3.1 Ice breaking with large bubble–ice distances

In this section, the bubble is initiated at a distance such that no contact occurs between the ice plate and the bubble during the first bubble collapse. In figure 2, a bubble is initiated at $\unicode[STIX]{x1D6FE}=1.43$ below a floating ice plate. The bubble is elongated in the vertical direction during the collapse and this will be discussed later in § 3.5. The bubble top is attracted to the ice while the bottom part contracts faster and turns into a jet. An initial shock wave is released when the jet impacts on the upper boundary of the bubble. This shock wave is shown in frame 6 and followed by a series of stronger shock waves in frames 7–8 that are emitted from the collapse of the bubble. The wave fronts appear as circles in the shadowgraphs. They are slightly non-concentric because they are emitted from a bubble ring formed after the jet penetration rather than a spherical bubble. Nevertheless, the emissions at the different parts of the ring are almost simultaneous. Two succeeding reflected waves appear after the shock wave incidence as in frames 9–10, figure 2. The first reflection is from the water–ice surface (the lower surface of the ice plate) and the second from the ice–air interface (the upper surface of the ice plate). They should be compression waves and expansion waves, respectively, given the properties of the reflecting interfaces. This is supported by the cavitation that appears after the second wave which indicates tension in the water. Subsequently, the ice breaks in frames 9–12. Interestingly, cracks only emerge after the reflection waves and develop from the top of the ice plate to the bottom. The development follows the same direction with the reflection waves. Since ice is much more vulnerable to tension than compression, this indicates that the cracks are caused by the tension due to the reflection of the shock waves at the ice–air interface.

The breaking up of the ice is also displayed in the bottom view (frames f–h) showing that the cracks radiate from the centre of the ice plate. This is also in alignment with the propagation of the reflection waves which are spherical. In this process, neither the bubble nor the jet make contact with the ice. Thus, the shock waves, or more precisely the reflected expansion waves, should be the predominant cause of ice breaking.

3.2 Ice breaking with medium ice–bubble distances

With the medium bubble–ice distance, a jet forms during the bubble collapse and impacts on the floating ice plate before the bubble reaches the minimum volume. Figure 3 shows another case with $\unicode[STIX]{x1D6FE}=1.06$ where only a small gap is left between the bubble and the ice plate before the jet forms. The jet penetrates the gap, impinges on the ice and is then deflected along the ice surface, as shown in frames 5–7 in figure 3(a), frame d in (b) and frames 7–8 in (c). However, the ice is not fractured immediately. Subsequently, the bubble (in a ring shape after the jet penetration) reaches a minimum volume and emits a series of shock waves from different parts of the ring in a rapid succession in frames 8–10 in (a).

Small cracks appear shortly after the jet tip reaches the ice plate, see frame 8 in (a) and frames 9–10 in (c). They further develop into larger cracks after the shock wave emission, see frames 9–12 in (a), d–f in (b) and 11–15 in (c). All visible cracks develop downwards from the top of the ice plate. The large cracks, as previously explained, should be attributed to the tension caused by the expansion wave formed by reflection of the shock waves at the ice–air interface. However, the small cracks appear before the bubble collapses to the minimum volume, hence they are possibly caused by the tension due to the reflection of pressure waves generated by the impact between the jet and the water or the ice, or caused by the hogging moment due to the jet impact. In addition, no fractures are observed at the centre of the ice plate’s bottom where the jet impacts, see frames 5–7 in (a) and 8–11 in (c).

3.3 Ice breaking with small ice–bubble distances

In this section, the bubbles are initiated close to the floating ice plates, and make contact with the ice before the collapse. In the experiment, shown in figure 4, $\unicode[STIX]{x1D6FE}$ is reduced to 0.47. The bubble expands over the ice surface and clings to it during the collapse phase that follows. Here the jet directly impacts on the ice plate without any cushioning since the liquid between the bubble and the ice has been squeezed away before the maximum bubble expansion. In frames d–g the ring bubble that forms after the jetting is shown. The jet, when impacting with the ice, appears as the opening in the centre of the bubble. The process of the jet penetrating through the bubble and impacting onto the ice can be seen clearly in figure 4(c), frames 6–10. After the impact, the jet liquid deflects and expands along the ice surface, and corrupts the ring bubble’s surface, see frames 11–15. Nevertheless, no fracture on the ice surface is observed where the impact occurred. The fractures only appear later from the other side of the ice plate. Subsequently, the ring bubble collapses and a series of shock waves are released, as visualized in frames 7–9, figure 4(a). The shock wave emissions are not simultaneous because different parts of the ring bubble reach the minimum volume with slight time differences which may be due to the irrigation caused by the deflected jet liquid.

As shown in figure 4(a), small cracks first develop from the upper side of the ice plate (frame 6). As explained before, this should be a result of tension caused by the reflection of pressure waves at the ice–air interface or by the hogging moment applied to the ice plate by the jet impact. Possible sources of the pressure waves include the impact between the jet and the ice as well as the collapse of a part of the ring bubble that occurs earlier than the other parts. Later, stronger shock waves are released from the collapse of the ring bubble (frame 7 in a) and the fracturing is extended (frames h–j, b and 16–20, c).

With a further reduced standoff distance, $\unicode[STIX]{x1D6FE}=0.31$ , figure 5 shows another experimental case with similar bubble behaviour but different characteristics of ice fracturing. Again, cracks are observed to form from the top of the ice plate and then expand, which can be attributed to the expansion waves formed by the reflection of the shock waves at the upper ice surface. The difference is, however, that fractures also appear at the bottom of the ice plate right after the incidence of the shock waves, as in frames 7–8. This indicates that the shock waves are able to directly crack the ice without turning into expansion waves, presumably due to their high strength. As a result, fractures grow from a point adjacent to the collapsed bubble (frame 8 and onwards), joining those developing from the top, and cause damage through the whole section of the ice plate. This enhanced damage effect can be attributed to the bubble collapse which is more violent than in the previous case. With a smaller $\unicode[STIX]{x1D6FE}$ , the jet formation is delayed. The splashing of the jet liquid and its interruption of the collapsing bubble is lessened. Therefore, the bubble can reach a higher collapse speed and compression to initiate stronger shock waves.

The jet impacts the ice and appears as the opening at the centre of the torus (frame b, figure 5). The impact will result in pressure wave generation and propagation (Philipp & Lauterborn Reference Philipp and Lauterborn1998) inside the ice, however, fractures are again not present at the impact site. The small cracks that appeared before the bubble rebounds (frames 7 and e) may be attributed to the reflection of this pressure wave. This is similar to the cases where $\unicode[STIX]{x1D6FE}=1.06$ and 0.47.

A more interesting scenario of bubble behaviour is shown with $\unicode[STIX]{x1D6FE}=0.07$ in figure 6. Near the end of the collapse, the bubble’s bottom part tends to split off from the main part instead of developing into a re-entrant jet as a result of the faster contraction on the bubble’s lateral parts, see frames 2–4. Repeated tests show that the split only occurs with the floating ice plates (figure 6) rather than the fixed ones (a comparison is shown in figure 8 b). The ice plate is moving towards the bubble centre before the bubble splits. It directs the horizontal fluid motion near the top of the bubble vertically downward, which causes the bottom of the bubble to pinch off. Besides, the images in frames b–c, figure 6 as well as frame d–e, figure 4(b) suggest that gas is detaching from the bubble as it contracts, probably due to friction, and this may contribute to the split as well. The liquid collides where the split occurs, releasing a shock wave (denoted by shock- $s$ , see frame 4).

Subsequently, the bottom part and the top part of the bubble collapse separately, each emitting a shock wave (marked as shock- $b$ and shock- $t$ and shown in frames 4–6, respectively). The shock wave from the bottom part may induce a jet towards the ice plate at the top part of the bubble. The jet impact may have contributed to the fracturing of the ice in frames 5–6. The stress caused by the shock wave incidence at the bottom of the ice plate is another likely reason for the fracturing. Also, the shock waves reflect into expansion waves and may cause ice fracturing by tension. Such fracturing initiates from the top, and is shown in the images with a higher frame rate in figure 8(a) (the first row). The combination of these effects causes severe damage throughout the ice plate.

Figure 7.

High-speed images of the crack development in the ice plates with thicknesses of 20 mm and 28 mm (i.e. the dimensionless thickness $k$ being 1.11 and 1.56, respectively). $\unicode[STIX]{x1D6FE}=1.06$ in both cases. Scale bar, 10 mm. Ice plate free floating. For the 20 mm plate, the initial cracks develop from the side opposite to the jet impact before the bubble reaches the minimum volume. For the 28 mm, such cracks are not observed. Larger fractures are induced after the bubble rebound in both cases.

3.4 Ice breaking with different thicknesses

In the above experiments, ice plates with a thickness of 20 mm were used to demonstrate the bubble behaviour and the ice fracturing. In the following sections, the thickness of the ice plate is varied to extend the observations. Here, we define the dimensionless ice thickness as

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle k=T/R_{m}, & \displaystyle\end{eqnarray}$$

where $T$ is the thickness of the ice plate and $R_{m}$ is the maximum bubble radius. In this section, $k$ is from 0.56 to 2.56 (i.e. the thickness is from 10 mm to 46 mm). Some representative results are shown in figures 7–9. The larger thickness increases the distance the shock waves propagate in the ice, therefore the tension caused by the reflected waves would be lower and less effective in fracturing the ice. A comparison is shown in figure 7. With $k=1.11$ , small cracks are observed at the top of the ice plate after the jet impacts on the ice, before the bubble collapses to the minimum volume. They are possibly induced by the reflection of the pressure waves from the impact (as is discussed in § 3.2), and they are no longer observed when the thickness is increased to $k=1.56$ and higher. Similarly, when the ice thickness further increases, it is hard to induce fractures at the top of the ice plate, even for the shock waves emitted by the bubble collapse. For example, for the ice plate with $k=1.89$ , fractures from the ice–air interface are absent within $\unicode[STIX]{x1D6FE}\approx 0.7{-}1.1$ . In this range, the bubble shape is affected by the jet and highly non-spherical when collapsing to the minimum volume. Therefore the shock wave emissions are not as concentrated. This, and the relatively longer travelling distance in the ice, weaken the wave strength, thus the reflected waves fail to cause ice fracture.

Figure 8.

High-speed images of the bubble collapse and the fracturing of the ice plates. (a) Floating ice plates with different thicknesses (22 and 28 mm, $k=1.22$ and 1.56, respectively). Shock- $s$ , - $t$ and - $b$ denote the shock waves emitted during the bubble split, the collapse of the top part and the collapse of the bottom part of the bubble, respectively. (b) Fixed ice plate, 28 mm in thickness, shock wave emitted only at the end of the bubble collapse. Scale bars, 10 mm.

Another change in the fracturing due to the variation in the thickness is shown in figure 8(a). As mentioned before, a bubble with a very small standoff distance ( $\unicode[STIX]{x1D6FE}<0.2$ , approximately) to the floating ice plate will split in the last stage of the collapse. Shock waves are emitted by the split and the subsequent collapses of the top and the bottom parts of the bubble (denoted as shock- $s$ , shock- $t$ and shock- $b$ , respectively). These shock waves are shown in figure 8(a), where they act upon two ice plates of different thicknesses. For the thinner ice plate, some of the cracks initiate from the top of the ice plate, which is possibly caused by the reflection of the earlier-emitted shock- $s$ , see the third frame in the first row. Spallation is observed (the last frame) at the ice–air interface, where ice fragments are ejected from the ice plate as a result of tension. However, for the thicker ice plate, the cracks initiate and expand merely from the bottom, because the shock wave decayed while travelling through the thicker ice and its reflection fails to fracture the ice from the top. In contrast, when the ice plate is fixed, the splitting and the multiple shock waves will no longer occur. An example with a fixed 28 mm-thick ice plate is given in figure 8(b), where the bubble collapses to a minimum volume without splitting and shock waves are only emitted by the final collapse.

When the thickness is further reduced, the ice plate fractures when subjected to the bubble expansion. An example is shown in figure 9, where the ice plate thickness is only 10 mm ( $k=0.56$ ). The rapid bubble expansion applies a hogging moment on the ice plate, and fractures are seen traversing the ice plate from the top to the bottom. The bubble subsequently collapses and migrates downward instead of being attracted towards the ice plate. This is similar to the bubble collapsing under a free surface, because the broken ice plate no longer acts as a solid wall for the bubble. Experiments showed that fracturing during the bubble expansion generally occurs with $\unicode[STIX]{x1D6FE}<1.5$ for $k=0.56$ . In addition, results showed that the patterns of the ice fracturing are essentially the same between the floating and the fixed ice plates. For example, compared to a floating ice plate ( $k=1.22$ ), the fracturing of a fixed ice plate with the same $k$ also develops from the top for large and medium $\unicode[STIX]{x1D6FE}$ , and from the bottom upon shock wave incidence for sufficiently small $\unicode[STIX]{x1D6FE}$ . One exception is that the fracturing induced by the shock waves from a splitting bubble is not repeated, because the split is only observed with a movable boundary in the experiment.

Figure 9.

High-speed image series of the bubble behaviour and the fracturing of a free floating 10 mm-thick ice plate ( $k=0.56$ ) with $\unicode[STIX]{x1D6FE}=1.44$ . Scale bar, 10 mm. The ice fractures when subjected to the expansion of the bubble.

3.5 Bubble characteristics and ice plate motion

In this section, bubble characteristics such as the jet speed, the bubble shape and the motion of the ice plate are analysed. Floating and fixed ice plates with various thicknesses (16–46 mm) are considered. The floating ice plate is driven by the bubble and heaves in the vertical direction, which in turn affects the bubble behaviour. Figure 10(a) depicts the time curves of the dimensionless heave motion, from bubble inception to the impact between the jet and the ice plate. The ice plate rises and falls with the expansion and collapse of the bubble. The bubble top is attracted towards the ice plate and its downward motion is reduced during the collapse. The maximum heaving displacement decreases with the thickness of the ice plate. The heaving is no more than 10 % of the maximum bubble radius, but there are obvious changes in the bubble shape and the jet speed compared to those with fixed ice plates.

Figure 10.

(a) Time curves of the vertical displacements of the floating ice plates with different thicknesses. For reference, the displacements of the top and the bottom of a bubble under a 22 mm-thick ice plate are also plotted. All displacements are scaled to the maximum bubble radius. $\unicode[STIX]{x1D6FE}=0.96$ . (b) The time curves of the bubble elongation with different ice plates. $\unicode[STIX]{x1D6FE}=1.36$ . The elongation is defined as the ratio of the vertical span of a bubble to the horizontal span. It increases as the bubble collapses but drops after the jet forms.

The bubble is elongated in the vertical direction during the collapse, see figures 2 and 3 for example. Here, the elongation is defined as the ratio between the vertical and the horizontal lengths of the bubble. The lateral boundaries of the bubble contract faster than the bottom, and the elongation continues to increase, until the jet forms at the bottom. Some examples of the elongation time curves are plotted in figure 10(b). The maximum elongation is larger when the ice plate is floating rather than fixed. It also increases with reduced ice plate thickness. The reason for the difference in the elongation may include the following. With a smaller thickness, the plate will be pushed further during bubble expansion and the bubble top will extend further. The plate is pulled back during the collapse and the counter-effect elongates the bubble top. The contraction of the bubble’s lateral boundaries is less hindered by the ice plate with a smaller thickness. All these contribute to the elongation. The maximum speed of the bubble jet also increases with smaller ice thickness. The maximum jet speed during the jet progression inside the bubble were measured at the jet tip and plotted in figure 11(b) with $\unicode[STIX]{x1D6FE}=0.93$ . The maximum jet speed is the lowest when the ice plate is fixed (approximately $100~\text{m}~\text{s}^{-1}$ ). Nevertheless, the speed is larger when the ice plate is floating, increasing drastically to around $300~\text{m}~\text{s}^{-1}$ for the 22 mm-thick ice plate and almost $450~\text{m}~\text{s}^{-1}$ for the 16 mm-thick ice plate. A possible explanation is as follows. The contraction from the bubble’s lateral part is greater when the ice plate is thinner and easier to heave. As a result, the bubble’s bottom becomes slimmer and so does the re-entrant jet (the jet widths are compared in figure 11 a). As pointed out in previous numerical studies (Brujan et al. Reference Brujan, Keen, Vogel and Blake2002; Zhang, Li & Cui Reference Zhang, Li and Cui2015; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016)), the jet is driven by the high pressure in the water near the far end of the bubble that is collapsing near a solid wall. Therefore, the slimmer jet gains a higher speed.

Figure 11.

(a) Magnified images of the high-speed jets with different widths piercing through the bubble, $\unicode[STIX]{x1D6FE}=0.93$ . Frame width, 2.88 mm. The ice plate thicknesses and conditions are marked. (b) The maximum jet speed $v_{jet}$ as a function of the thickness $T$ of the floating ice plate, measured at the tip of the jet during its penetration through the bubble. $\unicode[STIX]{x1D6FE}=0.93$ . Each filled box and error bar represent the average value and the standard deviation, respectively, of 30 measurements. The dash line represents the maximum jet speed with a fixed ice plate.

In this experiment, shock waves are emitted at different stages during the collapse of the bubble. The time of the shock wave emission is plotted as a function of the dimensionless standoff distance $\unicode[STIX]{x1D6FE}$ in figure 12. All emissions occur within the last 3 % of the first oscillation cycle of the bubble. First, shock waves are emitted by the impact between the jet and the opposite bubble wall (see figure 2 a, for example). The impact/emission time (denoted by $+$ ) is advanced as $\unicode[STIX]{x1D6FE}$ reduces. Then the ring bubble formed after jet penetration collapses. From the ring bubble, a series of shock waves are emitted consecutively over the time spans represented by the drop lines in figure 12. The emissions are not concentrated (see figure 4 a, for example) because the collapse of the different parts of the ring are not simultaneous. The time span of the shock wave emissions is the longest at approximately $\unicode[STIX]{x1D6FE}=0.6$ . For smaller $\unicode[STIX]{x1D6FE}$ , the bubble splits and then collapses at both the far end and the near end (see figure 8 a, for example). The shock wave emitted from the split is very close in time to that from the collapse at the far end of the bubble (not plotted), but is approximately $5{-}15~\unicode[STIX]{x03BC}\text{s}$ ahead of the shock wave emission at the near end.

Figure 12.

Shock wave emission time for different standoff distances. A 22 mm-thick floating ice plate was used. The time (vertical axis) is scaled to the duration of the first expansion and collapse of the bubble. The shock wave emissions are observed when the jet impacts on the opposite bubble wall (denoted by $+$ ), when the bubble splits (▵) and when the ring bubble, after being penetrated by the jet, collapses to a minimum volume (▴). The different parts of the ring bubble collapses asynchronously and a series of shock waves are emitted over a time span represented by the drop line.

3.6 Demarcation of different ice breaking patterns

In this section, the standoff distance ( $\unicode[STIX]{x1D6FE}$ ) limit for ice breaking is tested with differently sized bubbles, and the ice breaking patterns as functions of $\unicode[STIX]{x1D6FE}$ and the ice plate thickness are summarized. First, bubbles are generated by discharges from 200 up to 700 V with varying $\unicode[STIX]{x1D6FE}$ for a floating 20 mm-thick ice plate to find the upper limit of $\unicode[STIX]{x1D6FE}$ for successful ice breaking. The capacitance in the experiment is constant. The electric energy released by the discharge converts into light, heat and mechanical work, including an initial wave and the mechanical energy of the bubble, $E_{b}$ (Chahine et al. Reference Chahine, Frederick, Lambrecht, Harris and Mair1995). Since the ice breaking is observed only after the spark (in most cases, after the bubble collapse), we assume that the energy that causes ice breaking only comes from $E_{b}$ . $E_{b}$ can be estimated by the work done by the displaced water during the expansion of the bubble from initiation to the maximum radius, $R_{m}$ , against the ambient pressure, $P_{\infty }$ , as in the following equation (Buogo & Cannelli Reference Buogo and Cannelli2002; Tomita, Tsubota & Annaka Reference Tomita, Tsubota and Annaka2003), where $P_{v}$ is the saturated vapour pressure:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle E_{b}={\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}R_{m}^{3}(P_{\infty }-P_{v}). & \displaystyle\end{eqnarray}$$

As discussed before, the ice breaking with relatively large $\unicode[STIX]{x1D6FE}$ is due to the tension at the top of the ice plate caused by the reflection of the shock waves. Despite the different voltage, this ice breaking pattern is not changed. When $\unicode[STIX]{x1D6FE}$ increases, the reflection wave weakens until it is unable to cause ice fracturing. Thus there exists an upper limit of $\unicode[STIX]{x1D6FE}$ , denoted as $\unicode[STIX]{x1D6FE}_{max}$ , for successful ice breaking. With higher energy $E_{b}$ , the bubble collapse is able to crack the ice from a further distance and $\unicode[STIX]{x1D6FE}_{max}$ is larger. The result of ice breaking as a function of $E_{b}$ is plotted in figure 13. As is indicated, $\unicode[STIX]{x1D6FE}_{max}$ increases logarithmically with $E_{b}$ .

Figure 13.

Ice breaking test results as functions of the dimensionless standoff distance $\unicode[STIX]{x1D6FE}$ and the mechanical bubble energy $E_{b}$ , and an estimation on the variation of $\unicode[STIX]{x1D6FE}_{max}$ (the upper limit of $\unicode[STIX]{x1D6FE}$ for successful ice breaking) with $E_{b}$ . Ice plate thickness 20 mm, free floating.

Figure 14.

Ice fracturing patterns as a function of the dimensionless thickness of the ice, $k$ , and the dimensionless standoff distance, $\unicode[STIX]{x1D6FE}$ . ▫ ice fracturing due to the shock waves from the bubble split; ▴/▿ ice fracturing from the both sides/the top side due to the shock waves from the collapse of the ring bubble in contact with the ice; ▸/▹ ice fracturing due to the collapse of the ring bubble out of contact with the ice, with/without small cracks appearing at the top of the ice plate before the collapse-induced shock wave occurs; ▵ no fracturing, ring bubble collapsing in contact with the ice; ♢ ice fracturing when subjected to bubble expansion; ○/ $\times$  ice fracturing/no fracturing, no contact between the ice and the bubble or the jet.

The fracturing patterns of the floating ice plates are summarized in figure 14 as functions of the dimensionless thickness of the ice, $k$ , and the dimensionless standoff distance, $\unicode[STIX]{x1D6FE}$ . The experiment cases with no fracturing due to large bubble–ice distances are denoted by $\times$ . When $k$ is small, the ice plate will be fractured when subjected to the expansion of the bubble (denoted by ♢). Except that the bubble collapse and the shock waves are the main causes of the ice breaking. With a relatively large $\unicode[STIX]{x1D6FE}$ , the ice is fractured by the reflection of the shock waves emitted from the bubble collapse, before the jet impacts the ice plate (denoted by ○). With a relatively small $\unicode[STIX]{x1D6FE}$ , the ice is fractured from the bottom by the shock waves emitted upon the split and the subsequent collapse of the bubble (denoted by ▫). With medium $\unicode[STIX]{x1D6FE}$ , the ice plate is subjected to the impact of the bubble jet before the bubble reaches the minimum volume. The bubble then collapses in a ring shape and emits a series of shock waves. This corresponds to the region with triangles. In the region scattered with ▸ and ▹, the jet impacts on the ice after penetrating a gap between the bubble and the ice plate, while in the region with ▿, the bubble is in direct contact with the ice plate. In these cases the ice plate fractures from the top mainly as a result of the reflection of the shock waves emitted by the collapse of the ring bubble, except that in the cases denoted by ▸, small cracks form at the top of the ice plate before the bubble reaches the minimum volume. For the region scattered with ▴, the ring bubble ends up with a stronger collapse and a more concentrated shock wave emission that directly fractures the ice from the bottom. Finally, as the thickness increases, the ice plate will no longer be fractured by the reflection of the shock wave from the ring bubble. Such cases are denoted by ▵.