Thin jet, thick jet, and crown formation

Figure 3.4 clearly shows a typical dynamic of a shallow underwater oscillating bubble and a water jet with complete structure and morphology on the free surface.

The depth of the bubble from the free surface is 0.8 defined in stand-off parameter γ (h ∕ Rmax, h is initial distance between the center of the bubble and the free surface, Rmax is maximum bubble radius). As shown, the bubble extrudes a thin jet during its period of first expansion, as similar as described in recent works [1-3], and the thin jet continues to rise after the first expansion. At the end of the first expansion, the bubble starts to collapse due to the pressure decrease inside the bubble and moves downward, leading to the sinking around the surface on the bottom of the thin jet. This sinking surface was previously referred as surface depression [5]. When the sinking surface reaches its maximum depth, the surrounding water will flow toward the crater of the sinking surface, resulting in moving upward of the maximum depth point and the formation of the thick jet under the thin jet on the free surface [5]. Eruption of liquid jets from collapsing depressions has been observed in other circumstances, like standing Faraday-waves [6] and free-fall of a drop [7] on a liquid surface, and granular jets [8].

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Fig. 3.4

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It should be noted that, before the formation of the thick jet, the thin jet has already formed on the sinking surface. This is in contrast to the general cases of surface depression in which only a crater (without the thin jet) appears on the free surface by inserting an external force [6-8] or a burst cavity just below the surface [9].

To help see the detailed surface curvature of the surface depression with a thin jet, we have applied the boundary integral method assuming a pressure p p= _c+p V V0( 0 )^1.4 for a bubble in an inviscid, incompressible, and irrotational fluid domain, where pc is a constant vapor pressure, V is the volume of bubble, and 1.4 the ratio of specific heats.

Detailed numerical procedures of boundary integral method can be obtained from Appendix B and Refs. 2 and 3. Figure 3.5 shows the simulated dynamics of the bubble and free surface, along with the detailed schematic curvature of the surface depression with the thin jet. In fluid systems, a surface depression usually causes a singularity or near singularity of certain physical observables such as the divergences of velocity, surface curvatures, or pressure gradients at the minimum depth or pinch-off point [6]. The collapse of a surface singularity or near singularity leads to a jet formation. As shown in Fig. 3.5, the cross-sectional schematic image shows three singularities or near singularities as denoted by three solid dots. The central dot, obtained by linearly extrapolating the symmetric curves of the free surface without the thin jet, results in a thick jet as previously observed in collapsing depression [5]. The surface connected from the crater to the thin jet contributes to a curvature change which can induce two off-axis jets during the collapse of surface depression as indicated by the two arrows in Fig. 3.5. From the top view of the surface depression, this curvature change encircles around the thin jet, as shown in Fig. 3.6, and is called circular ring-shaped crater in the following discussions. It is the radially outward motion of the collapsing circular ring-shaped crater that results in the crown-like structure in Fig. 3.4.

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Fig. 3.5 The numerical simulation and curvature schematic of the surface depression are shown. The z and r of cylindrical coordinate are normalized to the Rmax of the bubble.

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Fig. 3.6 A surface schematic of the circular ring-shaped crater is shown, which is generated around the thin jet during the surface depression induced by the downward collapse bubble.

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When the bubble is very near the free surface for γ = 0.28, as shown in Fig.

3.7(a), the surface depression can be clearly seen with a segment of black line (layer of surface depression) generated on the horizontal free surface after the sinking around the bottom of the thin jet. The black line symmetrically shrinks toward the center axis from its opposite ends, and finally a thick jet forms under the thin jet, as depicted in the last picture in Fig. 3.7(a). The slow depression and collapse of the surface sinking are due to the extremely slow collapse of the bubble. The strong interaction between the upper boundary of the bubble and the free surface during the first expansion phase leads to some energy loss from the bubble similar as a venting effect mentioned in Ref. 10. Such an evolution about this venting effect was described more detail in the studies of laser ablation on a surface [11,12]. This venting effect could explain the unusual slowdown of the collapse of the bubble. Besides the slow collapse, the upper boundary of the bubble is so close to the free surface that it obstructs the downward depression of the surface sinking. Thus, the depression of the surface is not strong enough for the crown-like formation despite the appearance of the thin and thick jets. When γ = 0.48, the venting effect is diminished and the strength of the surface depression is restored to a level as the case of γ = 0.8 (near velocity of the thick jet). However, the thin jet experiences a violent disruption when going through the breach of the free surface by the expansion of the bubble at an initial time. The scraggly surface on the thin jet will disrupt the circular ring-shaped crater, and the crown-like structure is weak and unstable, as shown in Fig. 3.7(b).

When γ > 1.1, the weak interaction between the free surface and the bubble leads to a diminished thin jet shaped like a small hill and weak surface depression. Such a surface depression is not large enough to cause the circular ring-shaped crater to reach the threshold of surface topology change [6] to induce the crown-like formation.

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For c apt ion se e p. 67 .

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For c apt ion se e p. 67 .

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Fig. 3 .7 . T he f ra m e ra te is 80, 00 0 fps as sa m e as Fig. 3 .4 . F ig ur es 3.7 (a ) a nd ( c) il lus tra te the non -c ro w n for m at ion whe n γ- va lue is s m alle r t ha n 0. 5 a nd l ar ge r th an 1. 1, r es pe cti ve ly . F ig ur e 3.7 (b) sh ows an ot he r str uc tur e of th e cr own f or m at io n as co m pa re d to Fig. 3 .4 . Th is c rown is de no te d a s un sta ble c ro wn f or m at io n.

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In the second row of Fig. 3.7(c), we can see a vertical filament generated between the small hill on the free surface and the bubble, which is called secondary cavitation. This mechanism is discussed in the work by Tomita et al. [13] as follows.

When the bubble collapse to a very small volume, the pressure inside the bubble is increased, which can be several MPa depended on the minimum volume of the bubble at the collapse points. Due to the ultrahigh pressure, the bubble will release a shock wave, and this shock wave will be reflected by the free surface which is similar to a concave mirror, as shown in Fig. 3.8(a) (the Fig. 5 of the Ref [13]). Then, the reflected shock wave will focus on the central axis below the free surface, resulting a negative pressure which can induce a water cavitation for generating the vertical filament. An important study about the region with negative pressure was pointed by Blake et al.

[14] and Robinson et al. [3] from the theoretical simulation which shows a zero pressure region is formed during the first collapse of the bubble below the free surface, as shown in Fig. 3.8(b) (the Fig. 4(g) of the Ref [3]). Because the bubble is also generated by laser-induced “cavitation”, the cavitation generated by the reflected shock wave is called secondary cavitation. Additionally, in our experimental results, this secondary cavitation usually occurs when the bubble depth γ > 1. We consider that this reason is due to the minimum volume of the bubble at the first collapse.

Figure 3.7 shows that the volume of the bubble at the first collapse is smaller when the bubble depth is increased. As a result, the shock wave emitted from the bubble is increased when the bubble depth is increased for inducing the secondary cavitation.

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(a)

(b)

Fig. 3.8 (a) The Schlieren photograph about the interaction between the shock wave emitted from the bubble and the free surface [13, Fig. 5]. (b) A zero pressure region is formed between the bubble and the bottom of the thin jet [3, Fig. 4(g)].

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To see the evolution of bubble depth, Figs. 3.4 and 3.7 illustrate the morphologies of water jet for each γ-value of the above-mentioned four γ ranges.

Detailed correlation between the thin jet and crown-like structure is shown in Fig. 3.9 with increasing γ from 0.5 to 1.02. For γ = 0.5, the morphologies show the splash of water from the breached surface into multiple droplets. It is difficult to distinguish the shape of the thin jet from the upward splash. This defective structure of the thin jet results in an unstable crown, similarly as shown in Fig. 3.7(b). Increasing γ can reduce the influence of the water splash on the thin jet which begins to show a continuous line structure, as implied in the fourth frame of Fig. 3.9 for γ = 0.58. The crown-like jet is still unstable, and thus the circular ring-shaped crater resulted from the surface depression should be still defective. When γ ≥ 0.6, the splash is depressed and the crown structure becomes clear. When γ = 0.7, breached surface is absent and a pronounced growth of crown-like structure can be seen. The velocities of both the thin and thick jets decrease with increasing γ from 0.7 to 1.02 due to the depression of the surface interaction, and the height of the crown wall thus gradually decreases, and eventually disappears, as shown in Fig. 3.7(c). Figure 3.9 reveals that crown-like structure is always accompanied with a thin jet and becomes mature in structure when the thin jet is stabilized.

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Fig. 3.9 The structures of the thin jets and crowns with different γ-values which are labeled on the left top of each image. The fame rate is 30,000 fps but the time interval between each frame is 0.166 ms. The arrow shows the orientation of the crown wall, which rotates counterclockwise to vertical direction when γ is increased from 0.5 to 1.02.

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We have shown by Fig. 3.4 that the crown-like structure forms around the thick jet, which suggests a correlation between the crown formation and the origin of the thick jet. This correlation can be further confirmed by Fig. 3.9 in which, despite different orientations (defined by the arrows in Fig. 3.9) of the crown wall, the bottom of the crown-like structure always sits on the head of the thick jet. The orientation of the crown wall can be seen to gradually turn counterclockwise to vertical when γ is gradually increased from 0.5 to 1.02. The transition to a complete crown wall occurs when the arrow turns from downward (γ = 0.5) across the horizontal line (γ = 0.58) to upward (γ from 0.6 to 1.02). This feature shows a high regularity and correlation between the orientation of the crown wall and the depth of the bubble. For comparison, let us refer to Fig. 3.5 which shows the relation between the deformed surface depression (with the thin jet) and arrowed directions of the collapse of the circular ring-shaped crater. Based on Fig. 3.5, it is reasonable to infer that the directions of the circular ring-shaped crater should have a similar trend of rotation when the bubble depth is increased to change the surface depression. This comparability further suggests the crown-like formation is correlated to the surface depression with the thin jet (to induce the circular ring-shaped crater). From the continuous change of the orientation of the crown with increasing γ, the layer of water on top of the thick jet for 0.5 ≤ γ ≤ 0.6 should be considered as a crown-like structure similar to that for γ ≥ 0.6, and thus was denoted as “unstable crown formation” in the above discussions. Finally, as shown in Fig. 3.9, the top rim of the crown wall is nearly flat for 0.6 ≤ γ ≤ 1.02, which is significantly different to the two-arm splash on the crown wall induced by spark discharge [10]. This difference is probably related to the unavoidable mechanical influence due to the electrodes in spark discharge.

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