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International Journal of Thermal Sciences 59 (2012) 1e8

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Numerical simulation of the water bubble rising in a liquid column using thecombination of level set and moving mesh methods in the collocated grids

Lin Bu a, Jiyun Zhao b,*a Energy Research Institute @ NTU (ERIAN), Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, SingaporebDivision of Power Engineering, School of Electrical and Electronic Engineering, 50 Nanyang Avenue, Singapore 639798, Singapore

a r t i c l e i n f o

Article history:Received 31 August 2011Received in revised form25 February 2012Accepted 10 April 2012Available online 14 May 2012

Keywords:Level set methodMoving mesh methodCollocated gridsBubble rising

* Corresponding author. Tel.: þ65 67904508; fax: þE-mail address: [email protected] (J. Zhao).

1290-0729/$ e see front matter � 2012 Elsevier Masdoi:10.1016/j.ijthermalsci.2012.04.011

a b s t r a c t

The proper interpretation of bubble rising in a liquid column is critical in the investigation of thewaterevapour two phase flow and heat transfer. In this paper, the bubble behaviours are studied usingthe combination of Level Set Method (LSM) and moving mesh method in a collocated grid. Level setmethod is used to track the interface of the bubble, which has many advantages over other interfacetracking methods such as it can accurately calculate the curvature of the interface and easily expand tothree dimensions. But if the uniform grid was adopted, the bubble interface cannot adapt well withthe grid which may cause great numerical errors. Hence, moving mesh method is implemented in thispaper to increase the numerical accuracy. The numerical model developed in this paper is bench-marked with the experimental data. The results show that the grid distributions used in this paper cancatch the interface continuously in space and time. The process of bubble starting from deformationuntil break into three small bubbles is clearly shown from the numerical results. In addition, thechanging of bubble behaviour with the temperature and pressure is also investigated. It is found thatas the pressure and temperature increase, the deformation process will slow down and this tendencyaccelerates.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

Water either in single phase or waterevapour two phase iswidely used as working fluid in the industry. Especially in thenuclear industry, more than 90% power generated by nuclear isfrom water-cooled nuclear reactors either Pressurized WaterReactor (PWR) or Boiling Water Reactor (BWR). Water boils insidethe BWRs. Although water is in subcooled condition in the PWRs,the subcooled boiling occurs inside the reactor which makes theDeparture from Nucleate Boiling (DNB) a critical safety concern forthe operation of the PWRs. The supercritical water-cooled reactor(SCWR) is one of the six reactor types that are being investigated inthe GEN-IV international advanced reactor development program.Although the water is in supercritical condition and no phasechange or boiling will occur in the SCWR during the normal oper-ation, the subcritical two phase flow conditions will occur duringoff-normal operations [1,2]. Therefore, the proper interpretation ofbubble behaviour is critical for investigation of two phase flow andheat transfer, and finally the safety of the nuclear reactors.

65 67933318.

son SAS. All rights reserved.

There are many studies of the bubble or water droplet behav-iours under the atmospheric pressure both experimentally andnumerically. Experimental method is one of the most importantways in the study of the bubble or water droplet behaviours.Hepbasli et al. [3] used the experimental method to study thebubble behaviour at the free surface of a large three-dimensionalgas fluidized bed. Measurements were carried out to determinethe effects of bed height and excess air velocity on the bubbleeruption diameter, frequency and bubble fraction. All experimentswere performed at freely bubbling mode and the flow character-istics of bubbles were recorded by a video camera. Lee et al. [4]explores experimentally bubble behaviours in a single trapezoidmicrochannel with a hydraulic diameter of 41.3 m. Bubble nucle-ation, growth, departure size, and frequency are observed usinga high-speed digital camera and analyzed by the Image-Pro. Thesize of bubble departure from the microchannel wall is found to begoverned by surface tension and drag of bulk flow andmay be fairlycorrelated by a modified form of Levy equation. Wang and Dong [5]designed a suit of visualized experimental device to study thebubble rising in stagnant liquid in a vertical translucent rectangulartank. The high-speed video system combined with digital imageprocess methods was adopted. The movement of bubble rising in

Delta:1_given nameDelta:1_given nameDelta:1_surnameDelta:1_given namemailto:[email protected]/science/journal/12900729http://www.elsevier.com/locate/ijtshttp://dx.doi.org/10.1016/j.ijthermalsci.2012.04.011http://dx.doi.org/10.1016/j.ijthermalsci.2012.04.011http://dx.doi.org/10.1016/j.ijthermalsci.2012.04.011

Nomenclature

Fx body force in the x directonFy body force in the y directionJ Jacobean of transformationk curvatureP pressureU contravariant velocity normal to the xu velocity normal to xV contravariant velocity normal to the hv velocity normal to yx curvilinear coordinate_x the mesh moving velocity in the x direction_y the mesh moving velocity in the y direction

h curvilinear coordinater densitym viscosity

Subscriptl liquidv vapourx partial differentiation with respect to xh partial differentiation with respect to hs artificial time

Superscriptnþ1 the nþ1 time stepn the n time step

L. Bu, J. Zhao / International Journal of Thermal Sciences 59 (2012) 1e82

water was observed. And the behaviours of bubble were summa-rized with experimental analysing. The behaviour of a single largebubble flowing through a sudden constriction between a cylin-drical pipe and a channel of rectangular cross section is studiedexperimentally by Pacurar et al. [6]. Two types of constrictions areconsidered: an abrupt one and a smooth one. Image analysisdisplays the deformation of the large bubble generated in theupstream vertical pipe and flowing through each kind of constric-tion. Bai et al. [7] used a four-point optical fibre probe to study thebubble properties experimentally. The results show that the opticalprobe underestimates bubble properties, such as, bubble velocity,local void fraction. The presence of the probe in the bubble columninfluences the local flow conditions. Lucas et al. [8] use wire-meshsensors to investigate stationary upward airewater flows ina vertical pipe with an inner diameter of 195.3 mm. Due to thecombination of the new experimental procedure with the highspatial and temporal resolution of the wire-mesh sensor tech-nology the data have new quality especially regarding theirconsistency in the evolution with increasing L/D. The wire-meshsensors were used by Prasser et al. [9] to study the evolution ofthe bubble size distributions in a vertical two-phase flow. Thedifferent behaviours of small and large bubbles in respect to theaction of the lift force were observed in a mixture of small and largebubbles. Marco et al. [10] measured the rising velocities of gasbubbles in a still liquid and compared the results with availabletheories. Bubble size, aspect ratio, detachment frequency, velocityand frequency of shape oscillationsweremeasured by processing ofhigh-speed video images.

Owing to the advances in numerical methods and as well as incomputational hardware performance, numerical simulations havebecome increasingly popular in the study of complex multi-fluidsystems. There are various numerical methods available in litera-ture to compute bubble behaviour. Nagrath et al. [11] studied thethree-dimensional computation of bubble behaviour using a Levelset method. The continuum surface force (CSF) model was appliedin order to account for surface tension effects. To restrict theinterface from moving while re-distancing, an improved re-distancing scheme proposed in the finite-difference context isadapted for finite element discretization. Hence, the accuratecomputation can be carried out including the computation of thelarge viscosity difference and large density. A systematic compu-tational study of the behaviour of gas bubbles rising in a viscousliquid is presented by Smolianski et al. [12]. The numerical methodpresented in this paper can simulate a wide range of flow regimes,accurately capturing the shape of the deforming interface of thebubble and the surface tension effect, while maintaining the massconversation. Minsier et al. [13] studied the behaviour of a cavi-tating bubble in a viscous liquid near a solid surface. The effects of

the viscosity were taken into account. Gupta et al. [14] used LatticeBoltzmann method (LBM) to study the behaviour of bubble motionand bubble coalescence in liquids. Simulations of bubbles withdifferent sizes and in large periodic domain were studied. A singlegas bubble rising in a narrow vertical tube is investigated viaa numerical model on a 3-D axisymmetric computational domainbyWang et al. [15]. The transient governing equations are solved bya finite volume scheme with a two-step projection method. Theinterface between the liquid and gas phase is tracked by a coupledlevel set and volume-of-fluid (CLSVOF) method. It has been foundthat the bubble nose retains a relative stable shapewhile significantoscillations occur at the bubble bottom as it rises through the liquidfrom the static state.

Many other investigations on the bubble or water dropletbehaviours exist in the literature, however, most of them wereunder the atmospheric pressure [3e15]. In the present paper, Levelset method coupled with moving mesh method [16] was adoptedto investigate the bubble behaviour in the subcritical water atdifferent higher pressures. Level set method was first proposed byOsher and Sethian [17]. It is used to track the interface of thebubble in this paper. Level set method has the many advantagesover other interface tracking methods. It can accurately calculatethe curvature of the interface and the continuity of the interface isvery well. It also can be easily expand to three dimensions. But ifthe uniform grid was adopted, the bubble interface cannot adaptvery well with the grid, and great errors may appear. Hence,moving mesh method is implemented to increase the numericalaccuracy in this paper. The mesh changes continuously in spaceand time to adapt to the dynamic changes of time-evolving solu-tions. The numerical simulation is conducted by solving thecomplete incompressible NaviereStokes equations in an adaptivegrid system, and the surface tension force is modelled bya continuum surface force approximation. During the computa-tion, velocities of the fluids are corrected by using a velocity-correction formula, and the computation proceeds to conver-gence via a series of correction procedures so as to make thevelocity fields satisfy the continuity of the fluids.

2. Numerical model

2.1. Physical model and basic assumptions

In this paper, the rising process of a saturated bubble under thedifferent pressure conditions was studied. The model presented inthis paper is for a 2D two-phase flow. The initial bubble is staticallylocated at (0.1, 0.06), in the domain size 0.2 m � 0.2 m, as shown inFig. 1. The initial radius of the bubble is 0.01 m. The gravitationalacceleration is 9.8 m/s2. At the initial state, the water and the steam

Water

Steam

0 x(ξ)

y(η)u=0v=0

u=0v=0

u=0v=0

u=0v=0

Fig. 1. Physical model of the bubble and water ambience.

L. Bu, J. Zhao / International Journal of Thermal Sciences 59 (2012) 1e8 3

are in the thermal-equilibrium condition. The velocities on thewalls are all set as zero, as shown in Fig. 1.

The model was based on the following assumptions:

1. Since the Reynolds number is very low, the flow can be treatedas laminar flow.

2. Since the bubble and water are in thermal-equilibrium and theboundary conditions are adiabatic, the temperature changesare negligible in the whole system. Hence, the heat and masstransfer between the steam and the liquid was ignored in thispaper.

The water properties are calculated using the polynomialequations, which are obtained by curve fitting the data calculatedusing the ASME properties [19]. The detailed process on the waterproperties calculations can be found in the Appendix of this paper.

2.2. Governing equations

The governing equations include the NeS equations and theLevel set equations. All the equations were transformed into thecurvilinear coordinates. Geometry coefficients, a, b, g and theJacobean of transformation, J, may be used in the following equa-tions. Therefore, they must be predefined as

a ¼ x2h þ y2h (1)

b ¼ xxxh þ yxyh (2)

g ¼ x2x þ y2x (3)

J ¼ xxyh � yxxh (4)

2a

Dx2þ 2aDh2

þ 1Ds

!xnþ1ði;jÞ ¼

a

Dx2xnþ1ðiþ1;jÞ þ

a

Dx2xnþ1ði�1;jÞ þ

aDh2

xnþ1ði;j�1Þ þa

Dh2

þ �aþwn

Dx2þ s12Dx

!xnðiþ1;jÞ þ

�aþwn

Dx2�

þ�aþwn

Dh2� s22Dh

xnði;j�1Þ

Momentum equations can be transformed as follows [18]:� h i h i�

r J

vuvt

þux�u� _x�yh� �v� _y�xh þuh � �u� _x�yxþ �v� _y�xx

¼ �yhpxþyxphþ�m

J

�aux�buh

��xþ�m

J

�guh�bux

��hþ _Fx ð5Þ

r

�Jvv

vtþ vx

h�u� _x�yh��v� _y�xhiþ vhh��u� _x�yxþ�v� _y�xxi

�

¼ �xxphþxhpxþ�m

J

�avx�bvh

��xþþ

�m

J

�gvh�bvx

��hþ _Fy ð6Þ

Where _x and _y represent the velocities of the grid in x and ycoordinate, _F

xand _F

yrepresent the components of the body force in

x and y coordinate.The continuity equation can be transformed as follows:

Jnuxh�u� _x�yh��v� _y�xhiþuhh��u� _x�yxþ�v� _y�xxi

o¼ 0

(7)

The Level set equations can be transformed as follows:

JvF

vtþ Fx

h�u� _x�yh � �v� _y�xhiþ Fhh� �u� _x�yx þ �v� _y�xxi

¼ 0 ð8Þ

vF

vs¼ signðF0Þ

�1� J�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaF2x þ gF2h � 2bFxFh

q �(9)

2.3. Grid generating technique

The mesh generation method adopted in this article wasproposed by Ceniceros and Hou [18]. The grid generating equationscan be written as follows:

xt ¼ VðwVxÞ; yt ¼ VðwVyÞ (10)

Where w is the monitor function. Using the semi-implicit dis-cretization, we can get the following equations：

xnþ1 � xnDt

¼ aDxnþ1 þ V$ðwnVxnÞ � aDxn (11)

ynþ1 � ynDt

¼ aDynþ1 þ V$ðwnVynÞ � aDyn (12)

Where a ¼ maxwn, D and V are the operators of the Laplacian andthe gradient with respect to (x, h).

Eqs. (13) and (14) can be discretized as follows:

xnþ1ði;j�1Þ þ 2a� 2wn

Dx2þ 2a� 2w

n

Dh2þ 1Ds

!xnði;jÞ

s12Dx

!xnði�1;jÞ þ

�aþwnDh2

þ s22Dh

xnði;jþ1Þ (13)

2a

Dx2þ 2aDh2

þ 1Ds

!ynþ1ði;jÞ ¼

a

Dx2ynþ1ðiþ1;jÞ þ

a

Dx2ynþ1ði�1;jÞ þ

aDh2

ynþ1ði;j�1Þ þa

Dh2ynþ1ði;j�1Þ þ

2a� 2wn

Dx2þ 2a� 2w

n

Dh2þ 1Ds

!ynði;jÞ

þ �aþwn

Dx2þ s12Dx

!ynðiþ1;jÞ þ

�aþwn

Dx2� s12Dx

!ynði�1;jÞ þ

�aþwnDh2

þ s22Dh

ynði;jþ1Þ

þ�aþwn

Dh2� s22Dh

ynði;j�1Þ

(14)


2.4. Solving procedures

A collocated grid finite-difference scheme is used in the presentstudy. The scalar parameters are defined on the nodes of the gridsand a SIMPLER-based method is adopted to solve the solutions ofvelocities and the pressure. Cartesian velocity components are usedas the independent variables in the momentum equations, while

Initialize the grid

Update the mesh

Solving the level set equation and reinitialize the level set function

Update the fluid properties

Solve the momentum equations

Solve the pressure correction equation

Correct the velocity

Fig. 2. The computational procedure.

YL

XL

H

L

x0

y

Fig. 3. Computational domain.

the contravariant velocity components are used as the independentvariables in the continuity equation. The computation proceduresare shown as follows in Fig. 2:

3. Numerical model benchmarking

In order to verify the accuracy of the numerical model devel-oped in this paper, detailed comparison between the numerical

Fig. 4. The collapsing process of the water column. (a) t ¼ 0.0 s (b) t ¼ 0.06 s (c)t ¼ 0.12 s (d) t ¼ 0.18 s (e) t ¼ 0.24 s (f) t ¼ 0.30 s.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51

2

3

4

Experimental Results

2 g L

x/L

t

Numerical Results

Fig. 5. The comparison of experimental results [20] and numerical results.

Fig. 6. Grid distributions at the different time interv

Fig. 7. Level set profiles and the velocity fields for different time intervals. (a) t ¼ 40


results and the experimental data are carried out. A typicalbenchmarking example is used in this paper. The computationaldomain is: 0.584 m � 0.584 m. The liquid column is static in theinitial time. The height of liquid column is 0.292m, and the width is0.146m, as shown in Fig. 3. Driven by the gravity, the liquid columnstarts to collapse and finally bumped against the wall. Then theliquid rises along the right wall. The properties of the vapour andthe liquid are: rl ¼ 1000(kg/m3), rv ¼ 1(kg/m3), ml ¼ 0.5 kg/(m$s),mv ¼ 0.5 � 10�3 kg/(m$s), s ¼ 0.0755 (N/m).

The velocity boundary condition specified on walls in contactwith the free surface is not well posed due to the contradiction ofthe moving free surface and a no-slip condition on the boundingwall. For alternative free surface contact wall boundary conditionshave been adopted in the current work.

V u! ¼ 0 (15)The contact wall boundary problem has been overcome by

adopting a slip boundary condition on the free surface contactwalls such that the nonpermeability condition is maintained,whilst the tangential velocity is allowed to take a nonzero value by

als. (a) t ¼ 40 ms (b) t ¼ 80 ms (c) t ¼ 120 ms.

ms (b) t ¼ 80 ms (c) t ¼ 120 ms (d) t ¼ 160 ms (e) t ¼ 200 ms (f) t ¼ 240 ms.


implementing a Neumann boundary condition at the wall. Theboundary conditions were as follows:

�n!$ u! ¼ 0n!$V u! ¼ 0 (16)

Fig. 4 describes the collapsing process of the water column fromthe initial state. Martin and Moyce [20] reported the data of the xversus t based on the experiment. Fig. 5 shows the comparison ofthe experimental data and the numerical results obtained from the

523.5K-638.9K

t=0ms

523.5K

638.9K

t=80ms

638

t=160

a

c

e

Fig. 8. Level set profiles at different initial temperatures. (a) t ¼ 0

present paper. It can be seen that the experimental and thenumerical results are matched well. Therefore, the numericalmodel developed in this paper is accurate and valid.

4. Results and discussion

The numerical simulation results of the bubble dynamic processat a temperature of 523.5 K (pressure of 4 MPa) are shown in Figs. 6and 7. A sensitivity study is also performed for the differenttemperatures and pressures, and the results are shown in

523.5K

638.9K

t=40ms

523.5K

638.9K

t=120ms

523.5K

.9K

ms

b

d

ms (b) t ¼ 40 ms (c) t ¼ 80 ms (d) t ¼ 120 ms (e) t ¼ 160 ms.

450 500 550 600 650200

250

300

350

t /m

s

T /K

Fig. 9. The bubble collapsing time versus the initial temperature.


Figs. 8e10. Note that since the saturation and thermal-equilibriumconditions are simulated in this paper, once the temperatures arespecified, the pressures are set accordingly.

Fig. 6 shows the grid distributions at the time intervals of 40, 80and 120 ms under the normal gravity condition, when the initialtemperature is 523.5 K. Fig. 7 shows the velocity and the level setprofiles during the rising process of the bubble in the liquid, and theinitial temperature of the bubble is also at 523.5 K. From thecomparison of Fig. 6 (aec) and Fig. 7 (aec), we can see that the griddistributions in this paper can catch the interface continuously inspace and time.

As shown in Fig. 7, both the left side and the right side of thebubble begin to shrink and the upside of the bubble begins to rise,and the tendency continues as time going on until the big bubblecollapses and breaks into three small bubbles finally.

Fig. 8 shows the level set profiles at different temperatures andcorresponding pressures at the time intervals of 40 ms, 80 ms,120 ms and 160 ms. It can be seen from Fig. 8, as the temperatureand pressure of the bubble increasing, the deformation process ofthe bubble is slowing down, and the velocities of the bubble risingis decreased.

500 550 600 6500.08

0.09

0.10

0.11 40ms 80ms 120ms

Ygm

ax/m

T/K

Fig. 10. Bubble peak Ygmax versus the initial temperature.

In all the conditions studied in this paper, the bubble willcollapse. The big bubble will finally divide into three bubbles. Fig. 9shows the bubble collapsing time versus temperature. As the initialsaturated temperature increases, the bubble collapsing time willincrease, in other words, the bubble needs longer time to collapse.

Fig. 10 shows the peak of the bubble Ygmax versus the temper-ature. As the temperature increases, the peak of the bubble Ygmaxdecreases and this tendency accelerates.

5. Conclusions

In order to investigate the bubble behaviour for the water-vapour two phase flow condition, level set coupled with movingmesh method is adopted to develop the numerical model in thispaper. The model is benchmarked with the experimental data,which verified the accuracy and validity of the model. The bubblebehaviour is then investigated using the model developed. Thenumerical results show that the grids can catch the bubble profilesprecisely. It is shown that as the bubble rising in the liquid, both theleft side and the right side of the bubble begin to shrink and theupside of the bubble begins to rise. The process continues as timegoing on until the big bubble collapses and breaks into three smallbubbles. It is also found that as the temperature and pressure of thebubble increasing, the deformation process of the bubble will beslowing down, and the velocities of the bubble rising is decreased.It requires a longer time for bubble to collapse and the tendencyaccelerates if the temperature and pressure is increased.

Appendix A. The water properties calculations

The properties of the saturated water are calculated by thepolynomial equations, which are obtained by curve fitting the datacalculated using the ASME properties [19] based on IAPWS-IF 97,shown as follows:

rl ¼ �116403:4569210326� 1187:2268519176Tþ 4:852387025T2 � 0:0098500729T3 þ 9:9295601755E� 6T4 � 3:9805490227E � 9T5

(17)

rv ¼ �107845:8604256314þ 1109:2000537937T� 4:5297204456T2 þ 0:0091844859T3 � 9:2506529969E� 6T4 þ 3:7053629736E � 9T5

(18)

ml ¼ 0:0259376867� 2:3823767295E � 4T þ 8:9076924478E� 7T2 � 1:6792748598E � 9T3 þ 1:5903634852E � 12T4

� 6:0421265504E � 16T5(19)

mv ¼ �0:0088855426þ 9:1292780171E � 5T� 3:7214155045E � 7T2 þ 7:533699037E � 10T3

� 7:574042816E � 13T4 þ 3:0264242733E � 16T5(20)

The comparison of the properties of the saturated watercomputed from the equations and the IAPWS-IF 97 are shown inFig. A1. It can be seen that the two results matched well.

Fig. A1. Comparison of saturated water properties (a) the density of saturated water versus the temperature (b) the density of saturated vapour versus the temperature. (c) theviscosity of saturated water versus the temperature (d) the viscosity of saturated vapour versus the temperature.


References

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[3] A. Hepbasli, Investigation of the bubble behaviour at the free surface of a largethree-dimensional gas fluidized bed, International Journal of Energy Research22 (1998) 885e909.

[4] P.C. Lee, F.G. Tseng, C. Pan, Bubble dynamics in microchannels. Part I: singlemicrochannel, International Journal of Heat and Mass Transfer 47 (2004)5575e5589.

[5] H.Y. Wang, F. Dong, Moving character observation of bubble rising in verticalgas-liquid two-phase flow, in: L. Guo (Ed.), The 6th Symposium on MultiphaseFlow, Heat Mass Transfer and Energy Conversion, American Institute ofPhysics, 2009.

[6] C.B. Pacurar, C. Guigui, A. Linea, A. Tazi-Paind, Behaviour of a large bubbleflowing through a sudden constriction between a cylindrical pipe and a rect-angular cross section channel, Chemical Engineering Research and Design 89(2011) 921e929.

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classes using wire-mesh sensors, International Journal of Thermal Sciences 41(2002) 17e28.

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Numerical simulation of the water bubble rising in a liquid column using the combination of level set and moving mesh metho ...1. Introduction2. Numerical model2.1. Physical model and basic assumptions2.2. Governing equations2.3. Grid generating technique2.4. Solving procedures

3. Numerical model benchmarking4. Results and discussion5. ConclusionsAppendix A. The water properties calculationsReferences

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