Xin Li ,Pan Zhang ,Jianlong Li ,Weiwen Wang ,Guanghui Chen ,*

1 College of Chemical Engineering,Qingdao University of Science and Technology,Qingdao 266042,China

2 College of Electromechanical Engineering,Qingdao University of Science and Technology,Qingdao 266061,China

Keywords:Multiphase fl ow Bubble Interface Deformation analysis Internal fl ow pattern Rising behavior

ABSTRACT Gas-liquid multiphase fl ow is a signi fi cant phenomenon in chemical processes.The rising behaviors of single bubbles in the quiescent liquids have been investigated but the internal fl ow patterns and deformation rules of bubbles,w hich in fl uence the mass transfer ef fi ciency to a large extent,have received much less attention.In this paper,the volume of fl uid method was used to calculate the bubble shapes,pressure,velocity distributions,and the fl ow patterns inside the bubbles.The rising behavior of the bubbles w ith four different initial diameters,i.e.,3 mm,5 mm,7 mm and 9 mm w as investigated in four various liquids including water,61.23%glycerol,86.73%glycerol and 100%glycerol.The results show that the liquid properties and bubble initial diameters have great impacts on bubble shapes.Moreover,fl ow patterns inside the bubbles with different initial diameters w ere analyzed and classi fi ed into three types under the condition of different bubble shapes.Three correlations for predicting the maximum internal circulation inside the bubbles in 86.73%glycerol were presented and the R-square values were all bigger than 0.98.Through analyzing the pressure and velocity distributions around the bubbles,four rules of bubble deformation w ere also obtained to explain and predict the shapes.

1.Introduction

Gas-liquid multiphase fl ow is a common phenomenon in chemical processes[1].Operating ef fi ciencies of distillation columns,reactors,and absorbers are mainly determined by the interactions between gas phase and liquid phase.Now adays,the correlations,w hich are used to calculate some practical parameters in the engineering processes,are mainly based on experience and accuracy of these corrections needs to be improved under some conditions.Therefore,investigations about gas-liquid multiphase fl ow are fundamental and signi fi cant for the development of engineering processes.As a typical fi eld to investigate multiphase fl ow,the rising behavior of single bubbles in a quiescent liquid has been a hot issue of research for many years[2-4].Clarifying the in fl uences of liquid properties on the bubbles and the relationshipsbetw een thefeaturesof rising single bubbles,such as bubble dimension,bubble shape and rising velocity,issigni fi cant to deepen the understanding of the transfer processes[5,6].

Liquid properties such as density,viscosity and surface tension have impacts on the behavior of the rising bubbles[7].Bubble diameter and detachment time increase w ith the increase of liquid surface tension and viscosity,but decrease w hen the liquid density increases[8].The bubbleterminal velocity isrelated to density ratio and viscosity ratio between the liquid and the gas[9].With the density ratio increasing,the bubble terminal velocity increases,but its effect becomes insigni fi cant w hen the ratio is larger than 50.When the viscosity ratio increases,thebubbleterminal velocity increases.Liu et al.[10]captured thebubble rising trajectory by using PIV(Particle Image Velocimetry),and they found that the bubble rising trajectory changes from one-dimensional to three-dimensional as the liquid viscosity decreases.In the high viscosity liquid,the bubbles rise upward steadily following a straight line.In the moderate viscosity liquid,the bubbles move upw ard in a straight line initially and then change to a zigzag path quickly.While in the low viscosity liquid,bubbles rise in a spiral w ay with violent w obbling and rotational motion.

The parameters of single rising bubbles are affected by fl uid properties and some other factors.For instance,the bubble rising velocity and trajectory are related to the bubble dimension.In general,high rising bubble velocity can be obtained under the condition of large bubble dimension when other variables are constant[11,12].Wu and Gharib[13]found that w hen the bubble diameter changes in the range of 0.1-0.2 cm,there are tw o steady types of bubble shapes w hich are sphere and ellipsoid.For the spherical bubbles,when their diametersare larger than 0.152 cm,they rise up in a zigzag path.While for the ellipsoidal bubbles,w hen their diameters are larger than 0.171 cm,they move in a spiral path.The bubbles w ith similar diameters also show different shapes and rising behaviors.

The bubble behaviors are complex in the engineering equipment so that the above results under different conditions are precious and can supply theoretical foundation for the optimization of practical processes.The bubble shapesand internal fl ow patternsasw ell asthe bubble behavior parameters are signi fi cant to understand the transfer processes in the equipment[14]and the former ones are often ignored in engineering researches.Even for the theoretical studies,the relevant results are still insuf fi cient.

Different bubble shapes w ere exhibited in liquid solutions[15,16].The bubble shapes have effects on the fl ow pattern around the bubbles and may further in fl uence the mass transfer processes.Clift et al.[17]summarized thebubbleshapesunder different dimensionlessnumbers.Six types including spherical,ellipsoidal,w obbling,spherical-cap,skirted and dimpled ellipsoidal-cap bubbles are show n in their w ork.Funfschilling and Li[6]studied the fl ow fi elds around bubbles in Newtonian fl uid and non-New tonian fl uid.The fl uid fi eldsin the New tonian fl uid and non-Newtonian fl uid are different to some extent.B?hm et al.[4]then extended the fl ow patternsaround the bubbles to fi ve types by using the PIV.Zhang et al.[18]studied the fl ow structure around the bubble in shear-thinning fl uids by an improved level set approach and they found that the fl ow structure deviated qualitatively from that in Newtonian fl uids when the shear-thinning property becomes signi ficant.Nock et al.[19]used the change of bubble shape and bubble rising velocity to indicate the conditions for mass transfer rate.The above w ork focused on the bubble shapes,the surrounding fl ow patterns and their indicative role in re fl ecting the mass transfer rate.How ever,thereislittleanalysison how thebubbleshapeschange,w hich ishelpful to further understand the transport mechanism.

At present,researcherspay more attention to theliquid side transfer processes[20-23]or the fl ow patterns outside the bubbles in the liquid phase[4,24,25].Studies about fl ow patternsinside the bubble are insuffi cient.It is widely accepted that the fl ow patterns should be related to the mass transfer processes.Take the mass transfer process on distillation trays for example,when the bubbles rise through the liquid,if there is no circulation inside the bubbles,the mass transfer of gas substance mainly depends on diffusion.But if there is circulation inside thebubbles,theconvective masstransfer of gassubstanceexistsbesides diffusion and it w ould enhance the masstransfer coef fi cient.Now adays,someequipment ef fi ciency calculation equationsomit thein fl uencesinside the bubblesand the calculated results are usually not good enough.Hence,studies about fl ow patterns inside the bubbles are signi fi cant.Garner and Hammerton[11]and Filla et al.[26]experimentally observed a fl ow pattern inside the bubble in w hich the fl uid fl ow supward in thecenter and dropsdown along the sides.Li et al.[12]simulated single bubble rising behavior at elevated pressures and a similar fl ow pattern was shown.Moreover,liquid vortex formation and shedding of wakes w ere also show n clearly.Ansariand Nimvari[27]simulated the fl ow pattern inside the bubbles and studied the effects of viscosity ratio on maximum internal circulation velocity.Premlata et al.[28]investigated different streamline patterns caused by bubble deformation and several streamline patterns w ere show n under different bubble shapes.How ever,the above researches only show ed several internal fl ow patternsin the selected regular-shaped bubbles,w hile thedetailed changes of streamline patterns inside the consecutive deformed bubblesw erenot explained.Besides,the bubble shapescannot keep regular all the time in the practical devices,and hence the fl ow patterns inside the irregular-shaped bubbles deserve to be studied.

In thiswork,therising behaviorsof single bubbleswith different initial diameters of 3 mm,5 mm,7 mm and 9 mm w ere studied systematically in different fl uids by the commercial computational fl uid dynamics(CFD)software Fluent.The volume of fl uid(VOF)method w as applied to calculate the processes of bubble rise and track the gas-liquid phase interface.It is indicated that different bubble shapes,fl ow patterns and maximum internal circulation velocity w ereinvestigated in combination.The changes of fl ow patterns inside the bubbles w ere investigated,and three correlations betw een the maximum internal circulation and the aspect ratio w ere proposed.In addition,the changes of bubble shapes w ere analyzed based on the pressure and velocity distributions and rules about bubble deformation w ere deduced and their accuracies w ere veri fi ed.

Table 1 The physical properties of studied liquid phases[30]

2.Numerical Calculation

2.1.Governing equations

VOFis a surface capturing method to simulate the bubble rising behavior in thequiescent liquidsdue to the advantage of tracking the gas liquid phase interface[29].Both the gas and liquid phases obeyed the continuous equation and the momentum equation named the Navier-Stokes equation.The continuous equation is show n as follows:

Fig.1.Computational mesh structure.

Table 2 Dimensions of the three grid schemes

The Navier-Stokes equation is show n below:

w here u is the velocity vector;ρis density;p is pressure;μis viscosity;and FSrepresentsthesurfacetension sourceterm which istaken into account by the continuum surface force(CSF)model.The CSFmodel converts the surface tension into a volume force acting on the interface based on the divergence theorem.Therefore,the surface tension source term in the momentum equation can be expressed by:

Fig.2.The aspect ratio and rising velocity of bubbles in different grid schemes.

w here σ is surface tension;

The VOFmethod introducesthe volumefraction of each phasein the computational cell and sums the volume fractions of all phases to unity in thecontrol volume.If FVrepresentsliquid volume fraction,thephases and interface can be determined by the follow ing rules:Based on the volume fractions,the density and viscosity of the mixture are calculated by:

2.2.Solution methodsand mesh dependency

Pure w ater and three different solutionsw ere selected asthecontinuous phases to study the bubble rising behavior w hile air w as adopted as the gas phase.The physical properties are shown in Table 1 w hich are referenced from Jiang et al.[30].The dimension of study domain is 40 mm×80 mm.Initially the air bubble w as set asa sphere w hose center is at the middle of the x-axis and 7.5 mm high from the bottom.

Fig.3.Comparison of bubble shapesbetween experiment[13]and calculation at different times.

Table 3 Comparison of aspect ratios betw een the experiment[13]and simulation

A pressure-based solver w ith an unsteady implicit formulation w as utilized.Explicit formulation with the courant number 0.25 wasapplied here to calculate volume fraction parameters.The governing equation w as discrete using a quadratic upw ind interpolation for convection kinetics(QUICK)and the PRESTO!(PREssure STaggering Option)scheme w as used for the pressure interpolation.The Pressure-Implicit w ith Splitting of Operators(PISO)algorithm was employed to decouple the pressure and velocity.Besides,piecew ise linear interface construction(PLIC)w as applied to reconstruct the interface.

Fig.4.Rising bubbles with four different initial diameters in water-air system.

Agrid independence check wasperformed to ensure that the mesh w as suf fi ciently re fi ned not to in fl uence the results.Since the mesh of part Aasshow n in Fig.1 w ashighly sensitive to capture the fl uid behavior around the bubble rising trajectory accurately.Three different grid schemes were arranged and show n in Table 2.

The aspect ratio and rising velocity of bubbles w ith different grid schemes are compared as show n in Fig.2.The bubble aspect ratio of Grid 1 is a little larger than that of Grid 3 after 0.1 s w hile the aspect ratios of Grid 2 and Grid 3 are almost the same to each other.As for bubble rising velocity,the difference betw een Grid 2 and Grid 3 is smaller than that betw een Grid 1 and Grid 2.Therefore,taking the results of bubble aspect ratio and bubble rising velocity into account,the Grid 2 w hich has 880000 grids is adopted in the tests.

2.3.CFDmodel veri fi cation

Fig.3 show s the rising bubble shapes at different times w ith an initial diameter of 0.195 cm in a water-air system.The bubble changes from sphere to ellipse gradually and the calculated results are consistent w ell w ith the experimental results[13]in terms of bubble shapes.Table 3 also compares the calculated aspect ratios of the bubbles w ith the experimental results.The relative errors are no more than 10.5%.Therefore,these results indicate that the agreement is good.

3.Results and Discussion

3.1.Bubble shapes in different liquid-gas systems

The evolution of the bubblesw ith different initial diameters,which are 3 mm,5 mm,7 mm and 9 mm,w as investigated in four various liquids(shown in Table1).Theaspect ratio Arwasintroduced to re fl ect the variation degree of bubble shapes and it can be expressed as follow s:

w here dvrepresents vertical projected length and dhis horizontal projected length.

In the w ater-air system,the bubble shapes change more dramatically w hen the initial diameters increase(see Fig.4).When the initial bubble diameter is 3 mm,the bubble transforms to ellipse fi rstly and then becomes fl at,w hich is similar w ith that captured by a high resolution camera.After 0.08 s,both ends of the bubble fold upw ard.When the diameter is 5 mm,the bubble transforms to spherical-cap fi rstly and then becomes oblate quickly.After 0.17 s,both ends of the bubble fold upw ards and the center of the bubble rises slightly.While for the bubbles w ith the initial diameters of 7 mm and 9 mm,the shapes are complex after 0.06 s.The aspect ratios of the bubbles in the w ater-air system are shown in Fig.5a.The aspect ratios of bubbles with diameter of 3 mm and 5 mm reduce before 0.18 s and then increase slightly.For the larger bubbles,the variation tendencies are irregular because of the dramatic deformation.

In the 61.23%glycerol-air system,the deformations of bubbles are weaker than those in the water-air system(see Fig.6).The 3 mm bubble changes from sphere to ellipse and keeps stable during rising.The aspect ratio for the 5 mm bubble variesfrom 0.96 to 0.23 and thebubble shapes are shown as inverted spherical-cap instead of oblate which is exhibited in the w ater-air system after 0.07 s.Besides,the7 mm bubble and 9 mm bubble still deform greatly.In general,for small bubbles whosediameter equalsto 3 mm,thevariation amplitudeof aspect ratios for these bubbles in the 61.23%glycerol is smaller than those in the w ater.In other w ords,the bubble shapes in the former system are more stablethan thosein thelatter system.Whilefor thelarger bubbles,the aspect ratios are changed irregularly(see Fig.5b).The differences may result from the different liquid properties,especially for the viscosity.As show n in Table 1,the density and surface tension of the tw o systemsare approximate,and the viscosity of 61.23%glycerol is13.4 times higher than that of water.Hence,the viscosity may be the main reason for bubble deformation other than initial diameter.To further investigate the in fl uence of viscosity,tw o liquid phasessuch as86.73%glycerol and 100%glycerol are introduced.

Fig.6.Rising bubbles with four different initial diameters in 61.23%glycerol-air system.

Fig.7.Rising bubbles w ith four different initial diameters in 86.73%glycerol-air system.

For the 86.73%glycerol-air system,the viscosity is increased to 138.7 m Pa·s.In Fig.7,all the bubbles w ith different initial diameters can achieve stable shapes after 0.18 s.The bubbles w ith small diameter,i.e.,3 mm,are show n as sphere eventually.For the 5 mm bubble and 9 mm bubble,the bubble shapes are ellipse and spherical-cap,respectively.While for the 7 mm bubble,its fi nal shape is nearly sphericalcap.The aspect ratios of the bubbles are show n in Fig.5c.There is no dramatic variation for the aspect ratio of the 3 mm bubble.For larger bubbles,the aspect ratios reduce gradually and then keep stable.The bubble behaviors in the 100%glycerol-air system w ith a 1407.0 m Pa·s viscosity are also studied(see Fig.8).It is interesting that the shapes of all the bubbles change slightly,and the bubbles stretched vertically other than horizontally in other solutions.Fig.5d shows that the aspect ratios of the bubbles are all larger than 1 and those of small bubbles are more stable compared with large bubbles during the 0.2 s.The bigger the initial bubble diameter,the larger the aspect ratio is.Besides,the variation ranges of aspect ratios for the bubbles in the 100%glycerolair system are smaller than those in other systems.

The above results show that the evolutions of the bubble shapes are mainly dependent on the initial diameters and the liquid viscosity.

3.2.Flow patterns inside the bubble with different shapes

With the bubblesrising,the shapeschange and the fl ow patternsinside bubblesvary.Theoccurrenceof internal circulation resultsfrom the viscousdrag of the outer fl uid[11].However,the studiesabout the fl ow patterns inside the bubbles w ith different shapes are insuf fi cient,especially for irregular-shaped bubbles w hose shapes cannot be classi fi ed as sphere,ellipse,and so on.

An experiment w as designed to capture the images about internal circulation(see Fig.9).The bubble entry needle w ith a diameter of 3.5 mm is set at the bottom center of a PMMA vessel.The PMMAvessel is35 mm in thickness,500 mm in height and 120 mm in w idth.High purity glycerol(＞99.5%)w as applied as the liquid and its properties are show n in Table 4.The CCD camera(TSI,Model 630157,2006)w as applied to record the bubbles and the laser generator(Big sky laser,CFR PIV 120 PS2,2006)at 532 nm w avelength is used as the light source.The synchronizer(Laserpulse,Model 610035)w as applied to control the camera and the laser generator to w ork together.The left pump is operated at 100 ml·min-1during the w hole experiment to prevent the w ater from fl ow ing backward into the entry needle.In order to capture the internal circulation,the traditional tracer particle,such as hollow glass ball cannot be applied due to the large density difference betw een air and particles.By screening,smoke w as determined as the tracer.The gas with tracer w as pumped into the vessel by the right pump at 200 ml·min-1.The liquid level was 300 mm in height from the top of the entry needle and the temperature of high purity glycerol w as kept at(20 ± 1)°C.Fig.10 show s the internal circulation insides the spherical-cap bubble in the high purity glycerol(＞99.5%).It is clear that there are tw o vortexes inside the bubbles.How ever,in the low viscosity liquids,it is really hard to capture the clear images due to the dramatic deformation and the stringent experimental conditions.Hence,the fl ow patterns inside the bubbles with various diameters in the four liquids were investigated by simulations.

In the 100%glycerol,the aspect ratios of bubbles change slightly and their internal fl ow patterns are shown clearly(see Figs.8 and 11a).The gas inside the bubble rises in the center and fl ow s downward along the sides.Tw o main vortexes occurred inside the bubbles and this fl ow pattern can be called as “tw o main vortexes”pattern.While in the 86.73%glycerol,the similar internal fl ow pattern can be seen in the bubbles w ith diameters of 3 mm,5 mm and 7 mm.However,in the 9 mm bubble,there are tw o separation vortexes w hich occurred at the bottom(see Fig.11b)and their directions are countered to the fl ow of main vortexes w hen the aspect ratio is lower than 0.512(see Figs.5c and 7).

Fig.8.Rising bubbles with four different initial diameters in 100%glycerol-air system.

Fig.9.Experimental system:1-PMMA vessel;2-entry needle;3,4,8-valve;5-smoke collector;6,7-pump;9-smoke machine;10-CCDcamera;11-computer;12-laser generator.

In the 61.23%glycerol,the 3 mm bubble could almost keep itsshape after 0.07 s,and hence the internal fl ow pattern isalso stableat the“two main vortexes”pattern(see Fig.6).While for other bubbles w ith larger initial diameters,their shapes change dramatically and as a result,theinternal fl ow patterns are complex.For the 5 mm bubble,the internal fl ow pattern changes from “tw o main vortexes”pattern to “tw o main vortexes w ith separation vortexes”pattern.When the aspect ratio is lower than 0.570,theseparation vortexesstart to appear.Hence,the internal fl ow patterns are related to the bubble shapes.For the 7 mm and 9 mm bubbles,the shapes vary largely but they are axisymmetric.The“tw o main vortexes w ith separation vortexes”pattern can be also formed but the locations of the main vortexes and separation vortexes are changed with bubble deforming(e.g.the 9 mm bubble at 0.1 s,see Fig.11c).

Table 4 Properties of high purity glycerol

Fig.10.Internal circulation image captured by CCD camera.

In the w ater-air system,the 3 mm bubble show s a“tw o main vortexes”pattern.But in the 5 mm bubble,the fl ow pattern is not obvious after 0.15 s because the aspect ratio is too small and the bubble show s strip-shaped.The fl ow patterns in the irregular-shaped bubbles are complex and the inner vortexes cannot be classi fi ed simply as main or separation vortexes.Hence,the inner vortexes are called as“scattered vortexes”in this w ork.The 7 mm and 9 mm bubbles deform largely and they are not axisymmetric after 0.12 s.How ever,it can be seen that the scattered vortexes more likely occurred at the protrusions but the dimensions of the vortexes are different such as the 7 mm bubble at 0.13 s(see Figs.4 and 11d).

Fig.11.Internal fl ow patterns:(a)7 mm bubble at 0.10 sin the 100%glycerol;(b)9 mm bubbleat 0.13 sin the 86.73%glycerol;(c)9 mm bubble at 0.10 sin the 61.23%glycerol;(d)7 mm bubble at 0.13 s in the w ater.

Fig.12.Maximum internal circulation of different bubble in different systems:(a)water-air system;(b)61.23%glycerol-air system;(c)86.73%glycerol-air system;(d)100%glycerol-air system.

Fig.13.Comparison of simulated with predicted values obtained from maximum internal circulation correlations for:(a)5 mm bubbles;(b)7 mm bubbles;(c)9 mm bubbles.

It is clear that the internal fl ow patterns are related to the bubble shapes.The regular-shaped bubbles can be classi fi ed as spherical bubble,ellipsoidal bubble,spherical-cap bubble and so on,and the internal fl ow patterns can be classi fi ed as“tw o main vortexes”pattern and “tw o main vortexes w ith separation vortexes”pattern.As for the irregularshaped bubbles,the internal fl ow pattern is complex.How ever,regardlessof the kind of fl ow pattern,thegasnear the bubble boundary can be refreshed by the inner substancesquickly and it w ould prompt themass transfer processes.Internal circulation,w hich can re fl ect the ef fi ciency of inner gas substance contacting w ith the liquids,is a variable about the fl ow velocity inside the bubbles,w hich is obtained by subtracting the average velocity in the x and y directions from the velocity fi eld x and y values.The maximum internal circulation is related to the bubble shape parameters such as aspect ratio so if the correlationsof these can be fi tted,it w ill help speculate the ef fi ciency of inner gas substance to make contact w ith the liquids.For the w ater-air system,the maximum internal circulation changes dramatically,especially for the 7 mm bubble and 9 mm bubble(see Fig.12a).This is because the bubbles deform largely and the space for air to fl ow in fl uences the circulation.In the glycerol-air systems(see Fig.12b,c and d),the maximum internal circulation of the bubbles w ith the same diameters decreases w ith the increase of the glycerol content.Besides,the inverse relationship of maximum internal circulation and aspect ratio can be analyzed by comparing the tendencies of curves in Figs.5 and 12.The relationship is clearer in the 86.73%glycerol and three correlations betw een the maximum internal circulation and the aspect ratio w ere proposed in the bubbles with diameters of 5 mm,7 mm and 9 mm,respectively.

For 5 mm bubbles:

w here x represents the aspect ratio and y represents the maximum internal circulation.

The R-square values for Eqs.(7),(8),and(9)are 0.9985,0.9977,and 0.9838,respectively.The predicted and simulated values are show n in Fig.13.It can be seen that the tw o curves are fi tted w ell,and the correlations can predict the maximum internal circulations accurately.

3.3.Analysisof bubble deformation

The degree of bubble deformation w as usually re fl ected by the parameters such as aspect ratio[31,32]and circularity[33,34].Moreover,the relationship betw een bubble shapes and some dimensionless numbers was also investigated[13,24].When the bubbles rise up through the quiescent liquids,pressure and velocity distributions around the bubble w ould change,and the interactionsbetween the surrounding fl uids and bubbles themselves could cause the change of bubble shapes.

Fig.14 shows the pressure distribution and velocity vector around the bubble of the initial 9 mm diameter in the w ater-air system and 100%glycerol-air system at 0.01 s and 0.02 s,respectively.It can be seen that the pressure at the top of the bubble is higher than that at the bottom,and there is a low pressure zone below the bubble.However,the low pressure zone below the bubble in the w ater is smaller than that in the 100%glycerol.Because of the low pressure zone,the fl uid around the zone would fl ow into it and then push the bubble.In Fig.14b and e,the velocity vectors around the low pressure zone point to the bubble boundary.The directions of the arrow s represent the fl ow direction of the fl uid.In Fig.14b,the arrow s“traversed”the bubble boundary are almost assembled at the area of one sixth of the bottom in the bubble,and the directions are almost upward along the bubble boundary.While for Fig.14e,the arrows are almost assembled at theareaof half of the bottom,but thedirectionsare various.Some arrow s point dow nward w hile some arrow s point upward along the bubble.The differencemeans that the surrounding fl uidsact unequal forces on the bubble.As a result,the bubble in the water show s that there is a fl at areaat thebottom w hilethebubble in the100%glycerol isshow n as anearly spherewhose bottom becomespointed(see Fig.14c and f).The aspect ratios of the bubbles in Fig.14 are about 1.0.To further study the bubble deformation,other bubble shapes and relative fi elds were investigated.

Fig.14.The pressure distribution and velocity vectorsaround 9 mm bubble:(a)-(b)at 0.01 sin w ater-air system;(c)at 0.02 sin water-air system;(d)-(e)at 0.01 sin 100%glycerol-air system;(f)at 0.02 s in 100%glycerol-air system.

The 5 mm bubbles rising in the w ater at 0.03 s and 0.04 s w ere chosen(see Fig.15).The bubble is spherical-cap and tw o low pressure zones occurred at protrusions on the tw o sides of the bottom as shown in Fig.15a.There are tw o vortexes around the low pressure zones and only the tangential velocity vectors appeared.While at the bottom of the bubble,the arrow s point upw ard normal to the bubble boundary(see Fig.15b).Near the low pressure zones,the fl uid fl ow s around it instead of fl ow ing into it,and hence the pressure w ould keep low.How ever,w hen the pressure inside the bubble is higher than that in the liquid zones,the bubble boundary w ould be expanded by the pressure difference.Fig.15c show s the bubble shapes and pressure distribution at 0.04 s.Compared w ith the bubble at 0.03 s,the bubble at 0.04 s is longer in the horizontal direction and the radian of the bottom is larger.The changes are consistent w ith the pressure distribution and velocity vector described aforementioned.

From the analysis of Figs.14 and 15,the in fl uences of pressure and velocity distributions on bubble deformation can be concluded as follow s:

1.When the bubble rises,the pressure above the bubble is higher than that below the bubble.At the top of the bubble w here the pressure is higher,the fl uid w as moved by the impetus from the bubble.While at the bottom of the bubble where the pressure islow er,the changes of bubble boundary arein fl uenced by theforce from thesurrounding fl uid.Besides,there are low pressure zones that appeared at the protrusions.

Fig.15.The pressure distribution and velocity vectors around 5 mm bubble at 0.03 s and 0.04 s.

2.If the fl uid around the low pressure zones does not fl ow into the zones,w hich means the existence of vortexes,the differences betw een the pressure inside the bubble and that in the zones w ould lead to the expansion of bubble boundary.

3.The fl ow directions of fl uid near the bubble boundary can represent the force from the fl uid acted on the bubble.Hence,through analyzing the arrows“traverse”the boundary,the tendency of bubble deformation caused by fl uid can be predicted.

4.It can be seen in Figs.15a and 16 that there are high pressure zones and low pressure zones at the fi eld above the bubble.Due to the fact that the fl ow of the fl uid above thebubble is driven by the impetus from the bubble,the pressure distributions can re fl ect the stress conditions.At some moment,the high(low)pressure zones can refl ect that the large(small)force from the bubble acts on the fl uid in thesezonesand then thebubbleboundary shapesw ill changelargely(slightly)at the next moment.

To further verify the in fl uences of pressure and velocity distributions on the bubble deformation,7 mm bubbles w hose shapes are irregular at 0.18 s and 0.19 s are analyzed.Fig.16 show s the pressure contour and velocity vector around the bubble at 0.18 s and the bubble shape illustrated with the black outline at 0.19 s.At the top of the bubble,the bubble boundary in the high pressure fi eld expanded obviously;also,the bubble boundary in the low pressure zones expanded at the protrusions on the two sides of the bubble.While at the bottom right of the bubble,the arrow s “traverse”the boundary means the force from the fl uid acted on the bubble,which makes a boundary change.It should be noticed that the boundary changes in the w hite frame result from the relative motion between the boundary in this region and its adjacent regions instead of boundary expansion.The bubble deformation show n in Fig.16 follow s the above four rules.In a word,the rising behavior of bubbles promotes the fl ow of surrounding fl uid,and then the fl uid in turn in fl uences the bubble deformation.

4.Conclusions

The VOFmethod wasused to investigatetherising behavior of single bubbles in the quiescent liquids.Four liquid-gas systems,w hich are w ater-air,61.23%glycerol-air,86.73%glycerol-air,and 100%glycerolw ater,w ere applied to study the in fl uences of liquid propertieson bubblebehavior.In each liquid-gassystem,bubbleswith different initial diameters w ere also studied to investigate the effect of bubble diameters on its rising behavior.

In the same liquid-gas system,the bubbles w ith larger initial diameters deform more dramatically than those w ith smaller initial diameters.As for the bubbles w ith the same initial diameters,the aspect ratios change largely in the w ater-air system and 61.23%glycerol-air system.With the content of glycerol increasing,the viscosity increases and the variation amplitude of aspect ratio becomes smaller.Besides,the bubble shapes almost keep stable eventually for 0.2 s.

All thebubble shapesand internal fl ow patternsduringtherising behavior of 0.2 s were exhibited.The internal fl ow patterns can be classifi ed astw o types which are “tw o main vortexes”pattern and“tw o main vortexeswith separation vortexes”pattern for regular-shaped bubbles.While for irregular-shaped bubbles,the scattered vortexes are more likely occurred at the protrusions but the dimensions of the vortexes are different.Threecorrelationsw ereproposed to predict themaximum internal circulation inside the bubbles w ith diameters of 5 mm,7 mm,and 9 mm in 86.73%glycerol,and hence the conditions about inner gas substance contacting with the liquids can be re fl ected by bubble dimensions.The R-square values are all larger than 0.98,and the predictive accuracies are good.

Thepressureand velocity distributionsw ereanalyzed to explain and predict the bubble deformation.Four rules were concluded and further veri fi ed by analyzing the changes of the irregular-shaped bubbles.The results show that the rules are w ell suitable not only for regularshaped bubbles but also for irregular-shaped bubbles w hich are rarely investigated.

Nomenclature

Araspect ratio

dhhorizontal projected length,mm

dvvertical projected length,mm

FSsurface tension source term

FVvolume fraction

g gravity acceleration magnitude,m·s-2

n unit vector

P pressure,Pa

t time,s

u velocity,m·s-1

σ surface tension,N·m-1

μ viscosity,Pa·s

ρ density,kg·m-3

κ interface curvature

Fig.16.The pressure distribution and velocity vectors around 7 mm bubble at 0.18 s and 0.19 s.