Nikolaos A.Avgerinos*,Dionissios P.Margaris

Department of Mechanical Engineering and Aeronautics,Fluid Mechanics Laboratory,University of Patras,GR-26504 Patras,Greece

Keywords:Bubble Coalescence Computational fluid dynamics,CFD Methane Rise velocity Shape deformation

A B S T R A C T The hydrodynamic behavior of multiple bubbles rising upward is a field of ongoing research since various aspects of their interaction require further analysis.Shape deformation,rise velocity,and drag coefficient are some of the uncertainties to be determined in a bubble upward flow.For this study the predictions of the three-dimensional numerical simulations of the volume of fluid(VOF)CFD model were first compared with experimental results available in the literature,serving as benchmark cases.Next,28 cases of pairs of equal and unequal-sized in-line pairs of bubbles moving upwards were simulated.The bubble size varied between 2.0–10 mm.Breakthrough of the present study is the small initial distance of 2.5 R between the center of the bubbles.To provide a more practical nature in this study material properties were selected to match methane gas and seawater properties at deepsea conditions of 15 MPa and 4°C,thus yielding a fluid-to-bubble density ratio λ=7.45 and viscosity ratio n=100.46.This is one of the few studies to report results of the coalescence procedure in this context.The hydrodynamic behavior of the leading and trailing bubbles was thoroughly studied.Simulation results of the evolution of the rise velocity and the shape deformation with time indicate that the assumption that the leading bubble is rising as a free rising single one is not valid for bubbles between 2.0–7.0 mm.Finally,results of the volume of the daughter bubble exhibited an oscillating nature.

1.Introduction

One of the most common kinds of flow in industrial applications and research and development sector of oil and gas,pharmaceutical or refrigeration industries is the well-known bubbly flow[1–3].This kind of flow can be categorised either as monodispersed or polydispersed.Even in the simplest case of monodispersed flow,several complexities arise since single bubble hydrodynamic behavior is continuously changing resulting in different bubble shape and velocity,and at the same time bubbles interact between each other resulting in the breakup and coalescence phenomena.The plethora of these phenomena yields a continuously altering flow field[4].Another parameter of the complexity of the bubbly flows is the nature of the liquid phase,which can be either a Newtonian fluid or a non-Newtonian fluid[5–8].On top of this,the implementation of models originally developed for single bubble flow,thus omitting bubble interaction,add extra uncertainties when studying bubbly flow.All the above parameters are characteristic of the complexity of the behavior of a bubbly flow.In view of this,the need for more sophisticated correlations to shed light on bubble interaction emerged to further improve the processes design and accuracy.In the literature studies of mathematical,experimental and numerical nature try to elucidate the uncertainties of in-line bubble coalescence.A review of key studies related to these categories follows.

In general,if two particles are moving closer to each other,the flow structure around each one is quite different compared to the flow structure when moving as a free single rising particle.Even for the simplest case of a single bubble,hydrodynamic force prediction is usually based on correlations as a function of We,Mo,and Eo numbers.Bearing in mind that the bubble rise velocity is not taken into consideration in correlations based on Eo number,the uncertainty imposed in the accuracy of their prediction is significant.This became more evident in view of the latest studies which clearly present results of the periodic fluctuation of the rise velocity.These findings are in contrast with past studies assuming that the terminal rise velocity of a single bubble is stable[2,4,9,10].For that reason,the force balance analysis emerged as an alternative to study the behavior of the trailing bubble in a pair of spherical bubbles.Zhang and Fan[11]studied the drag force and the acceleration of a spherical trailing bubble,taking under consideration the buoyancy,gravity,drag,added mass and Basset forces at Re≤O(100).A similar model was introduced by Ramírez-Mu?oz et al.[12],incorporating an “artificial origin”parameter to better describe the velocity profile.This model performed well when the distance between the center of the bubbles was larger than 4.0 R.Another force model presented by Baz-Rodríguez et al.[13]with a modified fit-equation for the ‘a(chǎn)rtificial origin’exhibited good performance for in-line spherical bubbles rising at 50≤Re≤300 and separation distance≥5.0 R.

Early experimental measurements for particle in-line interaction using a micro-force balance system for 20≤Re≤130 were presented by Zhu et al.[14].The strong dependence of the hydrodynamic force on the trailing body with Re and the inter-particle distance was demonstrated.Results of the motion of nearly spherical in-line gas bubbles at 0.2≤Re≤200 were compared with predictions of a hydrodynamic force model[15,16].Zhang et al.[17]presented a correlation for total drag coefficient including mass and history force,as a function of Archimedes and Reynolds number,generating reliable predictions up to Re=100 when compared with experimental measurements of acceleration and steady motion of single bubbles.The periodic nature of the rise velocity of a single bubble for higher Reynolds number,550≤Re≤1700,was studied by Yan et al.[18].In their paper,a correlation for drag coefficient was proposed showing good agreement with experimental measurements,yet further modifications are needed to enable the study of in-line bubbles since it is based on single bubble experimental data.

Even though experiments can provide immediate and applicable results,important limitations are the difficultness in measuring the flow field surrounding the rising bubbles,thus not providing details of the pressure and velocity distributions, and the fact that the correlations based on specific liquid–gas system properties may not be applicable to other systems[19].In this regard,numerical simulation became a reliable alternative thanks to advances in numerical methods and available computational resources.Amongst past works,Yuan and Prosperetti[20]presented results regarding a pair of spherical in-line bubbles moving upwards at 50＜Re＜200 demonstrating the existence of a stable equilibrium distance between them.Chen et al.[21]studied single and multiple bubble motion and interaction inside a vertical pipe at Re≤20as a function of the initial separation of the bubbles. Flow characteristics of pairs of equal-sized spherical bubbles rising side by side at 0.02≤Re≤500 for separation distance between 2.25 and 20 were presented by Legendre et al.[22].Cheng et al.[19]examined the influence of the density ratio presenting results of the axial and oblique interaction of equal-sized bubbles for density ratio of 10,100 and 1000.

In this regard,it is evident that various parameters remain unexplored and observations need verification.Authors' intention is to present results of the numerical simulation on the in-line motion and interaction of in-line pairs of bubbles.In total,28 cases were simulated.Significant breakthrough of the present study is:(i)the bubble size utilised,(ii)the study of both equal and unequal-sized bubbles,and(iii)the small distance considered between each bubble pair.In more detail,while most of the past works studied relatively small-sized bubbles,in this work diameters between 3.0 and 10 mm are considered.These values are reported as the typical expected size of methane gas bubbles at a deep-sea bubbly flow,such as a blowout[23],offering a more practical aspect to the findings of this work. The corresponding bubble Reynolds number ranged between 340 and 4300 for fluid-to-bubble density ratio λ=7.45 and viscosity ratio n=100.46.Additionally,the analysis of the flow field and the coalescence process was addressed by observing the time-history of the rise velocity,and the shape deformation of both the leading and the trailing bubble as well as the distance between them and the necessary coalescence time.To our best knowledge,there is no previous study examining in-line bubble pair flow behavior under the scope of all these parameters.Therefore,this investigation will carry forward the research and implementation of findings in more precise correlations incorporated in numerical models.

The paper is organized as follows:Section 2 introduces the Computational Fluid Dynamics-VOF(CFD-VOF)method and validation.The framework of the computational analysis is presented in Section 3.Section 4 presents the results of the numerical analysis and, finally,Section 5 concludes the paper.

2.Computational Fluid Dynamics Model

2.1.Numerical model

The volume of fluid(VOF)model by Hirt and Nichols[24]was selected to analyse the in-line bubble coalescence process.The VOF model is formulated on the principle that two or more phases are not inter penetrating.For each phase incorporated to the model,its volume fraction in the computational cell is introduced.Without mass transfer,the continuity equation for the volume fraction of the qthphase is given by

where,ρq,αq,andare the density,volume fraction,and velocity of the qthphase,respectively.The volume fraction of all phases follows the constraint:

The momentum equation shared by all the phases is described by

The gas–liquid interface is tracked based on the distribution of αg,the volume fraction of gas in a computational cell,where αg=0.0 in the liquid phase and αg=1.0 in the gas phase.In this bubble simulation,F represents the surface tension force per unit volume.Computation of the surface tension force at the phases interface is based on the Continuum Surface Force(CSF)model by Brackbill et al.[25]:

where σ is the coefficient of surfacetension,κ is the radius of curvature,δ is the Dirac function and n is the unit normal vector on the surface.

2.2.Numerical methods

The motion and interaction of bubbles are simulated coupling the VOF and the CSF models.The primary phase is seawater and the secondary phase is methane gas.The flow field of bubbles is obtained by using commercials oft ware ANSYS Fluent 16.0.The pressure–velocity coupling is obtained by the SIMPLE algorithm,the pressure staggering option(PRESTO)and second order upwind scheme are used to discretize pressure and momentum discretization,respectively.The compressive scheme is selected for the interface reconstruction.A no-slip boundary condition is selected at the bottom of the computational domain,pressure outlet at the top of it and free-slip at the domain walls.The time step is 2.0×10-5s.

2.3.Numerical validation

To ensure the accuracy of the CFD model the predictions are compared with experimental data of Brereton and Korotney[26]for two separate cases:co-axial and oblique coalescence of two rising bubbles.

The computational domain in this work is a cuboid Lx×Ly×Lz,where Lxand Lyare the lateral dimensions of the domain in the horizontal direction,and Lzis the dimension in the vertical direction. Bubble-liquid system densities are ρg=10 kg·m-3and ρl=1000 kg·m-3.The experimental conditions considered are summarized in Table 1.

The results of the rising process of two in-line equal-sized bubbles are presented in Fig.1,at different time instants.Overall,a good agreement was achieved when compared with photographs of the experimental study.For t＞0.0 s the rapid deformation of the leading and the trailing bubbles can be observed. The initially spherical leading bubble changes to an ellipsoidal one, and the trailing bubble to bullet shape. At 0.012 s the trailing bubble touches the leading bubble which matches the fourth photograph in the study of Brereton and Korotney[26]included in the work of van Sint Annal and et al.[28]and Zhang et al.[29].For reason of clarity,the only difference is observed at the first-time instant.In the experimental work,the initial bubble shape is ellipsoid,while in our simulations is spherical.

Subsequently,the results of the oblique coalescence of two equalsized bubbles are given in Fig.2.It is clear that the duration of the process till the bubbles merge is different since more time is needed for the trailing bubble to catch-up with the leading one.Another aspect of the interaction is a different kind of shape deformation occurring at the trailing bubble,yielding much larger surface curvature compared to the leading bubble.Finally,another striking feature of the procedure is that the merged bubble is skirted-like and tilted to the right direction.Again,the CFD model could faithfully reproduce the flow field of the experimental study.

3.Computational Analysis Framework

The aim of the present analysis is to simulate a group of in-line pairs of equal and unequal-sized bubbles to observe the hydrodynamic behavior of the leading,the trailing,and the merged bubble.For this reason,a total of 28 configurations were selected,each one demanding a different computational domain and mesh resolution.Details are summarised in Table 2.

Selected properties of methane gas and seawater are the measured properties at 15MPa and 4.0°C,conditions matching deep-sea environment.Therefore,the density,surface tension and kinematic viscosity coefficients are ρl=1032.5 kg·m-3, ρg=138.58 kg·m-3, σ =0.0531 N·m-1,μl=0.0016576Pa·s and μg=1.71 ×10-5Pa·s,respectively,yielding a fluid-to-gas density ratio λ=100.46 and a viscosity ratio n=7.45.The distance between the center of the bubbles is 2.5 RT,where RTis the radius of the trailing bubble.

4.Results and Discussion

To facilitate the presentation of the simulation results of the hydrodynamic characteristics of the upward movement of the in-line pairs of methane bubbles they are divided into two sub-categories;equalsized bubbles and unequal-sized bubbles.The analysis includes figures of i)the rise velocity of the leading bubble and of a free rising single one of the same initial size against dimensionless time τ*ii)the dimensionless distance x*between the two bubbles and the velocity ratio of the two bubbles against τ*and,iii)the deformation A of the two bubblesagainst x*.The dimensionless distance is defined as x/RT,where x is the distance between the bubbles.The dimensionless time is given by τ*=where DTis the trailing bubble diameter.The velocity ratio is defined as uT/uL,where uTis the rise velocity of the trailing bubble and uLis the rise velocity of the leading bubble. The deformation A is defined as the ratio of the projection of the bubble to the vertical axis(h)and to the horizontal axis(w),i.e.h/w.

Table 1Parameters for the simulation of the co-axial and oblique coalescence

4.1.Rise velocity validation

The data presented in the experimental studies of Laqua et al.[27]were considered to validate the predictions of the CFD model for the rise velocity of small-sized methane bubbles.They presented results of the terminal rise velocity of methane bubbles of 2.0 and 3.0 mm diameter.The measured rise velocity was 0.29 m·s-1and 0.25 m·s-1,respectively.In the present work,predictions of the CFD model indicate a terminal velocity of 0.35 m·s-1and 0.3 m·s-1for a single methane bubble of initial diameter DS=2.0 and 3.0 mm.The predictions of the VOF-CFD model is in reasonable agreement with the experimental results,hence establishing the robustness of the prediction capabilities of the CFD model of this work.

4.2.Coalescence of equal-sized in-line bubbles

4.2.1.Leading bubble rise velocity

Curves of the rise velocity of the leading bubble are presented in Fig.3.All in all,two stages can be identified.The first one is the acceleration stage,for 0.0＜τ*＜1.0,where the leading bubble velocity rapidly increases until reaching a maximum value. In this stage, the rise velocity of the leading bubble is larger compared to the rise velocity of the free rising single bubble,which can be explained by the effect of the upward displacement of the flow field above the trailing bubble. This is more evident when the bubbles are either small or middle-sized;below 6 mm.This is clearly shown in Fig. 1(a),where the surplus of the rising velocity for DL=2.0,3.0and4.0mm is 0.15m·s-1,0.17m·s-1and0.05m·s-1,respectively.On the other hand,Fig.1(c)demonstrates that for bubbles between 7.0 and 10 mm the divergence between the values of the two curves at the acceleration stage is negligible suggesting that the behavior of the leading bubble is not affected by the trailing bubble.

After the acceleration stage,for τ* ＞ 1.0,the trailing bubble catches up with the leading bubble and they merge generating a daughter bubble.This is the collision stage and three cases are observed.The first refers to relatively small-sized pairs of bubbles with DL=2.0 and 3.0 mm,where the magnitude of the rise velocity of the leading bubble is smaller compared to the rise velocity of the free rising single bubble,as shown in Fig.1(a).In the second case,leading bubbles of DLbetween 4.0–6.0 mm rise 0.05 m·s-1faster compared to the free rising single ones,as shown in Fig.1(b).Finally,in the third case for bubbles of DLbetween 7.0 and 10 mm the rise velocity exhibits considerable fluctuation both for the leading bubble and the free rising single one,as shown in Fig.1(c).

In general,the trend of the curves for free rising single bubble size between 2.0 and 7.0 is similar to the one observed in the work of Dijkhuizen et al.(2005)[30].When bubble diameter is below 5.0 mm the rise velocity of the free rising single bubble exhibits minor oscillations.On the contrary,when bubble diameter is larger than 5.0 the rise velocity oscillates significantly.Additionally,results presented in this work agree with the results demonstrating that the rising velocity of a free rising single bubble does not exhibit a steady-state nature but rather oscillate,as recently reported by Yan et al.[18].Moreover,another striking effect observed in the in-line bubble pairs is that for small values of DLthe leading bubble did not behave as an isolated one and that the assumed steady-state nature of the velocity is not observed.Instead,rise velocity exhibits oscillations.According to the existing literature,it is believed that the trailing bubble has negligible influence on the movement of the leading bubble.In this regard,the leading bubble is expected to behave as a free rising single bubble of the exact same size.

Fig.1.Snapshots of the VOF model results of the co-axial coalescence at characteristic time instants.

4.2.2.Velocity ratio and bubble distance

Fig.4 illustrates the evolution of the dimensionless distance x*and the velocity ratio uT/uLas a function of dimensionless time τ*.As expected the distance between the in-line bubbles initially increases until it reaches a maximum value at x*=1.8.After this threshold,it decreases almost linearly.Correspondingly,the values of the velocity ratio uT/uLsuggest that shortly after τ*=0.0 the trailing bubble is accelerated at a lower rate.On the other hand,after τ*=2.0,the trailing bubble acceleration is sufficient to decrease the distance between the in-line bubbles.In more detail,Fig.4(a)displays that the maximum distance x*and the maximum values of uT/uLfor the pairs(DL×DT)=(2.0×2.0),(3.0×3.0)and(4.0×4.0)are 1.59,1.76,1.88,and 1.72,1.45,1.8,respectively.Fig.4(b)suggest that for(DL×DT)=(5.0×5.0),(6.0×6.0)the maximum distance x*is 2.03 and 2.16 and the maximum uT/uLis 1.57 and 1.61,respectively.Finally,for(DL×DT)=(7.0×7.0),(8.0×8.0),(9.0×9.0)and(10×10)the maximum distance x*is 1.83,1.77,1.46 and 1.6,while the maximumuT/uLis 1.98,2.05,2.5and 2.87,respectively.

4.2.3.Shape deformation

Fig.5 illustrates the shape deformation of the leading bubble(AL)and the trailing bubble(AT)plotted as a function of the dimensionless distance x*.In general,when x*reaches its maximum value the leading bubble suffers larger deformation in comparison with the trailing bubble.Moreover,the larger the leading bubble is the larger the deformation is.On the other hand,the deformation of the trailing bubble exhibits a more stable nature.In more detail,for initial bubble size between 2.0–4.0 mm when distance x*is maximum the minimum value of ALis 0.79,0.64,and 0.49,while the minimum value of ATis 1.0,0.7 and 0.49,respectively.For initial bubble size between 5.0 and 6.0 mm,the minimum value of ALis 0.48 and 0.47,while the minimum value of ATis 0.51 and 0.47,respectively.For initial size between 7.0–10 mm the minimum value of ALis 0.4,0.46,0.34 and 0.39,while the minimum value of ATis 0.49,0.52,0.48 and 0.85 respectively.

When x*decreases,just before bubbles collide and merge,ATis bigger compared to AL,especially for initial bubble size between 7.0 and 10 mm.This suggests that when the projected area of the bubble to the horizontal axis increases the velocity also increases.In more detail,for initial size between 2.0 and 4.0 mm,when the distance x*is minimum,ALis 1.0,0.73 and 0.63,while ATis 1.0,0.92 and0.63,respectively.For initial bubble size between5.0and6.0 mm ALis0.46and 0.56,while ATis 0.55 and 0.21,respectively.Finally,for initial bubble size between 7.0 and 10 mm ALis 0.35,0.33,0.4 and 0.33,while ATis 0.49,0.5,1.0 and 0.48,respectively.

Fig.2.Snapshots of the VOF model results of the oblique coalescence at characteristic time instants.

4.3.Coalescence of unequal-sized in-line bubbles

4.3.1.Leading bubble rise velocity

Results of the leading bubble rise velocity are plotted against τ*,as shown in Fig.6.In general,the acceleration stage is again observed for 0.0＜τ*＜1.0.Fig.6(a)illustrates that when DLis relatively small the maximum leading bubble rise velocity uLranges from 0.31 to 0.37 m·s-1at the acceleration stage and,for τ* ＞ 1.0 uLexhibits strong fluctuation.Fig.6(b)and(d)demonstrate that uLat the acceleration stage is greatly affected only when the trailing bubble initial size is DT=10 mm.For the rest of the cases,its value is nearly 0.32 m·s-1.Moreover,uLfluctuation after the acceleration stage diminishes and its magnitude is larger than the rise velocity of the corresponding single bubble.Finally,for DL=7.0–9.0 mm the acceleration of the leading bubble is proportionate with the acceleration of the corresponding isolated bubble.After τ* ＞ 1.0,the rise velocity gradually decreases and exhibits fluctuation.

4.3.2.Velocity ratio and bubble distance

The evolution of the distance x*and the velocity ratio uT/uLare presented in Fig.7.Maximum distance x*for pairs of(DL×DT)=(2.0×3.0),(3.0×4.0),(4.0×5.0),(4.0×6.0)and(4.0×7.0)is 2.03,1.33,1.83,1.59 and 1.51,while the maximum velocity ratio is 1.47,1.6,1.4,1.46 and 1.71,respectively,as shown in Fig.7(a).For pairs of(DL×DT)=(5.0×6.0),(5.0×7.0),(5.0×8.0),(5.0×9.0)and(5.0×10)the maximum value of x*is 1.24,1.34,1.43,1.34 and 1.4,while the maximum velocity ratio is 1.71,1.7,1.86,1.75 and 1.82,respectively.In comparison with results presented in Fig.5(a)the distance x*is decreased because of the bigger acceleration of the trailing bubble.For pairs of(DL×DT)=(6.0×7.0),(6.0×8.0),(6.0×9.0)and(6.0×10)the maximum value of x*is 1.27,1.49,1.48 and 1.38,while the maximum velocity ratio is 1.84,1.85,1.79 and 2.53,respectively.Finally,Fig.7(d)illustrates results for pairs of(DL×DT)=(7.0×8.0),(7.0×9.0),(7.0×10),(8.0×9.0),(8.0×10)and(9.0×10).The maximum value of x*is 2.33,1.5,1.49,1.7,1.46 and 1.78,while the maximum velocity ratio is 1.78,2.05,2.33,1.93,2.63 and 2.18,respectively.From the simulation results it is clear that an increase in DTfavors velocity ratio.Finally,it is evident that when DLis larger than 6.0 mm there is a time windowwhen distance x*remains almost constant and the velocity ratio is also stable.

Table 2Parameters for the simulation of the methane bubbles in-line coalescence

4.3.3.Shape deformation

The simulation results of the shape deformation of the in-line bubbles plotted against x*are presented in Fig.8.In general,when DLis between 2.0–4.0 mm and x*is maximum the divergence between the values of the shape deformation narrows. This is clearly demonstrated for pairs of(DL×DT)=(3.0×4.0),(4.0×5.0)and(4.0×7.0)mm in Fig.8(a).On the other hand,when the distance x* decreases before bubblemerge the leading bubble exhibit smaller shape deformation almost in every case. On top, the divergence between values of ATand ALis larger showing that the trailing bubble is approaching the leading one faster due to the larger projected area.For the following pairs of(DL×DT)=(2.0×3.0),(3.0×4.0)and(4.0×7.0)mm the difference between ATand ALis 0.13,0.43 and 0.18,respectively.

Results,when DLincreases to5.0 mm,are shown in Fig.8(b).In general,values of ALand ATare between 0.27–0.64and0.46–0.6,respectively,when x*is maximum.Before the collision,values of ALand ATare between 0.45–0.57 and 0.46–0.67,respectively.Apparently,the behavior of the unequal and equal-sized in-line bubbles is similar at both the acceleration and the collision stage keeping in mind the results presented in Fig.5(b).In the former,values of ALare smaller compared to ATand the divergence ranges from 0.1 to 0.26,while in the latter the divergence ranges from 0.07 to 0.18.Clearly,the larger the trailing bubble is the more the leading bubble is affected by the upward movement of the flow field above the trailing bubble.

Further increase of the leading bubble to 6.0 mm is studied and results are presented in Fig.8(c).In the acceleration stage values of ALand ATare between0.4–0.52and0.44–0.69,respectively.In the collision stage values of ALand ATare between 0.42–0.52 and 0.6–0.72,respectively.It is observed that the shape deformation of the leading bubble is stable after the acceleration stage while the trailing bubble increases its speed due to the increase in the projected area.

Fig.8(d)illustrates the simulation results for the rest of the cases with DLbetween 7.0 and 9.0 mm.In the acceleration stage values of ALand ATare between 0.34–0.36 and 0.46–0.57 for DL=7.0 mm,0.38–0.45 and 0.58–0.6 for DL=8.0 mm,respectively.In the collision stage values of ALand ATrange between 0.33–0.39 and 0.6–0.61 for DL=7.0 mm,0.27–0.37 and 0.5–0.67 for DL=8.0 mm,respectively.The results demonstrate that the shape deformation of the leading bubble at the acceleration stage is akin despite the size of the trailing bubble.On the other hand,at the end of the collision stage the discrepancy between ATand ALincreases extending between 0.22–0.3 for DL=7.0 mm and 0.23–0.35 for DL=8.0 mm.The impact of the upward movement of the liquid flow field above the trailing bubble is evident considering that the values of ALat the end of the acceleration and the collision stage are almost equivalent.

Fig.3.Evolution of the rise velocity of the leading bubble of equal-sized in-line pairs of bubbles and the corresponding free rising single bubble for λ=100.46 and n=7.45.

Fig.4.Evolution of the dimensionless distance x*and velocity ratio uT/uLof equal-sized in-line pairs of bubbles for λ=100.46 and n=7.45.

4.4.Flow field analysis

4.4.1.Maximum distance and coalescence time

From the flow field analysis,two distinctive cases are observed in terms of the effect of the size of the leading bubble to the distance evolution and the necessary coalesce time.In the first one,for DT=6.0 and 7.0 mm,when the leading bubble size increases maximum distance between the in-line bubbles increases.The range of the increase is between 16.35%–35.85%when DT=6.0 mm and between 10.45–73.88%when DT=7.0 mm.On top of this,the coalescence time increment is 12.5%when DT=6 mm and 3.64%when DT=7.0 mm.In the second case,when DTis between 8.0–10 mm the increase of maximum distance is accompanied by a decrease in coalescence time.Increment of x*ranges between 10.45% –32.09%when DT=8.0 mm,9.77%–27.07%when DT=9.0 mm,and 2.14%–27.14%when DT=10 mm.Additionally,coalescence time declines up to 3.45%when DT=8.0 mm,between 4.55%–30.3%when DT=9.0 mm and 2.31%–18.46%when DT=10 mm.

4.4.2.Vortex shedding

Fig.5.Evolution of the shape deformation A of equal-sized in-line pairs of bubbles before merge for λ=100.46 and n=7.45.

Fig.6.Evolution of the rise velocity of the leading bubble of unequal-sized in-line pairs of bubbles for λ=100.46 and n=7.45.

The coalescence process of two in-line bubbles for two distinctive cases is depicted in Figs.9 and 10.In the first case,DL=4.0 mm and DT=6.0 mm,bubbles start to rise at t=0.0 s due to buoyancy force.Subsequently,a wake is formed behind each bubble.The trailing bubble is immersed in the wake of the leading one and interacts with the wake flow.It can be observed in Fig.9(a)that the bottom of the leading and the trailing bubble is flattened whereas the top remains spherical.At that point,there are two vortex rings around each bubble.The acceleration of the leading bubble is clearly presented in Fig.9(b).The rise velocity of the leading bubble is larger than the rise velocity of the trailing one.It is also observed that the vortex rings of the leading bubble are gradually moving downwards.Next,the trailing bubble appears to catch-up with the leading bubble.The vortexes on the ipsilateral side start to interact with each other enhancing the coalescence progress,as shown in Fig.9(c).Another interesting feature is the shape of the bottom of the leading bubble,which is prolonged in the vertical axis.This sharp shape is caused by the low velocity of the flow field at the bottom of the leading bubble.The same feature is repeated in Fig.9(d)in the bottom surface of the trailing bubble as it gets close to the leading bubble and before they collide.It has to be noted that this kind of bubble shape is rarely presented in past literature results.

Fig.10shows the coalescence process of two bubbles with equal sizes,DL=DT=6.0 mm.They initially deform into an ellipsoidal shape,their bottom surface is flattened,and two vortexes appear on each side of the bubbles,as shown in Fig. 10(a). The trailing bubble rises faster and catches up with the leading bubble,which is flattened by the trailing bubble that dominates the coalescence process due to its large volume,high rise velocity and strong velocity field induced.At the acceleration stage,both bubbles retain their ellipsoidal shape,as shown in Fig.10(b).The thickness of the liquid film between the two bubbles is sufficient to prevent the strong interaction between the wake of the leading bubble and the upper surface of the trailing bubble.When this happens,the trailing bubble rise velocity is much larger and two vortexes on the ipsilateral sides of the bubbles appear,as presented in Fig.10(d).

Fig.7.Evolution of the dimensionless distance x*and velocity ratio uT/uLof unequal-sized in-line pairs of bubbles for λ=100.46 and n=7.45.

Fig.8.Evolution of the shape deformation A of unequal-sized in-line pairs of bubbles for λ=100.46 and n=7.45.

4.4.3.Bubble radius history

Another aspect of the evolution of the in-line bubble shape is the impact of the trailing bubble size to the deformation of the leading bubble.For that,time history results of the evolution of the projected diameter to the horizontal level of four indicative cases,when DL=4.0–7.0 mm,are illustrated in Fig.11.For the sake of comparison,time history of the projected diameter of the corresponding free rising single bubble is also plotted.This way it is easier to demonstrate the difference in the bubble shape evolution due to the presence of the trailing bubble.For the leading bubble, the projected diameter increases until reaching a maximum.After this threshold,a decrease is observed.On the contrary,the values of the free rising single bubble after a further short increase remain almost constant.This is clearly presented when DL=4.0 and 5.0 mm,and particularly after τ*=2.0.When DL=6.0 and 7.0 mm the above described behavior is observed only when the size of the trailing bubble is much larger,otherwise,the leading bubble projected diameter value oscillates.This is also evident for(DL×DT)=(6.0×6.0),(7.0×8.0)and(7.0×9.0).

Fig.9.Evolution of bubble shapes and instantaneous streamlines for the in-line coalescence of two bubbles with different size(DL=4.0 mm DT=6.0 mm)at times(a)t=0.02,(b)0.06,(c)0.12,(d)0.155 s.

Fig.10.Evolution of bubble shapes and instantaneous streamlines for the in-line coalescence of two bubbles with the same size(DL×DT)=(6.0×6.0)mm at times(a)t=0.025,(b)0.065,(c)0.0875,(d)0.125 s.

5.Conclusions

To the best of our knowledge experimental results of methane bubble rise velocity and shape deformation at high-pressure conditions are scarce.In the present study,the motion and coalescence process of equal and unequal-sized methane in-line bubbles were numerically investigated using the VOF method.In general,conclusions of the present work are as follows:

(1)A set of 28 cases of methane in-line bubbles rising at high-pressure conditions were studied. The terminal rising velocity of small-sized leading methane bubbles was compared with experimental data available in the literature and was found in reasonable agreement.

(2)The assumption that the leading bubble is moving like an isolated one is not valid form ethane bubbles of initial size between 2.0 and 7.0 mm at high-pressure conditions.When DL=2.0 and 3.0 the rise velocity of the leading bubble was smaller compared to the rise velocity of the corresponding isolated bubble.On top,shape deformation is less severe when DLis between 4.0 and 7.0 mm.It was also observed that for some cases leading bubble equivalent diameter exhibited oscillations.

Fig.11.Evolution of the equivalent diameter of the leading bubble when DL=4.0–7.0 mm.

(3)When DT=6.0 and 7.0 mm,the maximum distance between the in-line bubbles and the coalescence time increase if the leading bubble size increases.When DTis between 8.0 and 10 mm the coalescence time decreases.

(4)The trend of the projected diameter curve of the leading bubble is different from the trend of the corresponding free rising single bubble when DL=4.0 and 7.0 mm.On top,the larger the DLis the projected diameter exhibits oscillations.

Nomenclature

ALleading bubble deformation

ATtrailing bubble deformation

Dedrop equivalent diameter,mm

DLleading bubble,mm

DSFree rising single bubble,mm

DTtrailing bubble,mm

EoE?tv?s Number(Eo=gΔρ/σ)

g acceleration due to gravity,m·s-2

MoMorton number(Δρ/σ3)

n viscosity ratio n= μl/μg

R bubble radius

Re Reynolds number(Re= ρluDe/μl)

RTtrailing bubble radius

t time,s

uLleading bubble rise velocity,m·s-1

uTtrailing bubble rise velocity,m·s-1

x distance between the bubbles,mm

x* dimensionless distance x*=x/RT

Δρ density difference Δρ = ρl- ρg

λ fluid-to-bubble density ratio λ = ρl/ρg

μggas viscosity,Pa·s

μlliquid viscosity,Pa·s

ρggas density,kg·m-3

ρlliquid density,kg·m-3

σ interfacial tension,N·m-1