Collapse of cavitation bubbles near air bubbles ＊

Jing Luo, Wei-lin Xu, Rui Li

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065,China

Abstract: The cavitation erosion is one of the common damage modes in water engineering. The study of the interaction among cavitation bubbles, air bubbles and the wall is of great significance for understanding the mechanism of the air entrainment to alleviate the cavitation and to enhance the cavitation erosion mitigation effect of aeration. By using the high-speed camera, the regular patterns of the collapse of cavitation bubbles in the vicinity of the wall and the air bubbles are studied in this paper. It is shown that in the vicinity of air bubbles, the cavitation bubbles may only collapse towards or from air bubbles, while under the dual impacts of air bubbles and the wall, the direction of collapse of the cavitation bubbles depends on the combined vector of the impact forces of the air bubbles and the wall. When the air bubbles are very close to the cavitation bubbles, the air bubbles will be penetrated and stretched by the cavitation bubbles, when the distance between them is short enough, the cavitation bubbles and the air bubbles will connect to form gas-type cavitation bubbles, and the collapse strength will be significantly decreased, when there are two air bubbles near the cavitation bubbles, they may even be penetrated successively by the cavitation bubbles. Thus it can be concluded that during the air entrainment to alleviate the cavitation, the number of air bubbles is more important than the total air concentration.

Key words: Cavitation bubble, air bubble, high-speed photography, air entrainment to alleviate cavitation

Introduction

The cavitation is common in many engineering applications, such as large cavitation bubbles generated by underwater explosions, and millimeterscale cavitation bubbles that produce the cavitation erosion in ships, hydraulic machinery and hydraulic energy dissipation structures, as well as ultrasonicinduced millimeter-scale cavitation bubbles used in the biomedicine, the material preparation process and the chemical production to accelerate chemical reaction rates. Wang et al.[1]gave a very comprehensive review of the bubble dynamics and its applications, and pointed out the research difficulties and research directions in this field.

Philipp and Lauterborn[2]studied the collapse of a cavitation bubble near the wall and it was suggested that the jet plays a less important role in the cavitation damage, while the bubble-wall dimensionless distanceγ(γ=h/Rmax, whereRmaxis the maximal radius of the cavitation bubble,his the distance between the center of the cavitation bubble and the wall)determines whether the cavitation damage takes place or not. It would cause the cavitation erosion whenγis equal to or less than 2, but the jet can be only observed when the cavitation bubble is extremely close to the wall (withγequal to or less than 0.7).Furthermore, there were many other intensive researches of the mechanism of the cavitation erosion[3-5]. Shaw et al.[6]studied the interaction of a laser-generated bubble with a nearby solid boundary,and recorded the pressure increases for the cavities created with sizes between 0.56 and 1.5 times the maximum bubble radius from the solid boundary by a series of pressure transducer traces. However, in order to generate the cavitation bubble for different dimensionless distances, the maximum bubble radius should be varied, while the distance between the cavitation bubble and the wall should be fixed. With the high-speed photographic equipment with a camera speed of up to 108frames per second used to study the evolution properties of the cavitation bubbles around a wall, Lindau and Lauterborn[7]discovered the annular form during the collapse of a cavitation bubble, the formation of a contrajet and the shock wave.

In practical engineering applications, one sees often not isolated cavitation bubbles, but ones surrounded by other cavitation bubbles, particles and air bubbles. With the coupled effects between the cavitation bubbles，the multiple cavitation bubbles at different nascent moments and different sized cavitation bubbles, the study becomes extremely difficult[8-9]. Rungsiyaphornrat et al.[10]studied the interactions of explosion bubbles formed by underwater explosive of multiple underwater weapons at the same time or with some delay. Fong et al.[11]studied the coupling effect of two cavitation bubbles with different nascent moments by the method of a low voltage discharge. Luo et al.[12]found that the two cavitation bubbles collapse towards each other and then gradually merge, and the merged collapse body quickly moves to the wall. Pain et al.[13]studied jets in quiescent bubbles caused by a nearby oscillating bubble, and found that the jet in the initially stationary bubble can reach a velocity up to 250 m/s. Luo et al.[14]found that the low pressure formed in the opposite direction to the cavitation bubble jet stretches the air bubble in the low-pressure area into a step-like shape.Goh et al.[15]studied the interaction between a cavitation bubble and a non-oscillating air bubble attached to a horizontal polyvinyl chloride plate submerged in the de-ionized water, and found the ratio of the oscillation time and the wall-attached bubble oscillation time is an important parameter for determining the jet direction. The size of the air bubbles plays an outstanding role in the cavitation damage reduction[16]. In addition, the interaction between a cavitation bubble and a particle was researched by adopting a high-voltage discharge technology to induce the cavitation bubble[17]and a low-voltage underwater discharge technology to induce a cavitation bubble[18]. Wu and Gou[19]determined the critical size of the sand particles in the liquid by means of the special vibratory device, and it was found to be related to various effects on the cavitation damage. The bubble-particle interactions were found to be strongly dependent on two dimensionless parameters, the particle-bubble size ratio and the particle-liquid density ratio[20]. Four different interactions of a laser-induced bubble and a free-settling particle were studied by Wu et al.[21], a force balance model was built and it was shown that several non-dimensional factors involving the particle radius, the maximum bubble radius, the initial separation distance , the fluid viscosity and the particle and fluid densities influence the particlebubble dynamics. The interaction between the ice and the cavitation bubble was studied by Cui et al.[22], and the direction of the micro-jet and the propagation rate of the shock waves were captured.

With the development of the computer technology, the boundary element method (BEM) is widely used in numerical simulations of cavitation bubble dynamics for its inherent nature of reducing the dimension of the problem, and saving computing time, and also for its own unique advantages in dealing with three-dimensional and complex boundary surfaces (such as the cavitation bubble) or the free surface. Klaseboer et al.[23]simulated the ring stage of the cavitation bubble and the vortex surfaces, to study the dynamic characteristics of the cavitation bubble and the interaction with walls for different distances between the bubble and the wall, and the results are in agreement with the experimental ones. Wang and Khoo[24], and many others carried out numerical simulations for the cavitation bubbles.

The air bubbles are ubiquitous in the water flow in actual engineering. However, the behavior of a cavitation bubble near one or more air bubbles with a wall is not well studied. The focus of this paper is to predict the direction of the collapse of a cavitation bubble under the dual impacts of an air bubble and a wall. In the micro-level, this paper studies the cavitation bubble collapse near the air bubbles, the interaction among the cavitation bubble, the air bubbles and the wall, and the interacted process of the cavitation bubble with double air bubbles near a wall,as a study of the aeration mitigation cavitation.

1. Experimental setup and method

As shown in Fig. 1, several complex systems are used to investigate the interactions among cavitation bubbles, air bubbles and the walls, including a single-pulsed high voltage discharge cavitation bubble induced system, an air bubble release system and a high-speed dynamic data acquisition and analysis system.

Fig. 1 (Color online) The schematic diagram of experimental devices

When a single-pulsed high voltage discharge cavitation bubble system works, the voltage of the discharge device could reaches 0 kv-50 kv when the electric current passes through a step-up transformer,with the protected water resistance and the rectifier regulator of silicon stack. Once the voltage reaches or exceeds the air breakdown voltage threshold, the circuit is connected and the tungsten electrodes discharge, to immediately generate cavitation bubbles.By adjusting the distance between the discharge devices and the resistance value in the circuit, the desirable cavitation bubbles can be obtained to meet the experiment requirements. The plexiglass is used as the rigid wall material in the interaction experiments of the cavitation bubbles with the wall. Since the air bubbles are required to appear around the cavitation bubbles in a very short time for the study of the interaction of the cavitation bubble with the air bubbles, an air bubble release system is designed to be compatible with the cavitation bubbles. The air bubbles in the water are automatically pushed by the air injection pump to the pinholes (of 0.5 mm in inner diameter) pre-fixed in the water tank (0.4 m×0.3 m×0.4 m), which is filled with deionized water. In the experiment, the air bubble size is controlled by the diameter of the pinholes. In the high-speed data acquisition and analysis system, a certain area of the air bubble rising path is selected to monitor the change of the pixel RGB. If the pixel value changes in the selected area, the high-speed data acquisition and analysis system orders the pre-charged capacitor to discharge, and the high-speed camera to take pictures.The position of the electrode determines the position of the cavitation bubble, and the area in which the pre-selected air bubble passes controls the position of the air bubble and the angle between the air bubble,the cavitation bubble and the wall. Through coordinated work of above-mentioned several systems,we can have air bubbles around the cavitation bubbles.The water temperature keeps constant 22°C during the experiment for avoiding the evident influence of the outside temperature to the evolution of the cavitation bubbles.

Since the time range of the whole evolution is within a millisecond level, a high-speed dynamic data acquisition and analysis system is needed to study the very fast evolution process. The high-speed dynamic data acquisition and analysis system consists of a high speed camera, the lens assembly, a synchronization device, a lighting equipment, and computer components. The high-speed camera is the main equipment in the whole system, which is the MotionPro Y3-classic high-speed camera (Integrated Design Tools Inc., USA, Maximum data acquisition rate: 105fps). The cavitation bubbles and the air bubbles are video taped with a telephoto microscope lens Navitar (Zoom 6000, Navitar, USA) combined with the high-speed cameras. The high-speed camera’s fast filming speed, rigorous illumination condition for a short exposure time, combined with the illumination requirements of the microscope make more severe requirements to the lighting equipment when these two facilities are used together. Therefore,these experiments need a cold light source (Halogen,150 W) with better lighting and smaller heat.

2. Collapse of cavitation bubble near an air bubble

The detailed interaction between the air bubble and the cavitation bubble at different distances can be realized by keeping the constant distance of the discharge needle-plate in the circuit while changing the distance between the port that releases the air bubble and the electrodes. The voltage and the resistance of the four sets of experiments remain unchanged as shown in Fig. 2, Theoretically, the maximum radius of the cavitation bubbles in the four sets of experiments are the same, however, the maximum radius of the cavitation bubbles in groups c and d vary a lot compared with groups a and b. It is caused by the distance between the air and the cavitation bubbles, and the fact that the cavitation bubble has merged with the air bubbles in the expansion phase (see Figs. 2(c3) and 2(d3)). During the fusion of the two bubbles, the cavitation bubble contains a large amount of non-condensable gas, to form a gas-type cavitation bubble. In Fig. 2, the maximum expansion radiusRmaxof the cavitation bubble is selected as the characteristic parameter of the cavitation bubble,ε, the ratio of the initial air bubble radius to the characteristic parameter of the cavitation bubble, is the dimensionless air bubble radius,dis the minimum distance of the center of the cavitation bubble to the surface of the air bubble,δis the ratio of the nearest distance between the cavitation bubble center and the air bubble surface to the characteristic parameter of the cavitation bubble.

The first row of pictures in Fig. 2 show the air bubble positions from the electrode center before the cavitation bubble is incipient. In the second row of pictures, the bright white light is the water discharge electrode when the air bubbles in group a is farther away from the electrode center as compared with those in group d. The light intensity is strongest in the electrode center, and weaker outward. The bubbles in groups a and b have expanded to the maximum volume before the line 4, while those in groups c and d reach their maximum volume at the line 5 and line 6,respectively. At this stage, the bubbles in groups a and b farther away from the electrode centers are penetrated by the main shock wave generated by the expansion of the cavitation bubbles. Compared with that in group b, the bubble in group a have later breakdown time point (respecitvely Figs. 2(a3) and 2(b2)). For the bubbles in groups c and d, due to the close distance between the cavitation bubble and the air bubble, the cavitation bubble in group c breaks in the expansion phase much earlier, while the bubble in group d is fully integrated with the air bubble, to become a gas-type cavitation bubble. In group c, some air bubbles are split in the wake of the main shock wave and for the same reason, some of them, as precursors, form some breakdown split bubbles, as shown in c4, c5and c6.

Fig. 2 (Color online) The interaction between the air bubbles and the cavitation bubbles at different distances (Filming speed:19 700 fps, Exposure time: 48 μs, Frame size: 17.33 mm×7.36 mm)

After the contraction phase, the space is quickly released due to the cavitation bubble contraction and rapidly filled with the surrounding water, meanwhile,the air bubbles move slightly under the flow effect of the cavitation bubbles, as shown in group b.Compared with group a, the air bubbles of group b move more apparently to the center of the cavitation bubbles. The air bubbles are in dumbbell shape under the double actions of the collapse shock and the backfill flow, and the air bubbles in group a is farther away from the center of the cavitation bubbles as compared with the bubbles in group b, with a more obvious dumbbell shape. During the collapse process of the bubbles in groups a and b, the cavitation bubbles do not move to the air bubble, and the bubbles of both groups generate resilient bubbles after the complete collapse of the cavitation bubbles.Compared with the bubbles in groups a and b, the cavitation bubbles in groups c and d have longer expansion-collapse period. Under the same experimental conditions, this phenomenon is due to the fact that the air bubbles are closer to the cavitation bubble center, part or even all of the non-condensable gases of the air bubbles are mixed into the cavitation bubbles to form the gas-type cavitation bubbles in the expansion or collapse phase. As can be seen from the photos, the more non-condensable gas is mixed in the expansion phase, the longer the expansion-collapse period of the cavitation bubble will be.

Through the study of the interaction between the air bubbles and the cavitation bubbles at different distances, it can be found that with the gradually decreased distance between the cavitation bubbles and the center of the air bubbles, the air bubbles are more likely to break down in the expansion phase of the cavitation bubble. When the distance between the cavitation bubble and the air bubble is short enough,the cavitation bubble absorbs the air bubble to form a gas-type cavitation bubble. By generating the cavitation bubbles by means of underwater lowvoltage discharge, and changing the position of the cavitation bubbles to interact with the hemispherical bubbles adhered to the underwater plates, Goh et al.[15]found that when the bubble-wall dimensionless distance is in the range 1.5＜γ＜4.5, the direction of the jet flow has the tendency of facing to or deviating from the collapse. In this paper, the cavitation bubbles in Fig. 2(a) are collapsed away from the air bubble,while the cavitation bubbles in Fig. 2(b) are collapsed in situ. The energy of the cavitation is mainly concentrated in the first two expansion and contraction processes. Therefore, the critical bubblewall distance of the fusion in these two stages needs to be considered in combination with the shape and the size of the cavitation bubbles, In the experimental method of this paper, since the air bubble is free-moving, the distance between the air bubble and the cavitation bubble cannot be precisely controlled.As a result, we fail to obtain the critical condition for the fusion of free-moving air bubbles and cavitation bubbles.

2.1 Effect of air bubble on the collapse direction of cavitation bubble

A single cavitation bubble collapsing near a solid boundary is a well-studied phenomenon[1-6]. Under certain conditions, a liquid jet directed towards the boundary is formed during the collapse of the cavitation bubble. A large number of studies indicate that the wall strongly attracts the cavitation bubble in the collapse phase.

Figure 3 shows the interaction of the cavitation bubble with an air bubble: The black area in each picture on the left side is the air bubble, and the cavitation bubble is on the right side. The maximum radiiRmaxof the cavitation bubbles, respectively, are 2.76 mm and 3.35 mm, the dimensionless sizesεof the air bubbles are 0.39 and 0.28, and the minimum distancesδbetween the centers of the cavitation bubble and the surface of the air bubble are 2.29 and 2.35.

The pictures a12-a25and b13-b25in Fig. 3 show two cases of the collapse directions of the cavitation bubble, collapsing away from and to the air bubble,and it can be said that the air bubbles have the repulsion and attraction tendency for the collapse of the cavitation bubbles.

Fig. 3 (Color online) Repulsion and attraction of air bubbles on the cavitation bubbles (Filming speed: 197 00 fps, Frame size：17.33 mm×7.36 mm)

Under certain conditions, two different situations can be considered regarding to the bubble-bubble interaction: the cavitation bubble collapses towards and away from the air bubble. Furthermore, the rigid wall has an attraction to the collapse of the cavitation bubbles, while the air bubbles show certain repulsion and attraction to the collapse of the cavitation bubbles.By generating cavitation bubbles by means of underwater low-voltage discharge, and changing the position of the cavitation bubbles to interact with the hemispherical bubbles adhering to the underwater plates, Goh et al.[15]found that the dimensionless cavitation oscillation timeT′ is an important parameter that determines the collapse direction of the cavitation bubble (T′ is the ratio of the time in the first cycle of the cavitation bubble to the inverse of the resonance frequency of the air bubble). WhenT′＞ 1 .2, the cavitation bubble collapses toward the air bubble. WhenT′＜1.0, the cavitation bubble collapses away from the air bubble and the wall.WhenT′≈1.0, the cavitation bubble collapses and splits into two parts, to produce a jet toward the air bubble and a jet away from the air bubble. The interaction between the cavitation bubble and the air bubble studied in this paper has two main features different from those in the references. First, with the experimental method used in this paper, the motion of the air bubbles in the water is not affected by the outside world. Second, the shape of the air bubble interacting with the cavitation bubble is not a regular sphere. The air bubble generated by the experimental method used in the Ref. [15] is a hemispherical and still air bubble, similar to that in this paper whenθ=0 (θis the angle between the center line of two bubbles and the normal to the boundary). Therefore,these sections will study the interactions, respectively,when the air bubble locates between the cavitation bubble and the wall, and when the cavitation bubble locates between the air bubble and the wall.

In Fig. 4, a1and a2show the collapse directions of the cavitation bubble when the cavitation bubble locates between the wall and the air bubbles, a3, a4show the collapse directions of the cavitation bubbles when the air bubble locates between the wall and the cavitation bubble. As can be seen in experimental results, the cavitation bubbles move away from the air bubbles pushed by the repulsion forces of the air bubbles, meanwhile, the wall has attraction to the cavitation bubbles, which eventually collapse in the combined vector direction under the combined action of the two forces. The picture b1, b2in Fig. 4 present the collapse direction of the cavitation bubbles when the cavitation bubbles locate between the wall and the air bubbles, while picture b3, b4present the collapse direction of the cavitation bubbles when the air bubbles locate between the wall and the cavitation bubble. Similar to the experimental results in group a,the ultimate collapse direction of the cavitation bubbles depends on the combined vector direction when the air bubbles have an intensified attraction to the cavitation bubbles and the rigid wall. At present,our experiments are limited by the cavitation bubble induction methods and measurement techniques. The attraction of the wall towards the cavitation bubbles and the attraction or rejection of the air bubbles to the cavitation bubbles need further exploration, but the mechanism of the collapse direction of the cavitation bubbles under the action of the air bubbles and the walls is well studied.

2.2 Effect of air bubble on the collapse strength of cavitation bubble

Fig. 4 (Color online) Vector synthesis of the cavitation bubbles,air bubbles and the wall

Figure 5 shows the high-speed photographs of the cavitation bubbles which absorb and merge with the air bubbles in the interaction process under the same experimental conditions. In Fig. 5, the dimensionless radius of the air bubbles relative to the cavitation bubblesεranges from 0.21 to 0.22, the maximum radiusRmaxof the cavitation bubble varies from 3.57 mm to 4.06 mm, and the nearest dimensionless distance between the cavitation bubble center and the air bubble surfaceδranges from 0.72 to 1.01.The first column in Fig. 5 is the relative position of the electrode center and the air bubbles before the nascent cavitation bubbles. The cavitation bubbles are formed at the fourth column in each group experiment,while the cavitation bubbles in this process are in the expansion stage until the sixth column. At this moment, the air bubbles are broken down by the main shock generated by the cavitation bubble expansion.For instance, the air bubble in group b, due to the smallest dimensionless radius of the air bubble in the four groups of experiments, is not only broken down,but also generates a split part, (see Fig. 5(b6)). With a further expansion of the cavitation bubbles, the intensity of the main shock wave is reduced, and the outline shape of the air bubbles begins to grow due to the strong compressibility of the non-condensable gas inside the air bubbles, as shown in pictures from column 7 to column 9. When the cavitation bubbles are in the contraction phase as shown in pictures from column 10 to column 17, and the surrounding water fills the space released by the cavitation bubbles at once due to the rapid contraction, and the air bubbles are stretched by the action of the backfill. The main part of the air bubbles begins to grow from the tabular shape in the buffer stage into the slender shape, and with a further contraction of the cavitation bubbles,the water in the jets from the bubbles originally broken down moves faster to the center of the cavitation bubbles than the periphery water. At this moment, the air bubbles take tabular shapes at the original bubble breakdown position, mainly due to the uneasily stretched nature of the peripheral water with no air bubbles inside on one hand. And on the another hand, the strong compressibility of the noncondensable gas of the air bubbles makes themselves easily to be pulled to the center of the cavitation bubbles to take the tabular form. As can be seen from Fig. 4, sinceδin the group b is greater than that of the other three groups in the experiments, under the action of the backfill flow, the tabular shapes of the air bubbles develops significantly slower than the other three groups, which have merged with the cavitation bubbles in groups b, c and d in the later collapse.

From the expansion and the collapse of the cavitation bubbles and the morphology evolution of the air bubbles, it can be found that the surface of the air bubbles near the cavitation bubbles becomes concave under the action of the main shock wave,then the air bubbles break down to form a jet (see Fig.6(a)). Similar observations were made by Pain et al.[13]and Goh et al.[15]. The air bubble is stretched by the backfill water in the collapse process of the cavitation bubbles, and its surface near the center of the cavitation bubble becomes stepped (see Fig. 6(b)).From the shape analysis of the air bubbles at different stages, it can be found that the air bubbles are stretched into a stepped shape, which largely buffers the contraction and the collapse of the cavitation bubbles. On the other hand, the cavitation bubbles merge with the air bubbles, to form gas-type cavitation bubbles.

In the high-speed flow, the cavitation bubbles near the wall merge with the air bubbles during the development process when they are close enough, to have an incomplete collapse. The merged cavitation bubbles, containing non-condensable gas, have different collapse characteristics, as compared with other cavitation bubbles.

Fig. 5 (Color online) The breakdown jet and the backfill elongational flow (Filming speed: 19 700 fps, Exposure time: 48 μs,Frame size: 17.33 mm×7.36 mm)

Fig. 6 (Color online) Air bubble is penetrated and elongated

Figure 7 shows the merging process when the air bubble is between the cavitation bubble and the wall.The distance between the center of the cavitation bubble and the wall is 5.17 mm and 5.48 mm, and the dimensionless distanceγ(the ratio of the distance from the center of the cavitation bubble to the wall to the characteristic parameter of the cavitation bubble)between the centers and the wall is 1.68 and 1.78, both within the attraction range of the wall to the cavitation bubbles. Moreover, the bubble-boundary angleθis selected as the measure of the geometrical arrangement of a cavitation bubble and an air bubble near a rigid boundary. In Fig. 7, with the gradual increased volume of the cavitation bubble, the air bubble breaks down, to form a breakdown jet, as shown in a3and b3.The breakdown jet moves to the wall, and eventually rolls out along the flat wall by the blocking effect of the wall. In the contraction process of the cavitation bubble, the peripheral water quickly moves to the center of the cavitation bubbles, due to the breakdown air bubbles in contact with the wall, while the wall shows a strong adsorption to the air bubbles, then the air bubbles are stretched by the peripheral backfill flow, as shown in a8and b8. The cavitation bubble merges with the deformed air bubble in the collapse phase, to form a gas-type cavitation bubble, (see a12and b12in Fig. 7).

Figure 8 shows the merging process when the cavitation bubble locates between the air bubble and the wall. In the experiments, a constant distance is kept between the center of the cavitation bubble and the wall, and the size of the air bubbles is kept as 1.03 mm and 0.92 mm, and the dimensionless distances between the centers of the cavitation bubbles and the wall are 2.28 and 2.59.

In the two groups of experiments, the air bubble breaks down under the action of the main shock wave radiated by the cavitation bubble in the expansion phase, to form a breakdown jet flow. The length of the jet flow gradually decreases in the contraction phase of the cavitation bubble, and the air bubble is absorbed by the cavitation bubble in the collapse phase, with each cavitation bubble forming a gas-type cavitation bubble. Since the gas-type cavitation bubble contains a large amount of non-condensable gas, the cavitation bubble will not completely collapse, which would greatly relieve for the wall the destructive effects of the cavitation bubble collapse.

Figure 9 is the scatter plot of the maximum radiusRmaxof the cavitation bubbles with regard to dBmax(dBmaxis the peak noise level of the collapse of the cavitation bubble) in cases with and without the air bubbles. By adjusting the position of the air bubble releasing port over the holder in the experiment, the air bubbles can move freely to the water surface by the buoyant effect. Therefore, the cavitation bubble would interact with the air bubble in any position on the floating trajectory. After fixing the air bubble releasing port, the cavitation bubbles are generated by multiple discharging, in the experiments of the cavitation bubbles and the air bubbles of different distances.

In Fig. 9, it is seen that the noise level peak dBmaxradiated by the collapsed cavitation bubbles continues to rise with the gradual increase of the maximum radiusRmaxof the cavitation bubbles, but its growth trend gradually slows down. On the other hand, compared with the cases of no air bubbles, the noise peak level radiated by the collapsed cavitation bubbles gradually gets weaker when there is an air bubble around the cavitation bubble. Especially for the cavitation bubbles with the maximum radiusRmaxfrom 1.00 mm to 3.00 mm, the presence of the air bubbles will attenuate the noise peak level dBmaxmost significantly, and the impact of the air bubbles to the noise peak level gets softened with the gradually increased radiusRmaxof the cavitation bubble.

Fig. 7 (Color online) The merger when the air bubble is between the cavitation bubble and the wall (Filming speed: 19 700 fps,Exposure time: 48 μs, Frame size: 17.33 mm×7.36 mm)

Fig. 8 (Color online) The merger when the cavitation bubble is between the air bubble and the wall (Filming speed: 19 700 fps,Exposure time: 48 μs, Frame size: 17.33 mm×7.36 mm)

Figure 9 can be explained as follows: Firstly, the stretched air bubble, when in its contraction phase,could attenuate the powerful contraction of the cavitation bubble to some degree due to its compressibility.Secondly, when the air bubble and the cavitation bubble are close enough, the cavitation bubble merges with the air bubble into a gas-type cavitation bubble containing a large amount of non-condensable gas.The gas-type cavitation bubbles will have a non-complete collapse and largely decrease the collapse intensity. What is more, the air bubble size affects the buffering effect when the air bubble does not merge with the cavitation bubble. While forming gas-type bubbles, different sizes of the air bubble mean different amounts of non-condensable gas,which leads to varying collapse strength.

3. Collapse of cavitation bubble near two air bubbles

Figure 10 is the high-speed photographs of the interaction process of the cavitation bubbles with double air bubbles. The size of the air bubbles and the cavitation bubbles, and the relative positional relationship of the two are shown in Fig. 10, with the subscript 1 representing the air bubbles on the left side,and the subscript 2 representing the air bubbles on the right side.

In the expansion phase of the cavitation bubbles,whether the cavitation bubble locates on the same side of the air bubble or not, the double air bubbles near the cavitation bubbles will have an inner concave shape to varying degrees under the effect of the main shock, and the closer the air bubble to the cavitation bubble, the greater the inner concave level will be. In the contraction phase, the air bubble near the cavita-tion bubble forms a breakdown jet, and with further development, it will be detached from the air bubbles to form a single tiny bubble, as shown in a8and b6in Fig. 10.

Fig. 9 (Color online) The maximum radius Rmax of cavitation bubbles and the corresponding peak values of collapse noises

By generating the cavitation bubbles by means of underwater low-voltage discharge, Pain et al.[16]studied the jet phenomenon of the air bubbles wrapped inside the silicone oil under the action of the cavitation bubbles, and found that, the dimensionless distance from the center of the cavitation bubble to the center of the air bubble must be less than 2 to generate jets under the action of the cavitation bubbles.However, in Fig. 10(a), the distance from the air bubble to the closest cavitation bubble is greater than 2.33 (the bubble-bubble center distance is used as the reference, and the minimum distance from the cavitation bubble center to the air bubble surface is used in this paper), the jet can still be observed in the experiments in this paper. The main reason may be the difference in the method of obtaining the air bubbles in the experiment. The method used in the reference is to wrap the air bubble in silicone oil droplets, and in this paper a free-floating air bubble is generated in the water. Figure 11 is the high-speed photographs of the interaction process of the cavitation bubbles with double air bubbles near the wall. The size of the bubbles and the relative positional relationship of the two are shown in Fig. 11, with the subscript 1 representing the air bubbles on the left side and the subscript 2 representing the air bubbles on the right side.

Fig. 10 (Color online) The interaction of cavitation bubbles with double air bubbles (Filming speed: 19 700fps, Exposure time:48 μs, Frame size: 26.1 4mm×5.84 mm)

Fig. 11 (Color online) The interaction of cavitation bubbles near the wall with double air bubbles at the same side. (Filming speed:19 700 fps, Exposure time: 48μs, Frame size: 13.86 mm×7.36 mm)

As can be seen from Fig. 11, in the expansion stage of the cavitation bubbles, the two air bubbles experience a contraction deformation. As the radius of the cavitation bubble is further increased, the air bubbles near the cavitation bubbles first generate the jet, followed by the air bubbles near the wall with the jet flow being eventually developed towards the wall,as shown in a5, b5and c4. After the cavitation bubbles entering into the collapse phase, the space occupied by the cavitation bubbles is released, and the backfill flow is rapidly formed by the surrounding water. At this moment, the air bubbles are stretched into irregular shapes, while the air bubbles close to the wall is still attached to the wall.

4. Conclusions

With the experimental system of using high voltage pulse sparks to generate cavitation bubbles and a high-speed dynamic data acquisition and analysis system, a large number of experiments are conducted to study the interactions among the cavitation bubbles, the air bubbles and the wall. From the experimental results the following conclusions can be made:

(1) In the vicinity of the air bubbles, the cavitation bubbles may only collapse towards or from the air bubbles, while under the dual impacts of the air bubbles and the wall, the cavitation bubbles are subjected also to the attraction of the wall , along with the attraction or repulsion of the air bubbles, the final direction of the collapse of the cavitation bubbles depends on the combined vector of such two forces.

(2) If the cavitation bubbles are very close to the air bubbles, the cavitation bubbles in the expansion stage will penetrate the air bubbles, and the cavitation bubbles in the collapse stage will stretch the air bubbles. When the distance between the two is short enough, the cavitation bubbles and the air bubbles will connect to form gas-type cavitation bubbles, and then the collapse noise peak level will be significantly decreased, suggesting a significantly weakened collapse strength.

(3) By observing the interaction of two air bubbles and cavitation bubbles near the wall, it is found that the cavitation bubbles in the development process may penetrate the two air bubbles successively from the near to the distant ones.

The above results show that during the aeration and cavitation erosion mitigation, the number of air bubbles is more important than the total air concentration. These results have potential theoretical values for the anti-erosion design in hydraulic engineering, the erosion resistance of blades in hydraulic machinery,and the better use of the cavitation effect in ultrasonic cleaning technology.