Applied Mathematics and Mechanics Numerical simulation of bubble ...

Appl. Math. Mech. -Engl. Ed. 31(4), 449–460 (2010) DOI 10.1007/s10483-010-0405-z c Shanghai University and Springer-Verlag Berlin Heidelberg 2010

Applied Mathematics and Mechanics (English Edition)

Numerical simulation of bubble breakup phenomena in a narrow flow field∗ A-man ZHANG (), Bao-yu NI (), Bing-yue SONG (), Xiong-liang YAO () (School of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, P. R. China) (Communicated by Chuan-jing LU)

Abstract Based on the boundary integral method, a 3D bubble breakup model in a narrow flow field is established, and a corresponding computation program is developed to simulate the symmetrical and asymmetrical bubble breakup. The calculated results are compared with the experimental results and agree with them very well, indicating that the numerical model is valid. Based on the basic behavior of bubbles in a narrow flow field, the symmetrical and asymmetrical bubble breakup is studied systematically using the developed program. A feasibility rule of 3D bubble breakup is presented. The dynamics of sub-bubbles after splitting is studied. The influences of characteristic parameters on bubble breakup and sub-bubble dynamics are analyzed. Key words

bubble, breakup, narrow flow field, sub-bubble, narrow jet

Chinese Library Classification O351.2 2000 Mathematics Subject Classification

1

76B07

Introduction

Bubble breakup is a very common phenomenon in the nature. Unsteady bubble dynamics in the breakup process is always a hot subject for two-phase (gas-liquid) flow mechanics. The reason for bubble breakup is usually complex, and some main reasons are as follows: First, a bubble in a vortex field may split into small sub-bubbles because the vortex core radius, circulation, and pressure distribution vary along the length of the vortex[1]. Second, if the volume of a bubble is so large that the surface tension cannot sustain its shape, the collapsing or splitting will break out[2] . Third, a steady bubble will be separated by the disturbance of other objects; for reference, in [3], the breakup of a steady bubble was simulated when meeting different floating bubble breakers. Besides, a bubble near a rigid wall would split into two small sub-bubbles when the Bjerknes force equals the reverse buoyancy[4] . Much research has been done to explore the phenomena and essence of bubble breakup. Experimentally, Ishida et al.[5] simulated the expansion, necking, splitting, and rebound of a bubble between two horizontal walls in a water tank. Choi and Chahine[6] studied the splitting ∗ Received Oct. 9, 2009/ Revised Mar. 4, 2010 Project supported by the National Natural Science Foundation of China (No. 50779007), the International Science and Technology Cooperation Project (No. 2007DFR80340), the National Science Foundation for Young Scientists of China (No. 50809018), and the Postdoctoral Science Foundation of China (No. 200801104) Corresponding author Bao-yu NI, Ph. D., E-mail: [email protected]

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of a bubble between two vertical and oblique walls and observed the influence of buoyancy in the late necking phase. On numerical simulations, many researchers[7−9] all studied behaviors of a bubble between two parallel walls and found that the bubble may present a sandglass shape. Mohammad et al.[10] and Choi and Chahine[6] brought forward the axisymmetric bubble breakup rule and erected the symmetrical and unsymmetrical breakup models for the axisymmetric bubble. Wang et al.[11] analyzed different bubble breakups and coalescences with a computational fluid dynamics-population balance model (CFD-PBM) coupled model, in which the effect of gas-liquid transfer was involved. Based on the former work, the 3D bubble breakup phenomena in a narrow flow field are studied, and the symmetrical and asymmetrical bubble breakup models are established and a feasible 3D splitting rule is put forward. Especially, the change of flow and the dynamics of sub-bubbles are studied in this paper.

2

Theoretical background

The potential flow theory is applied to calculate the motion and breakup of a bubble in the narrow flow field formed by several close rigid walls. The detail procedure to simulate the bubble motion with the potential flow theory can be referred to [4, 12], and the changes of the physical quantities such as the potential and pressure are focused on in this paper. It is assumed that the bubble generated by electrical discharge contains non-condensable gas and liquid vapor. The non-condensable gas is assumed to satisfy a polytropic law with an exponent γ, and the surface tension is considered here. Thus, the pressure outside the bubble is  V γ 0 − σ∇ · n, (1) P = Pc + P0 V where Pc is the saturated vapor pressure of the condensable gas, P0 and V0 are the initial pressure and volume when the bubble is formed, γ is the ratio of specific heat of the gas, σ is the surface tension, and ∇ · n is the local surface curvature. The pressure inside the bubble just before the split is PB = Pc + P0 (V0 /VB )γ through the polytropic law. Here, the mass and energy losses in the splitting are ignored, and the pressure outside the sub-bubbles also satisfies the assumption of (1), that is, V  V γ  V γ γ  sub,A 0 sub,A − σ(∇ · n)sub = Pc + Pc + P0 − σ(∇ · n)sub , Psub = Pc + PB Vsub (t) VB Vsub where the subscript “B” represents the quantity just before the split and “A” for that just after the split, and the subscript “sub” represents the quantity of each sub-bubble after the split. The sum of Vsub,A over all sub-bubbles should be equal to VB , unless the process involves gas loss that is not considered in the present study. All the parameters are adopted in the dimensionless form and are scaled by Rmax for length, P∞ for pressure, Rmax (ρ/P∞ )1/2 for time, (P∞ /ρ)1/2 for velocity, and Rmax (P∞ /ρ)1/2 for velocity potential, respectively. Here, P∞ is the hydrostatic pressure at an infinite point in the same horizontal plane with the initial bubble center, Rmax is the maximum radius that the bubble would attain in an infinite fluid domain under P∞ , and ρ is the density of the fluid. The dynamic boundary conditions of bubble, sub-bubbles, and walls are given in the following dimensionless Bernoulli equations:  V γ dφ 1 0 = 1 + |∇φ |2 − δ 2 z  − ε − Pc + β∇ · n (bubble surface), (2)  dt 2 V       γ γ dφsub Vsub,A 1 V0  = 1 + |∇φsub |2 − δ 2 zsub − Pc + ε − Pc + β(∇ · n)sub dt 2 VB Vsub (sub-bubble surface), (3)

Numerical simulation of bubble breakup phenomena in a narrow flow field

1 dφ = 1 + |u  |2 − δ 2 rf − Pw dt 2

(wall surface),

451

(4)

where δ = (ρgRmax /P∞ )1/2 is the buoyancy parameter, ε = P0 /P∞ is the strength parameter, β = σ/(Rmax P∞ ) is the surface tension parameter, and rf = d/Rmax is the dimensionless distance between the initial bubble center and wall, namely, distance parameter. The dimensionless kinetic boundary condition of the bubble surface is dx /dt = ∇φ . To determine the initial condition, for underwater explosion bubbles, the strength parameter ε and the dimensional Rmax can be obtained through empirical formula[13] ; for cavitations, the initial parameters can be obtained through [14].

3

3D bubble breakup model

The numerical solving method of axisymmetric bubble breakup can refer to [6, 15]. A 3D breakup model is established in this paper and a splitting rule is presented. As described above, bubble breakup is a very complex and unsteady dynamic physical process because there exist gas diffusion[1] , gas-liquid transfer[11] , energy loss, and pressure discontinuity[6] in this process. In order to sustain the calculation, the actual physics of splitting is simplified; that is, when the nodes of bubble satisfy the breakup rule, the meshes and nodes are redistributed directly to form two sub-bubbles as shown in Fig. 1. The details are given as follows:

Fig. 1

Sketch map of bubble breakup rule

(a) Determine the local “symmetric axis”. As shown in Fig. 1, there exist an obvious “necking” part and a corresponding local “symmetric axis” in the splitting process (for the bubble near a wall, the local “symmetric axis” is normal to the wall). (b) Calculate the minimum distance. The dimensionless minimum distance r between the node and the axis is calculated each time-step until r  0.03, which is selected as the splitting critical value. The detecting procedure is undertaken by the computer automatically. (c) Concentrate the “breakup line” to a single point J. Once the node I corresponding to the minimum distance r is found, a circuit of nodes (named “breakup line”) including I are removed artificially, and a new node J is generated by taking the average value of these nodes. (d) Separate the point J into J1 and J2 . Two new nodes named J1 and J2 are produced through moving J a small distance 0.001 along the axis forward and backward. Connect the original separated nodes with nodes J1 and J2 , respectively. The node and segment indices as well as the bubble index are rearranged. (e) Calculate the physical quantities on the new points. Many variables including the potential and velocities need to be extrapolated to the new position, and a bubble is separated into two sub-bubbles successfully. The potentials and the locations of sub-bubbles are updated through (3) and dx /dt = ∇φ after splitting.

4

Results and discussions

4.1 Comparison between numerical and experimental results The validity of the numerical model is verified by comparing numerical results with experimental data in [5]. Experimental parameters are described briefly as follows: Two parallel

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A-man ZHANG, Bao-yu NI, Bing-yue SONG, and Xiong-liang YAO

round coppers that could move up and down were placed in the water tank, and the bubble was produced by the electrical discharge at the axis center of the two circular plates. Bubble behaviors were photographed by the high-speed movie camera at the framing rate of 41 000 flames per second. The 3D breakup model is adopted in the numerical simulation and the bubble surface is divided into 362 nodes and 720 elements, while each wall is divided into 1 100 nodes and 2 196 elements. The fluid in the horizontal direction is infinite and the boundary condition satisfies the infinite distance condition. 4.1.1 The symmetric bubble split The maximum radius of the bubble in this condition is Rmax = 6.93 mm, and the distance between the two plates is L = 15 mm, each 7.5 mm apart from the center of the initial bubble, namely, rf1 = rf2 = 1.08. Dimensionless parameters of the initial bubble are R0 = 0.078 8, ε = 600, δ = 0.026, and β = 0.000 1, and the calculation results are also given in the dimensionless form. In order to contrast better, the numerical results are compared with the experimental ones as simultaneously as possible. The specific process is shown in Fig. 2 with the corresponding dimensionless time marked below.

Fig. 2

Contrast between calculating results and experimental data[5] for symmetric bubble split

The contrast in Fig. 2 indicates that the calculated results are in good agreement with the experimental pictures at different stages, in which the two straight lines stand for the rigid walls. In Fig. 2(a), the bubble just starts to expand, motivated by the internal high-temperature and high-pressure gas. Figure 2(b) shows that the bubble continues to expand under the inertial effect. Figure 2(c) illustrates the bubble shape at t = 1.09; at this moment, the bubble volume reaches its maximum and the bubble shape is slightly flattened by the upper and lower walls. Figure 2(d) shows that the bubble begins to elongate in the vertical direction due to the Bjerknes force of the rigid walls. Figures 2(e)–2(g) show that the bubble begins to shrink rapidly in the mid-part while its vertical length remains almost unchanged. In Fig. 2(h), the bubble

Numerical simulation of bubble breakup phenomena in a narrow flow field

453

central “necking” gets to the smallest, forming a typical dumbbell-shaped bubble. Figure 2(i) illustrates the bubble shape at t = 2.43; at this moment, the bubble completely splits into two sub-bubbles, despite the fact that there is a fine water column link between the two subbubbles in the experiment, which may result from the surface tension. Figure 2(j) illustrates the bubble shape at t = 2.45; at this moment, the sub-bubble jet penetrates the other side to form a ring bubble. Unfortunately, there is no corresponding picture from the experiment to contrast. Besides, it is noteworthy that the sub-bubble jet is a typical “narrow-jet” and the jet top presents the teardrop-shape, which is different from the jet formed in a single bubble near the wall[4] . In short, the bubble moves away from and towards the wall during the expansion phase and the collapse phase, respectively. The bubble splits into two sub-bubbles symmetrically due to the equal Bjerknes force of the walls, and then, the sub-bubbles collapse and form the liquid jets, respectively. 4.1.2 The asymmetric bubble split The maximum radius in this condition is Rmax = 6.73 mm, and the distance between the upper and lower plates is L = 15 mm, with 6 mm and 9 mm apart from the center of the initial bubble, respectively, namely, rf1 = 0.89 and rf2 = 1.34. The dimensionless parameters of the initial bubble are the same as in Section 4.1.1, and the calculation results are also given in the dimensionless form. The specific process is shown in Fig. 3.

Fig. 3

Contrast between calculating results and experimental data[5] for asymmetric bubble split

Figure 3(a) shows that the bubble expands rapidly for the internal high temperature and high pressure. Figure 3(b) expresses that the bubble continues to expand and the upper part is slightly flattened. Figure 3(d) illustrates the bubble shape at t = 1.12; at this moment, the bubble volume reaches its maximum value. Figures 3(e) and 3(f) show that the upper bubble surface nearly sticks to the wall and the lower part begins to elongate in the vertical direction.

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A-man ZHANG, Bao-yu NI, Bing-yue SONG, and Xiong-liang YAO

Figure 3(g) illustrates the bubble shape at t = 2.25; at this moment, the lower part of the bubble begins to shrink and collapse rapidly. Figure 3(h) illustrates that the bubble “necking” gets to the smallest, forming a typical conic-shaped bubble. Figure 3(i) illustrates the bubble shape at t = 2.43; at this moment, the bubble splits into two sub-bubbles in different volumes. Magnify Fig. 3(i) locally in Fig. 3(j) to observe the sub-bubble shape clearly. It can be seen that the large sub-bubble forms a upward jet while the small one forms an opposite jet under the flow field pressure, despite that the small sub-bubble has already broken down in the experiment. In conclusion, the different strengthes of Bjerknes force cause the bubble to split asymmetrically, and the two different sub-bubbles collapse and jet under different dominant factors. 4.2 Bubble breakup and jet phenomena in the narrow flow field The strong non-linear behaviors during the bubble breakup and the dynamics of sub-bubbles are studied here by simulating axisymmetric bubble behaviors between two close walls. Some physical characteristic parameters are recorded and the basic physical phenomena are explained: 4.2.1 Symmetrical breakup The initial dimensionless parameters of the bubble are taken as R0 = 0.353, ε = 10, δ = 0, and β = 0.000 1. The dimensionless distances between the bubble and two walls are both rf1 = rf2 = 1. The numerical simulation of the bubble motion and the velocity and pressure contours around the narrow flow field are shown in Fig. 4.

Fig. 4

Velocity and pressure contours during the symmetrical bubble breakup

In Fig. 4, the arrows indicate the velocity vectors of the flow field, the contours indicate the non-dimensional pressure in the flow field, and the two thick lines stand for the walls. It can be seen in Fig. 4(a) that the high-pressure gas within the bubble promotes the fluid particles to expand rapidly. Figure 4(b) shows the bubble shape at its largest volume; at this moment, the excessive expansion makes the internal pressure less than the hydrostatic pressure, which causes the bubble to shrink. Figure 4(c) shows that the fluid near the walls stays static basically, while the fluid in the mid-field moves to the bubble quickly under the external high pressure and the bubble starts “necking”. Figure 4(d) illustrates that the bubble length in the vertical direction remains basically unchanged; meanwhile, the pressure near the middle part starts to rise and the fluid particles move inwards, which will lead to the breakup of the bubble inevitably. Figure 4(e) shows that the sub-bubble after the bubble breakup begins to form a liquid jet, whose velocity is pretty high, and the energy of the flow field is concentrated in the narrow jet area. Figure 4(f) shows that the high-speed jet of the sub-bubble continues to develop, and is about to penetrate the other side to form a ring bubble.

Numerical simulation of bubble breakup phenomena in a narrow flow field

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To further analyze the bubble dynamics of breakup, the time history curves such as the pressure on the walls and in the flow field, the flow field energy, and the sub-bubble jet speed are given below: Figure 5 illustrates the time history of the dimensionless pressure on the center point of the wall. The pressure of the upper wall is equal to that of the lower one due to the symmetrical breakup. The non-dimensional time when the bubble breakup happens is figured out in the circle. It can be seen from the local magnification that the bubble breakup will cause a pressure jump and the flow field will get stable again after the formation of sub-bubbles. Figure 6 illustrates the time history of the dimensionless pressure at the point (0, 1.5, 0) in the flow field. There is also a pressure jump in the flow field similar to that on the wall. The pressure jump phenomenon was also measured experimentally by pressure sensors in both [5] and [6]. Although the pressure peak is relatively small in comparison with that induced by the jet impact of the sub-bubble, it does exist. Besides, the change of the artificial bubble size in the numerical simulation of bubble breakup will result in an instantaneous pressure increase inevitably. Thus, the sudden pressure jump in this figure also partly results from the numerical treatment.

Fig. 5

Time history of pressure on the wall

Fig. 6

Time history of pressure in the flow field

Figure 7 is the dimensionless energy curves of the flow field, including the total energy, potential energy, and kinetic energy from top to bottom, respectively. It can be seen that there is only a transformation between the kinetic and potential energy in the phase of the bubble motion when the energy loss is ignored. When the bubble splits, there is a small loss of the kinetic energy, resulting in the total energy loss a little. Figure 8 is the the dimensionless velocity curve of the sub-bubble jet, showing that the sub-bubble jet velocity quickly reaches a peak before it tends to a stable value gradually.

Fig. 7

Energy curves of the flow field

Fig. 8

Velocity curve of sub-bubble jet speed

4.2.2 Asymmetrical breakup The initial dimensionless parameters of the bubble are taken same as Section 4.2.1. The dimensionless distances from the bubble to the upper and lower walls are taken as rf1 = 1.25

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A-man ZHANG, Bao-yu NI, Bing-yue SONG, and Xiong-liang YAO

and rf2 = 0.75, respectively. The numerical simulation results are shown in Fig. 9. Figure 9(a) illustrates the rapid outward expansion of the bubble. It can be seen that the pressure up and down is obviously unsymmetrical. Figure 9(b) illustrates the bubble at its maximum volume, and the lower surface of the bubble has become flattened and squeezes the fluid nearby outwards. Figure 9(c) shows that the fluid particles move towards the bubble driven by the external high pressure. Figure 9(d) illustrates the bubble necking to its smallest; at this moment, the pressure near the necking region starts to rise and the bubble will split under the fluid inertia. Figure 9(e) illustrates that the sub-bubbles begin to form jet with a pretty high speed, causing that the fluid velocity outside the jet region is relatively small. Figure 9(f) illustrates the high-speed jet of the sub-bubble continues to develop, and the upper sub-bubble’s jet velocity is a little faster than that of the lower one, which results in that the upper sub-bubble penetrates the other side prior to the lower one.

Fig. 9

Velocity and pressure contours during the asymmetrical bubble breakup

To further analyze the bubble dynamics of breakup, the pressure curves at the upper and lower walls, the Kelvin impulse curve as well as the sub-bubble jet velocity curve are given as follows, respectively. Figures 10 and 11 illustrate the time history curves of the dimensionless   and Pw2 on the center points of the upper and lower walls, respectively. It can pressure Pw1   be seen that Pw2 is larger than Pw1 initially due to the closer distance, but both of them decrease rapidly with the bubble expansion, falling to about 0.2 after the dimensionless time  increases faster and greater t < 0.5. Then, the pressure starts to rise after t > 3 and Pw1  than Pw2 . Similar to the symmetrical breakup, there is a sharp pressure peak, and at the time  is higher than the bubble splits. It can be seen from the figure that the pressure peak of Pw1  that of Pw2 , which is probably because the lower sub-bubble attaches to the lower wall and affects the propagation of the pressure disturbance. As the numerical simulation ceases until the jet penetrates the other side of the upper sub-bubble, while the lower sub-bubble jet is still   is significantly larger than Pw2 at the final time. developing, Pw1 Figure 12 illustrates the time history of the dimensionless Kelvin impulse K  in the flow field, and the Kelvin impulse can be referred to [16]. K  stays negative before the bubble breakup, expressing that the asymmetry of the bubble overall deflects to the bottom due to its bigger Bjerknes force. K1 and K2 of the upper and lower sub-bubbles are shown with a local magnification after the bubble breakup, respectively. It can be obtained that K1 > 0 and

Numerical simulation of bubble breakup phenomena in a narrow flow field

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K2 < 0. Figure 13 illustrates the time history of the dimensionless jet velocity of sub-bubbles,   Vjet1 for the upper one and Vjet2 for the lower one. It is evident that they both reach the peak   to 18.86 at t = 3.304, indicating that the sharply, Vjet1 rising to 15.84 at t = 3.300 and Vjet2 upper sub-bubble gets to the higher peak a little later than the lower one. After that, both jet velocities reduce and bottom out, approaching the same value 13.06 in this working condition.

Fig. 10

Fig. 12

Time history of pressure on upper wall

Time history of Kelvin impulse

Fig. 11

Fig. 13

Time history of pressure on lower wall

Time history of sub-bubbles jet speed

4.3 Influence of characteristic parameters on bubble breakup As mentioned above, the distance between the bubble and the walls has strong influences on the dynamics of bubble breakup. The influence rule is studied by changing the distance parameter rf systematically in this section. For the influence of other parameters, refer to [17] and [18]. 4.3.1 Symmetrical breakup Take ε = 10, β = 0.000 1, and δ = 0. Change rf1 = rf2 = rf from 0.75 to 2.0 referred to [10]. The curves of the bubble splitting time t1 , the sub-bubble jet impact time t2 , the bubble total volume, and the pressure inside the bubble Pb as well as the retarded flow velocity V  are given below (all the physical quantities are dimensionless). It is evident in Fig. 14 that the bubble splitting time t1 decreases with the increase of rf , which indicates that the closer the wall from the bubble, the stronger the restraining effect and the longer the bubble expansion and necking. The sub-bubble jet impact time t2 first declines and then ascends as rf increases. It is because that the side of the sub-bubble near the wall begins the secondary expansion when rf is too large, inducing the growth in the overall time. Numerical calculations show the turning point occurring somewhere in the vicinity of rf = 1.65. In Fig. 15, the arrows indicate the increasing direction of rf , and the six curves correspond to rf =0.75, 0.875, 1.0, 1.25, 1.5, and 2.0, respectively. It can be seen that the larger rf is, the

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earlier the bubble reaches its maximum volume. For rf = 2.0, the bubble reaches the minimum volume at t = 2.749 5 before it begins its secondary expansion. Figure 16 illustrates the time history of the pressure inside the bubble. The larger rf is, the earlier the pressure gets to the peak and the greater the peak is. If the energy loss and the effect of the wall are both ignored, the secondary peak pressure should keep same with the initial pressure. Taking the wall influence into account, the secondary peak becomes weaker as the wall gets closer. Figure 17 illustrates the retarded flow velocity at (0, 1.5, 0) in the field. It can be obtained that the direction of the retarded flow reverses with the bubble expansion and collapse, and the bubble breakup will not affect the direction of the retarded flow. Besides, the retarded flow velocity descends with the increasing rf , showing that the compression of walls will raise the speed of the narrow flow field.

Fig. 14

Curves of bubble splitting time and sub-bubble jet impact time

Fig. 15

Time history of bubble total volume(→ increasing direction)

Fig. 16

Time history of pressure inside the bubble (→ increasing direction)

Fig. 17

Time history of retarded flow velocity (→ increasing direction)

4.3.2 Asymmetrical breakup Take ε = 10, β = 0.000 1, and δ = 0. Keep rf1 + rf2 = rf = 2.0 and alter rf1 /rf2 from 0.5 to 1.0. The curves of the sub-bubble volume ratio, the bubble splitting time t1 , the jet impact time t2 , and the Kelvin impulse K  as well as the bubble center displacement Zc are given, respectively (all the physical quantities are dimensionless). In Fig. 18, V2b /V1b stands for the volume ratio of the smaller sub-bubble to the larger one at the time of splitting, and V2a /V1a represents the volume ratio at the time of jet impact. It can be obtained that the bubble splits more asymmetrically as rf1 /rf2 gets smaller. Figure 19 illustrates the changes of the bubble splitting time t1 and the jet impact time t2 . On the one hand, it shows that the larger rf1 /rf2 is, the earlier the bubble breakup happens. On the

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other hand, the jet impact time in this paper is defined until the small sub-bubble jet impact happens. Thus, the smaller rf1 /rf2 is, the earlier the jet impact occurs. In fact, the variation range of t1 and t2 is very small in different working conditions. Figures 20 and 21 illustrate the time histories of the Kelvin impulse K  and the vertical coordinate of the bubble center Zc , respectively. K  and Zc of the sub-bubbles are not figured due to the pretty short time after the bubble breakup. It can be seen in Figs. 20 and 21 that K  and Zc keep zero in the symmetrical breakup of rf1 /rf2 = 1. For the asymmetrical breakup, the bubble deflects to the upper wall more along with the descending rf1 /rf2 . The center of the overall bubble deflects towards the lower wall more because of the closer distance from the bubble to the upper wall, especially in the expansion phase. In addition, it is interesting to find that Zc returns to zero at the moment of the bubble splitting for any rf1 /rf2 .

Fig. 18

Fig. 20

5

Curves of sub-bubble volume ratio

Time history of Kelvin impulse

Fig. 19

Fig. 21

Curves of bubble splitting time and sub-bubble jet impact time

Time history of bubble center displacement

Conclusion

It can be obtained that the calculated results are in good agreement with the experimental data[5] through comparisons and analysis. Bubble behaviors of the symmetric and asymmetric breakup in a narrow flow field are studied using the developed program in this paper, and the main conclusions are obtained as follows: (i) Either symmetrical or asymmetrical breakup will induce a pressure peak in the flow field at the moment of bubble splitting. This pressure disturbance will decay and disappear rapidly after the formation of sub-bubbles. (ii) Jet formed in the sub-bubble is a typical “narrow jet” and presents a teardrop-shape in the tip. Jet velocity is usually very large, making it quite short from the bubble splitting to

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the sub-bubble jet impact. Thus, it is difficult to capture this jet velocity in experiments. (iii) Jet velocity of the sub-bubble reaches its summit sharply before it decays to a stable value gradually until the jet impacts. The small sub-bubble in the asymmetrical breakup has the bigger jet velocity and forms the ring bubble earlier. (iv) The distance parameter rf has great influences on bubble breakup dynamics. In terms of symmetrical breakup, as rf gets smaller, the bubble lifetime goes up, the retarded flow velocity ascends, and the pressure on the wall rises. With regards to the asymmetrical breakup, the greater the difference between rf1 and rf2 is, the higher the unsymmetrical degree of the bubble. However, the bubble will not split but form a jet pointing toward the closer wall if the difference becomes too large.

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