3.1. Experimental protocol and parameter settings

The main goal of this series of experiments is to investigate the role of condensable gases (vapour) during the impact of a boiling liquid (water), i.e. a liquid that is in thermal equilibrium with an environment that consists of its own vapour. This necessarily implies that once the equilibrium temperature is chosen, the equilibrium pressure follows directly from the vapour curve of the liquid. We have used six different temperatures, namely $T_0 = 20$ , $30$ , $40$ , $50$ , $60$ and $70$ $^\circ$ C. After setting the temperature in the autoclave, we allow the system (typically for a period of $12$ h) to equilibrate to the corresponding vapour pressures $p_0 = p_{V,0} = 23.3$ , $42.3$ , $73.6$ , $123.0$ , $198.6$ and $310.8$ mbar, respectively. Subsequently, we perform approximately $10$ experimental runs for three different wave shapes, characterised by the parameter $\alpha$ introduced in Ezeta et al. (Reference Ezeta, Kimmoun and Brosset2023), which corresponds to the wave steepness of the solitary wave before it reaches the beach, defined as $\alpha = A/h_l$ , where $A$ is the amplitude of the solitary wave and $h_l$ the (quiescent) liquid height inside the flume. For the purpose of this work, the liquid height was fixed to $h_l = 365$ mm, and measured as $h_l = 365.0 \pm 0.3$ mm over the entire experimental series of approximately $180$ experimental runs, where the error has been taken equal to the standard deviation of the sample. The chosen values for the wave steepness are $\alpha = 0.350$ , $0.385$ and $0.420$ , which correspond to a cross-sectional area of the entrapped vapour or gas pocket of $A_{{cross}} = 2.7 \pm 0.7$ , $9.5 \pm 0.9$ and $17.4 \pm 1.0$ cm $^2$ , respectively. Note that the relatively large error is mainly due to inaccuracies in tracing the pocket shape from the high-speed recordings.

The impact of the wave onto the impact wall is recorded with two high-speed cameras, located outside of the autoclave and imaging through two windows with backlighting from the other side, recording at $4000$ frames per second at a resolution of $1024$ by $1024$ pixels $^2$ . The side walls of the flume are made out of glass such that the wave can be imaged from the outside. One of the cameras captures a side view of the impacting wave, which allows us to reconstruct the shape of the wave, and the size of the captured vapour pocket as a function of time, making use of the quasi-two-dimensional shape of the wave. The other camera takes an oblique view of the impact, which serves to verify the quasi-two-dimensional character of the impact and in addition allows us to appreciate the dynamics of the entrapped vapour pocket in greater clarity.

The views from the two high-speed cameras allows us to verify that, for each value of $\alpha$ used, the wave shape is repeatable, and that they are reproducible for the different values of $T_0$ used. This is shown in more detail in Appendix A.

3.2. Comparison of a boilingliquid versus a water-air wave impact

Figure 2. Two snapshots from an experiment in water and air (a,c) at $T_0 \approx 20$ $^\circ$ C and atmospheric pressure $p_{{atm}} = 1$ bar, taken just before impact and $7.8$ ms later, compared with snapshots from a similar experiment with the same wave shape in a boiling liquid (i.e. water and water vapour at equilibrium); (b,d) at $T_0 \approx 20$ $^\circ$ C and $p_0 \approx p_V(T_0) = 23.3$ mbar. Note that the images are taken from an oblique viewpoint, looking through the wave at the wall of impact, which allows for good observation of the shape of the entrapped cavity, the free surface of which appears as the brightest object in the pictures. In both cases, a solitary wave with wave steepness $\alpha = 0.385$ has been used. Clearly, the wave shapes just before impact (a,b) are very similar, whereas $7.8$ ms later there is an entrapped air pocket visible in the air case (c), whereas the vapour pocket in the boiling liquid case has nearly completely vanished. (d). A sketch of the side view of the wave just before impact is provided in (e), and the oblique view – looking from the flume towards the impact wall – that is also taken in (a–d) is sketched in (f). Note that the wave can be observed through small windows in the autoclave since the side panels of the flume are made of glass. For a better visual comparison, see supplementary movie 1 comparing the impact in air and vapour available at https://doi.org/10.1017/jfm.2025.110.

In figure 2 we show two snapshots from a water-wave impact with wave steepness $\alpha = 0.385$ in air under atmospheric conditions at $T_0 \approx 20$ $^\circ$ C and $p_0 \approx 1.0$ bar in comparison with two snapshots at comparable times for a water-wave impact of the same wave shape in boiling liquid conditions at $T_0 \approx 20$ $^\circ$ C and $p_0 \approx p_V(T_0) = 23.3$ mbar. Note that the measured values of liquid and air temperature and air pressure were $T_l = 18.4$ $^\circ$ C, $T_g = 20.9$ $^\circ$ C and $p_0 = 1.015$ bar, respectively, in the atmospheric case and $T_l = 20.1$ $^\circ$ C, $T_g = 19.9$ $^\circ$ C and $p_0 = 26.1$ mbar, respectively, for the boiling liquid case. For $T_0 \approx 20$ $^\circ$ C, $p_0 = 25.9$ mbar was the closest we could get to the nominal vapour pressure $p_{V,0} = 23.3$ mbar, which is probably due to a combination of thermal inhomogeneities in the system and some residual non-condensable gas present in the system. Note that, although the smallest pressure that can be reached by our vacuum pump is about $4$ mbar (in a system devoid of water), the pump is pumping continuously for several hours, with water continuously supplied to the flume to keep the liquid height $h_l$ constant, such that the remaining partial pressure of non-condensable gases is likely much smaller than $4$ mbar. Throughout this work, we use $p_0$ for measured values of the ambient pressure and $p_{V,0}$ whenever referring to the expected vapour pressure, computed from the temperature $T_0$ , i.e. $p_{V,0} = p_V(T_0)$ . See Appendix C and figure 19 for an overview of these set points. Wherever we denote a pressure in capitals, it stands for the gauge pressure as it is measured by the pressure sensors, i.e. $P = p - p_0$ , where $p_0$ is the ambient pressure inside the autoclave.

In figures 2(a) and 2(b) we observe the two waves just before impact (at $\Delta t =$ $t - t_{{impact}} \approx -0.25$ ms, with $t_{{impact}}$ the time of impact and $t$ time), where we take the oblique view from the flume towards the impact wall that is also sketched in figure 2(e,f). It is clear that there are no obvious visual differences between the two snapshots, owing to the very good reproducibility of the wave shape, even with the greatly varying ambient pressures in the two systems ( $1$ bar vs $26$ mbar). So, just before impact the wave shapes are very similar and, as a result, the size of the gas/vapour pocket that is about to be entrapped in the two cases is very similar as well. At $\Delta t = 7.5$ , i.e. $7.8$ ms later, things look very different in the two cases. Whereas in the water–air case (figure 2 c) an air pocket has been entrapped, which at that time has reached its typical size; for the boiling liquid, one observes in figure 2(d) that the entrapped vapour pocket has almost completely vanished. This could partly of course be due to the large difference between the air ( $p_0 = 1015$ mbar) and vapour pressure ( $p_0 = 26.1$ mbar), which – for similar driving pressures generated by the impact – would allow the vapour pocket to contract much more than the air pocket, namely approximately a factor $4$ in radius. However, when looking at the pressure signals, it becomes clear that the picture is more complex than this.

Figure 3. Pressure signals measured for a wave with $\alpha = 0.385$ in the air case (left plots) and the boiling liquid case (right plots). (a,b) Time evolution of the pressure $P$ measured by the 32nd sensor at $z=455$ mm in the impact wall for (a) the air case and (b) the boiling liquid case. The moment $t_{{impact}}$ at which the wave crest impacts is indicated by a vertical dashed line in both plots. (c,d) Maximum pressure $P_{{max}}$ measured in each of the sensors, as a function of its height in the array for (c) the air case and (d) the boiling liquid case. The vertical location $z_{{impact}}$ at which the wave crest impacts is indicated by a vertical dashed line in both plots. (e,f) Time $t_{{max}}$ at which the maximum pressure was measured for (e) the air case (red dots) and (d) the boiling liquid case (blue dots). The black crosses indicate the moments in time that the crest impact pressures were measured. Note that pressures are reported as they are measured by the sensors, i.e. as gauge pressures $P = p -p_0$ (denoted with a capital), with $p_0$ the ambient pressure in the autoclave. This is true for all experimental figures in this work.

In figure 3(a) we plot the time evolution of the pressure measured in the 32nd sensor, located at a height $z=455$ mm, in the air case. (The 32nd sensor is the one that is first hit by the wave in the vapour case. In the air case, the first sensor to be hit is in fact the 34th sensor, and the measured pressure ( $\approx 0.7$ bar) is marginally higher than that in sensor 32.) At the moment of impact, the pressure rises steeply (the sharp peak at $t \approx 410$ ms) to a value denoted as $P_{{max}}$ , which is approximately $0.6$ bar, and which is subsequently followed by an oscillating signal with a frequency of approximately $f = 86$ Hz. Even if we measure it at the crest impact location in the liquid, i.e. at some distance from the entrapped pocket, this oscillating signal can be traced back to the oscillation of the air pocket that is entrapped below the impact point. Using Minnaert’s relation for the eigenfrequency of a cylindrical bubble of radius $R$ , namely $R f = 1.10$ m s^–1 (Ilinskii et al. Reference Ilinskii, Zabolotskaya, Hay and Hamilton2012), this frequency corresponds to a cylindrical bubble radius of $R = 1.28$ cm. This is about a factor two smaller than the actual effective radius of the entrapped cavity, defined as the radius of a hemicylindrical cavity with the same cross-sectional area (see also Appendix B), which was measured to be $2.45$ cm. (For comparison, Minnaert’s relation $af =$ $3.26$ m s^–1 for a spherical bubble with radius $a$ would have given $a = 3.79$ cm.) Note that the impact pressure is the highest pressure in this time series, which is not surprising, since we are measuring in the point that was first touched by the wave crest, which in air or other non-condensable gases usually gives the largest pressure in a wave impact time series. The surprise is however that when we turn to the equivalent series for the boiling liquid case (figure 3 b) we immediately observe that this is no longer true. In fact, the impact is still visible as the first peak in the time series, at $t \approx 406$ ms, and with $P_{{impact}} \approx 2$ bar is even considerably larger than that in the air case, but after that the pressure drops back to a very small value, indicating the absence of pressure building up in the entrapped vapour pocket. That is, until a second, higher peak ( $P_{{max}} \approx 4.6$ bar) is observed at $t \approx 413$ ms. Since no clear oscillation follows, or possibly extremely rapid ones, this suggests that the vapour pocket has collapsed and that at least a large part of it must have condensated in the process, leaving small gas or vapour fragments that regrow and subsequently collapse in a second, smaller event, or even series of events, starting at $t \approx 426$ ms. See supplementary movie 1 comparing the impact in air and vapour corresponding to figure 2.

The sequence of events suggested above is confirmed in figure 3(d), where the maximum pressure $P_{{max}}$ observed in the time series of each of the pressure sensor channels is plotted as a function of the sensor height $z$ . Here it becomes clear that the maximum pressure measured at the impact height is completely dwarfed by the maximum pressure measured in the system, which occurs at $z = 388$ mm and equals $P_{{max}}(z_{{max}}) = 78.1$ bar. This location nicely corresponds to the location of the collapse of the vapour pocket in the high-speed imaging recordings, available in the supplementary movies. Comparing this value to those obtained in the air case (figure 3 c), it is clear that pressures obtained in the boiling liquid case are up to two orders of magnitude larger than in the air case. In fact, in the latter the pressure that can be attributed to the wave crest impact is globally the largest in the system, and the pressure oscillations in and around the air pocket are smaller, even if they are quite uniformly spread over a larger area (corresponding to the size of the vapour pocket) and a larger time span. This is in sharp contrast to the boiling liquid case where the crest impact pressure corresponds not even to the maximum in its own time series (figure 3 b), and therefore, not even discernable in figure 3(d).

It is instructive to look at the times $t_{{max}}$ at which the pressure maxima are reached in the different sensors, which are plotted in figures 3(e) and 3(f), where we restricted ourselves to heights ${\lt } 480$ mm, since sensors above that height are in the gas or vapour phase where insignificant signals are obtained. For the air case, we observe that, for small heights (i.e. within the air pocket), the maximum pressure occurs at $t = 413$ ms, whereas around the crest impact point, the first maximum already occurs at $t = 409.9$ ms (red dots in figure 3 e). For the boiling liquid case, all pressure maxima occur at $t = 412.8$ ms (blue dots in figure 3 f), but due to the separation of the crest impact event and the vapour pocket collapse event in time, it is feasible to determine the impact pressure maxima by restricting the search to the time interval before the collapse. This yields the black crosses that indicate the time $t_{{impact}}$ when the crest impact pressures were recorded. In the air case, $t_{{impact}}$ coincides with $t_{{max}}$ , but for the boiling liquid case, $t_{{impact}} = 405.6$ ms, which is earlier than $t_{{max}}$ . Note that in air, the time span between impact and maximum air pocket compression (which can be equated with the maximum pressures measured in the area below the impact) is $t_{{impact}} - t_{{max}} \approx 3.1$ ms, which is considerably smaller than the corresponding time span in the boiling liquid case ( $t_{{impact}} - t_{{max}} \approx 7.2$ ms). This is because the time it takes for the air pocket to reach its minimum size is smaller than the time the vapour pocket needs to collapse to an almost vanishing length scale.

At this point it is good to stress that the origin of absolute time has no special significance since it depends on a rather arbitrary triggering of the cameras determined by the wavemaker stroke. The fact that the absolute bubble collapse/compression times are close in this case is therefore purely coincidental and has no physical significance.

Figure 4. (a) Pressure map $P(z,t)$ of the impact of the same breaking wave in water–air conditions at $T_0 = 20$ $^\circ$ C, with $\alpha = 0.385$ also reported in figures 2(a,c) and 3(a,c,e), focusing on the time span from $t = 400-460$ ms, containing crest impact, pressurisation of the air pocket and oscillations. Note that pressure is denoted by a logarithmic colour scale, where $p_{{atm}} = 1$ bar. Also note that all pressures smaller than $0.01$ bar are denoted by the same colour as $P=0.01$ bar, to avoid the noise level. (b) Pressure map $P(z,t)$ of the impact of the same breaking wave, but now in boiling liquid conditions (as reported in figures 2(b,d) and (3 b,d,f)), in the same time interval, containing crest impact and vapour pocket collapse. (c) Same data as in (b), but now horizontally zoomed in on impact (ELP1), jet propagation along the walls (ELP2) and the main vapour pocket collapse event (ELP3). The white dashed line, with approximately the same slope as the propagating downward jet, corresponds to a velocity $v_{{jet}} = 6.7$ m s^–1. (d) Same data as in (c), but now zoomed in on the vapour pocket collapse event. The slope of the white dashed line in this plot corresponds to the sound speed in water, $c = 1.5\times 10^3$ m s^–1. Note that the vertical axis is the same in all four plots and zooming was done exclusively along the time axis.

To finalise our discussion of the impact of a single wave in the water–air and boiling liquid cases we turn to the so-called pressure map, a space–time plot of the signal in the pressure sensor array in our impact wall, provided in figure 4. The upper plot shows the air water case: first, the direct impact of the wave crest as the leftmost point in which high pressures are observed in figure 4(a). This is the point that in wave impact literature has become known as ELP1, where the abbreviation ‘ELP’ is short for elementary loading process (Lafeber et al. Reference Lafeber, Bogaert and Brosset2012; Dias & Ghidaglia Reference Dias and Ghidaglia2018), and is associated with the combination of the incompressible, and even compressible, impact of the liquid mass onto the wall (Wagner Reference Wagner1932; Lesser Reference Lesser1981; Korobkin Reference Korobkin2004). Generally, due to its short duration in time and localisation in space, ELP1 is sometimes hard to resolve, but in this case we verified that, owing to the time resolution of our sensors ( $5$ $\mu$ s), we indeed have. From that point onwards, liquid jets start spreading upwards and downwards along the impact wall, a pressure signal known as ELP2, of which the upward one is clearly visible in the upper figure. The downward one is a bit obscured by the pressure build up in the air pocket, which in recent wave impact literature (Lafeber et al. Reference Lafeber, Bogaert and Brosset2012; Dias & Ghidaglia Reference Dias and Ghidaglia2018) became known as ELP3, a name that is also associated with the gas pocket oscillations in the air case. These are very visible in the pressure map, although one can only see the positive part of the oscillations due to the fact that for the logarithmic colour scale used, pressures below a threshold of $P=0.01$ bar are cut off to this value.

The second plot shows the corresponding boiling liquid case (figure 4 b). Again, the leftmost point in which high pressures are observed corresponds to the crest impact, but then the plot is very different from the water–air case. First, there is no pressurisation of the vapour pocket, which remains at a low pressure, which in turn makes the downward jet very visible. A bit more faint, but definitely also discernible is the motion of the lower three-phase-contact line of the entrapped vapour pocket (the so-called run-up), which culminates at the meeting point of the downward jet and run up in a vertical band of very large pressures throughout the entire system at $t = 412.8$ ms connected to the collapse of the vapour pocket. Visible at later times are a second collapse at $t \approx 426$ ms after a first rebound of the vapour pocket (or rather its fragmented remnants after regrowing in the rarefaction period that follows after collapse), and even a third one in the lower part of the system at $t \approx 440$ ms, both of which are also visible in the time series of figure 3(b). Most importantly, the oscillations that dominate the water–air case are absent.

When we zoom in onto the region between impact and vapour pocket collapse (figure 4 c), we clearly observe the high speed with which the liquid jet moves downwards along the impact wall, with a velocity $v_{{jet}} \approx 6.7$ m s^–1 (white dashed line) that exceeds the impact speed of the wave ( $v_{{impact}} \approx 2.1$ m s^–1) by a factor of $3$ , a factor that depends on the radius of curvature of the wave crest. The estimation of $v_{{impact}}$ is shown in Appendix B. Very different from what is observed in the air case, where the entrapped air pocket is pressurised as soon as the wave impacts the wall, the pressure inside the vapour pocket remains negligible during collapse, which is a strong indication that the vapour present in the pocket is in fact condensating instead of compressing. Following the signal in a single pressure sensor, e.g. at height $z = 420$ mm, the pressure remains small and close to the background noise level of the sensor until $t \approx 409.5$ ms. After this time, the sensor becomes wetted and the pressure suddenly rises to a level of about $0.2$ bar, where it remains until the vapour pocket collapses and the pressure rises to $10$ bar.

Finally, we zoom in even more in figure 4(d), where we focus on a time span of about 700 $\mu$ s around the collapse of the vapour pocket. The first thing to notice in this plot are the diagonal lines that radiate out from the collapse point, which is approximately located at $z = 388$ mm in space and at $t = 412.85$ ms in time. These diagonal lines have a slope that is consistent with the speed of sound in water $c = 1500$ m s^–1 indicated by the white dashed line in this plot. Therefore, we may conclude that these diagonal lines correspond to pressure waves emanating from the collapse point. Having said this, one may wonder why there are diagonal lines in the same plot that seem to originate at a point in time before the collapse of the cavity. To understand their origin it is good to realise that the system has a third dimension and that due to slight but unavoidable asymmetries along the horizontal axis parallel to the wall (the $y$ axis of our system), the collapse does not necessarily happen at the same point in time along that axis. So some nearby part of the cavity may already have collapsed, where the pressure is generated by the inertial collapse of the liquid onto part of that axis, which may induce large pressure signals already slightly before the collapse takes place in the region of the pressure sensor array. In fact, the one structure close to the dashed line with a larger slope than $c$ may be due to an event having taken place at some distance, such that the signal arrives at the pressure sensors more simultaneously as it would when the event had taken place on the sensor array.

In addition, we highlight other remarkable features in this plot. Firstly, when looking at subsequent sensor heights, i.e. subsequent horizontal lines in the diagram, one often observes a pattern that seems to repeat itself every third sensor and in both directions (upwards and downwards). This is likely connected to the threefold structure of the sensor arrangement (cf. Figure 1 c), where every third sensor is located exactly above (or below) the sensor from which one started, whereas distances in the $y$ direction are slightly larger. Secondly, there are clear oscillations in the signal, for which we may estimate the frequency to be of the order of $40$ kHz, i.e. considerably smaller than the acquisition rate of the pressure sensors ( $200$ kHz). This may be due to structural oscillations in the impact plate in which the sensors are embedded or small gas/vapour bubbles.

To conclude this subsection, we find that pressures measured in boiling liquid breaking wave impact may be up to two orders of magnitude larger than those in air. This enormous difference can be traced back to the collapse of the entrapped vapour pocket. Condensation of vapour is likely to strongly contribute to this phenomenon, since condensation depletes the vapour pocket of content, which facilitates its violent collapse.

Figure 5. Maximum pressure $P_{{max}}(z)$ measured as the maximum in the time series of each sensor and plotted as a function of the vertical sensor location $z$ at the impact wall under boiling liquid conditions, for a breaking wave with wave steepness $\alpha = 0.385$ and for six different ambient temperatures ranging from $T_0 = 20$ $^\circ$ C to $70$ $^\circ$ C, corresponding to (a) to (f), respectively. For each experimental setting, there were approximately $10$ repetitions of the experiment. In each panel, the blue-shaded dots correspond to the individual experiments, whereas the solid black line corresponds to their average. Note that the pressure range (vertical axis) is largest in (a,b) and decreases for (c,d) and (e,f). Movies comparing the wave impact in boiling liquid conditions for these six temperatures are available for $\alpha = 0.385$ (corresponding to this figure) and $\alpha = 0.420$ in the supplementary movies.

3.3. Comparison of pressure characteristics for different temperatures and wave shapes

We now turn to the description of the dependence of the vapour pocket and impact pressures on temperature and wave shape in figure 5, where we plot the maximum value $P_{{max}}(z)$ of the measured pressure time series for each of the pressure sensors, as a function of the vertical sensor position $z$ , and for each of the approximately $10$ experiments performed for every parameter setting (blue-shaded dots). Note that the data from figure 3(d) is also one of the $10$ experiments plotted in figure 5(a), and that we removed the data above $z = 550$ mm since those are in the vapour phase for each wave shape and not measuring anything worth showing. In each of the panels in figure 5 we also present the average of the approximately $10$ experiments, indicated by the black lines.

In addition, in the analysis below we make use of the fact that pressures caused by the crest impact and those caused by entrapped vapour pocket dynamics are temporally and spatially separated (cf. Figure 3 e,f), such that we can define the maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ as the maximum pressure associated with the vapour pocket dynamics (due to collapse or compression), averaged over all $10$ experiments performed at a single parameter setting and the maximum crest impact pressure $\langle P_{{crest}} \rangle$ , obtained by first averaging the three largest pressures measured during the impact stage for each experiment and subsequently averaging over the $10$ individual experiments for each parameter setting.

We start with the discussion of the temperature dependence of the maximum vapour pocket pressure and maximum crest impact pressure for a breaking wave with wave steepness $\alpha = 0.385$ , under boiling liquid conditions. In figure 5 we show the results for the maximum pressure $P_{{max}}(z)$ measured in each of the pressure sensors on the impact wall, as a function of their height $z$ , for temperatures increasing from $T_0 = 20$ $^\circ$ C to $70$ $^\circ$ C. At $T_0 = 20$ $^\circ$ C, with vapour pressure $p_{V,0} = 23.3$ mbar, the pressures are clearly the largest, with an average maximum value of $\langle P_{{pocket}} \rangle = 60$ bar (maximum of the black curve) and individual measurements reaching up to $88$ bar (blue-shaded dots). While the largest pressure occurs at the height where the pocket collapses, namely $z_{{coll}} \approx 388$ mm, nothing can be discerned at the location where the wave impact is expected, at $z_{{impact}} \approx 455$ mm. This is consistent with the fact that even there the recorded maximum pressures, with $P_{{max}}(z_{{impact}}) \approx 4{-}5$ bar, are much larger than the crest impact pressures ( $P_{{impact}} \lt 2$ bar).

Increasing the ambient temperature in liquid and vapour to $T_0 = 30$ $^\circ$ C, corresponding to a vapour pressure $p_{V,0} = 42.3$ mbar, the average maximum vapour pocket pressure clearly decreases to $\langle P_{{pocket}} \rangle = 27$ bar, but otherwise the data are similar as in the $T_0 = 20$ $^\circ$ C case, with pocket collapse being the dominant feature in the plot and no visible signal from the direct impact of the wave crest. Remarkable in both plots are the multiple peaks that are visible in the collapse region. Since these are both present in the average and in the individual data series (e.g. as in figure 3 d), this cannot be due to variability of the height at which the vapour pocket collapses: the collapse seems to occur at the same location ( $z_{coll}\approx 388 \ \text {mm}$ ) for all repetitions of the same conditions. In fact, the multiple peaks reveal again the apparent ‘threefoldness’ due to the sensor arrangement that we previously pointed out. This further supports the idea that the collapse pressures are not uniform in the spanwise ( $y$ ) direction along the impact wall.

At $T_0 = 40$ $^\circ$ C, corresponding to a vapour pressure $p_{V,0} = 73.6$ mbar, in figure 5(c) we see drastic changes: overall pressures are much smaller (note the change in the pressure scale from $0{-}90$ bar down to $0{-}10$ bar). In contrast to lower temperatures, the crest impact pressure is now discernible as a very clear peak at $z_{{impact}} \approx 455$ mm, where the maximum pressures reach up to about $P_{{impact}} \approx 2$ bar. Furthermore, looking at the collapse region, one sees a diversified picture: whereas in some experimental realisations hardly any collapse pressure is measured (lower blue-shaded data points), in other ones maximum collapse pressures reach up to $8$ bar thus revealing a large variability near the collapse region. The average maximum vapour pocket pressure decreases to $\langle P_{{pocket}} \rangle = 3.8$ bar.

This trend continues in the data of $T_0 = 50$ $^\circ$ C, corresponding to a vapour pressure $p_{V,0} = 123.0$ mbar, where the impact pressure is now even better discernible. Although the vapour pocket collapse is still a clear feature in most experimental realisations, with an average maximum pressure in the collapse region of $\langle P_{{pocket}} \rangle = 1.3$ bar, it is now for the first time smaller than the average maximum crest impact pressure, which equals $\langle P_{{crest}}\rangle = 1.5$ bar. Interestingly, this suggests the existence of a transition temperature in which both the collapse and impact pressure are nearly the same. For this wave shape, this temperature lies within $T\in [40, 50]$ $^\circ$ C. Note in addition that there is a very large crest impact pressure of approximately $7$ bar, which (without clear evidence) we tentatively attribute to the presence of a small entrapped vapour pocket during impact that upon implosion is responsible for the extreme impact pressure. It is good to note that such a large crest impact pressure has only been recorded in $1$ out of the total of $181$ impact experiments we conducted in boiling liquid conditions during this campaign.

Finally, figure 5(e,f), at $T_0 = 60$ $^\circ$ C (with $p_{V,0} = 198.6$ mbar) and $T_0 = 70$ $^\circ$ C ( $p_{V,0} = 310.8$ mbar), provides a similar picture: in these plots, where the vertical axis now only ranges from $0$ to $2$ bar, it is observed in both cases that the impact pressure is dominant, with average maximum crest impact pressures of $\langle P_{{crest}}\rangle = 1.2$ bar. In the region where for lower temperatures a vapour pocket collapse was observed ( $z \approx 360$ – $410$ mm), pressures are of the order of a few tenths of a bar, without the prominent peak that was observed for $T \leq 30$ $^\circ$ C.

It is good to stress that pressures measured in the boiling liquid case are so large that, when comparing the similar plot for the water–air impact (figure 3 c) to the data in figure 5, the data would almost coincide with the horizontal axis in figure 5(a,b), would reach up to 7 % of the height of the figure 5(c,d) and would only be clearly visible in the figure 5(e,f).

Figure 6. Maximum pressure $P_{{max}}(z)$ measured in each sensor as a function of the sensor location $z$ at the impact wall under boiling liquid conditions at an intermediate ambient temperature $T_0 = 40$ $^\circ$ C ( $p_{V,0} = 73.6$ mbar), for three breaking waves with wave steepness $\alpha = 0.35$ , $0.385$ and $0.42$ , corresponding to (a), (b) and (c), respectively. For each experimental setting, there were 10 repetitions of the experiment. In each panel, the blue-shaded dots correspond to the individual experiments, whereas the solid black line corresponds to their average. Note that the pressure range is largest in (a) and decreases towards (c).

Now, a first conclusion one can draw is that, as temperature and vapour pressure increase, the entrapped vapour pocket collapses in a less violent manner, leading to strongly decreasing collapse pressures as the temperature increases. This is however not yet the complete story. As soon as one considers the wave shape in the analysis, it becomes clear that the latter also plays a significant role.

In order to evaluate the wave shape dependence, we show in figure 6 the measurement of $P_{{max}}(z)$ at a fixed temperature ( $T_0 = 40$ $^\circ$ C) and for all three wave shapes ( $\alpha =0.350$ , $0.385$ , $0.420$ ). Here, we observe an extremely strong vapour pocket collapse for the $\alpha = 0.350$ case, with an average maximum vapour pocket pressure of $\langle P_{{pocket}} \rangle = 56$ bar, of the same order as that observed at the lowest temperature for $\alpha = 0.385$ . For the intermediate case ( $\alpha =0.385$ ), we find a very clear presence of both vapour pocket collapse and wave crest impact pressures, as already discussed above. In contrast, for $\alpha = 0.42$ in figure 6(c), the impact pressure is dominant, with an only very moderate but nevertheless clearly distinguishable average maximum vapour pocket pressure of $\langle P_{{pocket}} \rangle = 0.45$ bar. It is also worth noting that the crest impact and vapour pocket collapse locations are clearly changing, namely from $z_{{impact}} = 439$ mm through $455$ mm to $464$ mm, and from $z_{{coll}} = 407$ mm through $388$ mm to $390$ mm, respectively, from $\alpha = 0.35$ through $0.385$ to $0.42$ .

Figure 7. Maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ , averaged over all $10$ experiments performed at a single parameter setting, plotted as a function of ambient temperature $T_0$ for all three investigated wave shapes $\alpha = 0.35$ (blue), $0.385$ (red) and $0.42$ (yellow). The symbols represent the average over the individual experiments (data not shown), and the error bars are twice the standard deviation of the sample and would be symmetric on a linear scale.

In figure 7 we compare the average maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ for all three investigated wave shapes as a function of ambient temperature, where the pressure axis is logarithmic. For the wave shape that entraps the smallest vapour pocket ( $\alpha = 0.35$ ), collapse pressures remain very high ( ${\geq}50$ bar) up to $T_0 = 50$ $^\circ$ C, and only start to drop for the highest two ambient temperatures in our sample, where vapour pocket collapse pressures reach values below $10$ bar. From the very large error bars for these last two points one can conclude that there is quite a bit of variation between single realisations, and indeed for some experiments we observe collapse pressures close to $10$ bar, whereas in others a vapour pocket collapse appears to be completely absent (not shown).

For the intermediate vapour pocket case ( $\alpha = 0.385$ ) at the lowest ambient temperature ( $T_0 = 20$ $^\circ$ C), the average maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ is of the same order of magnitude as that for $\alpha = 0.35$ , but then almost exponentially decays with temperature, until a minimum value of $\approx 0.25$ bar is reached, where it subsequently remains. For the largest pocket ( $\alpha = 0.42$ ), the largest pressures are not even reached for $T_0 = 20$ $^\circ$ C. Here, the decay with temperature is similar to the other two wave steepnesses, but the minimal value is reached earlier and sustained up to the largest ambient temperature in our measurements.

Figure 8. Average maximum crest impact pressure $\langle P_{{crest}} \rangle$ as a function of ambient temperature $T_0$ for all three investigated wave shapes $\alpha = 0.35$ (blue), $0.385$ (red) and $0.42$ (yellow). The quantity $\langle P_{{crest}} \rangle$ has been obtained by first averaging the three largest pressures measured during the impact stage for each experiment (represented by grey dots) and subsequently averaging over the $10$ individual experiments for each parameter setting, denoted by the coloured stars. Note that we have horizontally shifted the data for the three wave shapes for clarity of presentation.

For completeness, in figure 8 we also present the average maximum crest impact pressures $\langle P_{{crest}} \rangle$ , measured during the crest impact stage. To define those, for each experiment, we first identify the three pressure channels containing the largest pressures measured during the crest impact stage, and subsequently average those three maximum pressures. These are represented by the grey dots. The coloured, star-shaped symbols in turn constitute the averages of those values for each experimental setting ( $\alpha$ , $T_0$ ). Note that we discarded the one instant of a very large impact pressure measured at $\alpha =0.385$ for $T_0 = 50$ $^\circ$ C discussed above in the average. As becomes clear from the data, $\langle P_{{crest}} \rangle$ shows much less variation with temperature as the maximum vapour pocket pressure, indicating that it is less affected by phase change. For all experiments, the averages lie between 0.8 and 1.3 bar. There may be a slight decreasing trend with temperature, but it is only convincing for the average data for $\alpha =0.385$ , and it is clear that the spread in the individual data points is too large to reliably corroborate the trend.

To conclude the section, we want to stress that in this work we have focused on the experimental characterisation of the pressure, instead of other parameters such as the force or the impulse during impact, which are also relevant parameters in many applications involving impacts. Nevertheless, we did numerically integrate the measured pressures over the height of the wall and multiplied with the width of the flume to obtain the force time series $F(t)$ , for which we could determine the maximum $F_{{max}}$ , the timing of which coincides with the vapour pocket collapse in the case such a collapse occurs. For $T_0 = 20$ $^\circ$ C, this leads to $F_{{max}} \approx$ $20$ - $50$ kN, and by integrating $F(t)$ around the maximum over a time period corresponding to the duration of the peak, we obtain an estimate for the associated impulse $I \approx$ $3$ – $5$ N s. Here, the ratio $I/F_{{max}} \sim$ $80$ – $120$ μs provides an estimate for the duration of the force peak (or an estimate for twice the rise time). This can be contrasted to the values observed for the water–air system, where the maximum in the force signal corresponds to the first pressurisation of the air pocket, leading to an order of magnitude smaller $F_{{max}} \approx$ $3$ – $5$ kN, but slightly larger $I \approx$ $5$ – $13$ N s, owing to the larger duration (rise time) of the air pocket pressurisation ( $I/F_{{max}} \approx$ $1$ – $4$ ms). One could say that (at small $T_0$ ) the slow momentum transfer occurring during the first air pocket pressurisation in the vapour case is postponed due to condensation and the corresponding momentum change is suddenly imparted onto the wall at the moment the vapour pocket collapses. Note that in all cases the crest impact is too localised in space to contribute significantly to the force.

4.1. A simplified model for the vapour pocket dynamics

Figure 9. (a) Sketch of the simplified model of the vapour pocket. A hemicylindrical bubble of radius $R(t)$ and length $W$ filled with vapour of density $\rho _V$ , temperature $T_V$ and pressure $p_V$ is in contact with a liquid of constant density $\rho _L$ and temperature $T_0$ . The pressure in the liquid is the sum of the equilibrium vapour pressure $p_{V,0}$ corresponding to the liquid temperature $T_0$ and the hydrodynamic pressure $\Delta p_0 = \rho _LV_{{wave}}^2$ in the liquid. The pressurisation of the vapour bubble triggers condensation, the latent heat of which will be transported into the liquid according to the temperature difference $T_V-T_0$ . (b) Sketch of the situation where the bubble is only partially covering the pressure sensor and the pressure $p_L$ in the surrounding liquid needs to be taken into account in addition to the vapour bubble pressure $p_V$ to estimate the measured pressure from the model; see also Appendix F.

Inspired by the seminal work of Prosperetti and many others in this field (Prosperetti & Plesset Reference Prosperetti and Plesset1978; Prosperetti Reference Prosperetti2017; Bergamasco & Fuster Reference Bergamasco and Fuster2017), we assume that in our case the heat transport into the liquid is the factor that limits phase change for the vapour pockets observed in the experiments. We assume that the vapour bubble is homogeneous for all relevant time scales, which is reasonable given the fact that the heat diffusivity in water vapour is typically at least two orders of magnitude larger than the heat diffusivity in the liquid. In addition, we assume that the bubble is in thermal equilibrium at all times, which stands to reason since the vapour density $\rho _V$ is much smaller than the liquid density $\rho _L$ , such that the heat capacity per unit volume $c_p/\rho$ is much larger for the liquid than for the vapour. If the bubble is pressurised, it will then strive to attain back to equilibrium by condensation, where the heat is to be conducted away into the liquid.

Assuming the bubble to be of hemicylindrical shape in an infinite medium with radius $R$ and length $W$ , as sketched in figure 9(a), neglecting the proximity of the free surface of the wave, we may thus write an energy-mass balance equation as

(4.1) \begin{equation} L \frac{{\rm d}m_V}{{\rm d}t} = S_w k_L\left .\frac {\partial T}{\partial r}\right |_{r=R(t)}, \end{equation}

where $m_V$ is the vapour mass contained in the bubble, $S_w = \pi R W$ the surface area of the hemicylindrical bubble, $L$ the latent heat of evaporation and $k_L$ the heat conductivity of the liquid. Here, we choose to neglect the volume changes in the liquid due to phase change, since the vapour density was assumed to be much smaller than that of the liquid. Also neglected is the sensible heat of the vapour bubble, in view of the expected smallness of the Jacob number $\textit{Ja} = c_{p,V}\Delta T/L$ , which is of order $8\times 10^{-3}$ or smaller for temperature changes below $\Delta T=10$ $^\circ$ C. Here, $c_{p,V}$ is the heat capacity of the vapour.

In addition, by writing down (4.1), we neglected the heat transport into the impact wall, which is assumed to be adiabatic in the model. This can be shown to correspond to the condition that the wall is dry (i.e. not wetted by the liquid phase) before impact. This assumption is discussed more extensively in Appendix D.

Writing $m_V = V \rho _V = ({1}/{2})\pi R^2 W \rho _V$ , where $V = ({1}/{2})\pi R^2 W$ is the volume of the bubble, in (4.1) leads to

(4.2) \begin{equation} \left (\frac {\partial \rho _V}{\partial p_V}\right )\frac {{\rm d}p_V}{{\rm d}t} = -\frac {2\rho _V}{R}\frac {{\rm d}R}{{\rm d}t} + \frac {2k_L}{LR}\left .\frac {\partial T}{\partial r}\right |_{r=R(t)}. \end{equation}

Physically, there is a competition between the two terms on the right-hand side: suppose, for example, because of a wave impact, that the bubble compresses, the first term states that the pressure will rise when the radius becomes smaller (noting that ${\rm d}R/{\rm d}t\lt 0$ ). Since the temperature then rises as well, the second term states that condensation heat will be transported away from the bubble, allowing some of the vapour to condensate and the pressure to decrease. Consequently, if the first term is dominant, the vapour bubble is expected to behave similar to a non-condensable gas bubble, which starts to oscillate after the wave impact, whereas dominance of the second term would lead to effective condensation and, hence, a collapse of the vapour bubble. We will return to this point in more detail in § 4.3.

Since the bubble is assumed to be on the vapour curve at all times, this implies we may choose any thermodynamic state variable to describe the other two, i.e. we may choose $\rho _V = \rho _V(p_V)$ and $T_V = T_V(p_V)$ . Using the Clausius–Clapeyron equation and the ideal gas law, the density and temperature of the gas may be expressed in terms of the bubble pressure $p_V$ together with the thermodynamic state variables at equilibrium, $p_{V,0}$ , $\rho _{V,0}$ and $T_0$ , which for convenience is summarised in Appendix C.

The energy-mass balance equation (4.2) is complemented by an equation of motion for the hemicylindrical bubble, for which we take the two-dimensional Rayleigh–Plesset equation (Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006; Ilinskii et al. Reference Ilinskii, Zabolotskaya, Hay and Hamilton2012)

(4.3) \begin{equation} \frac {\rm d}{{\rm d}t}\!\left (R\dot {R}\right )\log \!\left [\frac {R_\infty }{R}\right ] - \frac {1}{2}\dot {R}^2 + \frac {p_{V,0} - p_V(t)}{\rho _L} = 0, \end{equation}

which can be derived by integrating the axisymmetric, incompressible Euler equations from a point far away at a radial distance $R_\infty$ , where the velocity is assumed to be negligibly small, to a point on the bubble surface $R(t)$ , using a flow field that obeys the continuity condition $v_r(r) = R\dot {R}/r$ . Here, $\dot {R} = {\rm d}R/{\rm d}t$ denotes the time derivative of the bubble radius $R(t)$ and $p_V(t)$ is the pressure inside the bubble. Note that viscous and capillary effects can be included straightforwardly into (4.3) by adding the terms $+2\nu _L\dot {R}/R$ and $+\sigma /(\rho _LR)$ on the left-hand side of (4.3), respectively, where $\nu _L$ and $\sigma$ are the water kinematic viscosity and the surface tension of the water–vapour interface. Given the large Reynolds and Weber numbers of the problem we study, where with initial conditions $R_0\gt 1.0$ cm, $|\dot {R}_0|\gt 1.0$ m s^–1 we find that $\textit{Re}_0 = |\dot {R}_0|R_0/\nu _L\gt 10^4$ and $\textit{We}_0 = \rho _L\dot {R}^2_0R_0/\sigma \gt 10^2$ , we chose to neglect them here to not unnecessarily complicate the description. (Using the fact that $R(t) \sim (t_{{coll}} - t)^{0.5}$ for the collapse of the hemicylindrical bubble, one may easily verify that $\textit{Re}(t) \approx \textit{Re}_0$ and $\textit{We}(t) \sim \textit{We}_0 (t_{{coll}} - t)^{-0.5}$ , i.e. the local $\textit{Re}$ is approximately constant and $\textit{We}$ diverges during collapse.)

Finally, we need a closure that relates the unknown quantity in (4.2), namely $(\partial T/\partial r)_{r=R}$ in the liquid, to the other variables of the problem. The easiest way of formulating such a closure is to just write

(4.4) \begin{equation} \left .\frac {\partial T}{\partial r}\right |_{r=R(t)} \approx -\frac {T_V-T_0}{\sqrt {\pi \alpha _L t}}, \end{equation}

where we divide the temperature difference $\Delta T = T_V-T_0$ between the vapour bubble and the liquid far away from the bubble by a thermal boundary layer thickness $\delta _{{th}} = \sqrt {\pi \alpha _Lt}$ that starts growing at the moment the vapour bubble is entrapped at $t=0$ . Note that $\alpha_L = k_L/(\rho_L c_{p,L})$ is the thermal diffusivity of the liquid. Such an approach would be analytically correct in the limit of a constant temperature difference and for thin thermal boundary layers ( $\delta _{{th}} \ll R(t)$ ). Whereas the second condition is probably satisfied during most of the time evolution of the bubble, the first most certainly is not, and therefore, (4.4) can be a crude approximation at most. Fortunately, there is an approach that was proposed for the three-dimensional case by Plesset and Zwick (Plesset & Zwick Reference Plesset and Zwick1952), who formulated an integral equation relating $\Delta T$ to $(\partial T/\partial r)_{r=R}$ by formally integrating the spherically symmetric convective heat equation in the limit of a thin thermal boundary layer, an algebraically quite complex analysis that can, however, straightforwardly be reformulated for the axisymmetric case (Appendix E), leading to

(4.5) \begin{equation} T_V - T_0 = -\sqrt {\frac {\alpha _L}{\pi }}\int _{s=0}^t{\frac {R(s)\left .\frac {\partial T}{\partial r}\right |_{r = R(s)}}{\sqrt {\int _{w=s}^tR(w)^2{\rm d}w}}{\rm d}s} . \end{equation}

4.2. Results from the vapour bubble collapse model

The set of equations (4.2), (4.3) and (4.5) can be solved numerically, using the appropriate initial value conditions for the wave impact problem studied here, given by

(4.6) \begin{equation} R(0) = R_0 ;\quad \dot {R}(0) = -V_0 ;\quad p_V(0) = p_{V,0} + \rho _LV_{{wave}}^2 . \end{equation}

Here, $R_0$ and $V_0$ are obtained by determining the time evolution of the cross-sectional area $S(t)$ of the entrapped vapour bubble, where $R_0$ is the effective radius of the entrapped bubble at the moment of impact, i.e. when it becomes entrapped, defined as $R_0 = \sqrt {2S(t_{{impact}})/\pi }$ , consistent with the approximately hemicylindrical shape of the bubble, and $V_0$ is determined from the time rate of change of the cavity as $V_0 = -\dot {S}/\sqrt {2\pi S}$ , also determined at $t_{{impact}}$ . The final condition follows from the fact that the bubble is also pressurised by the water mass moving towards the wall. Although the wave impact is a highly unsteady process, the dynamics of the entrapped bubble are likely to be fast as compared with the overall pressurising motion of the wave towards the wall, such that we model it as the constant pressure inside a steady jet of velocity $v_{{jet}}$ hitting a wall, namely $p_{{jet}} = \rho _Lv_{{jet}}^2$ . If $V_{{wave}}$ is defined as the average horizontal velocity in the neighbourhood of the cavity, one may consequently estimate the pressure as $\Delta p_{{rise}} \approx \rho _LV_{{wave}}^2$ . The way in which these quantities are measured from the movies and the resulting values are further discussed in Appendix B, where we also argue that the inertial pressure scale is the appropriate scale to take.

Regarding $V_0$ and $V_{{wave}}$ it is worth noting that, although they are caused by the same physical phenomenon and have a similar magnitude, they are nevertheless causing distinctly different effects, that each of them separately may cause a cavity to collapse: if a quiescent cavity is created inside a liquid at a slightly higher pressure than the equilibrium vapour pressure, it will cause the cavity to shrink. If on the other hand a cavity is present in a liquid at equilibrium pressure but the cavity walls are given an initial inward velocity, the cavity will continue to shrink due to the converging action of the inertia present in the surrounding liquid. The first effect is connected to a diffuse, undirected pressure rise inside the liquid as a whole, whereas the second is the result of the already existing converging motion of the liquid.

Figure 10. Solution of the model system using the parameters observed for a wave of steepness $\alpha = 0.385$ ( $R_0 \approx 2.45$ cm, $V_0 \approx 1.8$ m s^–1 and $V_{{wave}} \approx 1.7$ m s^–1) and $R_\infty = W = 0.60$ m. (a) The vapour bubble radius $R(t)$ as a function of time $t$ for four different temperatures, $T = 20$ , $40$ , $60$ and $80$ $^\circ$ C. The horizontal dashed line indicates the radius of the pressure sensors used, $R_{{sensor}} = 2.75$ mm. (b) Same data as in (a) but zoomed in on the region until the first minimum occurs. The inset shows the same radius data but now as a function of the time $t_{{min}} - t$ remaining until the first minimum is reached, in a doubly logarithmic plot. (c) The corresponding vapour mass $m_V(t)$ divided by the original mass present in the bubble $m_{v,0} = m_V(0)$ , again as a function of time $t$ . (d) Same data as in (c) but zoomed in.

Taking the case $\alpha = 0.385$ , we find from the experiment that $R_0 \approx 2.45$ cm, $V_0 \approx 1.8$ m s^–1 and $V_{{wave}} \approx 1.7$ m s^–1 (Appendix B). In figure 10(a) we plot the bubble radius as a function of time, as obtained by the solution of the model system for four different values of the temperature $T_0$ . Here we observe that, for the lowest temperature ( $T_0 = 20$ $^\circ$ C), there is a full collapse of the vapour bubble. For $T_0 = 40$ $^\circ$ C, the bubble almost fully collapses but subsequently rebounds, whereas for the largest two temperatures ( $T_0 = 60$ and $80$ $^\circ$ C), the bubble appears to perform a damped oscillation. It is good to note that the latter is not due to viscous dissipation in the liquid phase, since this is neglected in (4.3), and of course also would be way too small to cause such a substantial dissipation. Instead, it is connected to condensation/evaporation cycles, where heat needs to respectively be stored in and retracted from the liquid.

Zooming in on the first 10 ms of the collapse (figure 10 b) it is clear that at first all bubbles follow the same path, which of course can be traced back to $V_0$ and $V_{{wave}}$ being the same in all cases, but soon the lowest temperature bubble appears as the fastest. Nevertheless, it is not the first to reach a minimum, which in fact is the highest temperature bubble. The $T_0=20$ $^\circ$ C bubble collapses as if no vapour was present in the bubble and the curve virtually overlaps with the one from an empty cavity using the same initial conditions, condensing all vapour present in the bubble in the process. The $T_0=40$ $^\circ$ C bubble collapses to an almost negligible radius of $R_{{min}} = 260$ $\mu$ m, corresponding to $0.011\, \%$ of the original volume, but as can be seen in the corresponding time evolution of the vapour mass in figures 10(c) and 10(d), defined as the product of the bubble volume and density, $m_V(t) = ({1}/{2})\pi W R(t)^2 \rho _V(t)$ , the minimum vapour mass is still substantial, at about 10 % of its initial value, and consequently, pressure and density at the minimum are high.

The two largest temperature bubbles however do not collapse but oscillate. The one at $T_0 = 60$ $^\circ$ C reaches its first minimum even slightly after the $40$ $^\circ$ C one, but that of $T_0 = 80$ $^\circ$ C even before the lowest temperature bubble has collapsed. This is remarkable, but can be traced back to the fact that these highest temperature bubbles oscillate very similar to a gas bubble, at an equilibrium pressure equal to $p_{V,0} + \rho _L V_{{wave}}^2$ , which is highest for the $80$ $^\circ$ C one and, therefore, leads to the largest oscillation frequency, i.e. the smallest period. Finally, comparing the time at which the minimum radius is reached with that at which the vapour mass is minimal, there is a clear shift: the minimum in the vapour mass is reached later than the minimum in radius. This can be traced back to the behaviour of the thermal boundary layer: as the minimum radius is reached, the vapour temperature becomes maximal and there is a thermal boundary layer transporting condensation heat into the liquid. This continues as the bubble starts to expand because the vapour temperature will still be higher than the far-field liquid temperature $T_0$ . In principle this could continue until the vapour temperature becomes equal to $T_0$ , but due to the fact that in the Plesset–Zwick formula the history of the thermal boundary layer is taken into account, this will happen earlier. The phase shift between the time evolution of the bubble and the structure of the thermal boundary layer in the liquid is ultimately also responsible for the damping of the oscillation.

Figure 11. Pressures in the model system using the same parameters as used in figure 10. (a) The vapour pressure $p_V(t)$ as a function of time $t$ . (b) Same data as in (a) but zoomed in on the region until the first minimum occurs. (c) The pressure $p_{{sensor}}(t)$ averaged over the sensor area and response period, again as a function of time $t$ . (d) Same data as in (c) but zoomed in. The inset in (c) shows the increase of $p_{{sensor}}$ towards the first minimum in $R$ (corresponding to the first maximum in $p_V$ ) in a doubly logarithmic plot. Note that in the experiment one may expect the sensor to measure $P_{\textit{sensor}} = p_{{sensor}} - p_{V,0}$ rather than $p_{{sensor}}$ itself. The inset in (d) shows a sketch of the pressure intensity (blue) on the sensor when the bubble radius is smaller than the sensor radius. The dashed square indicates the region taken to determine an estimate for the measured pressure in Appendix F. Note that in this theoretical figure absolute pressures are reported (in contrast to the gauge pressures of the experimental figures) to better separate the curves corresponding to different temperatures from one another.

Figure 11(a,b) shows the pressure $p_V(t)$ inside the vapour bubble as a function of time, again for the four ambient temperatures considered in figure 10. For the lowest two temperatures, the model pressures can reach highly unphysical values of thousands of bar (not shown), which are well beyond the critical point of water at $p=220.64$ bar, where the difference between vapour and liquid disappears and the model has lost its validity. For the highest temperatures, the vapour pressure remains very moderate with values of the order of $1$ bar maximally. This is also true for the first rebound of the $T_0 = 40$ $^\circ$ C case, where the maximum pressure equals $0.3$ bar.

Clearly, the computed vapour pressure would be only then equal to what is measured with the pressure sensors that we are using in the experiments, if the sensor is completely immersed inside the vapour bubble. As soon as the pressure sensor is partly covered by vapour and partly by liquid, the situation changes as sketched in figure 9(b) and more care needs to be taken. This case is treated in Appendix F, where an expression for the average pressure on the sensor area $p_{{sensor}}$ is derived, which is subsequently reported in figure 11(c,d), where we also take into consideration that the sensor has a finite response time of $\approx 5$ $\mu$ s (corresponding to the $200$ kHz acquisition rate), such that we report a time average over $5$ $\mu$ s. Here, we see that the maximum pressures reached on the sensor area for the lowest two temperatures are of the order of $100$ bar, i.e. of the same order as those measured in the experiment, especially for the $40$ $^\circ$ C case. For the higher temperatures, the sensor pressure is equal to the vapour pressure, as expected.

At this point it is good to ask how much of the observed high pressures in the model at lower temperatures are due to condensation and how much is just due to the fact that the equilibrium vapour pressure rapidly decreases with temperature? Clearly, also an adiabatic compression model would predict an increase of the pressure in the vapour pocket at lower ambient vapour pressures. Therefore, we numerically solved the Rayleigh–Plesset equation (4.3) with an adiabatic model for the vapour in the pocket (i.e. excluding phase change) and compared the result with the phase-change model presented in this section, using the same input parameters in both cases. The results, which are presented in more detail for the case $\alpha = 0.385$ in Appendix G, show that indeed there is a small effect, raising the pressure from $0.62$ bar at $T_0 = 70$ $^\circ$ C to $2.46$ bar at $T_0 = 20$ $^\circ$ C. Evidently, this effect is small and is dwarfed by the very large pressures ( $\sim 100$ bar) predicted by the phase-change model.

Finally, although in certain aspects the model agrees reasonably well with the experiments, there are clear dissimilarities as well. The most obvious ones have to do with the surface instabilities that occur on the surface of the vapour pocket, which essentially causes it to break up into a large bubble cloud rather than a single bubble after the first compression. This is clearly not captured by the model and will cause even more dissipation during oscillations. Nevertheless, the main features of the dynamics connected to phase change appear to be reasonably well captured.

4.3. Non-dimensionalisation of the model

To obtain additional insight, we non-dimensionalise the model of (4.2), (4.3) and (4.5) using the initial bubble radius $R_0$ as the length scale, the initial velocity $-V_0$ as the velocity scale and the equilibrium values $p_{V,0}$ , $\rho _{V,0}$ and $T_0$ as reference scales for the thermodynamic state variables. This leads to $\tilde {R} = R/R_0$ , $\tilde {t} = V_0t/R_0$ , $\dot {\tilde {R}} = \dot {R}/V_0$ (where for the dimensionless case the dot is understood to indicate a derivative with respect to dimensionless time $\tilde {t}$ ), $\tilde {p}_V = p_V/p_{V,0}$ , $\tilde {\rho }_V = \rho _V/\rho _{V,0}$ and $\tilde {T}_V = T_V/T_0$ . In addition, to simplify the discussion we use the approximate expression (4.4) for the temperature gradient at the bubble interface. The same conclusions can be drawn from the full set of equations, but the analysis is obscured by the complexity of (4.5).

The non-dimensionalised set of equations becomes

(4.7a) \begin{align} \left (\frac {d}{{\rm d}\tilde {t}}\left (\tilde {R}\dot {\tilde {R}}\right )\right )\log \left [\frac {\tilde {R}_\infty }{\tilde {R}}\right ] - \frac {1}{2}\dot {\tilde {R}}^2 - \frac {\tilde {p}_V-1}{\textit{Eu}_0} = 0 ,\end{align}

(4.7b) \begin{align} \frac {{\rm d}\tilde {p}_V}{{\rm d}\tilde {t}} = 2\left (\frac {\partial \tilde {\rho }_V}{\partial \tilde {p}_V}\right )^{-1}\left [\,-\tilde {\rho }_V\frac {\dot {\tilde {R}}}{\tilde {R}} \,- \Lambda \frac {\tilde {T}_V-1}{\tilde {R}\sqrt {\tilde {t}}}\right ] ,\end{align}

with initial conditions

(4.8) \begin{equation} \tilde {R}(0) = 1, \quad \dot {\tilde {R}}(0) = -1, \quad \tilde {p}_V(0) = 1 + \textit{Eu}_{{wave}} , \end{equation}

where we have defined the dimensionless groups

(4.9) \begin{equation} \textit{Eu}_0 = \frac {\rho _L V_0^2}{p_{V,0}}, \!\quad \textit{Eu}_{{wave}} = \frac {\rho _L V_{{wave}}^2}{p_{V,0}}, \!\quad \Lambda = \frac {k_LT_0}{\rho _{V,0}L\sqrt {\pi \alpha _LR_0V_0}} = \frac {\beta }{\sqrt {\pi }}\frac {\rho _L}{\rho _{V,0}}\frac {c_{p,L}}{\mathcal {R}_s}\sqrt {\frac {\alpha _L}{R_0V_0}} . \end{equation}

The first two groups represent Euler numbers based on the velocity scales $V_0$ and $V_{{wave}}$ , respectively, whereas the last one contains $\beta = \mathcal {R}_sT_0/L$ , the Péclet number $\textit{Pe}^{-1/2} = (R_0V_0/\alpha _L)^{-1/2}$ , the density ratio $\rho _L/\rho _{V,0}$ of liquid and vapour and the ratio of the specific heat of the liquid and the specific gas constant of the vapour, $c_{p,L}/\mathcal {R}_s$ .

By construction, all the dimensionless variables in (4.7) (namely $\tilde {R}$ , $\dot {\tilde {R}}$ , $\tilde {p}_V$ and $\tilde {T}_V$ ) are of order unity, at least shortly after the beginning of the dynamics, close to $t=0$ . Note that, for our experiments, since the Euler numbers are typically (considerably) smaller than $1$ , the initial condition for the pressure (4.8) is also of order one. This implies that we need to give special attention to the terms $\tilde {p}_V-1$ and $\tilde {T}_V-1$ in (4.7). Starting with the last one we may estimate

(4.10) \begin{equation} \tilde {T}_V-1 = \frac {T_V - T_0}{T_0} \approx \beta \frac {p_V - p_{V,0}}{p_{V,0}} = \beta (\tilde {p}_V-1) , \end{equation}

where we have used the linearised Clausius–Clapeyron relation (C4). For the term $\tilde {p}_V-1$ , one may estimate

(4.11) \begin{equation} \tilde {p}_V-1 \sim \tilde {p}_V(0) - 1 = \frac {p_V(0) - p_{V,0}}{p_{V,0}} = \frac {\rho _L V_{{wave}}^2}{p_{V,0}} = \textit{Eu}_{{wave}} \sim \textit{Eu}_0 , \end{equation}

such that a new order-one variable $\tilde {\unicode{x1D6E5} }$ can be introduced to replace $\tilde {p}_V-1$ as

(4.12) \begin{equation} \tilde {\unicode{x1D6E5} } \equiv \frac {\tilde {p}_V-1}{\textit{Eu}_0} . \end{equation}

With this definition our set of equations with initial conditions becomes

(4.13a) \begin{align} \left (\frac {d}{{\rm d}\tilde {t}}\left (\tilde {R}\dot {\tilde {R}}\right )\right )\log \left [\frac {\tilde {R}_\infty }{\tilde {R}}\right ] \,- \frac {1}{2}\dot {\tilde {R}}^2 \,- \tilde {\unicode{x1D6E5} } = 0 ,\end{align}

(4.13b) \begin{align} \frac {{\rm d}\tilde {\Delta }}{{\rm d}\tilde {t}} \approx \frac {2}{\textit{Eu}_0}\left (\frac {\partial \tilde {\rho }_V}{\partial \tilde {p}_V}\right )^{-1}\! \left [\,-\tilde {\rho }_V\frac {\dot {\tilde {R}}}{\tilde {R}} \,- \unicode{x1D6F1} \frac {\tilde {\unicode{x1D6E5} }}{\tilde {R}\sqrt {\tilde {t}}}\right ] ,\end{align}

(4.13c) \begin{align} \tilde {R}(0) = 1, \quad \dot {\tilde {R}}(0) = -1, \quad \tilde {\unicode{x1D6E5} }(0) = \frac {\textit{Eu}_{{wave}}}{\textit{Eu}_0} = \left (\frac {V_{{wave}}}{V_0}\right )^2 \equiv \kappa , \end{align}

where the term $\partial \tilde {\rho }_V/\partial \tilde {p}_V = \tilde {\rho }_V/\tilde {p}_V - \beta \approx 1$ , using (C5), could be further simplified. With respect to our discussion below (4.2), there is one key dimensionless parameter in this set of equations that determines the quality of the behaviour of the vapour bubble, namely $\unicode{x1D6F1}$ , defined as

(4.14) \begin{equation} \Pi = \beta \Lambda \textit{Eu}_0 = \frac {\beta ^2}{\sqrt {\pi }}\frac {\rho _L}{\rho _{V,0}}\frac {c_{p,L}}{\mathcal {R}_s}\textit{Pe}^{-1/2}\textit{Eu}_0 . \end{equation}

If $\unicode{x1D6F1} \gg 1$ , the second term in the dimensionless energy-mass balance equation is dominant, and a violent vapour bubble collapse is expected, similar to that predicted from the Rayleigh equation for an empty cavity. For the examples provided in figures 10 and 11, this would correspond to the $T_0=20$ $^\circ$ case, for which $\unicode{x1D6F1} = 4.0$ . The second-lowest temperature ( $T_0 = 40$ $^\circ$ C) is with $\unicode{x1D6F1} = 0.52$ in the crossover region, whereas the two highest temperatures ( $T_0 = 60$ and $80$ $^\circ$ C), for which $\unicode{x1D6F1} = 0.089$ and $0.020$ , respectively, are in the region where oscillations of the vapour bubble are dominant, similar to those of an oscillating non-condensable gas bubble that is entrapped during impact.

The other two dimensionless parameters in (4.13), namely $\textit{Eu}_0$ , which has influence on the overall time scale but not on the balance between the two terms on the right-hand side of (4.13b), and the order-one parameter $\kappa = V_{{wave}}^2/V_0^2$ , which appears in the initial value for the pressure variable $\tilde {\unicode{x1D6E5} }$ in (4.13c), are of smaller (although not negligible) consequence for the dynamics of the vapour bubble.

4.4. Measured maximum pressures in the light of model results

Figure 12. Average maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ data for all three investigated wave shapes $\alpha = 0.35$ (blue), $0.385$ (red) and $0.42$ (yellow), i.e. the same data as plotted in figure 7. When the data are plotted as a function of the key dimensionless parameter $\unicode{x1D6F1}$ , which characterises the qualitative behaviour of the entrapped vapour bubble, the data collapse between the different wave steepnesses $\alpha$ is unsatisfactory (inset), but when the modified parameter $\unicode{x1D6F1} ^*$ is used the data collapse is convincing (main plot). As in figure 7, the symbols represent the average over the individual experiments (data not shown), and the error bars are twice the standard deviation of the sample and would be symmetric on a linear scale. The vertical dashed blue lines indicate the condition $\unicode{x1D6F1} = 1$ (and $\unicode{x1D6F1} ^* = 1$ ), marking the boundary between the regions in which non-condensable gas-like behaviour (oscillations, $\unicode{x1D6F1} ,\unicode{x1D6F1} ^* \lt 1$ ) and vapour-like behaviour (collapse, $\unicode{x1D6F1} ,\unicode{x1D6F1} ^* \gt 1$ ) is expected.

Now that we have identified a single dimensionless parameter, $\unicode{x1D6F1}$ , to characterise the qualitative behaviour of the entrapped vapour bubble, we may use this knowledge to re-examine the maximum vapour pocket collapse pressure data from figure 7. To this end, we compute the value of $\unicode{x1D6F1}$ for each temperature and wave steepness and plot the result in the inset of figure 12. In this doubly logarithmic plot, we see, for each $\alpha$ separately, that the data shows the behaviour we expect: for small values ( $\unicode{x1D6F1} \ll 1$ ), the measured average maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ is small, of the order of 0.1–1 bar, connected to a relatively small amount of condensation (or evaporation) and close to non-condensable gas pocket behaviour, whereas for intermediate values ( $\unicode{x1D6F1} \sim 1$ ), the pressures suddenly rise until, for large values ( $\unicode{x1D6F1} \gg 1$ ), we find that the very large pressures connected to a full vapour pocket collapse, in the regime where the latent heat of condensation can be easily transported into the liquid. The intermediate region is clearly visible for $\alpha = 0.385$ , whereas the vapour pocket collapse region is particularly strong for $\alpha = 0.35$ and the small pressure response is dominant for $\alpha = 0.42$ . Unfortunately, the data for the three wave steepnesses seem shifted with respect to one another, such that the data collapse of the plot is unsatisfactory. Let us now look into the origin of this discrepancy. Note that $\unicode{x1D6F1}$ decreases monotonically, but nonlinearly with temperature.

The first cause is the fact that the entrapped vapour pockets are assumed to be hemicylindrical, whereas in fact they are rather flattened, especially in the case of small $\alpha$ . This means that the thermal contact in the experiment is larger than in the model. It is in fact straightforward to evaluate the consequences, if $\ell (t)$ is the measured wetted length of the vapour pocket, and $S(t) = ({1}/{2})R_{{eff}}^2$ is its cross-sectional area, then we can write the energy-mass balance (4.1), with $m_V = S(t)W\rho _V(t)$ , as

(4.15) \begin{equation} \frac {1}{2}\pi R_{{eff}}^2\frac {{\rm d}\rho _V}{{\rm d}t} = - \pi R_{{eff}}\dot {R}_{{eff}}-\frac {k_L \ell }{L}\left .\frac {\partial T}{\partial n}\right |_{\partial S} , \end{equation}

where $\partial T/\partial n$ is the normal derivative at the liquid–vapour interface $\partial S$ . Writing this equation in the form of (4.2) we obtain

(4.16) \begin{equation} \left (\frac {\partial \rho _V}{\partial p_V}\right )\!\frac {{\rm d}p_V}{{\rm d}t} \,\,= \,\,-2\rho _V\frac {\dot {R}_{{eff}}}{R_{{eff}}} + \frac {2k_L}{LR_{{eff}}}\left [\frac {\ell }{\pi R_{{eff}}}\right ]\left .\frac {\partial T}{\partial n}\right |_{\partial S} . \end{equation}

Note that if one replaces $R_{{eff}}$ by $R$ , we exactly retrieve (4.2), with the exception of the factor between square brackets in the second term, $\ell /(\pi R_{{eff}}) = \ell /\sqrt {2\pi S}$ , which multiplies the heat flux, and in the process of non-dimensionalisation leads to an additional mutliplicative factor $\ell /(\pi R_{{eff}})$ in $\unicode{x1D6F1}$ , that is not present in the analysis of the previous subsection. Here, it is good to note that the multiplication factor represents the ratio of the wetted length to that of a hemicylindrical cavity of the same area.

The second cause is due to the presence of $\kappa$ in the initial value condition (4.13c ) on the pressure variable $\tilde {\unicode{x1D6E5} }$ . If we compare our experiments to the ideal case $\kappa = 1$ , corresponding to $V_{{wave}}=V_0$ , then we see that in the first stages of the dynamics $\tilde {\unicode{x1D6E5} }(t) \sim \kappa$ , i.e. different from $1$ . To at least initially regauge the dynamics to the $\kappa =1$ case, we may simply introduce an effective value for $\unicode{x1D6F1}$ by dividing $\unicode{x1D6F1}$ by $\kappa$ .

Consequently, if we introduce a modified version $\unicode{x1D6F1} ^*$ of our key parameter, defined as

(4.17) \begin{equation} \unicode{x1D6F1} ^* \equiv \frac {1}{\kappa }\frac {\ell _0}{\pi R_{{eff},0}}\unicode{x1D6F1}, \end{equation}

we may hope for a better collapse of the data. And, indeed, this is corroborated from our experimental data, when looking at the very good collapse of the data in the main figure of figure 12, where we plot $\langle P_{{pocket}} \rangle$ as a function of $\unicode{x1D6F1} ^*$ as defined in (4.17). Note that the subscript $0$ in $\ell _0$ and $R_{{eff},0}$ indicate that these quantities are evaluated at the impact time $t_{{impact}}$ .

In conclusion, the model defined in this section is capable of capturing the essential physics of breaking wave impact in a boiling liquid. This is remarkable, since the large scale of the dynamics creates many additional dynamics already during vapour pocket compression or collapse, such as the development of surface instabilities; and definitely after reaching the first minimum, where the vapour pocket has broken up into a bubble cloud, the dynamics is expected to be significantly different. Also, even when taking utmost care in controlling the pressure inside the autoclave, we cannot exclude the presence of small amounts of non-condensable gases (air), especially that dissolved into the liquid phase, that may alter the dynamics of our system, e.g. in the form of small gaseous bubbles that may act as nuclei for vapour bubbles in the rarefaction stage that follows a complete collapse and will cause a secondary vapour bubble cloud to be created and collapse, forming the observed rebound.

Finally, since the data collapse observed in figure 12 is the result of dimensional analysis, one may argue that also the average maximum vapour pocket pressure $\langle P_{{pocket}} \rangle$ needs to be non-dimensionalised, most specifically using the vapour pressure $p_{V,0}$ . This, however, is only applicable for pressures smaller than a few bar, since the larger ones are all diverging and, therefore, crucially determined by the finite sensor size and response time (cf. the integration procedure outlined in Appendix F). These additional scales do interfere with the theoretically expected pressure scaling, but do not interfere with the qualitative behaviour of the vapour pocket, which is solely determined by the value of $\unicode{x1D6F1}$ (or $\unicode{x1D6F1} ^*$ ), as becomes clear from (4.13b). As a consequence, the value of $\langle P_{{pocket}} \rangle$ is a good measure of this qualitative behaviour.