Study on Bubble Cavitation in Liquids for Bubbles Arranged in a Columnar Bubble Group

applied sciences

Article Study on Bubble Cavitation in Liquids for Bubbles Arranged in a Columnar Bubble Group

Peng-li Zhang 1,2 and Shu-yu Lin 1,*

1 Institute of Applied Acoustics, Shaanxi Normal University, Xi’an 710062, China; [email protected] 2 College of Science, Xi’an University of Science and Technology, Xi’an 710054, China * Correspondence: [email protected]; Tel.: +86-181-9273-7031

Received: 24 October 2019; Accepted: 29 November 2019; Published: 4 December 2019

Featured Application: This work will provide a reference for further simulations and help to promote the theoretical study of ultrasonic cavitation bubbles.

Abstract: In liquids, bubbles usually exist in the form of bubble groups. Due to their interaction with other bubbles, the resonance frequency of bubbles decreases. In this paper, the resonance frequency of bubbles in a columnar bubble group is obtained by linear simplification of the bubbles’ dynamic equation. The correction coefficient between the resonance frequency of the bubbles in the columnar bubble group and the Minnaert frequency of a single bubble is given. The results show that the resonance frequency of bubbles in the bubble group is affected by many parameters such as the initial radius of bubbles, the number of bubbles in the bubble group, and the distance between bubbles. The initial radius of the bubbles and the distance between bubbles are found to have more significant influence on the resonance frequency of the bubbles. When the distance between bubbles increases to 20 times the bubbles’ initial radius, the coupling effect between bubbles can be ignored, and after that the bubbles’ resonance frequency in the bubble group tends to the resonance frequency of a single bubble’s resonance frequency. Fluent software is used to simulate the bubble growth, shrinkage, and collapse of five and seven bubbles under an ultrasonic field. The simulation results show that when the bubble breaks, the two bubbles at the outer field first begin to break and form a micro-jet along the axis line of the bubbles. Our methods and conclusions will provide a reference for further simulations and indicate the significance of the prevention or utilization of cavitation.

Keywords: dynamic equation; bubbles; resonance frequency; bubble number; distance

1. Introduction Cavitation is a typical hydrodynamic phenomenon that exists in many fields, such as hydraulic engineering [1], shipping [2], ultrasonic cleaning [3], underwater explosions [4], and the medical industry [5]. There are many reports on single and two cavitation bubble dynamics and the interaction between bubbles near a wall [6–8]. In fact, bubbles in liquids always exist in the form of a bubble group. The shape of bubble groups in liquids is very complicated and the vibration, translation, size, shape, and life of each bubble in a bubble group are not the same. Moreover, there is a weak mutual coupling between the bubbles in a bubble group [9]. It is of a great significance to study the utilization of cavitation and coupling between multiple bubbles [10–12]. Nicolas Bremond et al. [13] have investigated the dynamic behavior of cavitation bubbles on solid surfaces via experiments and numerical simulations. They believed that the pressure generated by each individual bubble should be considered in the Rayleigh–Plesset equation, and that the outer bubble movement period was short. Cui et al. [14] have studied the interaction of two bubbles of the same size in a free-field. An [15] has investigated the dynamic equation of bubbles in a columnar bubble group and analyzed the kinetic

Appl. Sci. 2019, 9, 5292; doi:10.3390/app9245292 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 5292 2 of 13 behavior of bubbles and the characteristics of multi-bubble sonoluminescence. The study of bubble dynamics is beneficial to regulating and controlling the intensity of ultrasonic cavitation. Ohl [16] and Luo Jing [12] have pointed out that the velocity of a bubble micro-jet is in the range 20~150 m/s. Tomita et al. [17] have observed through high-speed photography that when small bubbles are near a wall, these small bubbles under the action of a shock wave directly hit the wall, which may cause a certain degree of denudation of the structure. Han et al. [18] have conducted experimental and numerical studies on the interaction of two bubbles in a free field. Ida et al. [19] have found that injecting large bubbles in liquids can inhibit the occurrence and development of cavitation, and their research has mainly discussed the influence of the interactions between bubbles near a wall. The shape distribution of bubbles in water is very complicated, but with the modulation of ultrasonic waves, bubbles will aggregate into stable special structures such as “bubble grapes”, “cone-like bubble structures”, “Acoustic lichtenberg figures”, and “spherical” structures [20–24]. Studies have shown that the secondary Bjerkens force between bubbles is a major factor in bubble clusters [25–27]. Classical Bjerknes force theory states that two interacting bubbles can only exhibit the following behaviors: if the ultrasonic frequency is between the resonance frequency of two bubbles, the bubbles will repel each other; if the ultrasonic frequency is beyond the resonance frequency of two bubbles, the bubbles will attract each other. Hence, it is very important to learn about the resonance frequency of bubbles. Wang et al. [28] have simulated the coupling of two horizontally arranged bubbles under a free surface. However, the number of bubbles in a bubble group is very large, and the shape of the bubbles is very complicated. In order to analyze coupling effects between bubbles, the spherical and columnar bubble group models are widely used. S. k. Spratt et al. [29] concluded that the radius of bubbles in a champagne glass should be 0.94 mm according to the Minnaert frequency relationship by monitoring the radial acoustic frequency of bubbles at the bottom of a champagne glass. In fact, there is a weak coupling interaction between bubbles in a bubble group. This coupling effect is a kind of inhibitory effect on bubble growth, oscillation, and collapse; after this effect the resonance frequency of the bubble decreases [30,31]. The resonance frequency of bubbles in a bubble group is an important parameter with which to study the cavitation phenomenon, whether bubbles collapse in an ultrasonic field, and the secondary Bjerkens force between bubbles. However, research on bubble groups has mainly been focused on bubbles near walls or liquid surfaces, and most of these works are experimental studies with few theoretical explanations. There is little literature on the resonance frequency of bubbles in a bubble group. Hence, in this paper, the resonance frequency of bubbles in a columnar bubble group is obtained by linear simplification of the bubbles’ dynamic equation. The correction coefficient between the resonance frequency of bubbles in a columnar bubble group and the Minnaert frequency of a single bubble is given. Fluent fluid dynamics software is used to simulate bubble growth, shrinking, and collapse in five- and seven-columnar bubble groups under an ultrasonic field.

2. Theoretical Model If the effect of liquid turbulence is not considered, when the distance between bubbles and the size of the bubbles are much smaller than the wavelength of the ultrasonic waves in a liquid, it can be assumed that the bubbles are in the same ultrasonic field. When the distance between the bubbles is dij, the bubble dynamics equation of the bubble group under the ultrasonic field can be expressed as [32,33]

.. . 3k . 3 2 1 2δ Ri0 2δ 4u RiRi + Ri = [(p0 + pv)( ) + pv p0 Ri pa sin(ωt)] 2 ρ Ri0 − Ri − − Ri − Ri − N 2 2 P Rj d Rj dRj (1) [R + 2( ) ] d j dt2 dt −j=1 ij

The last item in the above equation reﬂects the interaction between cavitation bubbles. Ri is the bubble radius at any time, Ri0 is the bubble’s initial radius, ρ is the liquid density out of the bubble, p0 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 15

31 2δδR 24u RR+= R 23[( p +− p )(i0 )k +−−−− p p R  p sin(ω t )] ii iρ 00 v v i a 2 RRii0 RR ii (1) N RRRdd2 −+jjj[2()]R 2  j 2 j=1 dtij dd t

The last item in the above equation reflects the interaction between cavitation bubbles. Ri is the ρ bubbleAppl. Sci. radius2019, 9, 5292at any time, Ri0 is the bubble’s initial radius, is the liquid density out of3 ofthe 13 bubble, p∞ is the pressure of the liquid outside the bubble wall, u is the liquid viscosity coefficient, σ is the surface tension coefficient, and p is the gas pressure inside the bubble. is the pressure of the liquid outside the bubble wall, u isig the0 liquid viscosity coefficient, σ is the surface tension coefficient, and pv is the gas pressure inside the bubble. InIn fact, fact, the the movement movement of of the the bubbles bubbles under under the the ul ultrasonictrasonic field field is is a a forced vibration. The The main main parametersparameters that that affect affect the the forced forced vibration vibration of of th thee bubbles bubbles include include resonance resonance frequency, frequency, ultrasonic ultrasonic frequency,frequency, and and amplitude. amplitude. In In order order to to investigat investigatee the the relationship relationship between between the the resonance resonance frequency frequency ofof the the cavitation cavitation bubble bubble and and the the nu numbermber of of bubbles, bubbles, the the initial initial radius radius of of the the bubble, bubble, and and the the distance distance betweenbetween bubbles, bubbles, Equation Equation (1 (1)) has has been been used used as as a linear simplification. simplification. Assuming Assuming that that the the distance distance d betweenbetween bubbles inin aa columnarcolumnar group group is isd and and that that all bubbles all bubbles have have the same the initialsame initial radius, radius, the bubble the bubblewe investigate we investigate here is thehere first is the one. first At one. this At time, this the time, distance the distance between between any bubble any bubble and the and first the bubble first bubble can be expressed as dnd= , as shown in Figure 1. can be expressed as dj1 = nd, asj1 shown in Figure1.

d d

FigureFigure 1. 1.BubblesBubbles in ina columnar a columnar bu bubblebble group; group; the the spacing spacing between between the the bubbles bubbles is isdd..

Hence, the interaction term between bubbles in Equation (1) is simplified as Hence, the interaction term between bubbles in Equation (1) is simplified as . N 2 2 n 2 X Rj d Rj dRj X 1 d(R R) Nn[R 2 + 2( ) ] = 2  (2) RRRjjjjdd2 2 1d(R R ) rij[2()]R dt +=dt nd dt j=1 j 2 n=1 (2) jn==11rtij dd t ndt d According to Euler’s approximation from 1734 [34,35], Equation (2) can be changed to According to Euler’s approximation from 1734 [34,35], Equation (2) can be changed to n . . X 1 d(R2R) d(R2R) (ln(n + 1) + ξ) 22= (3) n 1d(ndRRdt ) d( RRdt  )(ln(d n++ 1)ξ ) n=1 =  (3) where ξ is a constant andn=1ξnd0.577218dd t. Let (ln(n + t1) + ξ)/d = A d, substituting (3) into (1), and the ≈ following can be obtained. where ξ is a constant and ξ ≈0.577218 . Let (ln(ndA++ 1)ξ ) / =, substituting (3) into (1), and the following can.. be obtained. . 3k . 2 3 2 1 2δ R0 2δ 4u (R + AR )R + ( + 2AR)R = [(p + pv)( ) + pv p R pa sin(ωt)] (4) 2 ρ 0 R − R − 0 − R − R − 312δδR 24u (R+AR22 )(2) R ++ AR R  = [( p + − p )()0 3k +−− p p − R  − p sin()]ω t In the case of small vibrationsρ of the00 bubble,vv ignoring the influence of the viscosity a coeffi cient(4) of the liquid 2 RR RR ...... R = R (1 + x) and x 1, R = R x, R = R x (5) In the case of small vibrations 0of the bubble, ignoring the0 influence0 of the viscosity coefficient of theand liquid R 3γ 1 1 ( 0 ) ( ) =+ 1 3γx,  =1 x  = (6) R Rx0 (1R ) and≈ −x  1R, ≈RRx−0  R, 0RRx0  (5) By substituting (5) and (6) into Equation (4), and considering x to be a small amount, we obtain and .. 1 1 x + [ω 2 p sin(ω t)]x = p sin(ω t) (7) r 2 3 a a 2 3 a a − ρ(R0 + AR0 ) ρ(R0 + AR0 )

In Equation (7), ωr is the circular resonance frequency of the bubble, and can be expressed as

1 2σ 2σ ω 2 = [3γ(p + p ) ] (8) r 2 3 0 v ρ(R0 + AR0 ) R0 − − R0 or as 1 1 3γ 2σ 2σ 2 = [ ( + ) ] fr 1 p0 pv (9) ρ R0 − − R0 2πR0(1 + AR0) 2 Appl. Sci. 2019, 9, 5292 4 of 13

From Equation (9) it is known that the resonance frequency of a bubble in a columnar bubble group is related to the distance between the bubbles and the number of bubbles in the bubble group. In order to describe the relationship between the bubble resonance frequency of the bubble group and the Minnaert frequency of a single bubble, a correction coeﬃcient M is given to the Minnaert frequency of the single bubble, where M = fr/ f0, i.e.,

1 1 3γ 2σ 2σ 2 f0 = [ (p0 + pv) ] (10) 2πR0 ρ R0 − − ρR0 and then 1 1 2 M = ( ) (11) 1 + AR0 Equation (11) is the correction relationship between the resonance frequency of the bubble in the spherical bubble group and the single bubble Minnaert frequency.

3. Study of Resonance Frequency of Bubbles in a Bubble Group From Equation (9) it can be seen that the resonance frequency of the bubbles in a columnar bubble group is related to the distance between the bubbles, the number of bubbles in the bubble group, and the initial radius of the bubbles. Hence, we have been able to numerically calculate the relationship between the distance between the bubbles, the number of bubbles, and the resonance frequency of the bubbles. From Equation (9) we obtain Figures2–4, where the values of the parameters are, respectively, 3 2 ρ = 999kg/m , γ = 1.33, σ = 0.072kg/s , p0 = 1bar, and pv = 2250Pa. The relationship between the number of bubbles and the resonance frequency of the bubbles is shown in Figure2. It can be seen from the ﬁgure that the more bubbles there are, the smaller the resonance frequency of the bubbles. When the bubble number increases, the resonance frequency of the bubblesAppl. Sci. changes 2019, 9, x quickly.FOR PEER REVIEW When the number of bubbles changes from two to twenty-ﬁve the5 of change 15 in bubble resonance frequency is most obvious; the resonance frequency changes from 135.8kHz to 117.8kHz135.8kHz, which to is a117.8kHz reduction, which of about is a reduction18kHz. After of about this, 18kHz the number. After ofthis, bubbles the number increases of bubbles and the increases and the resonance frequency tends towards a certain value. resonance frequency tends towards a certain value.

140

135 R = 20μm, d = 0.1mm 0

130

125 f (kHz) 120

115

110 0 10 20 30 40 50 n

FigureFigure 2. Relationship 2. Relationship between between the the number number of bubblesof bubbles and and the the resonance resonance frequencyfrequency of of each each bubble bubble in in a columnara columnar bubble bubble group. group.

The relationshipThe relationship between between the the distance distance between between bubbles bubbles andand the resonance resonance frequency frequency is given is given in in FigureFigure3, where 3, where the initial the initial radius radius of each of each bubble bubble is 20is 20µ mμm and and the the number number of of bubbles bubbles isis five.five. It It can can be seen frombe seen the figurefrom the that figure when that the when distance the distance between between the bubbles the bubbles increases increases and the and interaction the interaction between the bubblesbetween weakens, the bubbles after weakens, that the after resonance that the resona frequencynce frequency of the bubblesof the bubbles increases increases and and finally finally tends towardstends a fixed towards value a fixed which value is the which Minnaert is the Minnaert frequency frequency of a single of a bubble. single bubble. When theWhen distance the distance between between bubbles increases to 20 times the initial radius the interaction between the bubbles is almost negligible. According to Figures 2 and 3, it can be concluded that, with regard to the bubbles’ initial radius, when the distance between bubbles increases to 20 times the initial radius, or is equal to twenty bubbles in a columnar, the interaction between the bubbles is very weak and can be ignored.

160

R = 20μm 150 0

140

f (kHz) f 130

120

110 0 0.1 0.2 0.3 0.4 0.5 0.6 d (mm)

Figure 3. Relationship between the distance between bubbles and the bubble resonance frequency in a columnar bubble group， n = 5 . Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 15

135.8kHz to 117.8kHz , which is a reduction of about 18kHz . After this, the number of bubbles increases and the resonance frequency tends towards a certain value.

140

135 R = 20μm, d = 0.1mm 0

130

125 f (kHz) 120

115

110 0 10 20 30 40 50 n

Figure 2. Relationship between the number of bubbles and the resonance frequency of each bubble in a columnar bubble group.

The relationship between the distance between bubbles and the resonance frequency is given in Figure 3, where the initial radius of each bubble is 20 μm and the number of bubbles is five. It can Appl. Sci. 2019, 9, 5292 5 of 13 be seen from the figure that when the distance between the bubbles increases and the interaction between the bubbles weakens, after that the resonance frequency of the bubbles increases and finally tends towards a fixed value which is the Minnaert frequency of a single bubble. When the distance bubbles increases to 20 times the initial radius the interaction between the bubbles is almost negligible. between bubbles increases to 20 times the initial radius the interaction between the bubbles is almost According tonegligible. Figures According2 and3, it to can Figures be concluded 2 and 3, it can that, be concluded with regard that, with to the regard bubbles’ to the bubbles’ initial radius, initial when the distanceradius, between when bubbles the distance increases between to bubbles 20 times increase the initials to 20 radius,times the or initial is equal radius, to or twenty is equal bubbles to in a columnar, thetwenty interaction bubbles in between a columnar, the the bubbles interaction is betwee very weakn the bubbles and can is very be ignored.weak and can be ignored.

160

R = 20μm 150 0

140

f (kHz) f 130

120

110 0 0.1 0.2 0.3 0.4 0.5 0.6 d (mm)

Figure 3. RelationshipFigure 3. Relationship between between the distance the distance between between bubbles bubbles and the the bubble bubble resonance resonance frequency frequency in in a = columnar bubblea columnar group, bubblen = group5. ， n 5 .

Figure4Appl. shows Sci. 2019 that, 9, x FOR the PEER resonance REVIEW frequency of a bubble is a ffected by the initial6 radius of 15 of the bubbles, the distance between bubbles, and the number of bubbles in the bubble group. It can be seen Figure 4 shows that the resonance frequency of a bubble is affected by the initial radius of the bubbles, from Figurethe4a,b distance that when between the bubbles, space and between the number bubbles of bu increases,bbles in the thebubble inhibitory group. It ecanffect be seen between from bubbles becomes weakFigure and 4a,b the that bubblewhen the resonance space between frequency bubbles increases, increases the untilinhibitory the effect coupling between eff bubblesect disappears and finallybecomes returns weak to a and single the bubble bubble resonance resonance frequency frequency. increases until In Figure the coupling4 it can effect also disappears be observed and that for differentfinally initial returns radii to and a single diff erentbubble numbersresonance frequency. of bubbles In Figure after twenty4 it can also bubbles, be observed the couplingthat for effect different initial radii and different numbers of bubbles after twenty bubbles, the coupling effect between thebetween first bubble the first andbubble others and others becomes becomes very very weak. weak. Additionally, Additionally, when when the distance the distance between between bubbles increasesbubbles toincreases 20 times to 20 the times initial the initial radius radius of of the the bubbles, bubbles, the the interaction interaction between between bubbles bubbles can be can be neglected. Whenneglected. the When initial the radius initial radius of the of bubbles the bubbles is smaller is smaller and and the the bubbles are are closer closer to each to each otherother the coupling forcethe coupling between force bubbles between will bubbles be stronger. will be stronger.

310 700 (a) (b) 300 600 290 500 R = 5μm 280 0 R = 10μm 270 400 0 R = 20μm

f (kHz) f (kHz) 0 260 n = 10 n = 20 300 250 n = 30 200 240

230 100 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 d (mm) d (mm)

Figure 4. RelationshipFigure 4. Relationship between between resonance resonance frequency frequency and different different number number of bubbles of bubbles as well as well as different initialdifferent radius initial of radius bubbles. of bubbles. In (a ),In correspond(a), correspond to to didifferentfferent number numberss of bubbles of bubbles (respectively, (respectively, 10, 10, 20, and 30, where R =10μm ) and in (b) correspond to the different number of bubbles’ initial 20, and 30, where R0 = 10 µm0 ) and in (b) correspond to the different number of bubbles’ initial radii = (respectively,radii(respectively, 5 µm, 10 µm, and5 μm 20, 10µm, μm where, and 20n =μm10)., where n 10 ).

It has been pointed out that under a weak coupling effect between bubbles, bubbles maintain the shape of the hemisphere most of the time without contact and fusion, but, when the bubbles collapse, they form an interfacing jet [12]. We simulated the growth and collapse process of bubbles Appl. Sci. 2019, 9, 5292 6 of 13

4. Fluent Analysis Bremond et al. [36] have used a CCD (charge-coupled device camera) to capture morphological changes of the growth and collapse of single bubbles, two-bubble groups, and five-bubble groups. Their experimental results show that morphological changes during a multi-bubble collapse vary with the spacing of bubbles. At the same time, they found that the outer bubble movement period is shorter and that the jet of outer bubbles is completed first. This method can control the cavitation bubbles’ location, but cavitation bubbles can only form near the wall’s surface; in this case, the influence of the wall’s surface on cavitation bubble collapse cannot be ignored. Due to the influence of fluid properties, the micron scale of bubbles, instantaneous collapse, and other factors, it is impossible to accurately control bubbles in the bubble group, leading to the failure of bubble research under many conditions through experiments. With the development of computational fluid dynamics, numerical simulations have played an increasingly important role in the study of bubble dynamics. Simulation of the cavitation process by Fluent software can show the micro-bubble growth process and the rapid collapse process, as well as reflect the mutual coupling process of bubbles in the bubble group and the non-spherical change of the bubbles. It has been pointed out that under a weak coupling effect between bubbles, bubbles maintain the shape of the hemisphere most of the time without contact and fusion, but, when the bubbles collapse, they form an interfacing jet [12]. We simulated the growth and collapse process of bubbles for five and seven bubbles set in a line using Fluent software with an initial radius of 30 µm and a distance of 200 µm between bubbles. It can be found from Equation (9) that when the number of bubbles is five the resonance frequency of the bubbles is 89.8 kHz and that when the number of bubbles is seven the resonance frequency of the bubbles is 88.4 kHz. Using a volume of fluid (VOF) 2D axisymmetric analysis has the advantage of easy realization, small computational complexity, and high precision, and traces the volume of fluid in the grid, not the motion of the fluid particles [37]. For the simulation domain, shown in Figure5, the calculation field is 3mm 6mm, the number of cells is × 253,000, the number of nodes is 254,331, and the calculation step is 0.01µs. The liquid medium is water and the gas inside the bubble is nitrogen. In the process of simulation and calculation, the ultrasonic field is driven from the left side, for which Pa = pa sin(2π f t)—pa is the amplitude of the ultrasonic pressure and f is the ultrasonic frequency. The right side is the free interface and the outlet pressure is the hydrostatic pressure, which is an atmospheric pressure. The ultrasonic frequency f is 26 kHz, which is smaller than the resonance frequency of the bubble, and the ultrasonic pressure amplitude pa is 1.2 bar. Then we obtained Figures6–9. Figures6 and8 show that the bubble has grown well and remained spherical under the ultrasonic field. Due to the mutual restriction of the bubbles during the growth process, the two bubbles at the outer field can reach maximum volume first in the expansion process, and the maximum volume is larger than the inner bubble volume. In time, all bubbles begin to shrink simultaneously. In the process of shrinkage, the outer bubbles shrink slightly faster, becoming an ellipsoid first, and can maintain their ellipsoid shape while continuously shrinking. When contracted to a certain extent, the outer two bubbles begin to collapse first, forming a micro-jet and directing it towards the bubbles back and along the axial center of the wire-packing group to break the bubbles one by one until the group is completely torn open. At this time, due to the sharp change in the pressure gradient in the liquid domain, the intermediate bubbles are stretched from an ellipsoid to a gourd shape and stretched to split into two bubbles. Next, the micro-jet generated by the edge bubbles continues to break down the second and third bubbles in turn until all the bubbles are completely torn apart. Before the breakdown of the intermediate bubbles, because of the close distance between the bubbles, it is possible for the intermediate bubbles to fuse. The numerical simulation results are in good agreement with the experimental conclusions as given in Figure 10 by Bremond [36]; we replicate Figure 10 here again. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 15

for five and seven bubbles set in a line using Fluent software with an initial radius of 30 μm and a distance of 200 μm between bubbles. It can be found from Equation (10) that when the number of bubbles is five the resonance frequency of the bubbles is 89.8 kHz and that when the number of bubbles is seven the resonance frequency of the bubbles is 88.4 kHz . Using a volume of fluid (VOF) 2D axisymmetric analysis has the advantage of easy realization, small computational complexity, and high precision, and traces the volume of fluid in the grid, not the motion of the fluid particles [37]. For the simulation domain, shown in Figure 5, the calculation field is 3mm× 6mm , the number of cells is 253,000, the number of nodes is 254,331, and the calculation step is 0.01 μs . The liquid medium is water and the gas inside the bubble is nitrogen. In the process of simulation and calculation, the = π ultrasonic field is driven from the left side, for which Ppaasin(2 ft ) — pa is the amplitude of the ultrasonic pressure and f is the ultrasonic frequency. The right side is the free interface and the outlet pressure is the hydrostatic pressure, which is an atmospheric pressure. The ultrasonic Appl. Sci.frequency2019, 9, 5292 f is 26 kHz , which is smaller than the resonance frequency of the bubble, and the 7 of 13

ultrasonic pressure amplitude pa is 1.2 bar. Then we obtained Figures 6–9.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 15

ellipsoid first, and can maintain their ellipsoid shape while continuously shrinking. When contracted to a certain extent, the outer two bubbles begin to collapse first, forming a micro-jet and directing it towards the bubbles back and along the axial center of the wire-packing group to break the bubbles one by one until the group is completely torn open. At this time, due to the sharp change in the pressure gradient in the liquid(a domain,) the intermediate bubbles are(b )stretched from an ellipsoid to a gourd shape and stretched to split into two bubbles. Next, the micro-jet generated by the edge bubbles continues to break down the second and third bubbles in turn until all the bubbles are completely torn apart. Before the breakdown of the intermediate bubbles, because of the close distance between the bubbles, it is possible for the intermediate bubbles to fuse. The numerical simulation results are in good agreement with the experimental conclusions as given in Figure 10 by Bremond [37]; we replicate Figure 10 here again.

Figures 7 and 9 show that the pressure inside the bubbles decreases sharply during bubble expansion and that when the bubble expands to the maximum volume, the pressure inside the bubble is only 0.41 bar; in the period of bubble shrinkage and collapse, the pressure inside the bubble rises sharply. When the bubble collapses, the pressure inside the bubble reaches its maximum value of about 1.82 bar. After the bubble collapses in Figure 9f, the pressure inside and outside the bubble tends to be almost the same. Due to the micro-jet(c )generated by the bubble collapse, the pressure on the jet line is significantly greater than in other locations. In addition, on account of the mutual FigurecouplingFigure 5. betweenSimulation 5. Simulation the domainbubbles, domain andthe and resonance meshing: meshing: (frequency aa,b,b) )show show the of the calculationthe calculation bubbles of isthe reduced, of domain the domain and and the under and linear thethe linearsame bubbleacousticbubble in parameters the in liquid the liquid anda large and (c) shows(numberc) shows the theof third bubblesthird bubblebubble collapse and and mesh meshsooner in inthe andthe domain. domain.more easily. The pressure inside the bubble also changes more sharply. Figures 6 and 8 show that the bubble has grown well and remained spherical under the ultrasonic field. Due to the mutual restriction of the bubbles during the growth process, the two bubbles at the outer field can reach maximum volume first in the expansion process, and the maximum volume is larger than the inner bubble volume. In time, all bubbles begin to shrink simultaneously. In the process of shrinkage, the outer bubbles shrink slightly faster, becoming an

(a) t = 0 (b) t = 8 μs

Figure 6. Cont. Appl. Sci. 2019, 9, 5292 8 of 13

Appl.Appl. Sci. Sci. 2019 2019, 9,, x9 ,FOR x FOR PEER PEER REVIEW REVIEW 9 of9 of15 15

(e)( et) = t 30= 30 μ sμ s (f)( ft) = t 35= 35 μ sμ s

FigureFigureFigure 6. Bubble6. 6.Bubble Bubble volume volume volume fraction fraction fraction in inthe the columnar columnar columnar bubble bubblebubble group; group; n =n n 5.= = 5. 5.

(a)( at) = t 0= 0 (b ()b t) = t 8= μ8 sμ s

(c)( ct) = t 16= 16 μ sμ s (d()d t) = t 20= 20 μ sμ s

(e)( et) = t 30= 30 μ sμ s (f)( ft) = t 35= 35 μ sμ s

FigureFigureFigure 7. 7.Pressure 7.Pressure Pressure distribution distribution distribution in inthe the the bubble bubble bubble group; group;group; n =n 5.== 5.5. Appl. Sci. 2019, 9, 5292 9 of 13

Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15

(a) t = 0 (b) t = 10 μs

(e) t = 24 μs (f) t = 32 μs

FigureFigure 8. Bubble 8. Bubble volume volume fraction fraction in in thethe columnarcolumnar bubble bubble group; group; n =n 7.= 7.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15

(a) t = 0 (b) t = 10 μs

Appl. Sci. 2019, 9, 5292 (e) t = 24 μs (f) t = 32 μs 10 of 13 Figure 8. Bubble volume fraction in the columnar bubble group; n = 7.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 15

Appl. Sci. 2019, 9, x FOR PEER(a) REVIEWt = 0 (b) t = 10 μs 11 of 15

(e) t = 24 μs (f) t = 32 μs

Figure 9. Pressure distribution in the bubble group; n = 7. (e) t = 24 μs (f) t = 32 μs

FigureFigure 9.9. PressurePressure distributiondistribution inin thethe bubblebubble group;group;n n= = 7.7.

Figure 10. Comparison between experiment and simulation of the cavitation of ﬁve bubbles set on a Figure 10. Comparison between experiment and simulation of the cavitation of five bubbles set on a line with d = 200=µm and p0 = 1.4=− MPa, R0 = 4 µm= [36]. line with d 200μm and −p0a1.4MP , R0 4μm [37].

In orderFigure to 10. verify Comparison the reliability between ofexperiment the simulation and simulation results, of thethe cavitation relationship of five between bubbles set the on ﬁrst a bubble radius and time evolution= in the simulation=− results of Fluent= was obtained using the boundary tracking line with d 200μm and p0a1.4MP , R0 4μm [37].

In order to verify the reliability of the simulation results, the relationship between the first bubble radius and time evolution in the simulation results of Fluent was obtained using the boundary tracking method, and the tracking results were compared with the numerical calculation results of bubbleIn motion order obtainedto verify using the reliability Equation (1).of theThe simula comparisontion results, results theare shownrelationship in Figure between 11. The the initial first bubble radius and time evolution in the simulation results of Fluent was obtained using the boundary tracking method, and the tracking results were compared with the numerical calculation results of bubble motion obtained using Equation (1). The comparison results are shown in Figure 11. The initial Appl. Sci. 2019, 9, 5292 11 of 13 Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 15 method,radius of and the thebubbles tracking was results30 μm wereand the compared distance withbetween the numericalbubbles was calculation 150 μm . resultsFigure of11a bubble shows motionthe evolution obtained curve using of the Equation bubble (1).radius The with comparison time under results different are ultrasonic shown in frequencies. Figure 11. The Figure initial 11b radiusshows of the the evolution bubbles was curve 30 µofm theand bubble the distance radius between under bubblesdifferent was driving 150 µ multrasonic. Figure 11intensities.a shows the In evolutionFigure 11, curve the lines of the and bubble points radius are obtained with time by under Fluent di ﬀsoftwareerent ultrasonic simulation, frequencies. and the discrete Figure 11 pointsb shows are theobtained evolution by numerical curve of the solution bubble of radius Equation under (1). diﬀerent driving ultrasonic intensities. In Figure 11, the lines and points are obtained by Fluent software simulation, and the discrete points are obtained by numerical solution of Equation (1).

Figure 11. The evolution of bubble radius with time in a columnar bubble group. (a) shows different Figure 11. The evolution of bubble radius with time in a columnar bubble group. (a) shows different driving ultrasonic frequencies; the driving ultrasonic frequency is 26 kHz and 40 kHz, 46 kHz.(b) shows driving ultrasonic frequencies; the driving ultrasonic frequency is 26kHz and 40kHz , 46kHz . different ultrasonic intensities the driving ultrasonic intensity is 1.2Bar and 1.5Bar. (b) shows different ultrasonic intensities the driving ultrasonic intensity is 1.2Bar and 1.5Bar . When comparing these curves, it can be seen that the simulation results given by Fluent are consistentWhen with comparing the numerical these resultscurves, obtained it can be using seen Equationthat the simulation (1) for the bubble results growth given process,by Fluent only, are inconsistent the bubble with collapse the numerical stage there results is obtained a small di usingfference. Equation This (1) di ffforerence the bubble is caused growth by theprocess, action only, of ain micro-jet, the bubble because collapse the stage influence there ofis aa micro-jetsmall differenc cannote. beThis considered difference in is the caused numerical by the calculation action of a producedmicro-jet, usingbecause Equation the influence (1). This of shows a micro-jet the reliability cannot be of theconsidered simulation in results.the numerical By analyzing calculation the Fluentproduced calculation using Equation results, it (1). is further This shows shown the that reliability due to the of rolethe ofsimulation the micro-jet, results. the bubbleBy analyzing is broken the downFluent by calculation its own micro-jet results, whenit is further the radius shown is that greater due than to the or role equal of the to the micro-jet, initial radius the bubble of the is bubble. broken Indown addition, by its under own micro-jet the action when of di fftheerent radius acoustic is greater parameters, than or bubblesequal to canthe growinitial at radius the same of the time bubble. and reachIn addition, their maximum under the radius,action of but different the maximum acoustic expansionparameters, radius bubbles is slightlycan grow di ffaterent. the same This time can and be understoodreach their asmaximum that when radius, the bubble but the vibrates maximum under expansio the drivingn radius of ais sound slightly wave, different. the bubble This can is not be onlyunderstood the sound as scatteringthat when sourcethe bubble but also vibrates the transducer under the in driving the sound of a field.sound When wave, the the driving bubble sound is not intensityonly the increases,sound scattering the bubble source can but collect also more the transducer energy at the in samethe sound time, field. meaning When the the bubble driving can sound grow fasterintensity and increases, get a larger the maximum bubble can radius. collect The more process energy of bubble at the growthsame time is also, meaning the process the bubble of energy can collectiongrow faster until and the get micro-jet a larger generated maximum by radius. the collected The pr energyocess isof enoughbubble togrowth grasp is the also bubble, the process and then of theenergy bubble collection collapses. until the micro-jet generated by the collected energy is enough to grasp the bubble, and then the bubble collapses. 5. Conclusions and Discussion 5. Conclusions and Discussion In this paper, based on the dynamic equation of bubbles in a bubble group, the resonance frequency of bubblesIn this in paper, a columnar based bubbleon the groupdynamic has equation been obtained of bubbles by linear in a simplificationbubble group, of the the resonance bubbles’ dynamicfrequency equation. of bubbles The in correction a columnar coeffi bubblecient between group has the resonancebeen obtained frequency by linear of bubbles simplification in a columnar of the bubblebubbles’ group dynamic and theequation. Minnaert The frequency correction of coefficient a single bubblebetween has the been resonance given. frequency The vibration, of bubbles growth, in anda columnar collapse processbubble ofgroup five andand seven the Minnaert linear bubbles freque underncy theof a action single of bubble an ultrasonic has been field given. have been The simulatedvibration, bygrowth, using hydrodynamicsand collapse process software. of five The resultsand seven show linear that: bubbles under the action of an ultrasonic field have been simulated by using hydrodynamics software. The results show that: 1. There is a weak coupling interaction between bubbles in the bubble group which can change 1. theThere resonance is a weak frequency coupling of interaction the bubbles. between The resonance bubbles in frequency the bubble of bubblesgroup which is affected can change by many the factorsresonance such frequency as the initial of the radius bubbles. of the The bubbles, resonanc thee number frequency of bubblesof bubbles in a is bubble affected group, by many and thefactors distance such betweenas the initial bubbles. radius With of the regard bubbles to these, the parameters, number of thebubbles initial in radius a bubble of the group, bubbles and the distance between bubbles. With regard to these parameters, the initial radius of the bubbles and the distance between bubbles have the most significant effects on the resonance frequency Appl. Sci. 2019, 9, 5292 12 of 13

and the distance between bubbles have the most significant effects on the resonance frequency of the bubbles. For the same number of bubbles, when the distance between the bubbles is larger, the interaction between the bubbles becomes weaker. When the distance between the bubbles increases to 20 times the initial radius of the bubbles, the resonance frequency of the bubbles returns to that of a single bubble. In this case, the interactions between bubbles can be ignored. 2. The first bubble is affected by other bubbles behind it, and the resonance frequency reduces. The larger the number of bubbles, the stronger the interaction force between them, and the smaller the resonance frequency of the bubbles. However, when the number of bubbles increases to twenty and then continues to increase, the number of bubbles and the resonance frequency of the bubbles tends towards a fixed value. 3. Through Fluent simulation results it was found that due to the mutual restriction between bubbles in the expansion process, the energy ratio of the two bubbles at the edge reaches its maximum volume first in the expansion process. Later, the bubbles begin to contract. Through contracting, the bubbles slowly become, and remain, ellipsoid in nature. After this, the two bubbles begin to collapse, forming a micro jet which points to the back of the bubble and along the axis of the center of the linear bubble group, breaking the bubble until it is completely torn apart. 4. The maximum volume and collapse times of bubbles driven by sound parameters are different. The larger the driving frequency and the sound pressure are, the larger the maximum volume of the bubbles.

Author Contributions: Conceptualization, S.-y.L.; methodology, P.-l.Z.; software, P.-l.Z.; writing, P.-l.Z.; writing—review & editing, S.-y.L. Funding: This research was funded by the National Natural Science Foundation of China (grant nos. 50875132 and 60573172) and the Industrial Public Relation Project of Shaanxi Technology Committee, China (grant no. 2015GY182). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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