Transient movement of fluid spheres using Lattice

Boltzmann method

Luıs Orlando Emerich dos Santosa,∗, Savio Leandro Bertolib, AntonioGaspar Gerent Juniorb, Guilherme Eller Haverrotha

aCentro de Engenharia da Mobilidade, Universidade Federal de Santa Catarina,

89219-710, Joinville, Santa Catarina, BrasilbDepartamento de Engenharia Quımica, Universidade Regional de Blumenau, FURB,

SC, C.P. 1507, 89010-971 Blumenau, Brasil

Abstract

In this study, an immiscible lattice Boltzmann is applied to simulate bubble

dynamics at low Reynolds number. Several simulations were performed, con-

sidering, specially, the influence of the boundary conditions on the terminal

bubble velocity. The results of the simulations are compared with analytical

solutions and the differences encountered are evaluated, showing the condi-

tions under with the Lattice Boltzmann method can be applied to simulate

bubble dynamics.

Key words: Bubble dynamics, Lattice Boltzmann, Two-phase flow

PACS: 47.11.-j, 47.55.dd

1. Introduction

The lattice Boltzmann method (LBM) has increasingly been applied as

an effective method to simulate fluid dynamics, especially when the focus is

∗Corresponding author, Tel.: +55 48 3721-6452Email addresses: emerich@lmpt.ufsc.br (Luıs Orlando Emerich dos Santos),

savio@furb.br (Savio Leandro Bertoli)

Preprint submitted to Applied Mathematical Modelling July 28, 2011

on complex geometries or complex fluids. Unlike the traditional approach

of solving the macroscopic Navier-Stokes equations, the LBM is based in a

mesoscopic kinetic equation, e.g., a discretization of the Boltzmann trans-

port equation. This approach is particularly useful when the fluid flow in-

volves macroscopic behaviors of microscopic origins, like interfaces growth

and breakup, capillary flows and phase transition. The treatment of these

problems by traditional numerical methods is very difficult, usually involving

the use of empirical models, or it is even impossible [1]. In the LBM, however,

the microscopic interactions that originate interfaces and capillary flow can

be incorporated into the model in a natural manner, and many studies has

been published addressing these phenomena (for a review see [2]). Various

LBMs designed to simulate immiscible fluids were proposed [3, 4, 5, 6, 7, 8]

and bubble dynamic simulations were performed using some of them [9, 10].

These simulations are compared with experimental observations[9] or with

other numerical results [10]. In this work, we use the immiscible fluid LBM

proposed by Santos, Facin and Philippi (SFP) [8] to simulate bubble ascen-

sion dynamics and compare the results with recent analytical results obtained

by Bertoli[11] and with the experimental results of Haberman e Sayre[12], fo-

cusing, especially, under the influence of the walls and the periodic boundary

conditions. The paper is organized as follows. First, in section 2, we revisited

the SFP model, describing its dynamics and its main characteristics. In the

sequence, section 3, we discuss simulation aspects as the boundary conditions

and the way we introduce the forcing term, and presented a comparison of

SFP model and Shan-Chen (SC) model [5]. Section 4 presents simulation

results compared with analytical ones. Finally, in section 5, the results are

2

discussed concluding the paper.

2. The Santos-Facin-Philippi model

The SFP model was introduced in [8], where a complete analysis id also

presented, including the Chapman-Enskog analysis and the derivation of the

interfacial tension. Little improvements were introduced in [13], however,

the main aspects of the model were not changed. Three distribution func-

tions are used, Ri, Bi and Mi, the first two representing the fluids r and b.

The third distribution, namely, the mediator?s distribution function Mi, is

used to model the long range interaction, carrying out neighborhood infor-

mation, and being responsible for the segregation between the fluids. These

distributions are updated by two steps, namely:

(a) The local step, which is the particle collision process and the emission/

annihilation of the field mediators;

(b) The non-local step, i.e., the propagation step.

In what follows, X is the position vector and ci is a discrete velocity. In

the local step, particle distributions are updated at each time step T by a

collision process:

R′

i = Ri + ωrR0i (ρ

r,ur)− Ri

τ rr+ ωbR

0i (ρ

r, ~ϑb)−Ri

τm, (1)

B′

i = Bi + ωbB0i (ρ

b,ub)− Bi

τ bb+ ωrB

0i (ρ

b, ~ϑr)−Bi

τm, (2)

where

ρk =bm∑

i=0

Ki , uk =1

ρk

bm∑

i=1

Kici , (3)

are, respectively, the macroscopic density and the velocity of component k,

k = r, b. R0i and B0

i are the equilibrium distributions (see ref. [14]). The ω’s

3

are the mass fractions, ωk = ρk/ρ. The ϑ’s are the local velocities modified

by the action of mediators,

~ϑr = ur + Aum , ~ϑb = ub −Aum , (4)

and

um =

∑i Mici

|∑

i Mici|. (5)

The mediator’s distribution is updated locally by an emission/annihilation

step, defined by

M ′

i = αMi + βωr , (6)

where α and β are weights used for settling the interaction length[8]. The

propagation step is the only non-local step and is identical for all distribu-

tions,

Ki(X+ ci, T + 1) = K ′

i(X, T ) (7)

where K = R,B or M .

The best known and most applied LBM for immiscible fluids is the SC

model. The SFP has some similarities with SC model, which are important

to be mentioned. Both try to reproduce macroscopic behavior using a poten-

tial or intermolecular force, without reference to a thermodynamic potential

and in both models the long range interaction acts modifying the velocities

entering in the collision process. The main differences between models and

their importance are described in the sequence.

a) The use of mediators - although, in the Lattice Boltzmann context, the

SFP model be the only one to apply this concept, mediators could be intro-

duced without major changes in other models, not representing an essential

difference between the models. On the other hand, the use of mediators has

4

consequences in the behavior of the model, since the information velocity

becomes finite when using mediators.

b) Splitting of the collision operator - in SC model only one operator is

used in the collision process for each distribution. This process is divided in

the SFP model by the use of two collision operators for each distribution,

using, therefor, three relaxation times, instead of two.

c) Relaxation times depending on the mass fraction - the splitting of the

collision operator permits the use of variable relaxation times, it becoming

possible to consider properly the mass fraction gradients that are present in

the immiscible fluids.

3. Validation: a comparison between SFP and SC models

Considering that the SC model is a well-established method to simulate

immiscible fluids, we compare simulation of bubble ascension using both

methods (the 3D implementation of SC model is described in [15]). Although

this comparison was made simulating in two dimensions, the lattice used was

the D3Q19 one. The reason to do so is that the same computational codes

were used in the sequence to perform 3D simulations. Fig. 1 presents a

schematic view of the simulation, the bubble diameter and dimensions of the

lattice used.

Half-way bounce back boundary conditions were imposed on the solid

surfaces and periodic boundary conditions were imposed on the inlet and

outlet. These boundary conditions will be focused more detailed in com-

parisons with analytical and experimental results in the next section, here

it is only necessary that the same boundary conditions are impose in both

5

��������������

���������������

��������������

������

Figure 1: Schematic view of a 2D simulation

simulations. A buoyant force, f = g∆ρ, is applied changing the momentum

in each lattice site at every time step by the amount ωrg∆ρ, where ωr is the

mass fraction of the bubble fluid.

The ascension can be characterized the dimensionless numbers, Eotvos num-

ber (Eo), and Morton number (Mo), defined as:

Eo =g∆ρD2

σ, (8)

and

Mo =g∆ρµ4

L

ρ2Lσ3, (9)

where g∆ρ is the buoyancy force imposed in the bubble, µL is the bulk

viscosity of the liquid and σ is the interfacial tension. These numbers were

6

set Mo = 7.5 × 10−7 and Eo = 0.025 in both simulations presented in Fig.

2. This figure shows the agreement between both models confirming again

the capability of the SFP model to simulate multiphase flows. In the next

sections this model will be applied in 3D simulations.

0.00

0.00

0.00

0.01

0.01

0.01

0.01

0.01

0.02

5000 7000 9000 11000 13000 15000 17000 19000 21000 23000 25000

SFP model

SC model

Figure 2: Bubble ascension,comparison between SC model and SFP model.

4. Results

In addition to the boundary conditions used in 2D and 3D simulations,

the specular reflection boundary condition was used, as well. Given the

symmetries, it is not necessary to simulate the entire domain involved in

the problem, applying the specular reflection boundary condition in faces A

and B (see Fig. 3) only one quarter is effectively simulated. This kind of

7

boundary condition, although not appearing frequently in LB literature, has

already been applied successfully by one of the authors in ref. [13].

Figure 3: Simulation domain in the 3D simulations.

In order to examine the influence of the periodic boundary condition

applied in the inlet and outlet, the simulated results of terminal velocity were

compared with the experimental results considering wall effects obtained by

Haberman and Sayre [12]. They found that spheric fluid particles ascending

(or descending), at low Reynold numbers, inside cylindrical tubes, have the

terminal velocities, UT , affected by the wall in a way that the ratio between

8

the terminal velocity without wall effects, UT∞, and the terminal velocity,

K = UT∞/UT , can be expressed by

K =UT∞

UT

= (10)

=1 +

(2.2757λ5

−2.2757λ5κ2+3κ

)

1 +(−0.7017λ−2.1051λκ+2.0865λ3κ+1.1378λ5

−1.7067λ5κ−0.72603λ6+0.72603λ6κ1+κ

) ,

where κ = µ/µL is the viscosity ratio, and λ = D/Dcylinder is the ratio

between the diameter of the fluid sphere (D) and the diameter of the cylinder

(Dcylinder). The influence of the periodic boundary condition will depend on

the cylinder height, H , therefore this influence can be evaluated simulating

various heights, as presented in Fig. 4. In all these simulations λ = 2/9

(D = 22 and Dcylinder = 99), and κ = 0.5. As expected, the terminal velocity

is lower when the cylinder is higher, that is, when the influence of the walls

preponderates over the influence of the periodic boundary conditions. This

influence can be quantified using the correction in the terminal velocity given

by Eq. 10. According to this expression, we should have K = 1.455, while,

considering the case in whichH = 360, we haveK = 1.396. That means, even

when the cylinder height is many times greater then the bubble diameter,

the effect of the periodic boundary conditions is still significant, increasing

the expected terminal velocity. In these simulations, the Reynolds number

varies from Re = 0.99 (H = 360) to Re = 1.25 (H = 240).

As a matter of comparison, we present in Fig. 5 the results obtained

through the analytical solution of the transient movement of a fluid sphere

in a infinite newtonian fluid in the Stokes regime developed, originally, by

Bertoli [11], using Laplace transforms and a functional solution proposed by

9

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

0 5000 10000 15000 20000 25000 30000

step

U (

velo

cit

y)

H=240

H=280

H=320

H=360

Figure 4: Results of bubble ascension for different heights H .

Hackenberg[16]. The interested reader will find other details in the formal

solution of Chisnell [17], expressed in terms of improper integrals.

To minimize the effects of the periodic boundary condition in order to analyze

the wall effects in the simulations we set the ratio D/H = 0.061 (D =

22, H = 360). As would be expected the simulated results tend to the

analytical solution as the cylinder radius increases. The Reynolds number in

this simulations is about one.

The behavior of the bubble in the analytical and the simulated results

seems to be different, as it can be noted in the figures 5 and 4, the bubbles

in the simulations reaching a steady state earlier. But in fact, this is a

10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 5 10 15 20 25 30 35 40

t*

Simulated (r=99)

Simulated (r=89)

Simulated (r=79)

Theoretical

Figure 5: Comparison between theoretical and simulated results.

consequence of the presence of the walls as can be seen in Fig. 6. When we

increase the cylinder radius, r, the behavior of the bubble in the simulations

tends toward the analytical result.

5. Conclusion

In this work we present diverse simulations of bubble ascension using the

lattice Boltzmann method. These simulations are compared with analyti-

cal results from Bertoli [11] and considering the empirical corrections from

Haberman & Sayre [12]. Special attention is given in the influences of bound-

ary conditions: the influence of the walls and periodic boundary conditions

in the inlet and outlet. The differences between theoretical and analytical

11

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

steps

r=40

r=120

Figure 6: The bubble gets a steady state earlier ascending in cylinder with small r.

results are entirely explicable taking into account the boundary conditions

used and can be made smaller increasing the simulation domains.

6. Acknowledgments

The autors would like to acknowledge the finacial support of CNPq (Con-

selho Nacional de Desenvolvimento Cientıfico e Tecnologico) and to the pro-

gram PiPe /Art. 170.”

12

References

[1] Y. C. Chang, T. Y. Hou, B. Merriman, and S. Osher, “A level set formu-

lation of eulerian interface capturing methods for incompressible fluid

flows,” Jounal of Computational Physics 124(2), pp. 449–464, 1996.

[2] S. Chen and G. D. Doolen, “Lattice boltzmann method for fluid flows,”

ANNUAL REVIEW OF FLUID MECHANICS 30, pp. 329–364, 1998.

[3] A. K. Gunstensen, D. H. Rothman, S. Zaleski, and G. Zanetti, “Lattice

boltzmann model of immiscible fluids,” Phys. Rev. A 43, p. 4320, 1991.

[4] X. Shan and H. Chen, “Lattice boltzmann model for simulating flows

with multiple phases and components,” Phys. Rev. E 47, p. 1815, 1993.

[5] X. Shan and H. Chen, “Simulation of nonideal gases and liquid-gas

phase-transitions by the lattice boltzmann equation,” Phys. Rev. E 49,

p. 2941, 1994.

[6] M. R. Swift, W. R. Osborn, and J. M. Yeomans, “Lattice boltzmann

simulation of nonideal fluids,” Phys. Rev. Lett. 75(5), pp. 830–833, 1995.

[7] M. R. Swift, E. Orlandini, W. R. Osborn, and J. M. Yeomans, “Lattice

boltzmann simulations of liquid-gas and binary fluid systems,” Phys.

Rev. E 54(5), pp. 5041–5052, 1996.

[8] L. O. E. Santos, P. C. Facin, and P. C. Philippi, “Lattice-boltzmann

model based on field mediators for immiscible fluids,” Physical Review

E 68(5), p. Art. No. 056302 Part 2, 2003.

13

[9] X. Frank, D. Funfschilling, N. Midoux, and H. Z. Li, “Bubbles in a vis-

cous liquid: lattice boltzmann simulation and experimental validation,”

J. Fluid Mech. 546, p. 113?122, 2005.

[10] T. Inamuro, T. Ogata, and F. Ogino, “Numerical simulation of bubble

flows by the lattice boltzmann method,” Future Generation Computer

Systems 20, p. 959?964, 2004.

[11] S. L. Bertoli, Solucoes exatas para o movimento transiente de objetos

esfericos em fluido Newtoniano infinito no regime de Stokes. PhD thesis,

Federal University of Santa Catarina - Brazil, 2003.

[12] W. L. Haberman and R. M. Sayre, “Motion of rigid and fluid spheres in

stationary and moving liquids inside cylindrical tubes,” tech. rep., Dept.

of the Navy, David Taylor Model Basin, 1958.

[13] L. O. E. dos Santos, F. G. Wolf, and P. C. Philippi, “Dynamics of inter-

face displacement in capillary flow,” Journal of Statistical Physics 121,

pp. 197–207, 2005.

[14] Y. H. Qian, D. D?Humieres, and P. Lallemand, “Lattice bgk models for

navier-stokes equation,” Europhys. Lett. 17, pp. 479–484, 1992.

[15] N. S. Martys and H. Chen, “Simulation of multicomponent fluids in com-

plex three-dimensional geometries by the lattice boltzmann method,”

Physical Review E 53(1), pp. 743–750, 1996.

[16] C. M. Hackenberg, On the Unsteady Resistance of Submerged Spherical

Bodies. PhD thesis, University of Florida, 1969.

14

[17] R. F. CHISNELL, “The unsteady motion of a drop moving vertically

under gravity,” J. Fluid. Mech. 176, pp. 443–464, 1987.

15