Research ArticleNumerical Study of Bubble Rising and CoalescenceCharacteristics under Flow Pulsation Based on Particle Method

Ronghua Chen 1 Minghao Zhang1 Kailun Guo1 Dawei Zhao2 Wenxi Tian1

Guanghui Su 1 and Suizheng Qiu 1

1School of Nuclear Science and Technology Xirsquoan Jiaotong University No 28 Xianning West road Xirsquoan 710049 China2CNNC Key Laboratory on Nuclear Reactor Thermal Hydraulics Technology Nuclear Power Institute of China No 328Section 1 Changshun Avenue Chengdu 610213 China

Correspondence should be addressed to Ronghua Chen rhchenmailxjtueducn

Received 17 December 2018 Accepted 7 March 2019 Published 1 April 2019

Academic Editor Manmohan Pandey

Copyright copy 2019 Ronghua Chen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two-phase flow instability may occur in nuclear reactor systems which is often accompanied by periodic fluctuation in fluid flowrate In this study bubble rising and coalescence characteristics under inlet flow pulsation condition are analyzed based on theMPS-MAFLmethodTo beginwith the single bubble rising behaviorunder flowpulsation condition was simulatedThe simulationresults show that the bubble shape and rising velocity fluctuate periodically as same as the inlet flow rate Additionally the bubblepairsrsquo coalescence behavior under flow pulsation condition was simulated and compared with static condition results It is foundthat the coalescence time of bubble pairs slightly increased under the pulsation condition and then the bubbles will continue topulsate with almost the same period as the inlet flow rate after coalescence In view of these facts this study could offer theorysupport and method basis to a better understanding of the two-phase flow configuration under flow pulsation condition

1 Introduction

Two-phase flow phenomena could occur in nuclear reactorsystems (steam generators boiling water reactor cores con-densers etc) conventional power plants refrigeration andchemical and aerospace In a two-phase system when massflow density void fraction and pressure drop are coupledthe static flow instability (the flow is drifted by a slightdisturbance) and the kinetic instability (constantvariableamplitude flow oscillations at a particular frequency) arecollectively referred to as flow instability [1] These flowinstability phenomena are usually accompanied by periodicfluctuation in flow rate As is well known two-phase flowinstability can degrade the performance of the equipmentwhich will cause many safety problems for example themechanical forces caused by the flow oscillations will makethe components undergo harmful forced mechanical vibra-tions and continuous mechanical vibration could lead tofatigue failure of components In addition bubble dynamics isan indispensable part of the two-phase flow research and theintrinsic driving force of gas-liquid two-phase flow evolution

Therefore it is necessary to study the bubble behavior underflow pulsation condition

In experimental research a great deal of achievementson bubble dynamics have been made until now In thebubble coalescence visualization experiment of Sanada etal [2] the high-speed camera was used to capture therising and coalescence process of bubbles in the stationaryliquid phase and the microscopic characteristics such asthe trajectory velocity and interface configuration evolutionof the bubble coalescence process were obtained Liu andZheng [3] used the method of particle image velocimetry tostudy the motion behavior of bubbles in a stationary liquidwith a rectangular column and the diversity of bubble risingshapes in different viscosity liquids was finally observedSathe et al [4] used the discrete wavelet transform algorithmto reconstruct the PIV flow field distribution around thebubble which reduced the interference signal during thePIV flow field acquisition process and further obtained theflow field characteristics like phase drift velocity Basarovaet al [5] experimentally investigated the ascending motionbehavior of spherical bubbles in different concentrations of

HindawiScience and Technology of Nuclear InstallationsVolume 2019 Article ID 2045751 13 pageshttpsdoiorg10115520192045751

2 Science and Technology of Nuclear Installations

water ethanol and water propanol and obtained characteristicvalues such as bubble rising velocity and bubble size whichproved the effect of the concentration of the two solutionson the liquid properties and the gas-liquid interface Mal-donado et al [6] experimentally studied a single bubble of25mm diameter rises in different liquids They establisheda method for calculating the aspect ratio and the risingvelocity of bubble in experiment Meanwhile they foundthat the bubble aspect ratio has a unique relationship withthe bubble rise velocity under different conditions Liu et al[7] performed experiments to study the rise of bubbles instationary water and glycerol aqueous solution Specificallythe experiment detected that the bubble shape has a strongconnection and interaction with the terminal velocity At thesame time the inertial force and the surface tension mainlyaffect the rising shape of the bubble in water but in theaqueous glycerin solution the influence of the viscous forceneeds to be considered Raymond and Rosant [8] performedexperiments on the movement of moderate deformationbubbles in a viscous liquid mixed with glycerin and waterwhere the dimensionless Reynolds number and the Webernumber range are [1100] and [05] respectively The bubbleaspect ratio and the bubble terminal velocity which describethe bubble motion characteristics were obtained by theseexperiments and the experimental resultswere also comparedwith numerical simulation Sugrue et al [9] investigatedthe influences of channel orientation on bubble dynamicsBesides the bubble dynamics experiments under differentfluid conditions [10ndash13] have been further conducted

Due to the uncertainty of the experiment and the limi-tations of the experimental conditions numerical simulationand mechanism studies play an increasingly important rolein study of bubble dynamics Tomiyama et al [14] inves-tigated the bubble terminal velocity in an unlimited largestatic liquid dominated by surface tension and proposed atheoretical model applicable to predict the terminal velocityof a twisted spherical bubble with a high nondimensionalReynolds number Based on the balance of all forces actingon the bubble Wang et al [15] analyzed the related forces ofbubble and illustrated the bubble interface and the shape ofthe bubble Moreover by theoretical analysis a conclusionwas drawn that the mass and heat transfer in the gas-liquid phase interface provides the possibility of continuousmotion of the bubble Gumulya et al [16] performed anumerical analysis on the rising behavior of bubbles inviscous liquids On the basis of the three-dimensional VOFmethod the correlation between bubble aspect ratio andseveral dimensionless numbers in different bubble shapeswasemphatically studied Additionally they also paid attentionto the formation and expansion features of bubble wakes Yuet al [17] used the new kind of lattice Boltzmann method tosimulate a pair of bubbles with different initial positions andinitial velocities and the interactions between bubbles havebeen successfully obtained such as attraction and repulsionSanada et al [18] principally studied the effects of viscosityon coalescence of a bubble upon impact with a free surfaceit was concluded that both the liquid film and the bubblewake have a significant impact on the bubble coalescence orbounce

In the 1990s a meshless method called the Moving Parti-cle Semi-implicit (MPS) method was proposed by Koshizukaand Oka [19] It is applicable to the simulation of incom-pressible fluid especially those with free surface In order tosolve the problem of particle tracking in the presence of inletand outlet flows in fluid mechanics Yoon et al [20] came upwith a hybrid method called MPS-MAFL At this stage two-phase simulation based on the MPS-MAFL method has beenwidely used [21ndash25] because of its outstanding advantages incapturing phase interface

Under the condition of inlet flow pulsation the gas-liquidphase interface changes rapidly and dramatically due to thefluctuation of flow velocity When using the traditional gridmethod to simulate the bubble motion characteristics underthe condition of velocity pulsation the mesh quality and thestability of the calculation near the phase interface are verychallenging Nowadays most of the researches on bubbledynamics were carried out under the static condition andthe study of bubble behavior under flow pulsation conditionis insufficient Therefore it is reasonable and feasible to studythe bubble rising and bubble pairs coalescence characteristicsunder flow pulsation condition by using the predominanceof MPS-MAFL in tracking phase interface Furthermore thisresearch will also provide a strong theoretical support forunderstanding the two-phase flow configuration

2 Numerical Method and Program Validation

21 Governing Equations In MPS-MAFL method the con-tinuity equation and the Navier-Stokes equation for theincompressible viscous fluid are as follows

nabla sdot u = 0 (1)

120588(120597u120597119905 + (u minus u119888) sdot nablau) = minusnabla119901 + 120583nabla2u + 120590120581 sdot n + 120588g (2)

where u 120590 120581 n 120583 and u119888 are velocity surface tensioncoefficient steam liquid interface curvature unite vector ofphase interface viscosity and the motion of a computingpoint that is adaptively configured during the calculationrespectively Due to the permission of arbitrary calculationbetween fully Lagrangian (u119888 = u) and Eulerian (u119888 = 0)calculations the sharp fluid front is calculated accuratelyby moving the computing points in Lagrangian coordinateswhile the computing point is fixed at the boundaries that aredescribed with Eulerian coordinates [20]

22 Particle Interaction Models InMPSmethod the govern-ing equations are solved by particle interaction models suchas gradient model Laplacian model and divergence modelwhere a particle interacts with nearby particles covered witha weight function 119908(119903 119903e) where 119903 is the distance betweentwo particles and 119903e is the radius of interaction area In thisstudy the kernel function is adopted as weight function andits expression is as follows

119908 (119903) = 119903119890119903 minus 1 (0 le 119903 lt 119903119890)0 (119903119890 le 119903) (3)

Science and Technology of Nuclear Installations 3

No-Slip Wall

002m

004m

003

7m

008m

047

m

Free surface

U = U + ΔUＭＣn(2t)

(a) Single bubble

No-Slip Wall

00018m

0004m

000

2m001m

000

8m

Free surface

0002m

U = U + ΔUＭＣn(2t)

(b) Bubble pairs

Figure 1 Initial geometry and boundary arrangement under flow pulsation condition

The gradient and Laplacian of physical quantity 120601 onparticle 119894 are expressed by the summation of physical quan-tities over its neighboring particles 119895 according to the weightfunction

The gradient model and Laplacian model are as followsrespectively

⟨nabla120601⟩119894 = 119889119899119894sum119895 =119894[[

120601119895 minus 12060111989410038161003816100381610038161003816r119895 minus r119894100381610038161003816100381610038162

(r119895 minus r119894)119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)]]

(4)

⟨nabla2120601⟩119894= 2119889

120582119899119894 sum119895 =119894 [(120601119895 minus 120601119894)119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)] (5)

where 119889 represents spatial dimension and 120582 is diffusioncoefficient and can be determined as

120582119894 = sum119895 =119894 [119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816) 10038161003816100381610038161003816r119895 minus r119894

100381610038161003816100381610038162]sum119895 =119894 [119908 (r119895 minus r119894)] (6)

In the MPS-MAFL method the divergence model isshown in (7) The velocity divergence between two particlesis defined by (u119895 minus u119894)(r119895 minus r119894)|(r119895 minus r119894)|2 and the velocity

divergence at one particle is calculated from the weightedaverage of the individual velocity divergences

⟨nabla sdot u⟩119894 = 119889119899119894sum119895 =119894[[

(u119895 minus u119894) sdot (r119895 minus r119894)10038161003816100381610038161003816r119895 minus r119894100381610038161003816100381610038162

119908(10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)]]

(7)

23 Computational Domain and Boundary Condition In thisstudy the calculation domain of the single bubble risingprocess under flow pulsation condition is a 008m wide and047m high two-dimensional region as shown in Figure 1(a)The initial radius of the bubble is 001m and the bubble islocated in the center of the channelThe distance between thecenter of the initial bubble and the inlet of the flow channel is0037m This geometric setting ensures that the influence ofthe wall can be neglected during bubble motion The vaporphase property is given based on ideal gas state equationand the liquid area is described by using discretized particleswith nonuniform scheme And the particle number densitydecreases as the distance from the phase interface increasesin the liquid area

As for the boundary condition setting the left and theright sides of the flow path are set as nonslip boundary con-ditions and the upper side of the flow path adopts a free

4 Science and Technology of Nuclear Installations

Table 1 Physical property parameters of the gas-liquid phase

Gasliquid phase ViscosityPasdots Densitykgsdotmminus3 Surface tensionNsdotmminus1Air (T = 8∘C) 175times10minus5 1250 Air (T = 29∘C) 185times10minus5 1170 Water (T=8∘C) 138times10minus3 9998 0074Water (T=29∘C) 086times10minus3 9967 0072

R = 001m

Bubbleterminal

shape(t = 03s) 18R 19R 110R 111R

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

01 02 0300time (s)

particle size = 18Rparticle size = 19R

particle size = 110Rparticle size = 111R

Figure 2 Particle size sensitivity analysis (Re=Bo=10)

surface boundary condition The bubble interface is set asfree surface and the pressure values at various points onthe interface are obtained by solving the ideal gas stateequation Moreover at the inlet boundary of the channel aflow pulsation condition is introduced The flow at the inletchanges in a sinusoidal cycle over time as shown in (8) Theequation contains three variables the initial flow rate of theliquid phase 119880 the relative amplitude of the flow pulsationΔ119880119880 and the flow pulsation period 120591

119880 = 119880 + Δ119880 sin (2120587119905120591 ) (8)

The calculation domain of bubble pairs coalescence underflow pulsation condition is set in a similar way as shownin Figure 1(b) The calculation domain is 001m wide and0008m high and the two bubbles are symmetrically dis-tributedThe boundary conditions and particle placement arethe same as single bubble conditions

In the process of bubble dynamics simulation using MPSmethod the particle size (the distance between neighboringparticles) could affect the simulation efficiency and conver-gence Specifically the optimal particle size could be figuredout by sensitivity analysis In this study the simulation ofsingle bubble rising in a static fluid was performed withdifferent particle size As shown in Figure 2 the interfacialparticle size was set to 18 19 110 and 111 of the bubbleradius respectively It can be seen that the bubble centroidvelocity calculation results have good convergence under

minus02

minus01

00

01

02

03

04

05

Bubb

le te

rmin

al v

eloci

ty (m

s)

JamialahmadiLiu T=8∘CLiu T=29∘C

MPS-MAFL T=8∘CMPS-MAFL T=29∘C

7 8 9 10 11 126Bubble diameter (mm)

Figure 3 Terminal rising velocity of the bubble under differentinitial bubble diameter conditions

these particle sizes At the same time the shape change duringthe bubble rising progress was obtained It is found that theterminal shapes are nearly identical in each case Thereforebased on the sensitivity analysis results the interfacial particlesize was set to 110 of the initial bubble radius to ensure theefficiency of the program simulation and the convergence ofthe calculation

24 Program Validation The single bubble rising in staticwater was simulated by MPS-MAFL and the simulationresults were compared with the experiment of Liu et al [26]The physical property parameters of the relevant cases arelisted in Table 1 In MPS-MAFL the calculation conditionsand parameters were set as same as the experiment andthe terminal rising velocity of the bubble under differentinitial bubble diameter conditions was obtained as shownin Figure 3 where scatter points respectively representexperimental data points for two temperatures

As shown in Figure 3 the calculated bubble terminalvelocity value is slightly lower than the experimental onewith the error about 15 One reason is that the lateralmovement of the bubbles existing in the experiment hadbeen simplified in the two-dimensional simulations of thisstudy Within the range of bubble diameter larger than 6mmthe bubble terminal rising velocity slightly increases with the

Science and Technology of Nuclear Installations 5

Reyn

olds

num

ber (

Re)

Eotvos number (Eo)

Log M10

5

104

103

102

101

100

10minus1

103

102

101

100

10minus1

10minus2

-14-13-12-11-10

-9

-8-7-6-5-4-3-2-101

2 3 4 5 6

Figure4Gracersquos graphical correlation for predicting terminal shapeof bubble

bubble diameter which is consistent with the experimentalconclusion of Liu et al [26] Thus the overall trend ofthe simulation results is in reasonable agreement with theexperiment

In the process of single bubble rising in a static liquidthe stable shape of the bubble is another important charac-teristic that should be concerned On the basis of previousexperimental results Grace [27 28] proposed a graphicalcorrelation that predicts the final stable shape of the risingbubble in an infinite static liquid It can be seen from thisdiagram that the terminal shape of the bubble can be roughlydivided into four regions namely Sphere region Ellipsoidregion Dimples region and Spherical-cap region In order toverify the accuracy of the simulation the following four casesare respectively set Re = 8854 Eo = 0392 Re = 8854 Eo =392 Re = 10 Eo = 100 and Re = 1000 Eo = 300 Moreoverthe stable shape in the bubble rising process in each case isobtained and the stable shape of the bubble is marked at thecorresponding position according to the Reynolds numberand the Eotvos number (also called Bond number) as shownin Figure 4 In view of these facts it is completely consistentwith the results predicted by Grace

This study aims to discuss the bubble rising and coa-lescence characteristics under inlet flow pulsation Therebyit is necessary to verify the accuracy of the added inletflow pulsation condition before the simulation First a flowchannel only with liquid is set and its geometric parametersare the same as those for studying single bubble rising underflow pulsation condition as shown in Figure 5 Then a flowrate that changes periodically with timewas added to the inletof the flow path The initial flow rate of the liquid phase theflow pulsation period and the relative amplitude of the flowpulsation were 005ms 02s and 07 respectively Figure 5shows the velocity values of the particles at the inlet and outlet

U = 005 + 0035Sin(314lowasttime)the inlet particlethe outlet particle

005 010 015 020000time (s)

000

002

004

006

008

010

Part

icle

velo

city

(ms

)

Figure 5 The velocity of inlet and outlet particles under flowpulsation condition

of the flow channel when the flow stability is reached Asshown in Figure 5 the velocity values of the outlet particlesare the same as those of the inlet particles at any of the samemoments and they all satisfy the analytical formula of theadded flow pulsation condition

3 Results and Discussions

The Reynolds number (Re) and the Bond number (Bo)are commonly adopted to study the bubble dynamics Theexpressions are respectively as follows

Re = 120588119887radic1198921198633120583119887

(9)

119861119900 = 1205881198871198921198632120590 (10)

where 120588119887 is the density of liquid 120583119887 is liquid viscosity and119863 is the bubble diameter 120590 represents the surface tensionThrough these two dimensionless expressions the Reynoldsnumber represents the ratio of the inertial force to the viscousforce of the bubble and the Bond number represents the ratioof the gravity of the bubble to its surface tension During themovement of the bubble these four forces have a significanteffect on the bubble Besides the density ratio and viscosityratio of the gas-liquid two phases also have an impact In thisstudy the density ratio and viscosity ratio of liquid phase togas phase are set as 10001 and 1001 in all bubble simulationsrespectively

31 Single Bubble Rising under Flow Pulsation Condition Inthis study the influence of the pulsation condition on therising motion characteristics of the bubble is investigatedDue to the addition of the pulsating flow in the form of(8) three independent variables are added Together with

6 Science and Technology of Nuclear Installations

Table 2 Cases simulated in Section 31

Case ParameterRe Bo 119880(ms) Δ119880119880 120591(s)

a 5 10 005 07 02b 10 10 005 07 02c 20 10 005 07 02d 50 10 005 07 02e 5 30 005 07 02f 5 50 005 07 02119892 5 10 0075 07 02h 5 10 01 07 02i 5 10 005 07 01j 5 10 005 07 025k 5 10 01 01 02l 5 10 01 03 02m 5 10 01 05 02n 5 10 01 07 02o 5 10 01 1 02

the two dimensionless numbers mentioned above thereare actually five independent variables that may affect themotion behavior of the bubble under the inlet flow pulsationcondition Therefore the control variable method can beused to discuss the bubble deformation process and thesurrounding flow field changes under the flow pulsationcondition The specific analysis is as follows

311 Single Bubble Deformation Process The cases of study-ing the single bubble rising process under flow pulsationcondition are listed in Table 2 Initially a case under staticcondition is taken as a reference example while its dimen-sionless number and other conditions were as same as case119886 in Table 2 And the bubble rising process under staticcondition was shown in Figure 6(a) with red-dashed line Tosum up in this case the bubble will only gradually stabilizeduring the ascent and the terminal shape is consistent withthe prediction of Grace [27]

By simulating the various cases set in Table 2 it is possibleto observe the change of the shape and the surrounding flowfield during the bubble rising in the liquid under the periodicvariation of the inlet flow rate Figure 6 shows the changeprocess of the rising bubble under cases 119886 and 119887 in two cycles(t = 04s) and the flow field velocity vector information andpressure distribution in the red frame (around the bubble) aregiven by Figure 7 The bubble has a wake during the ascentand there are two symmetric vortices around the bubble Itcan also be observed in Figure 6 that there is a significantshape and volume fluctuation in each cycle of the bubblerising process due to the introduction of the inlet periodicchange flow rate This is one of the differences between thebubble deformation processes under the inlet flow pulsationcondition and the static condition

At the same time in order to numerically verify thefluctuation behavior of the bubble under pulsation conditionthe bubble equivalent diameter at each moment is calculatedThe trend of the bubble equivalent radius over time under

cases 119886 and 119887 is given by Figure 8 It can be seen from thecurve change in Figure 8 that during the whole process ofbubble rising the bubble equivalent radius almost fluctuatesin the same cycle (120591 = 02s) as the flow pulsation conditionIt appears that the periodic fluctuation of the equivalentradius also corresponds to the volatility of the bubble shapeand volume in one cycle In addition the findings indicatethat the pressure at the location of the bubble will decreaseas it gets closer to the upper free surface during the risingprocess Based on the ideal gas state equation the bubblevolume and equivalent radius tend to increase throughoutthe process Moreover since the bubble rises faster in caseb the overall trend of the bubble equivalent radius curvein case 119887 is higher than that in case a as shown inFigure 8

312 Single Bubble Rising Velocity The bubble rising velocityis another important characteristic that should be concernedto reflect the bubble motion state Under static conditionsetting the same dimensionless number and other conditionsas case b-d in Table 1 the change of the bubble centroidvelocitywith time is obtained as shown in Figure 9 It appearsthat the single bubble rising velocity gradually increasesto a certain stable value and then fluctuates in a smallrange for different dimensionless number of cases Howeverwhen the inlet flow rate changes periodically it will havea significant impact on bubble rising velocity According tothe simulation results of each case in Table 1 the influenceof five independent variables on the bubble rising velocityunder flow pulsation condition is found In general thebubble centroid velocity during the bubble rising processvaries sinusoidally in the same cycle as the inlet flow rateunder pulsation condition as shown in Figure 10 This isanother difference from the bubble rising process under staticcondition

The effects of each variable on bubble rising velocityunder flow pulsation are discussed below Based on the

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

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The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 3

No-Slip Wall

002m

004m

003

7m

008m

047

m

Free surface

U = U + ΔUＭＣn(2t)

(a) Single bubble

No-Slip Wall

00018m

0004m

000

2m001m

000

8m

Free surface

0002m

U = U + ΔUＭＣn(2t)

(b) Bubble pairs

Figure 1 Initial geometry and boundary arrangement under flow pulsation condition

The gradient and Laplacian of physical quantity 120601 onparticle 119894 are expressed by the summation of physical quan-tities over its neighboring particles 119895 according to the weightfunction

The gradient model and Laplacian model are as followsrespectively

⟨nabla120601⟩119894 = 119889119899119894sum119895 =119894[[

120601119895 minus 12060111989410038161003816100381610038161003816r119895 minus r119894100381610038161003816100381610038162

(r119895 minus r119894)119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)]]

(4)

⟨nabla2120601⟩119894= 2119889

120582119899119894 sum119895 =119894 [(120601119895 minus 120601119894)119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)] (5)

where 119889 represents spatial dimension and 120582 is diffusioncoefficient and can be determined as

120582119894 = sum119895 =119894 [119908 (10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816) 10038161003816100381610038161003816r119895 minus r119894

100381610038161003816100381610038162]sum119895 =119894 [119908 (r119895 minus r119894)] (6)

In the MPS-MAFL method the divergence model isshown in (7) The velocity divergence between two particlesis defined by (u119895 minus u119894)(r119895 minus r119894)|(r119895 minus r119894)|2 and the velocity

divergence at one particle is calculated from the weightedaverage of the individual velocity divergences

⟨nabla sdot u⟩119894 = 119889119899119894sum119895 =119894[[

(u119895 minus u119894) sdot (r119895 minus r119894)10038161003816100381610038161003816r119895 minus r119894100381610038161003816100381610038162

119908(10038161003816100381610038161003816r119895 minus r11989410038161003816100381610038161003816)]]

(7)

23 Computational Domain and Boundary Condition In thisstudy the calculation domain of the single bubble risingprocess under flow pulsation condition is a 008m wide and047m high two-dimensional region as shown in Figure 1(a)The initial radius of the bubble is 001m and the bubble islocated in the center of the channelThe distance between thecenter of the initial bubble and the inlet of the flow channel is0037m This geometric setting ensures that the influence ofthe wall can be neglected during bubble motion The vaporphase property is given based on ideal gas state equationand the liquid area is described by using discretized particleswith nonuniform scheme And the particle number densitydecreases as the distance from the phase interface increasesin the liquid area

As for the boundary condition setting the left and theright sides of the flow path are set as nonslip boundary con-ditions and the upper side of the flow path adopts a free

4 Science and Technology of Nuclear Installations

Table 1 Physical property parameters of the gas-liquid phase

Gasliquid phase ViscosityPasdots Densitykgsdotmminus3 Surface tensionNsdotmminus1Air (T = 8∘C) 175times10minus5 1250 Air (T = 29∘C) 185times10minus5 1170 Water (T=8∘C) 138times10minus3 9998 0074Water (T=29∘C) 086times10minus3 9967 0072

R = 001m

Bubbleterminal

shape(t = 03s) 18R 19R 110R 111R

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

01 02 0300time (s)

particle size = 18Rparticle size = 19R

particle size = 110Rparticle size = 111R

Figure 2 Particle size sensitivity analysis (Re=Bo=10)

surface boundary condition The bubble interface is set asfree surface and the pressure values at various points onthe interface are obtained by solving the ideal gas stateequation Moreover at the inlet boundary of the channel aflow pulsation condition is introduced The flow at the inletchanges in a sinusoidal cycle over time as shown in (8) Theequation contains three variables the initial flow rate of theliquid phase 119880 the relative amplitude of the flow pulsationΔ119880119880 and the flow pulsation period 120591

119880 = 119880 + Δ119880 sin (2120587119905120591 ) (8)

The calculation domain of bubble pairs coalescence underflow pulsation condition is set in a similar way as shownin Figure 1(b) The calculation domain is 001m wide and0008m high and the two bubbles are symmetrically dis-tributedThe boundary conditions and particle placement arethe same as single bubble conditions

In the process of bubble dynamics simulation using MPSmethod the particle size (the distance between neighboringparticles) could affect the simulation efficiency and conver-gence Specifically the optimal particle size could be figuredout by sensitivity analysis In this study the simulation ofsingle bubble rising in a static fluid was performed withdifferent particle size As shown in Figure 2 the interfacialparticle size was set to 18 19 110 and 111 of the bubbleradius respectively It can be seen that the bubble centroidvelocity calculation results have good convergence under

minus02

minus01

00

01

02

03

04

05

Bubb

le te

rmin

al v

eloci

ty (m

s)

JamialahmadiLiu T=8∘CLiu T=29∘C

MPS-MAFL T=8∘CMPS-MAFL T=29∘C

7 8 9 10 11 126Bubble diameter (mm)

Figure 3 Terminal rising velocity of the bubble under differentinitial bubble diameter conditions

these particle sizes At the same time the shape change duringthe bubble rising progress was obtained It is found that theterminal shapes are nearly identical in each case Thereforebased on the sensitivity analysis results the interfacial particlesize was set to 110 of the initial bubble radius to ensure theefficiency of the program simulation and the convergence ofthe calculation

24 Program Validation The single bubble rising in staticwater was simulated by MPS-MAFL and the simulationresults were compared with the experiment of Liu et al [26]The physical property parameters of the relevant cases arelisted in Table 1 In MPS-MAFL the calculation conditionsand parameters were set as same as the experiment andthe terminal rising velocity of the bubble under differentinitial bubble diameter conditions was obtained as shownin Figure 3 where scatter points respectively representexperimental data points for two temperatures

As shown in Figure 3 the calculated bubble terminalvelocity value is slightly lower than the experimental onewith the error about 15 One reason is that the lateralmovement of the bubbles existing in the experiment hadbeen simplified in the two-dimensional simulations of thisstudy Within the range of bubble diameter larger than 6mmthe bubble terminal rising velocity slightly increases with the

Science and Technology of Nuclear Installations 5

Reyn

olds

num

ber (

Re)

Eotvos number (Eo)

Log M10

5

104

103

102

101

100

10minus1

103

102

101

100

10minus1

10minus2

-14-13-12-11-10

-9

-8-7-6-5-4-3-2-101

2 3 4 5 6

Figure4Gracersquos graphical correlation for predicting terminal shapeof bubble

bubble diameter which is consistent with the experimentalconclusion of Liu et al [26] Thus the overall trend ofthe simulation results is in reasonable agreement with theexperiment

In the process of single bubble rising in a static liquidthe stable shape of the bubble is another important charac-teristic that should be concerned On the basis of previousexperimental results Grace [27 28] proposed a graphicalcorrelation that predicts the final stable shape of the risingbubble in an infinite static liquid It can be seen from thisdiagram that the terminal shape of the bubble can be roughlydivided into four regions namely Sphere region Ellipsoidregion Dimples region and Spherical-cap region In order toverify the accuracy of the simulation the following four casesare respectively set Re = 8854 Eo = 0392 Re = 8854 Eo =392 Re = 10 Eo = 100 and Re = 1000 Eo = 300 Moreoverthe stable shape in the bubble rising process in each case isobtained and the stable shape of the bubble is marked at thecorresponding position according to the Reynolds numberand the Eotvos number (also called Bond number) as shownin Figure 4 In view of these facts it is completely consistentwith the results predicted by Grace

This study aims to discuss the bubble rising and coa-lescence characteristics under inlet flow pulsation Therebyit is necessary to verify the accuracy of the added inletflow pulsation condition before the simulation First a flowchannel only with liquid is set and its geometric parametersare the same as those for studying single bubble rising underflow pulsation condition as shown in Figure 5 Then a flowrate that changes periodically with timewas added to the inletof the flow path The initial flow rate of the liquid phase theflow pulsation period and the relative amplitude of the flowpulsation were 005ms 02s and 07 respectively Figure 5shows the velocity values of the particles at the inlet and outlet

U = 005 + 0035Sin(314lowasttime)the inlet particlethe outlet particle

005 010 015 020000time (s)

000

002

004

006

008

010

Part

icle

velo

city

(ms

)

Figure 5 The velocity of inlet and outlet particles under flowpulsation condition

of the flow channel when the flow stability is reached Asshown in Figure 5 the velocity values of the outlet particlesare the same as those of the inlet particles at any of the samemoments and they all satisfy the analytical formula of theadded flow pulsation condition

3 Results and Discussions

The Reynolds number (Re) and the Bond number (Bo)are commonly adopted to study the bubble dynamics Theexpressions are respectively as follows

Re = 120588119887radic1198921198633120583119887

(9)

119861119900 = 1205881198871198921198632120590 (10)

where 120588119887 is the density of liquid 120583119887 is liquid viscosity and119863 is the bubble diameter 120590 represents the surface tensionThrough these two dimensionless expressions the Reynoldsnumber represents the ratio of the inertial force to the viscousforce of the bubble and the Bond number represents the ratioof the gravity of the bubble to its surface tension During themovement of the bubble these four forces have a significanteffect on the bubble Besides the density ratio and viscosityratio of the gas-liquid two phases also have an impact In thisstudy the density ratio and viscosity ratio of liquid phase togas phase are set as 10001 and 1001 in all bubble simulationsrespectively

31 Single Bubble Rising under Flow Pulsation Condition Inthis study the influence of the pulsation condition on therising motion characteristics of the bubble is investigatedDue to the addition of the pulsating flow in the form of(8) three independent variables are added Together with

6 Science and Technology of Nuclear Installations

Table 2 Cases simulated in Section 31

Case ParameterRe Bo 119880(ms) Δ119880119880 120591(s)

a 5 10 005 07 02b 10 10 005 07 02c 20 10 005 07 02d 50 10 005 07 02e 5 30 005 07 02f 5 50 005 07 02119892 5 10 0075 07 02h 5 10 01 07 02i 5 10 005 07 01j 5 10 005 07 025k 5 10 01 01 02l 5 10 01 03 02m 5 10 01 05 02n 5 10 01 07 02o 5 10 01 1 02

the two dimensionless numbers mentioned above thereare actually five independent variables that may affect themotion behavior of the bubble under the inlet flow pulsationcondition Therefore the control variable method can beused to discuss the bubble deformation process and thesurrounding flow field changes under the flow pulsationcondition The specific analysis is as follows

311 Single Bubble Deformation Process The cases of study-ing the single bubble rising process under flow pulsationcondition are listed in Table 2 Initially a case under staticcondition is taken as a reference example while its dimen-sionless number and other conditions were as same as case119886 in Table 2 And the bubble rising process under staticcondition was shown in Figure 6(a) with red-dashed line Tosum up in this case the bubble will only gradually stabilizeduring the ascent and the terminal shape is consistent withthe prediction of Grace [27]

By simulating the various cases set in Table 2 it is possibleto observe the change of the shape and the surrounding flowfield during the bubble rising in the liquid under the periodicvariation of the inlet flow rate Figure 6 shows the changeprocess of the rising bubble under cases 119886 and 119887 in two cycles(t = 04s) and the flow field velocity vector information andpressure distribution in the red frame (around the bubble) aregiven by Figure 7 The bubble has a wake during the ascentand there are two symmetric vortices around the bubble Itcan also be observed in Figure 6 that there is a significantshape and volume fluctuation in each cycle of the bubblerising process due to the introduction of the inlet periodicchange flow rate This is one of the differences between thebubble deformation processes under the inlet flow pulsationcondition and the static condition

At the same time in order to numerically verify thefluctuation behavior of the bubble under pulsation conditionthe bubble equivalent diameter at each moment is calculatedThe trend of the bubble equivalent radius over time under

cases 119886 and 119887 is given by Figure 8 It can be seen from thecurve change in Figure 8 that during the whole process ofbubble rising the bubble equivalent radius almost fluctuatesin the same cycle (120591 = 02s) as the flow pulsation conditionIt appears that the periodic fluctuation of the equivalentradius also corresponds to the volatility of the bubble shapeand volume in one cycle In addition the findings indicatethat the pressure at the location of the bubble will decreaseas it gets closer to the upper free surface during the risingprocess Based on the ideal gas state equation the bubblevolume and equivalent radius tend to increase throughoutthe process Moreover since the bubble rises faster in caseb the overall trend of the bubble equivalent radius curvein case 119887 is higher than that in case a as shown inFigure 8

312 Single Bubble Rising Velocity The bubble rising velocityis another important characteristic that should be concernedto reflect the bubble motion state Under static conditionsetting the same dimensionless number and other conditionsas case b-d in Table 1 the change of the bubble centroidvelocitywith time is obtained as shown in Figure 9 It appearsthat the single bubble rising velocity gradually increasesto a certain stable value and then fluctuates in a smallrange for different dimensionless number of cases Howeverwhen the inlet flow rate changes periodically it will havea significant impact on bubble rising velocity According tothe simulation results of each case in Table 1 the influenceof five independent variables on the bubble rising velocityunder flow pulsation condition is found In general thebubble centroid velocity during the bubble rising processvaries sinusoidally in the same cycle as the inlet flow rateunder pulsation condition as shown in Figure 10 This isanother difference from the bubble rising process under staticcondition

The effects of each variable on bubble rising velocityunder flow pulsation are discussed below Based on the

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

4 Science and Technology of Nuclear Installations

Table 1 Physical property parameters of the gas-liquid phase

Gasliquid phase ViscosityPasdots Densitykgsdotmminus3 Surface tensionNsdotmminus1Air (T = 8∘C) 175times10minus5 1250 Air (T = 29∘C) 185times10minus5 1170 Water (T=8∘C) 138times10minus3 9998 0074Water (T=29∘C) 086times10minus3 9967 0072

R = 001m

Bubbleterminal

shape(t = 03s) 18R 19R 110R 111R

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

01 02 0300time (s)

particle size = 18Rparticle size = 19R

particle size = 110Rparticle size = 111R

Figure 2 Particle size sensitivity analysis (Re=Bo=10)

surface boundary condition The bubble interface is set asfree surface and the pressure values at various points onthe interface are obtained by solving the ideal gas stateequation Moreover at the inlet boundary of the channel aflow pulsation condition is introduced The flow at the inletchanges in a sinusoidal cycle over time as shown in (8) Theequation contains three variables the initial flow rate of theliquid phase 119880 the relative amplitude of the flow pulsationΔ119880119880 and the flow pulsation period 120591

119880 = 119880 + Δ119880 sin (2120587119905120591 ) (8)

The calculation domain of bubble pairs coalescence underflow pulsation condition is set in a similar way as shownin Figure 1(b) The calculation domain is 001m wide and0008m high and the two bubbles are symmetrically dis-tributedThe boundary conditions and particle placement arethe same as single bubble conditions

In the process of bubble dynamics simulation using MPSmethod the particle size (the distance between neighboringparticles) could affect the simulation efficiency and conver-gence Specifically the optimal particle size could be figuredout by sensitivity analysis In this study the simulation ofsingle bubble rising in a static fluid was performed withdifferent particle size As shown in Figure 2 the interfacialparticle size was set to 18 19 110 and 111 of the bubbleradius respectively It can be seen that the bubble centroidvelocity calculation results have good convergence under

minus02

minus01

00

01

02

03

04

05

Bubb

le te

rmin

al v

eloci

ty (m

s)

JamialahmadiLiu T=8∘CLiu T=29∘C

MPS-MAFL T=8∘CMPS-MAFL T=29∘C

7 8 9 10 11 126Bubble diameter (mm)

Figure 3 Terminal rising velocity of the bubble under differentinitial bubble diameter conditions

these particle sizes At the same time the shape change duringthe bubble rising progress was obtained It is found that theterminal shapes are nearly identical in each case Thereforebased on the sensitivity analysis results the interfacial particlesize was set to 110 of the initial bubble radius to ensure theefficiency of the program simulation and the convergence ofthe calculation

24 Program Validation The single bubble rising in staticwater was simulated by MPS-MAFL and the simulationresults were compared with the experiment of Liu et al [26]The physical property parameters of the relevant cases arelisted in Table 1 In MPS-MAFL the calculation conditionsand parameters were set as same as the experiment andthe terminal rising velocity of the bubble under differentinitial bubble diameter conditions was obtained as shownin Figure 3 where scatter points respectively representexperimental data points for two temperatures

As shown in Figure 3 the calculated bubble terminalvelocity value is slightly lower than the experimental onewith the error about 15 One reason is that the lateralmovement of the bubbles existing in the experiment hadbeen simplified in the two-dimensional simulations of thisstudy Within the range of bubble diameter larger than 6mmthe bubble terminal rising velocity slightly increases with the

Science and Technology of Nuclear Installations 5

Reyn

olds

num

ber (

Re)

Eotvos number (Eo)

Log M10

5

104

103

102

101

100

10minus1

103

102

101

100

10minus1

10minus2

-14-13-12-11-10

-9

-8-7-6-5-4-3-2-101

2 3 4 5 6

Figure4Gracersquos graphical correlation for predicting terminal shapeof bubble

bubble diameter which is consistent with the experimentalconclusion of Liu et al [26] Thus the overall trend ofthe simulation results is in reasonable agreement with theexperiment

In the process of single bubble rising in a static liquidthe stable shape of the bubble is another important charac-teristic that should be concerned On the basis of previousexperimental results Grace [27 28] proposed a graphicalcorrelation that predicts the final stable shape of the risingbubble in an infinite static liquid It can be seen from thisdiagram that the terminal shape of the bubble can be roughlydivided into four regions namely Sphere region Ellipsoidregion Dimples region and Spherical-cap region In order toverify the accuracy of the simulation the following four casesare respectively set Re = 8854 Eo = 0392 Re = 8854 Eo =392 Re = 10 Eo = 100 and Re = 1000 Eo = 300 Moreoverthe stable shape in the bubble rising process in each case isobtained and the stable shape of the bubble is marked at thecorresponding position according to the Reynolds numberand the Eotvos number (also called Bond number) as shownin Figure 4 In view of these facts it is completely consistentwith the results predicted by Grace

This study aims to discuss the bubble rising and coa-lescence characteristics under inlet flow pulsation Therebyit is necessary to verify the accuracy of the added inletflow pulsation condition before the simulation First a flowchannel only with liquid is set and its geometric parametersare the same as those for studying single bubble rising underflow pulsation condition as shown in Figure 5 Then a flowrate that changes periodically with timewas added to the inletof the flow path The initial flow rate of the liquid phase theflow pulsation period and the relative amplitude of the flowpulsation were 005ms 02s and 07 respectively Figure 5shows the velocity values of the particles at the inlet and outlet

U = 005 + 0035Sin(314lowasttime)the inlet particlethe outlet particle

005 010 015 020000time (s)

000

002

004

006

008

010

Part

icle

velo

city

(ms

)

Figure 5 The velocity of inlet and outlet particles under flowpulsation condition

of the flow channel when the flow stability is reached Asshown in Figure 5 the velocity values of the outlet particlesare the same as those of the inlet particles at any of the samemoments and they all satisfy the analytical formula of theadded flow pulsation condition

3 Results and Discussions

The Reynolds number (Re) and the Bond number (Bo)are commonly adopted to study the bubble dynamics Theexpressions are respectively as follows

Re = 120588119887radic1198921198633120583119887

(9)

119861119900 = 1205881198871198921198632120590 (10)

where 120588119887 is the density of liquid 120583119887 is liquid viscosity and119863 is the bubble diameter 120590 represents the surface tensionThrough these two dimensionless expressions the Reynoldsnumber represents the ratio of the inertial force to the viscousforce of the bubble and the Bond number represents the ratioof the gravity of the bubble to its surface tension During themovement of the bubble these four forces have a significanteffect on the bubble Besides the density ratio and viscosityratio of the gas-liquid two phases also have an impact In thisstudy the density ratio and viscosity ratio of liquid phase togas phase are set as 10001 and 1001 in all bubble simulationsrespectively

31 Single Bubble Rising under Flow Pulsation Condition Inthis study the influence of the pulsation condition on therising motion characteristics of the bubble is investigatedDue to the addition of the pulsating flow in the form of(8) three independent variables are added Together with

6 Science and Technology of Nuclear Installations

Table 2 Cases simulated in Section 31

Case ParameterRe Bo 119880(ms) Δ119880119880 120591(s)

a 5 10 005 07 02b 10 10 005 07 02c 20 10 005 07 02d 50 10 005 07 02e 5 30 005 07 02f 5 50 005 07 02119892 5 10 0075 07 02h 5 10 01 07 02i 5 10 005 07 01j 5 10 005 07 025k 5 10 01 01 02l 5 10 01 03 02m 5 10 01 05 02n 5 10 01 07 02o 5 10 01 1 02

the two dimensionless numbers mentioned above thereare actually five independent variables that may affect themotion behavior of the bubble under the inlet flow pulsationcondition Therefore the control variable method can beused to discuss the bubble deformation process and thesurrounding flow field changes under the flow pulsationcondition The specific analysis is as follows

311 Single Bubble Deformation Process The cases of study-ing the single bubble rising process under flow pulsationcondition are listed in Table 2 Initially a case under staticcondition is taken as a reference example while its dimen-sionless number and other conditions were as same as case119886 in Table 2 And the bubble rising process under staticcondition was shown in Figure 6(a) with red-dashed line Tosum up in this case the bubble will only gradually stabilizeduring the ascent and the terminal shape is consistent withthe prediction of Grace [27]

By simulating the various cases set in Table 2 it is possibleto observe the change of the shape and the surrounding flowfield during the bubble rising in the liquid under the periodicvariation of the inlet flow rate Figure 6 shows the changeprocess of the rising bubble under cases 119886 and 119887 in two cycles(t = 04s) and the flow field velocity vector information andpressure distribution in the red frame (around the bubble) aregiven by Figure 7 The bubble has a wake during the ascentand there are two symmetric vortices around the bubble Itcan also be observed in Figure 6 that there is a significantshape and volume fluctuation in each cycle of the bubblerising process due to the introduction of the inlet periodicchange flow rate This is one of the differences between thebubble deformation processes under the inlet flow pulsationcondition and the static condition

At the same time in order to numerically verify thefluctuation behavior of the bubble under pulsation conditionthe bubble equivalent diameter at each moment is calculatedThe trend of the bubble equivalent radius over time under

cases 119886 and 119887 is given by Figure 8 It can be seen from thecurve change in Figure 8 that during the whole process ofbubble rising the bubble equivalent radius almost fluctuatesin the same cycle (120591 = 02s) as the flow pulsation conditionIt appears that the periodic fluctuation of the equivalentradius also corresponds to the volatility of the bubble shapeand volume in one cycle In addition the findings indicatethat the pressure at the location of the bubble will decreaseas it gets closer to the upper free surface during the risingprocess Based on the ideal gas state equation the bubblevolume and equivalent radius tend to increase throughoutthe process Moreover since the bubble rises faster in caseb the overall trend of the bubble equivalent radius curvein case 119887 is higher than that in case a as shown inFigure 8

312 Single Bubble Rising Velocity The bubble rising velocityis another important characteristic that should be concernedto reflect the bubble motion state Under static conditionsetting the same dimensionless number and other conditionsas case b-d in Table 1 the change of the bubble centroidvelocitywith time is obtained as shown in Figure 9 It appearsthat the single bubble rising velocity gradually increasesto a certain stable value and then fluctuates in a smallrange for different dimensionless number of cases Howeverwhen the inlet flow rate changes periodically it will havea significant impact on bubble rising velocity According tothe simulation results of each case in Table 1 the influenceof five independent variables on the bubble rising velocityunder flow pulsation condition is found In general thebubble centroid velocity during the bubble rising processvaries sinusoidally in the same cycle as the inlet flow rateunder pulsation condition as shown in Figure 10 This isanother difference from the bubble rising process under staticcondition

The effects of each variable on bubble rising velocityunder flow pulsation are discussed below Based on the

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

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The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 5

Reyn

olds

num

ber (

Re)

Eotvos number (Eo)

Log M10

5

104

103

102

101

100

10minus1

103

102

101

100

10minus1

10minus2

-14-13-12-11-10

-9

-8-7-6-5-4-3-2-101

2 3 4 5 6

Figure4Gracersquos graphical correlation for predicting terminal shapeof bubble

bubble diameter which is consistent with the experimentalconclusion of Liu et al [26] Thus the overall trend ofthe simulation results is in reasonable agreement with theexperiment

In the process of single bubble rising in a static liquidthe stable shape of the bubble is another important charac-teristic that should be concerned On the basis of previousexperimental results Grace [27 28] proposed a graphicalcorrelation that predicts the final stable shape of the risingbubble in an infinite static liquid It can be seen from thisdiagram that the terminal shape of the bubble can be roughlydivided into four regions namely Sphere region Ellipsoidregion Dimples region and Spherical-cap region In order toverify the accuracy of the simulation the following four casesare respectively set Re = 8854 Eo = 0392 Re = 8854 Eo =392 Re = 10 Eo = 100 and Re = 1000 Eo = 300 Moreoverthe stable shape in the bubble rising process in each case isobtained and the stable shape of the bubble is marked at thecorresponding position according to the Reynolds numberand the Eotvos number (also called Bond number) as shownin Figure 4 In view of these facts it is completely consistentwith the results predicted by Grace

This study aims to discuss the bubble rising and coa-lescence characteristics under inlet flow pulsation Therebyit is necessary to verify the accuracy of the added inletflow pulsation condition before the simulation First a flowchannel only with liquid is set and its geometric parametersare the same as those for studying single bubble rising underflow pulsation condition as shown in Figure 5 Then a flowrate that changes periodically with timewas added to the inletof the flow path The initial flow rate of the liquid phase theflow pulsation period and the relative amplitude of the flowpulsation were 005ms 02s and 07 respectively Figure 5shows the velocity values of the particles at the inlet and outlet

U = 005 + 0035Sin(314lowasttime)the inlet particlethe outlet particle

005 010 015 020000time (s)

000

002

004

006

008

010

Part

icle

velo

city

(ms

)

Figure 5 The velocity of inlet and outlet particles under flowpulsation condition

of the flow channel when the flow stability is reached Asshown in Figure 5 the velocity values of the outlet particlesare the same as those of the inlet particles at any of the samemoments and they all satisfy the analytical formula of theadded flow pulsation condition

3 Results and Discussions

The Reynolds number (Re) and the Bond number (Bo)are commonly adopted to study the bubble dynamics Theexpressions are respectively as follows

Re = 120588119887radic1198921198633120583119887

(9)

119861119900 = 1205881198871198921198632120590 (10)

where 120588119887 is the density of liquid 120583119887 is liquid viscosity and119863 is the bubble diameter 120590 represents the surface tensionThrough these two dimensionless expressions the Reynoldsnumber represents the ratio of the inertial force to the viscousforce of the bubble and the Bond number represents the ratioof the gravity of the bubble to its surface tension During themovement of the bubble these four forces have a significanteffect on the bubble Besides the density ratio and viscosityratio of the gas-liquid two phases also have an impact In thisstudy the density ratio and viscosity ratio of liquid phase togas phase are set as 10001 and 1001 in all bubble simulationsrespectively

31 Single Bubble Rising under Flow Pulsation Condition Inthis study the influence of the pulsation condition on therising motion characteristics of the bubble is investigatedDue to the addition of the pulsating flow in the form of(8) three independent variables are added Together with

6 Science and Technology of Nuclear Installations

Table 2 Cases simulated in Section 31

Case ParameterRe Bo 119880(ms) Δ119880119880 120591(s)

a 5 10 005 07 02b 10 10 005 07 02c 20 10 005 07 02d 50 10 005 07 02e 5 30 005 07 02f 5 50 005 07 02119892 5 10 0075 07 02h 5 10 01 07 02i 5 10 005 07 01j 5 10 005 07 025k 5 10 01 01 02l 5 10 01 03 02m 5 10 01 05 02n 5 10 01 07 02o 5 10 01 1 02

the two dimensionless numbers mentioned above thereare actually five independent variables that may affect themotion behavior of the bubble under the inlet flow pulsationcondition Therefore the control variable method can beused to discuss the bubble deformation process and thesurrounding flow field changes under the flow pulsationcondition The specific analysis is as follows

311 Single Bubble Deformation Process The cases of study-ing the single bubble rising process under flow pulsationcondition are listed in Table 2 Initially a case under staticcondition is taken as a reference example while its dimen-sionless number and other conditions were as same as case119886 in Table 2 And the bubble rising process under staticcondition was shown in Figure 6(a) with red-dashed line Tosum up in this case the bubble will only gradually stabilizeduring the ascent and the terminal shape is consistent withthe prediction of Grace [27]

By simulating the various cases set in Table 2 it is possibleto observe the change of the shape and the surrounding flowfield during the bubble rising in the liquid under the periodicvariation of the inlet flow rate Figure 6 shows the changeprocess of the rising bubble under cases 119886 and 119887 in two cycles(t = 04s) and the flow field velocity vector information andpressure distribution in the red frame (around the bubble) aregiven by Figure 7 The bubble has a wake during the ascentand there are two symmetric vortices around the bubble Itcan also be observed in Figure 6 that there is a significantshape and volume fluctuation in each cycle of the bubblerising process due to the introduction of the inlet periodicchange flow rate This is one of the differences between thebubble deformation processes under the inlet flow pulsationcondition and the static condition

At the same time in order to numerically verify thefluctuation behavior of the bubble under pulsation conditionthe bubble equivalent diameter at each moment is calculatedThe trend of the bubble equivalent radius over time under

cases 119886 and 119887 is given by Figure 8 It can be seen from thecurve change in Figure 8 that during the whole process ofbubble rising the bubble equivalent radius almost fluctuatesin the same cycle (120591 = 02s) as the flow pulsation conditionIt appears that the periodic fluctuation of the equivalentradius also corresponds to the volatility of the bubble shapeand volume in one cycle In addition the findings indicatethat the pressure at the location of the bubble will decreaseas it gets closer to the upper free surface during the risingprocess Based on the ideal gas state equation the bubblevolume and equivalent radius tend to increase throughoutthe process Moreover since the bubble rises faster in caseb the overall trend of the bubble equivalent radius curvein case 119887 is higher than that in case a as shown inFigure 8

312 Single Bubble Rising Velocity The bubble rising velocityis another important characteristic that should be concernedto reflect the bubble motion state Under static conditionsetting the same dimensionless number and other conditionsas case b-d in Table 1 the change of the bubble centroidvelocitywith time is obtained as shown in Figure 9 It appearsthat the single bubble rising velocity gradually increasesto a certain stable value and then fluctuates in a smallrange for different dimensionless number of cases Howeverwhen the inlet flow rate changes periodically it will havea significant impact on bubble rising velocity According tothe simulation results of each case in Table 1 the influenceof five independent variables on the bubble rising velocityunder flow pulsation condition is found In general thebubble centroid velocity during the bubble rising processvaries sinusoidally in the same cycle as the inlet flow rateunder pulsation condition as shown in Figure 10 This isanother difference from the bubble rising process under staticcondition

The effects of each variable on bubble rising velocityunder flow pulsation are discussed below Based on the

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

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Shock and Vibration

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Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

6 Science and Technology of Nuclear Installations

Table 2 Cases simulated in Section 31

Case ParameterRe Bo 119880(ms) Δ119880119880 120591(s)

a 5 10 005 07 02b 10 10 005 07 02c 20 10 005 07 02d 50 10 005 07 02e 5 30 005 07 02f 5 50 005 07 02119892 5 10 0075 07 02h 5 10 01 07 02i 5 10 005 07 01j 5 10 005 07 025k 5 10 01 01 02l 5 10 01 03 02m 5 10 01 05 02n 5 10 01 07 02o 5 10 01 1 02

the two dimensionless numbers mentioned above thereare actually five independent variables that may affect themotion behavior of the bubble under the inlet flow pulsationcondition Therefore the control variable method can beused to discuss the bubble deformation process and thesurrounding flow field changes under the flow pulsationcondition The specific analysis is as follows

311 Single Bubble Deformation Process The cases of study-ing the single bubble rising process under flow pulsationcondition are listed in Table 2 Initially a case under staticcondition is taken as a reference example while its dimen-sionless number and other conditions were as same as case119886 in Table 2 And the bubble rising process under staticcondition was shown in Figure 6(a) with red-dashed line Tosum up in this case the bubble will only gradually stabilizeduring the ascent and the terminal shape is consistent withthe prediction of Grace [27]

By simulating the various cases set in Table 2 it is possibleto observe the change of the shape and the surrounding flowfield during the bubble rising in the liquid under the periodicvariation of the inlet flow rate Figure 6 shows the changeprocess of the rising bubble under cases 119886 and 119887 in two cycles(t = 04s) and the flow field velocity vector information andpressure distribution in the red frame (around the bubble) aregiven by Figure 7 The bubble has a wake during the ascentand there are two symmetric vortices around the bubble Itcan also be observed in Figure 6 that there is a significantshape and volume fluctuation in each cycle of the bubblerising process due to the introduction of the inlet periodicchange flow rate This is one of the differences between thebubble deformation processes under the inlet flow pulsationcondition and the static condition

At the same time in order to numerically verify thefluctuation behavior of the bubble under pulsation conditionthe bubble equivalent diameter at each moment is calculatedThe trend of the bubble equivalent radius over time under

cases 119886 and 119887 is given by Figure 8 It can be seen from thecurve change in Figure 8 that during the whole process ofbubble rising the bubble equivalent radius almost fluctuatesin the same cycle (120591 = 02s) as the flow pulsation conditionIt appears that the periodic fluctuation of the equivalentradius also corresponds to the volatility of the bubble shapeand volume in one cycle In addition the findings indicatethat the pressure at the location of the bubble will decreaseas it gets closer to the upper free surface during the risingprocess Based on the ideal gas state equation the bubblevolume and equivalent radius tend to increase throughoutthe process Moreover since the bubble rises faster in caseb the overall trend of the bubble equivalent radius curvein case 119887 is higher than that in case a as shown inFigure 8

312 Single Bubble Rising Velocity The bubble rising velocityis another important characteristic that should be concernedto reflect the bubble motion state Under static conditionsetting the same dimensionless number and other conditionsas case b-d in Table 1 the change of the bubble centroidvelocitywith time is obtained as shown in Figure 9 It appearsthat the single bubble rising velocity gradually increasesto a certain stable value and then fluctuates in a smallrange for different dimensionless number of cases Howeverwhen the inlet flow rate changes periodically it will havea significant impact on bubble rising velocity According tothe simulation results of each case in Table 1 the influenceof five independent variables on the bubble rising velocityunder flow pulsation condition is found In general thebubble centroid velocity during the bubble rising processvaries sinusoidally in the same cycle as the inlet flow rateunder pulsation condition as shown in Figure 10 This isanother difference from the bubble rising process under staticcondition

The effects of each variable on bubble rising velocityunder flow pulsation are discussed below Based on the

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 7

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(a) Re = 5 Bo = 10119880= 005ms Δ119880119880= 07 120591 = 02sRe = 5 and Bo = 10 (static condition red-dashed line)

t = 02Ｍ t = 03Ｍ t = 04Ｍ t = 05Ｍ t = 06Ｍ

(b) Re = 10 Bo = 10119880= 005ms Δ119880119880= 07 and 120591 = 02s

Figure 6 Shape change of single bubble rising process under pulsation condition

Static condition t = 05Ｍ

3500

3000

4000

3500

3000 2500

Pres

sure

(Pa)

Pres

sure

(Pa)

Pulsation condition(case a) t = 05Ｍ

Figure 7 Speed vector of the area around bubble under static and pulsation condition

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

8 Science and Technology of Nuclear Installations

case acase b

0005

0010

0015

0020

Bubb

le eq

uiva

lent

radi

us (m

)

02 04 06 08 1000time (s)

Figure 8 Equivalent radius of bubble under flow pulsation condi-tion

Re=10 Bo=10Re=20 Bo=10Re=50 Bo=10

01 02 0300time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Figure 9 The change of bubble centroid rising velocity with timeunder static condition

Reynolds number expression of (9) the Reynolds num-ber increases as the viscosity of the corresponding liquiddecreases while other conditions are equal When the viscos-ity of the corresponding liquid becomes smaller the bubblerises faster in the liquid Therefore the results revealed thatthere is a positive correlation between the Reynolds numberand the pulsation amplitude of the bubble velocity underflow pulsation condition as shown in Figure 10(a) Anotherdimensionless Bond number characterizes the ratio of gravityto surface tension As can be seen fromFigure 10(b) the Bondnumber has slight effect on the bubble centroid velocity

Figure 10(c) shows that the larger the initial flow velocityof the liquid phase is the greater amplitude of periodic changein bubble centroid velocity becomes Based on the result of

Figure 10(d) it reveals that the fluctuation amplitude of thebubble centroid velocity becomes greater as the amplitudeof the inlet flow pulsation increases Figure 10(e) illustratesmore information about the flow pulsation period whenthe flow pulsation period is different the bubble risingvelocity fluctuates as almost the same pulsation period asthe respective inlet flow rate The velocity variation curvesrepresenting the respective pulsation periods are interlacedin the figures

313 Single Bubble Aspect Ratio In addition to calculatingthe bubble equivalent radius at each moment as aforemen-tioned to numerically verify the fluctuating behavior of thebubble under flow pulsation condition the variation of thebubble aspect ratio (the ratio of width to height of the bubble)with time under different variables is acquired as shown inFigure 11 From all the results the aspect ratio of the bubblealso changes periodically under flow pulsation condition

Figure 11(a) shows that the baseline of periodic fluctua-tion of the bubble aspect ratio increases with the Reynoldsnumber when other conditions are the same This is becausethe degree of lateral elongation of the bubble increases withthe Reynolds number Actually this conclusion could also bederived from the study of Grace [27] and Hua [29] When itcomes to Bond numbers it has slight effect on bubble aspectratio as in Figure 11(b) Figure 11(c) shows the influence ofthe initial flow rate of the liquid phase on the aspect ratioof the bubble Specifically the larger the initial flow rate ofthe liquid phase is the greater periodic change effects on thebubble becomesThis will give rise to amore dramatic changeto the bubble aspect ratio The effect of the flow pulsationamplitude on the aspect ratio of the bubble is similar to theinitial flow rate of the liquid phase as shown in Figure 11(d)Similar to the aforementioned results of the flow pulsationperiod affecting the bubble rising velocity Figure 11(e) showsthat the bubble aspect ratio fluctuates as almost the samepulsation period as the respective inlet flow rate

32 Bubble Pairs Coalescence under Flow Pulsation ConditionIn this study the coalescence process of bubble pairs underthe condition of inlet flow pulsation is also simulated Thegeometric and boundary conditions are shown in Figure 1(b)Other related parameter values are set as Re = 400 Bo =04 119880= 003ms Δ119880119880= 1 and 120591 = 5ms Firstly the bubblecoalescence process under flow pulsation condition is showninFigure 12 Furthermore in order to compare the differencesof bubble coalescence process under flow pulsation conditionand static condition a bubble pair coalescence case understatic condition is set in which the geometric conditionsand the ReBo numbers are the same as those of pulsationcondition This bubble coalescence process under staticcondition is depicted in Figure 12 with red-dashed line

As shown in Figure 12 numerical simulations in bothstatic and flow pulsations cases illustrate the appearance ofgas bridges between the bubbles during coalescence Thelarge surface tension of the gas bridge drives the coalescencebubble to expand rapidly in the vertical direction and toshrink in the horizontal directionThen the bubblewill shrinklongitudinally and the lower surface will accelerate to the

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 9

Re=5Re=10

Re=20Re=50

02 04 06 08 1000time (s)

00

01

02

03

04Bu

bble

cent

roid

velo

city

(ms

)

(a) The effect of Re on the velocity (case a-d)

Bo=10Bo=30

Bo=50

02 04 06 08 1000time (s)

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(b) The effect of Bo on the velocity (case a e f )

U = 01ＧＭ

U = 0075ＧＭ

U = 005ＧＭ

02 04 06 08 1000time (s)

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

(c) The effect of 119880 on the velocity (case a 119892 h)

ΔUU = 1

ΔUU = 07

ΔUU = 05

ΔUU = 03

ΔUU = 01

02 04 06 08 1000time (s)

minus01

00

01

02

03

04

05

06Bu

bble

cent

roid

velo

city

(ms

)

(d) The effect of Δ119880119880 on the velocity (case k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

minus01

00

01

02

03

04

05

Bubb

le ce

ntro

id v

eloci

ty (m

s)

02 03 04 05 06 07 08 09 1001time (s)

(e) The effect of 120591 on the velocity (case a i j)

Figure 10 The change of bubble centroid rising velocity with time under the influence of each independent variable

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

10 Science and Technology of Nuclear Installations

Re=5Re=10

Re=20Re=50

08091011121314151617181920

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(a) The effect of Re on the aspect ratio (cases a-d)

Bo=10Bo=30

Bo=50

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

02 04 06 08 1000time (s)

(b) The effect of Bo on the aspect ratio (cases a e and f )

U = 0075ＧＭ

U = 005ＧＭ U = 01ＧＭ

02 04 06 08 1000time (s)

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

(c) The effect of 119880 on the aspect ratio (cases a 119892 and h)

ΔUU = 01

ΔUU = 03

ΔUU = 05

ΔUU = 07

ΔUU = 1

07

08

09

10

11

12

13Bu

bble

aspe

ct ra

tio

02 04 06 08 1000time (s)

(d) The effect of Δ119880119880 on the aspect ratio (cases k-o)

= 01Ｍ = 02Ｍ

= 025Ｍ

07

08

09

10

11

12

13

Bubb

le as

pect

ratio

01 02 03 04 05 06 07 08 09 1000time (s)

(e) The effect of 120591 on the aspect ratio (cases a i and j)

Figure 11 The change of bubble aspect ratio with time under the influence of each independent variable

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 11

t = 0ＧＭ t = 250ＧＭ t = 534ＧＭ t = 724ＧＭ t = 793ＧＭ

t = 996ＧＭ t = 1253ＧＭ t = 1774ＧＭ t = 2213ＧＭ t = 2930ＧＭ

Figure 12 Shape change of bubble pairs coalescence process under pulsationstatic condition (red-dashed line)

upper surface However the bubble change progress is notcompletely consistent under the two conditions Under staticcondition the bubble will gradually reach a steady stateand continue to rise As to pulsation condition due to theexistence of a strong pressure field caused by the inlet flowpulsation condition the volume of both two bubbles shrinksduring the initial rising phase as shown in Figure 12 (t =250ms 534ms)Then the twobubbles get closer and coalescetogether This makes the bubble coalescence slightly delayedSimilar to the static condition the shape of the bubble willalso undergo a series of changes due to the surface tensionafter coalescence However under the influence of the inletpulsation condition cyclical fluctuation in shape and volumewill still occur during the final stage of bubble rising asshown in Figure 12 (t = 1774ms sim 2930ms) which is anotherdifference in the bubble coalescence processes under the twoconditions

Moreover the bubble centroid velocity during bubblecoalescence under two conditions is obtained as shown inFigure 13 It can be seen that the two bubbles have a lowvelocity before coalescence under the static condition butin the initial rising phase of the pulsation condition thebubble centroid velocity fluctuates periodically After the twobubbles coalesce and experience a short-term fluctuationthe bubble velocity is finally stable under static condition(after 0023ms) However the bubble centroid velocity stillfluctuates under the flow pulsation condition At the sametime the effect of the flow pulsation condition about thebubble coalescence timementioned above can also be verifiedfrom Figure 13

4 Conclusions

In the present study the single bubble rising and coales-cence of bubble pairs under flow pulsation condition are

minus02

minus01

00

01

02

03

Bubb

le ce

ntro

id v

eloci

ty (m

s)

Left bubble(Static condition)Right bubble(Static condition)Merger bubble(Static condition)Left bubble(Pulsation condition)Right bubble(Pulsation condition)Merger bubble(Pulsation condition)

0005 0010 0015 0020 0025 0030 00350000time (s)

Figure 13 The change of bubble centroid velocity with time undertwo initial conditions

numerically investigated by using the MPS-MAFL methodThe key conclusions of the present work are as follows

Different from the single bubble rising process understatic condition the bubble shape and rising velocity will haveperiodic fluctuation under the flow pulsation condition andthe fluctuation period is almost the same as the flow pulsationperiod Correspondingly the shape and velocity of bubblewill only gradually stabilize during the rising process underthe static condition For the coalescence process of bubble

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

12 Science and Technology of Nuclear Installations

pairs due to the introduction of flow pulsation conditionthe bubble pairs begin to coalesce slightly later than the staticcondition and after coalescence the bubbles will continue tofluctuate with the same cycle as the inlet flow pulsation

The calculation and analysis results of this study couldhelp to reveal the bubble behavior under flow pulsationcondition and they establish the basis for further applicationof MPS method to the two-phase flow analysis

Nomenclatures

119861119900 Bond number119863 Bubble diameter m119889 Dimension number119864119900 Eotvos numberg Gravity acceleration ms2n Unit vector of phase interfacer Particle location vector119903 Particle distance m119877119890 Reynolds number119905 Time su Particle velocity vector119908 Weight function

Greek Letters

120582 Diffusion coefficient120581 Local curvature120583 Viscosity Pasdots120588 Density kgm3120590 Surface tension coefficient Nm

SuperscriptSubscript

119894 119895 Particle number

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present study is supported by the National NaturalScience Foundation of China (nos 11875217 11505177 and11505134) the National Natural Science Foundation of Chinaoutstanding youth fund (11622541) and the Young EliteScientists Sponsorship Program by CAST (2018QNRC001)

References

[1] J A Boure A E Bergles and L S Tong ldquoReview of two-phaseflow instabilityrdquo Nuclear Engineering and Design vol 25 no 2pp 165ndash192 1973

[2] T Sanada A Sato M Shirota and M Watanabe ldquoMotion andcoalescence of a pair of bubbles rising side by siderdquo ChemicalEngineering Science vol 64 no 11 pp 2659ndash2671 2009

[3] Z Liu and Y Zheng ldquoPIV study of bubble rising behaviorrdquoPowder Technology vol 168 no 1 pp 10ndash20 2006

[4] M J Sathe I HThaker T E Strand and J B Joshi ldquoAdvancedPIVLIF and shadowgraphy system to visualize flow structurein two-phase bubbly flowsrdquo Chemical Engineering Science vol65 no 8 pp 2431ndash2442 2010

[5] P Basarova J Pislova J Mills and S Orvalho ldquoInfluence ofmolecular structure of alcohol-water mixtures on bubblebehaviour and bubble surface mobilityrdquo Chemical EngineeringScience vol 192 pp 74ndash84 2018

[6] M Maldonado J J Quinn C O Gomez and J A Finch ldquoAnexperimental study examining the relationship between bubbleshape and rise velocityrdquo Chemical Engineering Science vol 98pp 7ndash11 2013

[7] L Liu H Yan andG Zhao ldquoExperimental studies on the shapeand motion of air bubbles in viscous liquidsrdquo ExperimentalThermal and Fluid Science vol 62 pp 109ndash121 2015

[8] F Raymond and J-M Rosant ldquoA numerical and experimentalstudy of the terminal velocity and shape of bubbles in viscousliquidsrdquo Chemical Engineering Science vol 55 no 5 pp 943ndash955 2000

[9] R Sugrue J Buongiorno and T McKrell ldquoAn experimentalstudy of bubble departure diameter in subcooled flow boilingincluding the effects of orientation angle subcoolingmass fluxheat flux and pressurerdquo Nuclear Engineering and Design vol279 pp 182ndash188 2014

[10] G E Thorncroft J F Klausner and R Mei ldquoAn experimentalinvestigation of bubble growth and detachment in verticalupflow anddownflowboilingrdquo International Journal ofHeat andMass Transfer vol 41 no 23 pp 3857ndash3871 1998

[11] J F Klausner R Mei D M Bernhard and L Z Zeng ldquoVaporbubble departure in forced convection boilingrdquo InternationalJournal of Heat and Mass Transfer vol 36 no 3 pp 651ndash6621993

[12] R A M Al-Hayes and R H S Winterton ldquoBubble diameteron detachment in flowing liquidsrdquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 223ndash230 1981

[13] S Bovard H Asadinia G Hosseini and S A Alavi FazelldquoInvestigation and experimental analysis of the bubble depar-ture diameter in pure liquids on horizontal cylindrical heaterrdquoHeat and Mass Transfer vol 53 no 4 pp 1199ndash1210 2017

[14] A Tomiyama G P Celata S Hosokawa and S YoshidaldquoTerminal velocity of single bubbles in surface tension forcedominant regimerdquo International Journal ofMultiphase Flow vol28 no 9 pp 1497ndash1519 2002

[15] Z Wang X F Peng and J M Ochterbeck ldquoDynamic bubblebehavior during boiling in bead-packed structuresrdquo Interna-tional Journal ofHeat andMassTransfer vol 47 no 22 pp 4771ndash4783 2004

[16] M Gumulya J B Joshi R P Utikar GM Evans andV PareekldquoBubbles in viscous liquids Time dependent behaviour andwake characteristicsrdquoChemical Engineering Science vol 144 pp298ndash309 2016

[17] Z Yu H Yang and L-S Fan ldquoNumerical simulation of bubbleinteractions using an adaptive lattice Boltzmann methodrdquoChemical Engineering Science vol 66 no 14 pp 3441ndash34512011

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Science and Technology of Nuclear Installations 13

[18] T Sanada M Watanabe and T Fukano ldquoEffects of viscosityon coalescence of a bubble upon impact with a free surfacerdquoChemical Engineering Science vol 60 no 19 pp 5372ndash53842005

[19] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[20] H Y Yoon S Koshizuka and Y Oka ldquoA Mesh-free numericalmethod for direct simulation of gas-liquid phase interfacerdquoNuclear Science and Engineering the Journal of the AmericanNuclear Society vol 133 no 2 pp 192ndash200 1999

[21] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical simulation on void bubble dynamics using movingparticle semi-implicitmethodrdquoNuclear Engineering andDesignvol 239 no 11 pp 2382ndash2390 2009

[22] W Tian Y Ishiwatari S Ikejiri M Yamakawa and Y OkaldquoNumerical computation of thermally controlled steam bub-ble condensation using Moving Particle Semi-implicit (MPS)methodrdquo Annals of Nuclear Energy vol 37 no 1 pp 5ndash15 2010

[23] R Chen W Tian G H Su S Qiu Y Ishiwatari and YOka ldquoNumerical investigation on bubble dynamics during flowboiling using moving particle semi-implicit methodrdquo NuclearEngineering and Design vol 240 no 11 pp 3830ndash3840 2010

[24] R H ChenW X Tian G H Su S Z Qiu Y Ishiwatari and YOka ldquoNumerical investigation on coalescence of bubble pairsrising in a stagnant liquidrdquo Chemical Engineering Science vol66 no 21 pp 5055ndash5063 2011

[25] J Zuo W Tian R Chen S Qiu and G Su ldquoTwo-dimensionalnumerical simulation of single bubble rising behavior in liquidmetal using moving particle semi-implicit methodrdquo Progress inNuclear Energy vol 64 pp 31ndash40 2013

[26] L Liu H Yan G Zhao and J Zhuang ldquoExperimental studieson the terminal velocity of air bubbles in water and glycerolaqueous solutionrdquo ExperimentalThermal and Fluid Science vol78 pp 254ndash265 2016

[27] J R Grace ldquoShapes and velocities of bubbles rising in infiniteliquidsrdquo Transactions of the American Institute of ChemicalEngineers vol 51 pp 116ndash120 1973

[28] J R Grace ldquoShapes and velocities of single drops and bubblesmoving freely through immiscible liquidsrdquo Chemical Engineer-ing Research and Design vol 54 pp 167ndash173 1976

[29] J Hua and J Lou ldquoNumerical simulation of bubble rising inviscous liquidrdquo Journal of Computational Physics vol 222 no2 pp 769ndash795 2007

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Nuclear InstallationsScience and Technology of

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Power ElectronicsHindawiwwwhindawicom Volume 2018

Advances in

CombustionJournal of

Hindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Renewable Energy

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Solar EnergyJournal of

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Submit your manuscripts atwwwhindawicom