THE COUNTER-JET FORMATION IN AN AIR BUBBLE INDUCED BY THE IMPACT OF SHOCK WAVES*

BAI Li-xin

Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China, E-mail: lixin.bai@gmail.com

XU Wei-lin

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

LI Chao

Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

GAO Yan-dong

School of Electronic and Information Engineering, Liaoning Technical University, Huludao 125105, China

THE COUNTER-JET FORMATION IN AN AIR BUBBLE INDUCED BY THE IMPACT OF SHOCK WAVES*

BAI Li-xin

Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China, E-mail: lixin.bai@gmail.com

XU Wei-lin

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

LI Chao

Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

GAO Yan-dong

School of Electronic and Information Engineering, Liaoning Technical University, Huludao 125105, China

The interaction of an air bubble (isolated in water or attached to a boundary) with shock waves induced by electric sparks is investigated by high-speed photography. The interaction is closely related to the counter-jet induced by the impact of shock waves. The formation of a counter-jet in an air bubble is related to the liquid jet formed in the same air bubble, but themechanism is different with that of the counter-jet formation in a collapsing cavitation bubble. The formation of a counter-jet in an air bubble is related to discharge energy, air bubble size and radius of shock wave. W ith a given energy of the spark discharge, the formation of a counter-jet in an air bubble is related to δ/ε (the ratio of the dimensionless bubble-bubble distance to the dimensionless air bubble radius). The counter-jet w ill only be produced when δ/ε is in the range of 1.2-2.2. The counter-jet in an air bubble is of an important nuclei-generatingmechanism.

cavitation bubble, shock wave, counter-jet, high-speed photography

Introduction

Inmost practical situations, cavitation bubbles do not occur in isolation but coexist with cavitation bubbles or air bubbles in large numbers[1,2]. These cavitation bubbles grow and collapse continuously. The shock waves released by the cavitation bubblesmay interact with boundaries, air bubbles or other cavitation bubbles[3,4]. The interaction of shock waves with air bubbles was investigated as issues in the area ofmedical treatment and hydrodynamics. Brujan[5,6]measured the shock wave emission after the optical breakdown in water experimentally. Philipp et al.[7]investigated the shock wave-induced collapse and jet formation of pre-existing air bubbles using high-speed photography. Kodama and Tom ita[8]investigated the interaction of an air bubble attached to a gelatin surface with a shock wave. The interaction of an air bubble with a shock wave produces a liquid jet, with a speed of tens ofm/s, towards the gelatin surface. Ding and Gracewski[9]and Jamaluddin et al.[10]simulated the air bubble responses to shock waves. It is found that for a weak shock (P＜30 MPa ), the air bubble w ill collapse spherically without jet formation, for a strong shock (500 MPa

Fig.1 Schematic diagram of the experimental setup

The abovementioned investigations focus on the liquid jet formation and the damage effect on the boundary, but the subsequent dynamic process of bubbles after the liquid jet formation is ignored. In fact, the bubbles w ill not break into pieces immediately. Kodama and Tomita[8]found a counter-jet in the opposite direction to themain jet in the experiment of the interaction of an air bubble with a shock wave. The counter-jet in an air bubble induced by the interaction of a shock wave is sim ilar to the counter-jet in a collapsing cavitation bubble near a solid boundary[17,18]. Kodama suggested that the bubble interacting with the shock wave expands after the rebound, while the liquid jet penetrating into the gelatin returns back to the upper stream to produce the counter-jet.

In this article, an experimental investigation is carried out on the counter-jet formation in an air bubble (isolated in water or attached to a boundary) with shock waves induced by electric sparks bymeans of high-speed photography. Themechanism and the conditions of the counter-jet formation are investigated in detail. It is found that the counter-jet is not produced by the bounce of the penetrating liquid jet on the boundary as Kodama suggested, and the counter-jet is also different from the counter-jet in a collapsing cavitation bubble near a solid boundary. The counter-jet in an air bubble is of an important nucleigeneratingmechanism, in which the air bubble is split into smaller bubbles (nuclei) even if the air bubble is already as small as the cavitation nuclei and can not be split by a turbulent fluctuation. The research w ill provide some food of thought in the utilization of the cavitation phenomena and in the prevention of the cavitation erosion.

1. Experimental facilities

The experimental setup consists of a cavitation bubble generation system, imaging and illumination devices, and a device to adjust the relative position of air bubbles and electrodes. Cavitation bubbles are generated in a glass chamber (170mm3× 50mm3× 90mm3, (length × w idth × height)) filled with deionized water (at room temperature, T=291K , under atmospheric pressure, P =101.5 kPa ) by using highvoltage spark discharge.

Air bubbles are carefully positioned on the surface of amovable waxed thread bymeans of a syringe. Because of the small size of air bubbles (with diameters generally smaller than 5mm), the deformation is small. The relative position of the air bubble and the cavitation bubble can be adjusted arbitrarily.

Themovements of the cavitation bubble and the air bubble are recorded with a CMOS HG-LE highspeed camera (Redlake MASD Inc. USA) equipped with a long distancemicroscope (Zoom 6000, Navitar, USA). The pictures are taken in a framing rate of 10 000 fps (256 × 256 pixels and 12 μm pixel size) or 3 000 fps (512 × 512 pixels) and a exposure time of 28 μs-42 μs. The frames are illum inated with a cold light resource (150 W) and fibre bundle. A schematic diagram depicting the experimental arrangement used for investigating the interaction of an air bubble and shock waves is shown in Fig.1.

2. The counter-jet in an isolated bubble

The formation of the counter-jet induced by the interaction of a shock wave is shown in Fig.2. The shock waves are induced by electric sparks. When a cavitation bubble is generated, a shock wave termedthe primary shock wave is generated. When a cavitation bubble rebounds, a secondary shock wave is produced. Under the action of the primary shock wave, the bubble wall near the electrode shrinks rapidly and the bubble wall on the other side stays almost unchanged (Fig.2(a2) and Fig.7(a2)). The air bubble as a whole shrinks asymmetrically to produce a liquid jet on the side of the bubble in the direction of the shock wave propagation. The jet penetrates into the bubble and goes out from the far surface. When the cavitation bubble begins to collapse, the air bubble rebounds with an enlarged size (Fig.2(a3)). The secondary shock wave is produced when the cavitation bubble collapses. The air bubble splits into two air bubbles under the impact of the secondary shock wave. The air bubble near the electrode is a crater-shaped bubble (Fig.2(a4) and (Fig.4(b1))), which becomes an air jet with a speed as high as 5m/s within 0.6ms. The Fig.2(a) is sim ilar to Fig.2(b) except that the cratershaped bubble becomes a cone-shaped bubble in Fig.2(b4).

Fig.2 Counter-jet formation induced by the interaction of a shock wave (Frame rate 3 000 fps. Exposure time 42 μs)

Fig.3 Counter-jet formation induced by the interaction of a shock wave (Frame rate 10 000 fps. Exposure time 42 μs)

In order to investigate in detail the formation ofthe crater-shaped bubble and the cone-shaped bubble, high-speed photographs (with frame rate of 10 000 fps) covering the whole period of the bubble-shock wave interaction are taken (as shown in Figs.3(a), 3(b)). Because of the high framing rate, the size of the pictures is small, and only part of the bubble can be seen in the pictures. The experiment condition and bubble dynamics as shown in Fig.3 are the same as those in Fig.2, so we can refer to these figures to get a full view. The first collapse occurs in Fig.3(a7) and the second collapse occurs in Fig.3(a9) and Fig.2(a4). The air bubble deformation in Fig.2(a4) is due to the secondary shock wave induced by the cavitation bubble collapse (as shown in Figs.3(a7), 3(a8)). The shape of the split bubble near the electrode in Fig.2(a4) is the same as what shown in Fig.3(a10). The coneshaped bubble grows up towards the electrode with its bubble wall on the left side remaining almost unchanged. It grows up into a crater-shaped bubble and then a tower-shaped bubble (as shown in Fig.2(a5) and Fig3(a12)) and then breaks up with a small bubble being ejected (as shown in Fig.2(a6) and Figs.3(a13), 3(a14)). The air bubble deformation in Fig.3(b) is the same as that in Fig.3(a). The cone-shaped bubble (as shown in Fig.3(b8)) becomes a tower-shaped bubble (as shown in Fig.3(b9)) after 100 μs -200 μs.

Fig.4 Details of the counter-jet formation

The pictures in Fig.2 and Fig.3 are small, where details can not be seen, so Fig.3(b7) and Fig3(b8) aremagnified as Fig.4(a1) and Fig.4(a2), and Fig.2(a4) and Fig.2(a5) aremagnified as Fig.4(b1) and Fig.4(b2). Figure 4(a1) shows that under the impact of the secondary shock wave, the air bubble tends to change the shape and becomes a cone-shaped bubble right in the root of the liquid jet. Then the second sock wave hits the air bubble, the bubble begins to contract. Because of the cone-shaped liquid jet in the air bubble, the air bubble contracts along the liquid jet and becomes a coreshaped bubble, whichmay have a liquid core. The air bubble collapses rapidly, and the fluid outside the bubblemoves towards the bubble and collides at the cone tip with a high speed. The cone tip becomes a high-pressure region (as shown in Fig.4(a2)). When the shock wave passes across the bubble and the bubble begins to rebound, the high pressure at the cone tip w ill push the cone-shaped bubble towards the electrode to become a crater-shaped bubble (as shown in Fig.4(b1)) and then a tower-shaped bubble (as shown in Fig.4(b2)).

Figure 4(b1) shows that the left side of the cratershaped bubble is flat with sharp edges, however, the right side is concave with blur edges, whichmeans that the two sides of the cone-shaped bubble have been collided already. W ith the same exposure time,more blur edgesmeans higher speed of the bubble wall. The left side of the crater-shaped bubble stays unchanged and the right sidemoves rapidly in the negative direction of the shock wave. The counter-jet is thus formed. By the way, the tower-shaped bubble in Fig.4(b2) consistsmaybe of two parts: a vortex ring bubble on the left and a tower-shaped bubble with a liquid jet inside on the right.

Figure 5 shows schematic diagrams of the air bubble deformation impacted by shock waves. The primary shock wave is induced by the electric spark in the early stage of the cavitation bubble generation. The primary shock wave hits the air bubble. The spherical air bubble shrinks asymmetrically to produce a liquid jet, formed by one side of the bubble in the direction of the shock wave propagation (as shown in Fig.5(b)). The jet penetrates the cavity and goes out from the far surface. When the shock wave passes across the bubble, the bubble rebounds in an enlarged size (as shown in Fig.5(c)). The secondary shock wave hits the bubble and the bubble shrinks again. Because of the cone-shaped liquid jet in the bubble, the dynam ics of the bubble is different from that of a spherical bubble. The air bubble contracts along the liquid jet to become a cone-shape small bubble (as shown in Figs.5(d), 5(e))). The air bubble collapses rapidly, and the fluid outside of the bubblemoves towards the bubble and collides at the cone tip at a high speed. The cone tip becomes a high-pressure region. When the secondary shock wave passes across the bubble and the bubble begins to rebound, the high pressure at the cone tip pushes the cone-shaped bubble towards the negative direction of the shock wave propagation (as shown in Fig.5(e)) to become a cratershaped bubble (as shown in Fig.5(f)) and then a tower-shaped bubble (as shown in Fig.5(g)). Finally the bubble breaks up with a small bubble, which is ejected (as shown in Fig.5(h)).

Fig.5 Conceptual illustration of counter-jet formation induced by shock wave

Fig.6 Deformation of an air bubble attached to a boundary impacted by a shock wave (Frame rate 3 000 fps. Exposure time 28 μs)

Fig.7 The critical distance for the counter-jet formation (Frame rate 10 000 fps. Exposure time 42 μs)

3. The counter-jet in a bubble attached to a boundary

The deformation of an air bubble attached to a boundary impacted by a shock wave is similar to that of an isolated bubble. But the boundarymakes a little difference. The experiment condition and the bubble dynam ics in Fig.6(a) are the same as those in Fig.6(b), but the startingmoment of taking photos is different with photos in Fig.6(b) being taken earlier in time so we can cross-reference to get a full view. The primary shock wave is induced by the electric spark. It hits the air bubble, which contracts without liquid jet inside (as shown in Fig.6(b2)). The liquid jet is formed when the cavitation bubble begins to collapse and the air bubble begins to grow up (as shown in Fig.6(a2)), which is different from the air bubble in Fig.3 and Fig.5, whose liquid jet is formed in the early stage of cavitation bubble generation. This is because the air bubbles have different compressibility. Hit by the same shock wave, a big air bubble w ill deform slightly and slow ly because of the cushion effect as compared with a small air bubble. When the volume of the air bubble is infinitely large, i.e. it becomes a cavitation bubble, to grow and collapse near the free surface of liquid, the liquid jet is also formed in the collapse stage of the cavitation bubble[19]. The liquid jetmoves towards the boundary and forms a full developed liquid jet (as shown in Fig.6(b3)). The secondary shock wave is induced when the cavitation bubblecollapses. Under the impact of the secondary shock wave, the air bubble begins to contract to form a convex shape at the root of the liquid jet (as shown in Fig.6(a3) and Fig.6(b4)), as is comparable with the deformation of the air bubble in Fig.5(d). The shock wave is weak for the air bubble, that is why a coneshaped bubble with a sharp edge is not formed. The air bubble begins to rebound after the shock wave passes away (as shown in Fig.6(a4) and Fig.6(b5)). A smaller convex bubble is formed on the surface of the former convex bubble under the inertia and due to the oscillations of pressure (as shown in Fig.6(a5) and Fig.6(b6)). Figures 6(b6), 6(b9) show the process of ejecting a small bubble.

4. The form ing conditions of counter-jet

The counter-jet as a special physical phenomenon was largely neglected in consideration. Maybe the counter-jet was not observed in experiments inmany studies. The formation of a counter-jet requires some necessary conditions. The shock-wave pressure impulse and the compressibility of the air bubble influence the formation of the counter-jet. The formation of the counter-jet is related to the diameter of the air bubble with a given shock wave. A counter-jet w ill not be formed if the air bubble is too large. For example, when the volume of the air bubble is infinitely large, as a cavitation bubble grows and collapses near the free surface of liquid, no counter-jet w ill be formed. On other hand a counter-jet w ill not be formed if the air bubble is too small. For example, the air bubble is too small relative to the pressure impulse of the shock wave in Philipp’s experiment[3]and is too large relative to the pressure impulse of the shock wave in Kodama’s experiment[3]. Figure 7 shows the critical distance for the counter-jet formation when the diameter of the air bubble and the energy of the spark discharge are constant. The distance between the electrode and the bubble wall is 3mm in Fig.7(a) and 5mm in Fig.7(b). The distance is a little too large in Fig.7(a) and a little too small in Fig.7(b) to form a counter-jet. A embryonic form of cone-shape is formed in Fig.7(a7) and Fig.7(b7) like in Fig.5(d), but no counter-jet is formed.

If the formation condition for the cavitatoin bubble is the same, i.e., the energy of the spark discharge is constant, the formation of the counter-jet is related to the radius of the shock wave1r (the distance between the electrode and the bubble wall) and the radius of the air bubble r2. The ratio of1r to r2defines the condition of the counter-jet formation. W ith dimensionless parameters as often used in describing the relative position of cavitation bubbles[20], a dimensionless formula for the condition of the counter-jet formation is,

where R is themaximum cavitation bubble radius, δ the dimensionless bubble-bubble distance, and ε the dimensionless air bubble radius.

More than seventy experiments were carried out to determine the conditions of the counter-jet formation. The Fig.8 shows that a liquid jet w ill be formed in an air bubble impact by a shock wave when δ/ ε＜6, and no liquid jet is formed when δ/ε＞8. A counter-jet w ill only be produced when a liquid is formed and δ/ε is in the range of 1.2-2.2.

Fig.8 Conditions of counter-jet formation

5. Conclusion

The interaction of an air bubble (isolated in water or attached to a boundary) with shock waves induced by electric sparks is investigated by high-speed photography. These events are related to the counter-jet induced by the impact of shock waves. A counter-jet w ill only be produced when a liquid-jet is present, but themechanism is different from that of a counter-jet formation in a collapsing cavitation bubble. When a cavitation bubble is generated, the primary shock wave impacts the air bubble surface, which contracts and yields a liquid jet inside the bubble. When the cavitation bubble collapses, the secondary shock wave impacts the air bubble, which contracts again. Because of the influence of the cone-shaped liquid jet inside the air bubble, the deformation process is different from a spherical air bubble. The air bubble tends to attach to the liquid jet during the contraction. Because of the liquid jet, a cone-shaped small air bubble is produced with another big air bubble behind it. A high pressure is produced in the conic node. The coneshaped small air bubble is pushed in the opposite direction of the shock wave propagation and a counter-jet is produced. The formation of a counter-jet in an air bubble is related to discharge energy, air bubble size and radius of shock wave. W ith a given energy of the spark discharge, the formation of a counter-jet in anair bubble is related to δ/ε (the ratio of the dimensionless bubble-bubble distance to the dimensionless air bubble radius). The counter-jet w ill only be produced when δ/ε is in the range of 1.2-2.2. The counter-jet in an air bubble is of an important nucleigeneratingmechanism. The studymay help the utilizetion and prevention of cavitation.

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January 6, 2011, Revised March 29, 2011)

10.1016/S1001-6058(10)60150-3

* Project supported by the National Science and Technology Support Plan of China (Grant No. 2008BAB29B04), the National Basic Research Program of China (973 Program, Grant No. 2007CB714105), the National Natural Science Foundation of China (Grant No. 11174315) and the National Science and Technology Major Project of China (Grant No. 2011ZX05032-003).

Biography: BAI Li-xin (1977-), Male, Ph. D.

XU Wei-lin, E-mail: xu_w l@sina.com