Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-014-0821-z

Coalescence of two at-rest equal-sized drops in static vapor of the same material:

A lattice Boltzmann approach† Ehsan Amiri Rad*

Department of Mechanical Engineering, Hakim Sabzevari University, PO Box: 9617976487, Iran,

(Manuscript Received November 19, 2013; Revised March 6, 2014; Accepted June 15, 2014)

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Abstract We studied the coalescence of two stationary equal-sized droplets in static vapor using a lattice Boltzmann approach. The non-ideal

behavior of one-component, two-phase flow is coupled with BGK lattice Boltzmann by defining a suitable free energy function, which produces the correct equilibrium conditions of the flow. The accuracy of developed model is confirmed by calculating droplet surface tension for different conditions and comparing that with theoretical results. Finally, the coalescence process of two equal-sized drops is modeled and effective parameters on critical gap of coalescence are investigated. The results show that for two at-rest and equal-sized drops in a static flow, the critical gap of coalescence only depends on thickness of the interface, and other parameters such as droplet radius, density ratio and surface tension do not have influences on that directly.

Keywords: Droplet coalescence; Static flow; Critical gap; Lattice Boltzmann; Interface thickness; Surface tension; Density ratio ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Coalescence problems represent a class of fluid flows where separate bodies of fluid merge. The dynamics of droplet coa-lescence plays an important role in many engineering proc-esses. For example, merging or break up of small nucleated droplets in supersonic steam flow can affect the flow specifi-cations. Coalescence of two drops into a single droplet of the same total volume reduces the total surface energy of the sys-tem, and so should be energetically favorable. But sometimes the coalescence will not happen. Actually, coalescence is initi-ated when two droplets come into contact and form a liquid bridge, which then starts to grow due to surface tension. This growth is typically either opposed by viscous dissipation or inertial forces. Therefore, coalescence depends on the balance of surface tension, viscosity and inertia. Indeed, for two sta-tionary droplets suspended in static bulk vapor, if the distance between droplet boundaries is larger than a certain value, the surface tension cannot dominate on viscosity and inertia and eventually the coalescence will not happen. This certain dis-tance is named critical gap in this paper. The coalescence process has been investigated experimentally or numerically by many researchers [1-5], but none of these researchers in-vestigated the critical gap. Conventionally, two-phase flows can be simulated by solution of Navier-Stokes (N-S) equations

coupled with an interface capture equation. Some famous methods such as volume of fluid (VOF) use this approach. However, they may encounter some numerical difficulties in the treatment of topological deformation of interface coalesc-ing. The lattice Boltzmann method is a novel approach that can simulate complicated and multiphase flows more pre-cisely. Many researchers use this method for multiphase sys-tem simulation [6-10]. In this paper a vapor - liquid system is modeled by defining an appropriate free energy function and its integration with a lattice Boltzmann algorithm. This method introduces the non-ideal interaction of a two-phase system into LBM through a pressure term. Using the devel-oped model, the coalescence process of two stationary equal-sized droplets in static bulk vapor is modeled, and then the critical gap of two droplets and its dependency on important parameters of the system is studied. This study is limited to isothermal systems in which mass and momentum are con-served in the collision, and energy conservation is abandoned in favor of a constant temperature requirement.

2. Model description

2.1 Thermodynamics aspect

Free energy, chemical potential, and pressure are key ther-modynamic concepts to understand the phase behavior of a system. The phase separation of a system is determined by the shape of its bulk free energy density function. An initial ho-mogeneous system of density 1r and bulk free energy den-

*Corresponding author. Tel.: +98 5144410104, Fax.: +98 5144410104 E-mail address: Ehsanamech@gmail.com

† Recommended by Associate Editor Gihun Son © KSME & Springer 2014

3598 E. A. Rad / Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603

sity 1y separates into two phases of densities Ar and Br with the average bulk free energy density 2y to minimize

the total free energy of the system. Fig. 1 illustrates the phase separation behavior of a one component system. The total free energy of a two-phase system is composed of the bulk part and the interfacial part known as a Landau free energy and is expressed as [11]:

( ) ( )2, ,2

T dVky r ræ öY = + Ñç ÷è øò (1)

where y is the bulk free energy density and k is capillarity coefficient. This function can describe the equilibrium proper-ties of a one-component, two-phase fluid. To study the phase behavior near the critical point, it is convenient to express the free energy density as [12]:

( ) ( ), , ,b bT T py r a r rm= + - (2)

where ( ) ( )22, cT pa r a bt= - is the excess free energy density; ( )4 1 /b c cpm bt r= - is the bulk chemical potential; and ( )21b cp p bt= - is the bulk pressure. b is a constant related to compressibility; ( ) /c ca r r r= - is the nondimen-sional density; ( ) /c cT T Tt = - is the nondimensional tem-perature; and cT , cp and cr are the critical temperature, critical pressure, and critical density, respectively.

The free energy enters the lattice Boltzmann algorithm via the pressure tensor that can be defined as [13]:

2

20( ) ,

2ij iji j

P px x

k r rkr r r d k ¶ ¶= - Ñ - Ñ +

¶ ¶

r (3)

where 0p rr y y= ¶ - is the equation of state of the fluid.

2.2 Lattice Boltzmann algorithm

The lattice Boltzmann equation is defined as a discrete Boltzmann equation. The LBE with a single relaxation time from the BGK model is expressed as [14]:

( ) ( ) ( )0, , ,i i i i if x c t t t f x t f fw+ D + D - = - -r r r

(4)

where xr is the lattice position vector; ic

r is particle veloc-ity; t is time; and w is collision frequency. ( ),if x t

r denotes

the particle distribution associated with the discrete velocity icr ; and 0

if indicates the local equilibrium distribution. The discrete velocity ic

r is chosen such that the ic tDr is a lattice

vector. Using a suitable equilibrium distribution function, Eq. (4) can describe the dynamics of a non-ideal, one-component fluid. In this paper a two-dimensional square lattice with nine velocity vectors (D2Q9 Lattice) is used. Eq. (4) is solved nu-merically in two consecutive steps as the following:

1st step: Collision: ( ) ( ) ( ) ( )( )0, , , , .i i i if x t t f x t f x t f x tw+ D = - -r r r r% (5)

2nd step: Streaming: ( ) ( ), , .i i if x c t t t f x t t+ D + D = + Dr r r% % (6)

The collision frequency ω is defined by:

1

2 0.5 .sC tuw

-æ ö

= +ç ÷ç ÷Dè ø (7)

In the above equation u is viscosity and sC denotes the

lattice sound speed, which is defined as:

.3s

CC = (8)

With the lattice speed C:

.xCt

D=D

(9)

It is shown that the lattice Boltzmann equation can lead to

continuity and Navier-Stokes equations as the following [15]:

. 0,utr r¶+ Ñ =

¶r

(10)

,

i j iji

j i

ji kij i

j j i k

u u Put x x

uu u ax x x x

rr

u r d r

¶ ¶¶+ = - +

¶ ¶ ¶

æ öæ ö¶¶ ¶¶ ç ÷ç + + ÷ +ç ÷ç ÷¶ ¶ ¶ ¶è øè ø

(11)

where a is the acceleration. The main parameters of these equations are r and ur

rwhich are related to distribution func-

tion as the following:

, .i j i iji i

f u f cr r= =å å (12)

The domain of solution is fully periodic as Fig. 2 and

boundary conditions are presented below:

Fig. 1. Phase separation of a one component system.

E. A. Rad / Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603 3599

1 1 5 5 8 8

3 3 6 6 7 7

2 2 5 5 6 6

4 4 7 7 8 8

; ; ,

; ; ,

; ; ,

; ; ,

W E W E W E

E W E W E W

S N S N S N

N S N S N S

f f f f f f

f f f f f f

f f f f f f

f f f f f f

= = =

= = =

= = =

= = =

(13)

where W, E, S. And N refer to west, east, south and north boundaries. The key point of simulating non-ideal, two-phase flow by LBM is finding a suitable equilibrium distribution function associated with thermodynamic aspects of this flow.

Equilibrium distribution function can be written as a sec-ond-order expansion of velocity:

0 2 .i j ij j k ij ik jk ij ikf A Bu c Cu Du u c c G c c= + + + + (14)

Since the Eq. (12) is always conserved, the zeroth and first

moments of equilibrium distribution function are as the fol-lowing equations:

0 0, .i i ij j

i i

f f c ur r= =å å (15)

The next moment is chosen such that the continuum macro-

scopic equations correctly describe the hydrodynamics of a one-component, non-ideal fluid [17]. This gives:

0

.

i ij ik jk j ki

j k jk mk j m

f c c P u u

u u ux x x

r

r r ru d

= + +

æ ö¶ ¶ ¶ç + + ÷ç ÷¶ ¶ ¶è ø

å (16)

jkP is the pressure tensor that is introduced in Eq. (3). The

first formulation of the model omitted the third term in Eq. (16) and was not Galilean invariant. It is shown that the addi-tion of this term led to any non-Galilean invariant terms being of the same order as finite lattice corrections to the Navier-Stokes equations [16]. To fully constrain the coefficients a fourth condition is needed, which is [18]:

( )2

0 .3i ij ik im j km k jm m jk

i

cf c c c u u ur d d d= + +å (17)

Subscribing Eq. (14) in the relations Eqs. (15), (16) and (17), the unknown coefficients of equilibrium distribution function will be found and the equilibrium distribution function of a two phase system is determined completely.

3. Results and discussion

3.1 Grid resolution dependence

A grid resolution analysis was conducted using a grid size as a function of the droplet radius 5 4R R´ for 0.2bt = and

0.01k = . The size of the grids used in this analysis was 200 160´ , 400 320´ and 800 640´ . The effect of grid resolution on the surface tension is shown in Table 1. Surface tension can be derived using the density profile obtained from LBM, as the following [19]:

2

0

.d ddrs k hh

¥æ ö

= ç ÷è øò (18)

Zero denotes the center of the droplet. Considering the re-

sults provided by this analysis, it is determined that using the 200 160´ grid size would provide reasonable accuracy and avoid high computational cost.

3.2 Accuracy of the model

A single liquid droplet surrounded by vapor is considered in a rectangular domain (160´ 200) and LBM surface tension is compared to theoretical one. Theoretically, the surface tension of such a fluid can be derived as [17]:

( )34 2 .3 c cps r bt k= (19)

Regarding the theoretical formula, surface tension is inde-

pendent of droplet radius. To check the independency of sur-face tension and droplet radius, simulation is initialized with 0.01k = , 0.4bt = and different droplet radiuses. After 20000 time steps equilibrium conditions are reached. Fig. 3 shows the surface tension for different radii obtained by LBM and the theoretical surface tension. It is obvious that the values of surface tension are independent of the radius and LBM results are close enough to theoretical surface tension. An important parameter that governs the surface tension is the capillarity coefficient (k ) that is related to substance. Fig. 4

Fig. 2. D2Q9 lattice model illustrated in the solution domain.

Table 1. LBM surface tension for different grid sizes.

Grid size LBM surface tension

Theoretical surface tension Error (%)

200 160´ 0.02131 0.02087 2.09

400 320´ 0.02132 0.02087 2.06

800 640´ 0.02129 0.02087 2.01

3600 E. A. Rad / Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603

shows the LBM and theoretical surface tension for different capillarity coefficients. The LBM results in this figure are very close to theoretical results and averaged value of errors is less than 5 percent. Regarding the theoretical formula, bt is a parameter that can influence surface tension. On the other hand, the density ratio of saturated liquid and vapor is gov-erned by bt as the following:

1

.1

l

g

btrr bt

+=

- (20)

Fig. 5 shows the LBM and theoretical surface tension for

different values of bt . Similarly to Figs. 3 and 4, the LBM results in this figure are very close to theoretical results and mean value of errors are less than 2 percent. These results validate the accuracy of LBM density profile.

3.3 Coalescence process

In this section, two stationary equal-sized droplets are sus-pended in static vapor of the same material with fully periodic boundary conditions. Fig. 6 shows the schematic geometry of the domain. The problem is initialized with different cases of capillarity coefficient, droplet radius and bt . For each case, the problem is solved for different values of gap (W ) to find the critical gap ( cW ) that is the maximum distance between outer droplet surfaces where coalescence will happen. In Fig. 7 the process of coalescence is shown.

Surface tension, interface thickness and density ratio are important parameters they may affect the critical gap of coa-lescence. Thickness of interface can be found from the follow-ing equation [8]:

2

.4

c

ch

pkrbt

= (21)

Regarding Eqs. (19)-(21), for the same material with con-

stant critical properties, surface tension, density ratio and in-terface thickness are governed by two independent coeffi-cients k and bt . Increment of capillarity coefficient ( k ) increases the surface tension and interface thickness simulta-

Fig. 3. Surface tension for different radiuses ( 0.01, 0.4k bt= = ).

Fig. 4. Surface tension for different capillarity coefficient ( 0.4bt = ).

Fig. 5. Surface tension for differentt bt ( 0.01k = ).

Fig. 6. Schematic geometry of coalescence problem.

Fig. 7. Coalescence process.

Fig. 8. Variations of critical gap with k ( 0.4bt = ).

E. A. Rad / Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603 3601

neously, but does not affect the density ratio. Fig. 8 shows the variations of critical gap with capillarity coefficient. Regard-ing the diagram, the critical gap grows with the capillarity coefficient, and eventually greater surface tension and inter-face thickness increase the critical gap as Figs. 9 and 10.

bt is a non-dimensional parameter that depends on liquid and vapor density. Increasing this parameter increases the density ratio and surface tension but decreases the interface thickness. Fig. 11 shows the variations of critical gap with bt . According to this graph, increasing of bt decreases the criti-cal gap. Variations of critical gap with surface tension, inter-face thickness and density ratio in this case are shown as Figs. 12-14. Comparison of these graphs with Figs. 9 and 10 shows that the critical gap is proportional to interface thickness in both cases. But variations of critical gap with surface tension are not consistent in Figs. 9 and 12. Actually, in the case of capillarity coefficient variations, k growth leads to higher interface thickness that increases the critical gap. But in the case of bt variations, bt growth leads to lower interface thickness that decreases the critical gap. Of course, variations of bt and k change the surface tension. But it is a side effect and surface tension does not affect critical gap inde-pendently. Similarly, the density ratio does not influence the critical gap directly. The critical gap of coalescence is found for six test cases where different values of bt and k lead to different surface tension and density ratio. But, in all cases interface thickness is constant.

Figs. 15 and 16 show the variation of critical gap with sur-

face tension and density ratio for constant interface thickness. Obviously, in a wide range of surface tension and density ratio the critical gap of coalescence is almost constant when the

Fig. 9. Variations of critical gap with surface tension ( 0.4bt = ).

Fig. 10. Variations of critical gap with interface thickness ( 0.4bt = ).

Fig. 11. Variations of critical gap with bt ( 0.01k = ).

Fig. 12. Variations of critical gap with surface tension ( 0.01k = ).

Fig. 13. Variations of critical gap with interface thickness ( 0.01k = ).

Fig. 14. Variations of critical gap with density ratio ( 0.01k = ).

3602 E. A. Rad / Journal of Mechanical Science and Technology 28 (9) (2014) 3597~3603

interface thickness is fixed. The influence of droplet radius on critical gap was investigated and the results are shown in Fig. 17. Regarding the results, it is well obvious that the critical gap is independent of droplet radius. This conclusion can be explained by considering the independency of interface thick-ness from droplet radius.

4. Conclusion

The coalescence of two equal-sized droplets, standing close to each other in a one-component, two-phase system is simu-lated using a free energy lattice Boltzmann approach. To con-firm the accuracy of the model, surface tension of a single

droplet was obtained by LBM and results compared to theo-retical surface tension for different conditions. In the case of constant critical properties, all important parameters such as density ratio, surface tension and interface thickness were controlled by k and bt . By comprehensive review of the results it is concluded that the main parameter that governs the critical gap of coalescence is interface thickness and other parameters such as droplet radius, surface tension and density ratio do not affect the critical gap independently. Actually, the effects of density ratio and surface tension influence the criti-cal gap through interface thickness.

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Fig. 15. Variations of critical gap with surface tension in the case of constant interface thickness.

Fig. 16. Variations of critical gap with density ratio in the case of con-stant interface thickness.

Fig. 17. Variations of critical gap with droplet radius ( 0.01k = ,

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Ehsan Amiri Rad he is an assistant professor of Mechanical engineering at Hakim Sabzevari University. He re-ceived the B.Sc. in Mechanical Engi-neering from Iran University of Science and Technology in 2005 and M.Sc. in Energy Conversion from Ferdowsi Uni-versity of Mashhad-Iran in 2007. His

Ph.D. is from Ferdowsi University of Mashhad-Iran in 2011.