ ·( ∇φ

|∇φ|)

where n represents the outward unit normal to the interface.

2.4. Numerical methodFor the numerical treatment of Eqs. (6)-(7), we have used

an explicit Navier-Stokes solver (Salih [22]) on a MAC-typestaggered grid based on Chorin’s projection method [3] as de-scribed in Peyret and Taylor [19]. Though the projection algo-rithm described in Peyret and Taylor [19] is based on a finitedifference formulation, we have adapted it to the finite vol-ume formulation for present study. The temporal discretizationhas been done using the explicit forward Euler time stepping.The use of explicit time integration introduces a restriction ontime step and hence �t must obey the Courant-Friedrichs-Lewy(CFL) condition due to convective terms, which asserts that thenumerical wave should propagate at least as fast as the physi-cal waves. The time step restriction must also include the stiff

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0.3

0.6

0.9

1.2

1.5

Time

Rey

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32 x 4848 x 7272 x 108

0 1 2 3 4 5 60.9975

0.998

0.9985

0.999

0.9995

1

Time

Nor

mal

ized

vol

ume

of b

ubbl

e

FIG. 5. Time evolution of rise Reynolds number (top) and normalized volumeof bubble (bottom) for 2-D rising bubble with Re = 5.623.

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terms due to body force (gravity), surface tension, and vis-cous terms (see Sussman et al. [24]). The convection termsare treated by a second-order upwind scheme ([16]) and theviscous terms by standard central differencing schemes. Thenumerical solution of the pressure Poisson equation resultingfrom the projection step is obtained using preconditioned con-jugate gradient method with SSOR preconditioner (Golub andvan Loan [8]). We remark here that the use of staggered gridavoids the explicit implementation of pressure boundary con-dition while solving the pressure Poisson equation numerically(see for example, Gresho and Sani [9]). For validating the im-plementation of surface tension force using CSF model, wehave simulated the Young-Laplace law for a static drop andfound that the pressure jump across the interface is correctlycaptured [23].

The numerical solution of the level set Eq. (1) and the reini-tialization Eq. (5) are carried out by the high resolution fifth-order accurate WENO scheme introduced by Jiang and Peng[15]. For the efficient implementation of the WENO scheme,we have used grid staggering for the level set function. We havelocated φ at the vertices of the main control volumes. This kindof grid staggering avoids nonuniform grids adjacent to the com-putational boundary and consequently no special treatment ofWENO stencil is required near the boundary.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Rey

nold

s N

umbe

r

20 x 6030 x 9045 x 135

0 1 2 3 4 50.992

0.994

0.996

0.998

1

1.002

Time

Nor

mal

ized

vol

ume

of b

ubbl

e

FIG. 6. Time evolution of rise Reynolds number (top) and normalized volumeof bubble (bottom) for axisymmetric rising bubble with Re = 5.623.

2.5. Mass ConservationAs pointed out earlier, the major drawback of the level set

method lies in accurately preserving individual fluid masses (orvolumes for incompressible flows). The main reason for thevolume loss can be attributed to the straying of the zero levelset from the initial position during the reinitialization process.The volume loss can be reduced to a great extent by usinghigh-order shock capturing schemes like the fifth-order WENOfor both advecting the level set function and for reinitializing it.However, in the present study it was found that even with the useof WENO-5 scheme considerable loss of bubble volume takesplace during long time computations. In an effort to controlthe volume loss during the level set computations, Salih [22]introduced a volume-reinitialization scheme. In the proposedscheme, the volume correction is accomplished by performinga volume-reinitialization process after every time step by solvingthe following equation

∂φ

∂τ+ (Vb0 − Vb(τ ))|κ/κmax | = 0 (8)

with the initial condition

φ(x, τ = 0) = φτo(x)

0 1 2 3 4 5 6 70

5

10

15

20

Time

Rey

nold

s N

umbe

r

32 x 9648 x 14472 x 216

0 1 2 3 4 5 6 70.994

0.995

0.996

0.997

0.998

0.999

1

1.001

Time

Nor

mal

ized

vol

ume

of b

ubbl

e

FIG. 7. Time evolution of rise Reynolds number (top) and normalized volumeof bubble (bottom) for 2-D rising bubble with Re = 29.91.

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0 1 2 3 4 5 6 70

5

10

15

20

25

30

Time

Rey

nold

s N

umbe

r

20 x 12430 x 18445 x 276

0 1 2 3 4 5 6 70.994

0.995

0.996

0.997

0.998

0.999

1

1.001

Time

Nor

mal

ized

vol

ume

of b

ubbl

e

FIG. 8. Time evolution of rise Reynolds number (top) and normalized volumeof bubble (bottom) for axisymmetric rising bubble with Re = 29.91.

where τ is another (different from the one used in reinitializationEq. (5)) pseudo-time variable, Vb0 the initial volume of bubble,Vb(τ ) the volume of the bubble corresponding to the level setfunction φ(x, τ ) during the volume-reinitialization process, andφτo(x) the level set function at the beginning of the currentvolume-reinitialization process. The κ is the local curvatureof the interface and κmax the maximum value of curvature in

0 1 2 3 4 5 6 70.75

0.8

0.85

0.9

0.95

1

Time

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ubbl

e

32 x 9648 x 14472 x 216

FIG. 9. Time evolution of normalized volume of bubble for 2-D rising bubble(Re = 29.91) without volume-reinitialization scheme applied.

the neighborhood of the interface. The local curvature is beingnormalized using κmax in order to avoid excessive change in φ

where the curvature has a high value. The volume of the bubble,Vb(τ ), during the volume-reinitialization process is computedusing the following equation

Vb(τ ) =n∑

k=1

m∑i=1

Hε(φ)i,k �x�z (9)

where m and n, respectively, represent the number of cellsin x and z directions. It may be noted that Eq. (9) for bubblevolume is valid only for the 2-D Cartesian coordinate system.In the axisymmetric, cylindrical coordinate system the expres-sion can be suitably modified to take care of the radius factor. Itcan be seen from Eq. (8) that in the term, Vb0 − Vb(τ ) > 0,the zero-level set function φ would change in such a waythat the fluid in the bubble region expands. The reverse hap-pens if Vb0 − Vb(τ ) < 0. We shall apply this technique whilecomputing the bubble motion in various shape regimes. Foractivating the volume-reinitialization scheme we prescribe a

32 x 9648 x 14472 x 216

32 x 9648 x 14472 x 216

FIG. 10. Grid refinement analysis of 2-D rising bubble (Re = 29.91) at timet = 7 with basic level set method (top) and with volume-reinitialization schemeapplied (bottom).

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0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 0.6

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 3.0

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 5.0

FIG. 11. Evolution of a rising bubble in spherical regime (Eo = 1, Mo = 1 × 10−3, Re = 5.623, λ = 1/820, η = 1/55).

tolerance value δV . It means that during computation, when-ever the bubble volume changes more than δV% of the initialvolume of the bubble, the volume-reinitialization scheme getsactivated.

0 1 2 3 4 50

0.5

1

1.5

Time

Rey

nold

s N

umbe

r

0 1 2 3 4 50.392

0.394

0.396

0.398

0.4

0.402

Time

Bub

ble

diam

eter

FIG. 12. Time evolution of rise Reynolds number (top) and vertical bubblediameter (bottom) corresponding to Figure 11.

3. SOME PRELIMINARY NUMERICAL RESULTS3.1. Bubble Motion without Surface Tension

We first consider the case of very slow motion (Stokes flowregime) of the bubble through an ambient fluid. This problemwould serve as a useful validation test for the numerical modelwe have developed. In the absence of surface tension, and ne-glecting inertia force, an exact solution of the Navier-Stokesequation can be obtained for the motion of a spherical bubbledriven by buoyancy force (see Clift et al. [4]). Accordingly, theterminal velocity of the bubble can be determined from the bal-ance between buoyancy force and drag force. For a sphericalbubble of density ρb and viscosity µb rising through the ambi-ent fluid of density ρo and viscosity µo, the terminal velocity isgiven by the Hadamard-Rybczynski equation

Wb = 2

3

R2g(ρb − ρo)

µo

µo + µb

2µo + 3µb

(10)

where R is the radius of the bubble and g the gravitational ac-celeration. The above equation is valid only for low Reynoldsnumber flows. The reference velocity used in the Reynolds num-ber is obtained by defining U = √

gdb . In order to compare thenumerical solution using the present level set formulation withthe analytical solution, we solve the case of an axisymmetricbubble rising through the ambient fluid. The Froude number isset equal to 1.0 and the Reynolds number is taken as 0.5. Theinitial setup consists of an axisymmetric bubble of diameterdb = 0.3 placed at the center of a cylindrical domain of radius0.5 and height 1.5. The axis of the cylinder is assumed to bevertical. The initial velocity and pressure field are set equal tozero. The density and viscosity ratios between the bubble andthe background fluid are set equal to 0.05. In order to reducethe effect of the computational boundaries on bubble motion

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0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 0.6

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 1.0

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 2.0

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 3.0

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 4.0

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t = 5.0

FIG. 13. Evolution of a rising bubble in ellipsoidal regime (Eo = 10, Mo = 1 × 10−3, Re = 31.62, λ = 1/820, η = 1/55).

free-slip conditions are imposed on all boundaries. Tolerancelimit for activating volume-reinitialization scheme is taken as0.3. Computation has been carried out until a dimensionlesstime t = 1.5 (sufficient to reach the terminal velocity) on gridsof 16 × 48, 24 × 72, and 36 × 108. Figure 1 displays the bubbleshape at the end of computation and the instantaneous stream-line pattern generated by the bubble motion on a 36 × 108 gridresolution. Note that the stream function was calculated fromthe velocity field defined in a reference frame moving with thebubble centroid. It is clear from the figure that the bubble retainsits spherical shape during its motion. The streamline plot displaythe presence of internal circulation, similar to Hill’s sphericalvortex, within the bubble. The internal circulation leads to the

movement of the bubble surface with respect to the bubble cen-troid. This is characteristically different from buoyancy drivenmotion of a solid sphere, where the surface of the sphere remainsstationary with respect to its center.

In Figure 2 we display the time history of dimensionless risevelocity and the normalized bubble volume. From the figureit is clear that the bubble reaches the terminal velocity in avery short period of time. A good grid convergence behaviorof rise velocity can be observed. The plot of transient bubblevolume shows that only for the coarsest grid is the volumereinitialization scheme activated. The dimensionless terminalrise velocity calculated from Eq. (10) is 0.038 while the presentnumerical solution of the full Navier-Stokes equation gives the

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value 0.033 on 36 × 108 grid resolution. The slightly lower valueof the numerical prediction can be attributed to the difference inboundary conditions between the two models; the Hadamard-Rybczynski model assumes that the bubble rises through aninfinite expanse of fluid, whereas in our model the bubble isenclosed in a finite and closed cylindrical domain.

3.2. Bubble Motion with Surface TensionWhen the bubble motion is in the non-Stokes regime, where

the inertia effects also play a major role in the bubble dynam-ics, no exact solution of the Navier-Stokes equation is available.Also, the presence of surface tension offers additional difficultyin solving the Navier-Stokes equation analytically. Thus, forcomparing the results of bubble motion with surface tension innon-Stokes regime we take the case of a 2-dimensional risingbubble problem solved by Esmaeeli and Tryggvason [6] usingfinite difference/front-tracking method. The following dimen-sionless numbers, Eo = 1.0, Mo = 1 × 10−3, λ = 0.05, andη = 0.05, were used in [6]. For the above combination of dimen-sionless numbers, the Reynolds number, Re, is found to be 5.623and the rising bubble is expected to retain its initial circular shapeand quickly reach its terminal velocity. The initial setup consistsof a 2-dimensional cylindrical bubble of diameter db = 0.39placed in computational domain = [0, 1.0] × [0, 1.5]. Theinitial location of the center of the bubble is at a height of0.4 from the bottom boundary and is equidistant from the sidewalls. The initial velocity and pressure field are set equal tozero. Free-slip conditions are imposed on all walls. Tolerancevalue δV for activating volume-reinitialization scheme is takenas 0.23. Computation has been carried out until a dimensionless

0 0.5 10

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1

1.2

1.4

1.6

1.8

2

t = 5.0

FIG. 14. The instantaneous velocity field generated by the rising bubble inellipsoidal regime at t = 5.

time t = 5.5 (reference time is taken as db/U ) on grids of 32 ×48, 48 × 72, and 72 × 108.

The left frame in Figure 3 displays the location and shape ofbubble at the end of the computation. The circle at the bottomdenotes the initial position of the bubble. We have also solvedthe case of an axisymmetric spherical bubble rising through theambient fluid considered by de Sousa et al. [5]. The physical andcomputational parameters considered in this case are the same asthat of the 2-dimensional bubble except that the computationaldomain is now a cylindrical space with radius equal to 0.5. andcomputation has been carried out until a dimensionless timet = 5.0. The right frame in Figure 3 displays the initial and finalposition of the bubble. Again, we see that the bubble retains itsinitial spherical shape during the upward motion.

To further demonstrate the capability of the algorithm wehave also solved the case of bubble rise at moderate Reynoldsnumber. The parameters for the 2-D case were taken from Es-maeeli and Tryggvason [7]. They are Eo = 2.0, Mo = 1×10−5,Re = 29.91, λ = 0.1, and η = 0.1. The bubble diameter, db, istaken as 0.0024. The same parameters are used for computingthe rise of a 3-D bubble by de Sousa et al. [5]. For both the 2-Dand axisymmetric case, the computations were carried out untila dimensionless time t = 7.0, which is sufficient enough for thebubble to acquire a steady rising speed. In Figure 4 we display

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1

1.2

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1.6

1.8

2

t = 5.0

FIG. 16. Evolution of a rising bubble in dimpled ellipsoidal regime (Eo = 125, Mo = 5, Re = 25, λ = 1/820, η = 1/55).

the shape and location of the bubble at the end of computation.It can be seen that, during the translation of the bubble, it ac-quires an elliptic shape in 2-D and an ellipsoidal shape in theaxisymmetric case.

In order to compare the results of our computation with thatof Esmaeeli and Tryggvason [6, 7] and de Sousa et al. [5],it is necessary to calculate the rise speed of the bubble ex-pressed as the rise Reynolds number. The rise Reynolds number,Reb = ρodbWb/µo, is where Wb is the rise speed of the cen-troid of bubble. For 2-D bubble, we compare our results withthat of Esmaeeli and Tryggvason [6, 7] whereas, for axisym-metric bubble, our results are compared with that of de Sousaet al. [5]. Table 1 displays the summary of the comparison ofrise Reynolds numbers. It can be seen that our results compare

well with the results of de Sousa et al. [5] for both low andmedium Reynolds numbers. Also, for the case of 2-D bubble,the rise Reynolds number obtained in the present experiment is

TABLE 1Comparison of predicted rise Reynolds numbers

Reb (2-D) Reb (axisymmetric)

Esmaeeli and de SousaTryggvason[6]∗ [7]∗∗ Present et al. [5] Present

Re = 5.623 1.41∗ 1.40 1.39 1.35Re = 29.91 20.5∗∗ 19.1 25.0 24.5

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in close agreement with that of Esmaeeli and Tryggvason [6, 7].Figures 5–8 display the transient evolution of Reb for variousgrid resolutions and the normalized volume of the bubble for allfour cases discussed above. It is clear from the plots of Reb thatthe terminal rise velocity is indeed reached in all the cases. Todemonstrate the grid, independence of Reb, they are plotted forthree different grid resolutions. It can be seen that the initial un-steady motion is well resolved even on the coarsest grid. Smalloscillations in Reb can be observed on the coarsest grid, whichwas eliminated when the grid is refined. The volume histories ofthe bubble in all the cases show that the volume-reinitializationscheme is activated at a relatively early time of simulation andthus one can expect that considerable amount of volume wouldhave been lost if the volume-reinitialization scheme is not ap-plied. For instance, in the case of 2-D bubble with Re = 29.91,21.0% of volume has been lost on the coarsest grid of 32 ×96, 16.2% has been lost on the intermediate grid of 48 × 144,and 12.1% has been lost on the fine grid of 72 × 216 (seeFigure 9 for time history of bubble volume). With our volume-reinitialization scheme applied, the loss of volume was restrictedto about 0.55% for all grid resolutions. Finally, in Figure 10 wedisplay the grid sensitivity test for bubble shape with and withoutvolume-reinitialization scheme applied. Excellent grid conver-gence behavior can be observed when volume-reinitializationscheme is applied. The bubble shape essentially remains thesame for all three grid resolutions. On the other hand, the effecton volume loss on bubble shape can be clearly seen when novolume-reinitialization scheme is applied.

4. NUMERICAL RESULTS FOR DIFFERENT SHAPEREGIMESThe gas-liquid two-phase flows can appear in quite different

topological or morphological configurations. However, it is cus-tomary to classify all possible flow patterns into several differentgroups, called shape regimes, characterized by the assumptionthat within each regime the bubble remains nearly the same inits shape. A particular representative shape of the bubble canbe realized only some time after the initially spherical bubblestarts its shape evolution process. It is also possible that at a latertime the bubble shape may change significantly so that transitionfrom one shape regime to another can occur. Thus, the notionof shape regime is somewhat vague, though it has been provedextremely useful for classification of different types of bubblebehavior. The special significance of the flow regime arises fromthe fact that the various physical transfer processes taking placeacross the phase-interface strongly depend on the flow regime.

Clift et al. [4] displays a general diagram of bubble shapesin dependence on the Reynolds number and Eotvos number fora single bubble rising in an infinite expanse of fluid under theinfluence of gravity. Generally, the bubbles are grouped intothe following six basic categories: viz., spherical, ellipsoidal,dimpled ellipsoidal, skirted, spherical-cap, and wobbling. Thebubble dynamics associated with each of these bubble-shape

regimes are characteristically different. For low Eo, bubblesremain nearly spherical and rise in a steady-state manner. Forlarge Eo, bubbles attain a spherical-cap shape and also rise ina steady-state fashion. For intermediate Eo, bubble shape andmotion are also a function of the Morton (or Reynolds) number.Bubbles with large Mo become ellipsoidal before a spherical-cap shape is adopted but continue to have a well-defined steady-state motion. For low Mo the bubbles also become ellipsoidal,but their motion is unsteady.

Although direct quantitative comparison of 2-D numericalresults with 3-D experimental results is precluded, it is rea-sonable to expect a qualitative comparison between the two.For the purpose of numerical simulation of bubbles in dif-ferent shape regimes we consider a 2-dimensional cylindri-cal bubble of diameter db = 0.4 placed in Cartesian domain = [0, 1.0] × [0, 2.0]. The initial center of the bubble is lo-cated at a height of 0.5 from the bottom wall and is equidistantfrom the side walls. The initial velocity and pressure field areset equal to zero. Free-slip conditions are imposed on all thewalls. The density and viscosity ratios, λ and η, between thebubble and the background fluid are set equal to 1/820 and1/55, respectively, so that they correspond to the property ratiosbetween air and water at standard temperature and pressure.

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TABLE 2Combination of dimensionless numbers used to obtain various shape regimes

spherical ellipsoidal dimpled ellipsoidal skirted spherical-cap wobbling

Eo 1.0 10.0 125.0 800.0 200.0 3.0Mo 1 × 10−3 1 × 10−3 5.0 40.0 1 × 10−5 1 × 10−11

Re 5.62 31.62 25.0 59.8 945.7 1281.8

The particular combination of dimensionless groups Eo, Re,and Mo are selected based on the chart given in Clift et al. [4]to obtain different bubble-shape regimes. They are shown inTable 2.

4.1. Spherical BubbleThe motion in spherical shape regime can be realized when

the interfacial tension and viscous force dominates over inertiaforces. This is the case when both Eotvos number and Reynolds

0 0.5 10

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t = 5.0

FIG. 18. Evolution of a rising bubble in skirted regime (Eo = 800, Mo = 1, Re = 150, λ = 1/820, η = 1/55).

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number are small. In spherical regime the bubble is expected torise in nearly spherical (cylindrical in 2-D) form. As a rule ofthumb, the bubble may be termed as spherical if the minor tomajor axis ratio of the bubble is greater than 0.9. In Figure 11 wedisplay a rising bubble in the spherical shape regime (Eo = 1,Re = 5.62) at various instants of time. The computations werecarried out on a 48 × 96 grid. It is clear from the figure thatfor the given combination of dimensionless numbers the shapeof the bubble remains the same as it rises. The figure alsodisplays the streamline patterns generated by the bubble motionin a frame attached to the rising bubble. The presence ofinternal circulation can be clearly seen from these plots. Nopronounced wake is visible behind the bubble. The wake inthis case is defined as the distance from the bottom of thebubble down to the point where the vertical velocity has a zerovalue relative to a frame attached to the moving bubble. Theflow pattern within the bubble, after a small transient period,does not seems to change during its upward motion and hencethe entire flow can be regarded as steady relative to a frameattached to the bubble centroid. In Figure 12 we display thetime evolution of rise Reynolds number and the “vertical”bubble diameter (measured along the vertical line throughthe center of computational domain). It can be seen that the

bubble reaches its terminal velocity during a short period oftime after it is set in motion. The minor to major axes ratiowhen “steady” state is reached is found to be about 0.97. Themotion of the bubble front is almost linear after the initialtransients.

4.2. Ellipsoidal BubbleWith the same Morton number as that of a spherical bubble,

the ellipsoidal bubble shape can be realized for the higher Eotvosnumber. Ellipsoidal bubbles are oblate with a convex shapearound the entire bubble surface. The symmetry about the majoraxis may not be present. In general, after the initial transients,an ellipsoidal bubble is expected to rise in a steady state manner.In Figure 13 we display a rising bubble in the ellipsoidal shaperegime (Eo = 10, Re = 31.62) with instantaneous streamlinepattern around the bubble. The computations were carried outon a 72 × 144 grid, which is good enough to obtain a gridindependent results. The convergence to the ellipsoidal shape atthe later stages of motion can be clearly seen. The presence ofinternal circulation can be seen in this case also. However, theyare not symmetric with respect to the bubble shape. A relativelylarge wake with a pair of counter rotating weak vortex can beseen behind the bubble. In Figure 14 the instantaneous velocityfield generated by the rising bubble at the end of computation(t = 5.0) is displayed. In Figure 15 we display the time evolutionof rise Reynolds number and the vertical bubble diameter. It canbe seen that the bubble reaches its terminal velocity after a slightovershoot.

4.3. Dimpled Ellipsoidal BubbleWe now consider the case of a bubble rising in the dim-

pled ellipsoidal regime. In Figure 16 we display the evolutionof such a bubble computed on a 96 × 192 grid. The dimpledellipsoidal shape can be obtained for relatively large values ofMorton number and Eotvos number. In the present study wehave used Mo = 5, Eo = 125, and the corresponding Reynoldsnumber, Re = 25. When the shape is fully evolved, a perma-nent indentation at the bottom side of the bubble is established,and resulting shape is often called dimpled ellipsoidal shape.The figure also displays the instantaneous streamline patternsgenerated by the bubble motion. The wake pattern developedbehind the bubble in this case is more or less similar to that ofthe ellipsoidal bubble but is relatively stronger. In Figure 17 wedisplay the time evolution of rise Reynolds number and verticalbubble diameter. It is clear that the bubble undergoes a slightoscillation during the shape evolution.

4.4. Skirted BubbleNext we take up the case of a bubble rising in skirted regime.

With further increase in the Morton number and Eotvos num-ber from the dimpled ellipsoidal bubble regime, a discontinuityappear, near the sharp edge. For suitable combination of di-mensionless numbers a cusp–shaped edge appears and forms

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2

t = 5.0

FIG. 20. Evolution of a rising bubble in spherical-cap regime (Eo = 200, Mo = 1 × 10−5, Re = 946, λ = 1/820, η = 1/55).

a skirt around the bubble. In the present study we have usedMo = 40, Eo = 800, and the corresponding Reynolds number,Re = 59.8. Note that the bubble in this regime requires a highergrid resolution than the case previously considered. A grid in-dependence test was carried out and it was found that a resolu-tion of 128 × 256 grid is reasonable enough to obtain a grid-independent solution. In Figure 18 we display a rising bubble inthe skirted regime at various instants of time. The correspondingstreamline patterns are displayed in the figure. It can be seen thatwhen the bubble begins to rise, a jet of liquid forms at the rearend of the bubble, which pushes the bottom surface of the bubbleforward to the bubble front leading to large deformation of thebubble. As time progresses, the action of the jet causes the for-mation of a skirt along the edges of the bubble which eventually

pinches off the bubble. The pinched off portions of the bubble gettrapped in the vortex pair behind the main bubble and undergorotation as they move up. The main bubble then develops skirt–ike edges displaying the features of a typical bubble rising in theskirted regime. This qualitative picture of the bubble break-upand consequent skirt formation is in full agreement with experi-mental observations of Hnat and Buckmaster [12]. In Figure 19we display the time evolution of rise Reynolds number and ver-tical bubble diameter. It is clear that the rise Reynolds numbervaries in a nonlinear fashion and has not reached a steady statevalue during the simulation time considered. From Figure 18 itcan be seen that, as in the other shape regimes, the upper surfaceof the bubble is relatively unaffected by the dynamics inside thebubble.

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4.5. Spherical-cap BubbleThe motion of a bubble in spherical-cap regime can be real-

ized for relatively large values of Eotvos number and intermedi-ate values of Morton number. In the present study we have usedMo = 1 × 10−5, Eo = 200, and the corresponding Reynoldsnumber, Re = 945.7. As in the case of simulation of skirtedbubbles, a higher grid resolution is required here. A grid of

128 × 256 is found to be good enough to capture the essen-tial phenomena associated with rising bubble in spherical-capregime. In Figure 20 we display a rising bubble in the ellipsoidalshape regime at various instants of time. The correspondingstreamline patterns are displayed in the figure. It can be seenthat the shape evolution of the bubble at the initial stages is sim-ilar to that of the bubble in skirted regime. As described earlier,the action of the jet causes the formation of a skirt along theedges of the bubble, which begins to roll up with time and fi-nally breaks-off from the main bubble. However, subsequent tothe detachment of secondary bubbles, the main bubble does notdevelop the skirt–like edges as in the case of bubble in skirtedregime. Instead, after the shredding of secondary bubbles, thebottom surface gets flatter, giving the characteristic cap-likeshape. It may be noted that the actual size of the shredded bub-bles and shape of main bubble within the spherical-cap regimedepends on the particular combination of Eotvos and Mortonnumbers. The plot of stream function indicates the presence ofa strong wake behind the main bubble. In Figure 21 we displaythe time evolution of rise Reynolds number and vertical bub-ble diameter. It is clear that the rise Reynolds number variesin a nonlinear manner and has not reached a steady state valueduring the simulation time considered. The transient variationof the vertical bubble diameter is similar to that of the bubbleconsidered in the skirted regime.

4.6. Wobbling BubbleA combination of very low Morton number and low enough

Eotvos number results in the motion of bubbles in wobblingregime. Typical range for the Reynolds number correspondingto the wobbling motion is approximately the same as that forthe spherical-cap regime. In this regime the bubble undergoesoscillation as it rises and the motion is truly unsteady. Though thebubble does not break up owing to the high value of interfacialtension, the simulation of rising bubble in the wobbling regimeis more challenging than the bubble in any other regimes. In thepresent study a combination of Mo = 1 × 10−11 and Eo = 3

32 x 74 48 x 110 72 x 166 108 x 248 110 x 252

FIG. 22. Wobbling bubble (Eo = 3, Mo = 1 × 10−11, Re = 1282, λ = 1/820, η = 1/55) computed on different grids.

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0 0.5 10

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FIG. 23. Evolution of a rising bubble in wobbling regime (Eo = 3, Mo = 1 × 10−11, Re = 1282, λ = 1/820, η = 1/55).

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0 1 2 3 4 5−200

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FIG. 24. Time evolution of rise Reynolds number, vertical bubble diameter, kinetic energy of the bubble, and normalized bubble volume corresponding toFigure 23.

has been used. The corresponding Reynolds number, Re =1281.8. It was found that, even in the early periods of simulation,the bubble shape undergoes drastic changes with successiverefinement of grid. In Figure 22 we display transient bubbleshapes for various grid resolutions (32 × 74, 48 × 110, 72 ×166, 108 × 248, and 110 × 252). It can be seen that the gridindependent shape is obtained only up to a relatively short timeof about 2 units. At later times the bubble shapes are not similarfor the first four grid resolutions. To get an idea of whethergrid independency can be obtained by further refining the grid,we increase the grid resolution marginally from 108 × 242 to110 × 252. A comparison of bubble profiles on these gridsshows that they are not similar. The computation on a very finegrid turned out to be very expensive because of the limitedcomputational resources available with us. The gross changeof bubble shapes associated with change of grid resolution isnot surprising because the bubble in this regime is known tobe unstable (see for example, Clift et al. [4], and Bhaga andWeber [1]) and we believe that a grid independent bubble shapein this regime may not be possible at all. We remark here thatdue to large flattening of the bubble around a time, t = 3,the surrounding wall would have had some effect on further

evolution of the bubble. This precludes a direct comparisonwith experimental results. Also, to the best of our knowledge,not much published numerical results of bubble in the wobblingregime with high density and viscosity comparable to that of airand water are available in the literature. In a recent paper, Haarioet al. [10] used a finite difference formulation in conjunctionwith level set method for the simulation of bubbles in differentshape regimes, including wobbling bubble. They provide someinteresting results, such as the wake pattern and vortex sheddingbehind the bubble. However, the authors do not provide theinformation about how the bubble shape evolves during theupward motion. A grid independent study, which is essential forany simulation of rising bubble, especially when the bubble isin the wobbling regime, is also not available as expected.

In Figure 23 we display the shapes of rising bubble in thewobbling shape regime (computed on a 110 × 252 grid) at se-lected instants of time. The corresponding streamline patternsare displayed in the figure. The shape evolution the bubble un-dergoes in a relatively short period of time is quite interesting.The bubble remains almost spherical at about t = 0.6 and itdevelops a flat surface when t = 1.0. Then at t = 2.0 it takes theshape of a crescent. The crescent shape gets further elongated

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at about t = 3.0. A slight change from the crescent shape canbe observed at t = 3.5. Then, within a very short period oftime, the bubble quite remarkably transforms to an altogetherdifferent shape at about t = 4.0 and the rapid morphologicalchange continues to takes place through time t = 4.5 and att = 5.0 it adopts a shape of an elongated circle. These suddenchanges occur because of the interplay between the interfacialtension force and the inertia force. The viscous force is not verysignificant here. An examination of streamline pattern generatedby the bubble will reveal that even at the very early stages of theevolution there are three vortex pairs present inside the bubble.However, no significant wake is seen at this point of time. Asthe time progresses and the shape gets evolved, the number andstructure of the vortices inside the bubble change in a quickmanner. This type of sudden change of flow pattern inside thebubble is not seen in any of the other bubble regimes. It can alsobe seen that the vortex shedding and the wake instability occurtowards the later part of the time period considered. In Figure 24we display the time evolution of rise Reynolds number, verticalbubble diameter, kinetic energy of the bubble, and the normal-ized bubble volume. From these plots it is clear that the bubbleundergoes a very unstable motion. We mention here that morework needs to be done for successful simulation of a bubble inthe wobbling regime.

5. CONCLUSIONSWe have presented the results of simulation of two-

dimensional bubble dynamics. For validating the numericalmethod against this class of problems, we have simulated aspherical bubble rising in Stokes regime in the absence of surfacetension. The volume change of the bubble during the simulationremains small and only for the coarsest grid was the volume-reinitialization scheme activated. In this regime the bubble es-sentially remains spherical and it attains a terminal velocityrather quickly. The numerically estimated terminal velocity andthe theoretical value predicted by the Hadamard-Rybczynskiequation (Eq. (10)) agree well. For further validation, we haveconsidered the bubble rise in non-Stokes regime with surfacetension force included. Both two-dimensional and axisymmet-ric cases were considered with volume-reinitialization schemeemployed. Excellent agreement between the present results andpreviously published results was observed.

We have then considered the simulation of bubbles in variousshape regimes. The bubble shape depends on the Reynolds num-ber and Eotvos number for a single bubble rising in an infiniteexpanse of fluid under the influence of gravity. Various shapesidentified are spherical, ellipsoidal, dimpled ellipsoidal, skirted,spherical-cap, and wobbling. In all the cases, simulation re-quires the application of the volume-reinitialization scheme tomaintain the bubble volume under control. For some bubble-shape regimes a clear terminal velocity is reached, whereas insome other shape regimes the fragmentation of the bubble takesplace. The wobbling regime is found to be hard to simulate.

For this highly unstable bubble no grid independent shape isachieved. All these simulation results show the enhanced ca-pabilities of the level set methods by employing the simplevolume-reinitialization strategy.

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Documents

Numerical Simulation of Buoyancy-driven Bubble Motion Using Level Set Method