4.1.1 Shadowgraph images

Figures 6(a) and 6(b) show shadowgraph images of two- and one-phase flow, respectively. They are for $T_{b}-T_{on}\simeq 10$  K, corresponding to $Ra\simeq 1.9\times 10^{10}$ . Corresponding movies can be found in the supplementary material. One can see that the thermal plumes and any large-scale flows had different structures and dynamics in the presence and absence of vapour bubbles. In the two-phase flow (figure 6 a and the corresponding supplementary movie 3) hot thermal plumes, mainly produced at the heated area of the bottom plate, drove and were carried by a LSC. They moved laterally to the right just above the bottom plate and then rose along the side of the cell that was higher due to the deliberately introduced small tilt of the cell. Due to thermal dissipation they became weaker along their path, with only a fraction reaching the top plate. Cold plumes were emitted along the entire top plate. They moved to the left under the influence of the tilt, and then descended along the left sidewall. They too were diminished along their path due to thermal dissipation, and very few reached the bottom plate. Only a small number of plumes were found near the sample centre. The structure of the LSC is qualitatively similar to that observed for classical RBC where both plates are heated or cooled over their entire area (Xi et al. Reference Xi, Lam and Xia2004). There is, however, a quantitative difference in that the hot plumes generate a very intense rising flow near the cell wall while the cold ones produce a much weaker descending flow just inside the opposite part of the wall.

The rising bubbles (small black solid dots), which nucleated at the central heated area of the bottom plate where the nucleation sites were located, rose with a speed that was much larger than the characteristic LSC speed. Although near the plate they were dragged sideways a short distance by the LSC, over their entire path they moved more nearly in a vertical direction and did not follow the lateral motion of the plumes and the LSC. In this particular run a large number of bubbles reached the top plate. We present it here because the rising bubbles are clearly identifiable. Under our standard experimental conditions (air concentration between 20 % and 25 % by volume, see Narezo Guzman et al. (Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015)) in both sets of apparatus, most bubbles fully vanished (condensed and/or dissolved) before they came close to the top plate. A longer lifetime was due to a higher concentration of dissolved air in the liquid, which nevertheless remained below the saturation concentration (see § 4.1.2). The air concentration did not affect the qualitative features of the LSC.

Although heating took place only over the central area $A_{h}$ , lateral heat flow through the silicon wafer and polycarbonate ring (see figures 1 and 2) assured that the temperature difference between the top plate and all lateral positions of the bottom plate was large enough to drive turbulent flow. Just below the sidewall this difference was measured to be about $0.6{\rm\Delta}T$ . Thus, especially in one-phase flow, we observed hot plumes emitted over the entire bottom plate, although their abundance was greater over the central area $A_{h}$ . This can be seen in figure 6(b) and the corresponding supplementary movie 4. In our experiment, where heating was localized over the central area $A_{h}$ , one-phase flow did not produce a well-defined LSC as can also be seen in the figure and the movie. This finding contrasts with the well-known single-roll structure of the LSC for classical ${\it\Gamma}\simeq 1$ RBC where the heat input is uniformly distributed over the bottom plate (see, for instance, Xi & Xia (Reference Xi and Xia2008) and Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009)).

Figure 7.

Dynamics of bubbles ( $l=0.60$  mm) as they rose from $z/L=0$ to $z/L\simeq 0.4$ : (a) diameter $d_{x}$ as a function of time; (b) vertical velocity component $U_{z}$ (same colour and symbol represent the same bubble as in (a)) as a function of $t$ ; (c) same as (b), but with the horizontal scale expanded; (d) horizontal velocity component $U_{x}$ as a function of $t$ .

Interestingly, for two-phase flow and a superheat of about 6 K or more a well-defined single-roll LSC was found (see figure 6 a and the supplementary movie 3), similar to that of classical RBC. The transition from disordered flow to a single-roll LSC in the two-phase case provides an explanation for relatively sharp changes in some other quantities reported previously by Narezo Guzman et al. (Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015). These authors found that the skewness $S$ of the local temperature probability distribution measured at the two locations $z/L=0.28$ and $z/L=0.50$ along the vertical axis of the cell was modified by vapour bubbles. The skewness was positive for one-phase flow, being slightly larger closer to the bottom plate. For superheat values larger than about 6 K $S$ was reduced in two-phase flow relative to the single-phase value, with a larger reduction closer to the plate. For the largest superheat values $S$ became negative. Shadowgraph visualization provided insight into these results. The location of the thermistors used by Narezo Guzman et al. (Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015) to measure local temperature time series is indicated in figure 6(a) (red dots). The shadowgraph movie for one-phase flow shows that the number of plumes passing by the thermistor at $z/L=0.28$ was larger than that passing $z/L=0.50$ . This explains the larger value of $S$ closer to the bottom plate. The shadowgraph movie for two-phase flow explains why $S$ was reduced in the presence of bubbles, as a LSC roll swept the hot plumes sideways close to the bottom plate and reduced the number of plumes rising vertically across the cell centre where the thermistors were located. In two-phase flow the probability of hot plumes passing by the thermistor at $z/L=0.28$ was larger than it was at $z/L=0.50$ . If only hot plumes determined $S$ in two-phase flow, one would expect a larger (positive) $S$ closer to the bottom plate than at mid-height, opposite to the results of Narezo Guzman et al. (Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015). This supports their conclusion, based on the argument of Lakkaraju et al. (Reference Lakkaraju, Toschi and Lohse2014), that bubbles homogenized the temperature field in the bulk, especially close to the bottom plate where the bubbles are bigger (see § 4.1.2) and thus had a larger heat capacity.

4.1.2 Size of bubbles after departure

In figure 7(a) the temporal evolution of the bubble diameter $d_{x}$ of 15 non-merging bubbles after detaching from the bottom plate is shown. Note that the initial bubble-diameter values shown in figure 7(a) ( $d_{x}(t=0)\sim 0.5$  mm) are consistent with the width $d_{x,max}$ right before departure obtained in § 4.2 below. Two of the tracked bubbles had vanished (condensed and dissolved) almost completely at $t\simeq 0.5$  s. The rest of the bubbles continued their rising motion outside of the viewed region. Within about 0.05 s after a bubble detached from the plate, its diameter $d_{x}$ typically was reduced by about $1/3$ of its initial value. The vertical location of bubbles at that point was between 4 and 5 mm above the surface. However, it was within about the first 0.01 s after departure and within about 1 mm above the surface that $d_{x}$ decreased at the highest rate. We estimated that the average volume rate of change of a bubble was about $3~\text{mm}^{3}~\text{s}^{-1}$ during the first 0.01 s. For $t>0.01$  s the bubbles volume continued to decrease, albeit more slowly, with a nearly linear dependence of their diameter on time. We estimated an average volume change rate during the time interval $t\geqslant 0.05$  s to be $0.06\pm 0.02~\text{mm}^{3}~\text{s}^{-1}$ per bubble.

The initial ( $t\lesssim 0.05$  s and a vertical bubble location less than $z/L\simeq 0.05$ ) and later ( $t\gtrsim 0.05$  s and $0.05\lesssim z/L\lesssim 0.4$ ) volume change rates differed from each other by nearly two orders of magnitude.

We believe that the initial volume change rate is largely due to the rapid vapour condensation, as the bubbles are subjected to a dramatic change of the thermal environment along their path. Initially, they reside in part in the thin thermal boundary layer where the temperature drops by about 70 % of ${\rm\Delta}T$ (Narezo Guzman et al. Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015). At later times the bubble experiences a nearly uniform thermal environment approximately at $T_{b}-0.7{\rm\Delta}T$ . We note that the locally defined Jakob number varies from $Ja\simeq 19$ at the bottom plate ( $Ja\propto T_{b}-T_{on}$ ) to about $-1$ ( $Ja\propto (T_{b}-0.7{\rm\Delta}T)-T_{on}$ ) in the bulk of the fluid above the thermal boundary layer. Theoretically (Okhotsimskii Reference Okhotsimskii1988) and experimentally (Chen & Mayinger Reference Chen and Mayinger1992) $Ja$ has been shown to be a relevant parameter in bubble condensation. A large decrease of the bubble diameter was also found in numerical simulations (Yang & Prosperetti Reference Yang and Prosperetti2008) of a bubble encountering a subcooled region of liquid, both of similar size. From their simulations Legendre, Borée & Magnaudet (Reference Legendre, Borée and Magnaudet1998) concluded that for a condensing bubble, the evolution of $d_{x}$ and the heat transfer rate crucially depend on the ratio between bubble surface radial velocity ${\dot{d}}_{x}=\text{d}(d_{x})/\text{d}t$ and the streamwise velocity $v$ (relative velocity between fluid and bubble). If $|{\dot{d}}_{x}/v|>1$ , then the bubble condenses at the same rate as if $v=0$ . However, when $|{\dot{d}}_{x}/v|<1$ , the thermal boundary layer around the bubble is controlled by the stream velocity and the bubble condenses faster than for a bubble at rest. Neither the thermal profile in our experiment nor the streamwise velocity are known and a more quantitative comparison with the simulations do not seem warranted.

Since some air will have diffused into the vapour bubble during its formation process, once the vapour closest to the bubble wall condenses right after departure, one may argue that any remaining vapour in the core of the bubble will be surrounded by an air layer or shell. However, the diffusion timescale ${\it\tau}_{g}$ in which air and vapour inside the bubble mix is

where $D_{g}$ is the binary diffusion coefficient of gases, which is typically of the order of $0.2\times 10^{-4}~\text{m}^{2}~\text{s}^{-1}$ . Comparing ${\it\tau}_{g}\sim 2\times 10^{-3}$  s with the time of the order of 0.05 s over which the volume changed at the largest rate, we conclude that vapour and air inside the bubble were well mixed.

Next to the condensation of the remaining vapour, we associate the later small volume change rate of the rising bubbles for $t\gtrsim 0.05$  s to the slow diffusion of air inside the bubble into the surrounding undersaturated liquid. The timescale for an air bubble to diffuse into an undersaturated still liquid around it is given by (see e.g. Lohse & Zhang Reference Lohse and Zhang2015)

where $D_{l}$ is the diffusion coefficient of air in the liquid (typical values are in the order of $D_{l}\sim 10^{-5}~\text{cm}^{2}~\text{s}^{-1}$ ), ${\it\rho}_{g}$ is the density of the gaseous phase at the bubble boundary (with values around ${\it\rho}_{g}\sim 10^{-3}~\text{g cm}^{3}$ for nitrogen), $c_{s}$ is the air saturation concentration, and $c_{\infty }$ the air concentration far away from the bubble. If the bubble is moving with respect to the surrounding liquid, the concentration boundary layer around it is thinner than for the case of a bubble in a quiescent liquid, i.e. the diffusion rate is increased. Similarly as done by Zhang et al. (Reference Zhang, Lu, Tan, Bao, He, Sun and Lohse2015) for a drop, we assume that the concentration boundary layer around the bubble is of Prandtl–Blasius–Pohlhausen type (Grossmann & Lohse Reference Grossmann and Lohse2004). Then its thickness scales as ${\sim}d_{x}/\sqrt{Pe}$ , where the Péclet number of the moving bubble is given by

Here $U_{z}\simeq 10~\text{cm s}^{-1}$ is the vertical velocity component of the rising bubble (see § 4.1.3), resulting in a Péclet number in the order of $Pe\sim 4\times 10^{4}$ . For such a moving gas bubble the dissolution time scale is then given by

From (4.4) for our experimental data we find that the timescale for complete bubble dissolution should be of the order of 7 s, which is larger than the dissolution time of the order of about 1 s observed in figure 7(a), but it is of the same order of magnitude. In experiments with a larger air concentration in the liquid we found that the volume rate of change for $t\gtrsim 0.05$  s was smaller than that of figure 7(a) (see appendix B), which qualitatively is consistent with (4.4). For a more quantitative treatment of the role of dissolved air in bubble condensation we refer to the work of Shpak et al. (Reference Shpak, Stricker, Versluis and Lohse2013), who found a minimum amount of gas necessary in order to prevent the complete collapse of a bubble mostly filled with vapour when driven by ultrasound.

4.1.3 Velocity of bubbles after departure

In figure 7(b) the vertical velocity component $U_{z}$ (see (3.1)) of all 15 bubbles is plotted as function of time after departure. Figure 7(c) shows some of the same data, but on an expanded horizontal scale. All curves have a minimum in the range $0.01~\text{s}\lesssim t\lesssim 0.03$  s, which could be due to an interaction with the horizontal drag force exerted by the LSC (see the next paragraph) together with the fluid velocity gradient across the kinetic boundary layer adjacent to the bottom plate. At later times ( $t\gtrsim 0.03$  s) $U_{z}$ increased again, reached a local maximum near $t\simeq 0.06$  s, and then decreased at a nearly constant (but perhaps very slowly decreasing) rate.

In figure 7(d) the corresponding horizontal velocity component $U_{x}$ is plotted. The horizontal motion implied by a non-zero $U_{x}$ is seen very well in the movies in the supplementary material of the paper by Narezo Guzman et al. (Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015). In the range $0.01~\text{s}\lesssim t\lesssim 0.03$  s, $U_{x}$ reached a positive maximum. Given the angle of tilt (by less than $1^{\circ }$ ) of the cell, a positive $U_{x}$ corresponded to bubble displacement in the same direction as that of the preferred LSC orientation right above the bottom plate. Thus, we associate the horizontal motion with the horizontal drag exerted on the bubbles by the LSC. This drag, and thus $U_{x}$ , are largest near the plate where the horizontal LSC velocity is largest (see, for instance, Qiu & Tong Reference Qiu and Tong2001). As the bubbles rose further, they encountered the bulk flow where the LSC was weaker (see § 4.1.1 and figure 6 a and the corresponding supplementary movie 3) and $U_{x}$ attained a nearly constant positive value near $7~\text{mm s}^{-1}$ . A small contribution to this value can be attributed to the small tilt of the cell relative to the horizontal camera alignment. With the tilt angle of $1^{\circ }$ and the typical vertical velocity of, say, $100~\text{mm s}^{-1}$ (see figure 7 b), a contribution to the measured $U_{x}$ of about 1 or 2  $\text{mm s}^{-1}$ would be expected. This is smaller than the measured $U_{x}$ at large $t$ , indicating that a small lateral motion of the bubbles persists even well away from the bottom plate. This motion, we assume, is caused by the LSC which, although much weaker, does persist in the bulk with a velocity (in the time average) which vanishes only at one point on the vertical axis of the sample.

A detailed analysis of the forces acting on the rising bubbles with decreasing size, and in some cases with changing shape (mostly bubbles were spherical but some bubbles were ellipsoidal briefly after departure), would be desirable but, we believe, is too difficult to carry out reliably for our turbulent system. There is a very large body of work (Magnaudet & Eames Reference Magnaudet and Eames2000) on force-balance models which consider drag, added mass and buoyancy forces. They reproduce the experimentally observed dynamics of gas bubbles rising in quiescent liquids. Moore (Reference Moore1965) studied the drag of ellipsoidal bubbles, the work by de Vries, Luther & Lohse (Reference de Vries, Luther and Lohse2002) focused on the added-mass force of a rising bubble when it experiences shape oscillations and the study by Ohl, Tijink & Prosperetti (Reference Ohl, Tijink and Prosperetti2003) investigated the added mass of an expanding bubble. To carry out similar analyses for the present work, information about the turbulent fluid-velocity field around a bubble is necessary in order to quantify added-mass and drag forces. The contribution to the added-mass effect due to both the relative acceleration and the pressure gradient in the liquid, as well as the drag-force dependence on the relative velocity between liquid and bubble, would be needed. We lack most of this information for this complex turbulent system.

4.1.4 Velocity of bubbles across the whole cell

Similar to the results in § 4.1.2, we tracked 33 bubbles starting at a few frames after they detached from the surface and continuing until they condensed and dissolved, or reached the top plate (see figure 8 d). We tracked both bubbles which merged and others that did not. The vertical and horizontal velocity components are plotted as a function of the vertical coordinate $z/L$ in figures 8(a) and 8(b), respectively. The large majority of bubbles did not reach the top plate ( $z/L=1$ ). There was no common maximal value of $z/L$ where the bubbles vanished, and the value of $z/L$ where $U_{z}$ reached its largest value varied over a wide range. We believe that the large variation in the bubble paths is related to the occasional merging of bubbles, as well as to the random thermal and dynamical history along each path. Bubbles that merged did so right after departure most of the times, determining the bubble volume and thus its vertical speed from a very early stage on until they vanished.

Figure 8.

(a) The vertical velocity component $U_{z}$ of the rising trajectories of 33 bubbles as a function of height $z/L$ . (b) The horizontal velocity component $U_{x}$ of the same bubbles as in (a) as a function of $z/L$ . (c) The vertical velocity component $U_{z}$ for all bubbles in (a) and (b), scaled by its maximal value $\max (U_{z})$ , as a function of $z$ scaled by the height $\max (z)$ at which the bubble condensed and dissolved completely. The data collapse fairly well onto a unique curve, especially for $z/\max (z)\gtrsim 0.3$ . (d) A snapshot of the two-phase flow with $l=0.60$  mm taken from the side. The etched area with nucleating cavities can be seen as the bright ellipse near the bottom of the image. Superimposed on the photograph are the trajectories (in the form of circles of different colour for different bubbles and of uniform size) of some of the tracked bubbles from the moment they detached until they vanished. See the corresponding supplementary movie 5.

It is interesting that figure 8(b) indicates a small positive value of $U_{x}$ for a large range of $z/L$ , but indicates for the one bubble that reached the top of the cell that $U_{x}$ became negative as it approached the top plate. This would be consistent with a lateral drag exerted by the LSC in our sample, which is toward positive $x$ in the bottom and toward negative $x$ in the top part of the cell (see supplementary movie 3).

The vertical trajectories of all bubbles had similar shapes. In figure 8(c) the $U_{z}$ values of a given bubble are scaled by their maximal value $\max (U_{z})$ and plotted as a function of height $z/\max (z)$ where $\max (z)$ is the vertical distance that each bubble travelled before it vanished or reached the top plate. The curves collapse onto a unique curve relatively well for $z/\max (z)>0.3$ , showing the correlation between the rising velocity and the lifetime of a bubble.

The characteristic speed of the bubbles at early times was much larger than that of any LSC or plume motion. For $t\simeq 0.07$  s for instance one sees in figure 7(c) that $U_{z}$ was near $120~\text{mm s}^{-1}$ . Inspection of the supplementary movies indicates that the LSC speed $U_{LSC}$ in the two-phase system was near $40~\text{mm s}^{-1}$ , about a factor of three smaller. The plume motion $U_{pl}$ in the one-phase system, where there was no LSC, was even slower, near $15~\text{mm s}^{-1}$ . The larger value of $U_{z}$ indicates that the bubbles significantly drive the fluid flow.

4.1.5 Bubble aspect-ratio before departure

The vertical (height) and horizontal (width) sizes of the bubbles $d_{z}$ and $d_{x}$ were measured for the $l=0.60$  mm wafer immediately before they departed from the plate. The results had no significant superheat dependence in the range from 7.8 to 9.8 K. This is consistent with the measurements of $d_{x}$ at departure obtained from top visualization over a wider superheat range from ${\sim}3$ to 10 K for the same cavity separation (see § 4.2.1). From the 70 departing bubbles selected for $T_{b}-T_{on}=9.8$ , 8.8 and 7.8 K we estimated the mean and standard deviation of the bubble sizes to be $d_{x}=45.0\pm 2.0$ pixels and $d_{z}=57.1\pm 3.5$ pixels, giving an aspect ratio ${\it\Gamma}_{b}\equiv d_{x}/d_{z}=0.79\pm 0.03$ . For the case with $l=1.0$  mm and $T_{b}-T_{on}=10.6$  K, based on 33 bubbles at departure, we similarly found ${\it\Gamma}_{b}=d_{x}/d_{z}=0.72\pm 0.03$ .

For a spherical bubble ${\it\Gamma}_{b}=1$ . We note that the departure of ${\it\Gamma}_{b}$ from unity is due to the extension, in the shape of a down-pointing triangle, of the bubbles toward their attachment point which can be seen in figure 5(a). Almost immediately after detachment the bubbles became essentially spherical, except for a few cases where the shape of the free bubbles was elliptical with a horizontal major axis and ${\it\Gamma}_{b}\simeq 1.2$ for a brief time after departure.

4.1.6 Bubble volume before departure

Bubble volumes right before departure were computed from their contours as described in § 3.6.3. Mean values $V$ of all selected bubbles with a clear contour were obtained for $l=0.60$ and $l=1.0$  mm (see § 3.6.3). They were compared with the theoretical maximal volume of a bubble growing at a wall under equilibrium conditions (slow growth) $V_{max}$ when only surface tension and buoyancy forces are considered (Fritz Reference Fritz1935). For a spherical bubble $V_{max}$ is given by

where ${\it\beta}$ is the contact angle between the surface and the bubble, and $l_{c}$ is the capillary length

Here ${\it\gamma}$ is the surface tension, ${\it\rho}$ was evaluated at $T_{b}$ and ${\it\rho}_{v}$ at the vapour partial pressure. Fritz (Reference Fritz1935) gave ${\it\varphi}({\it\beta})$ for nine ${\it\beta}$ values. For the working liquid in our experiments $l_{c}$ ranged from 0.97 to 0.95 mm because the surface tension decreased with increasing bottom-plate temperature. From side visualization we found ${\it\beta}=40^{\circ }\pm 5^{\circ }$ for $l=0.60$  mm and ${\it\beta}=45^{\circ }\pm 5^{\circ }$ for $l=1.0$  mm (see § 4.1.5), with corresponding values ${\it\varphi}(40^{\circ })=0.105$ and ${\it\varphi}(45^{\circ })=0.150$ , resulting in $V_{max}=0.26~\text{mm}^{3}$ and $V_{max}=0.36~\text{mm}^{3}$ , respectively, for $T_{b}=38.5\,^{\circ }\text{C}$ .

In figure 9 the ratio between $V$ and $V_{max}$ is presented as a function of both superheat and $Ja$ for the data obtained for the wafers with $l=0.60$  mm and $l=1.0$  mm. The vertical bars indicate the uncertainty of $V$ . One sees that there is only little variation with $T_{b}-T_{on}$ .

Figure 9.

The ratio between estimated bubble volume $V$ and the maximal bubble volume $V_{max}$ of spherical bubbles in an equilibrium state balancing surface tension and buoyancy forces only, plotted as a function of superheat and  $Ja$ . The vertical bars indicate the uncertainty of  $V$ . Cavity separation shown in label.

Og̃uz & Prosperetti (Reference Og̃uz and Prosperetti1993) developed a model that predicted $V/V_{max}$ of detaching gas bubbles from a needle, expressed in terms of the gas flow rate through the needle and into the bubble. This process has some similarity to bubble departure in boiling, but also differs in many ways from our case. When the growth was dominated by the flow rate rather than by inertia, and for flow rates below a certain critical value, they found that the bubbles detached from the needle with a volume $V$ smaller than $V_{max}$ given by (4.5), with $V/V_{max}$ varying from 0.8, when the flux was two orders of magnitude smaller than the critical value, to $V/V_{max}\simeq 1$ at the flux critical value. They reported that the criterion for departure for flow rates below the critical value is met before $V_{max}$ is achieved and the bubble departure is limited by surface tension, which is overcome when $V\approx V_{max}$ in the case of a spherical bubble. However for small flow rates and slow bubble growth, sphericity is hardly met since the bubble surface is subjected to the effects of gravity and surface tension, thus deviations from this criterion ( $V\approx V_{max}$ ) need to be considered. The much smaller value $V/V_{max}\simeq 0.3$ displayed in figure 9 indicates that there are additional mechanisms at play in our system. Presumably the LSC exerts a drag on the attached bubbles, promoting an early detachment.

4.2.1 Bubble growth and detachment

Time series of bubble widths $d_{x}$ for growing bubbles on single nucleation sites of the $l=0.60$  mm wafer (see § 3.6.2) at two different superheat values are shown in figure 10(a,b). They were obtained in apparatus 1 by viewing from above (see § 3.2). In figure 10(a), with $T_{b}-T_{on}=4.69$  K, five departure events, each followed by a short delay between departure and the beginning of the next bubble growth, and four full cycles (growth from $d_{x}=0$ to $d_{x,max}$ ) are shown. Based on the imaging results (as in figure 10) we estimate the delay between departure and the onset of new growth to be about 0.5 s for $T_{b}-T_{on}=4.69$  K and about 0.05 s for $T_{b}-T_{on}=8.59$  K. All bubbles which grew and departed within the 14 s of the figure showed similar behaviour, reaching about the same $d_{x,max}$ right before departing from the surface. The vapour-bubble growth rate shown in figure 10(b) for the larger superheat $T_{b}-T_{on}=8.59$  K is much larger than that for figure 10(a) with $T_{b}-T_{on}=4.69$  K. Within the 3.5 s covered by the figure, six bubbles departed.

Figure 10.

Time series of the width $d_{x}$ and approximate volume ${\rm\pi}d_{x}^{3}/6$ of growing bubbles obtained from high-speed movies of a single nucleating site ( $l=0.60$  mm) viewed from the top using apparatus 1: (a,c)  $T_{b}-T_{on}=4.69$  K; (b,d)  $T_{b}-T_{on}=8.59$  K. (a,b) The width $d_{x}$ . (c,d) The approximation ${\rm\pi}d_{x}^{3}/6$ to the bubble volume. The maximal (departure) width $d_{x,max}$ and the time difference between consecutive bubble departures ${\it\delta}t$ are indicated in (a).

Visualization from the top cannot yield the precise bubble volume since the main deviation from a spherical shape occurs underneath the bubble (see § 4.1.5). Thus, in figures 10(c) and 10(d), we display the approximation ${\rm\pi}d_{x}^{3}/6$ to the bubble volume based on the assumption of a spherical shape that corresponds to the time series of $d_{x}$ shown in figures 10(a) and 10(b), respectively. This approximation serves well to illustrate the time evolution of the bubble volume. For both superheat values ${\rm\pi}d_{x}^{3}/6$ increased about linearly with time.

The linear growth of ${\rm\pi}d_{x}^{3}/6$ contrasted with existing models for the growth of vapour bubbles in a quiescent fluid (Forster & Zuber Reference Forster and Zuber1954; Plesset & Zwick Reference Plesset and Zwick1954), which predict that, for a spherical bubble, the volume should grow $\propto t^{3/2}$ . The models are valid in the asymptotic stage of large radius, when growth is controlled by thermal energy inflow and liquid inertia is not a limiting factor. The model assumes growth occurs in stagnant and uniformly superheated liquid, and is only valid for sufficiently large liquid superheats (Prosperetti & Plesset Reference Prosperetti and Plesset1978). It has successfully reproduced bubble growth in water and $\text{CCl}_{4}$ and at moderate and high pressures in ethanol and water (Sluyter et al. Reference Sluyter, Slooten, Copraji and Chesters1991). In our experiments the growth conditions differ from those assumed in the models; here the temperature field around a growing bubble had a steep gradient across the thermal boundary layer at the bottom plate. An estimate of the boundary layer thickness in one-phase flow is ${\it\lambda}_{t}\simeq 90~{\rm\mu}\text{m}$ for $Ra=2\times 10^{10}$ and $L$ the same as here (Narezo Guzman et al. Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015). Thus, during the very early stage of the bubble growth $d_{x}<{\it\lambda}_{t}$ and the bubble is growing entirely within this steep temperature gradient, while at later times the bubble presumably becomes much larger than ${\it\lambda}_{t}$ and is exposed primarily to the turbulent fluid at the bulk temperature which is approximately $T_{b}-0.7{\rm\Delta}T$ and less that $T_{on}$ . To summarize, major differences between the models and our experiment included non-spherical bubbles (see § 4.1.5), bubble interactions with neighbours, growth in a varying temperature field and turbulent rather than quiescent flow. Thus, it is no surprise that the bubble growth rate is very different from that expected under the uniformly superheated conditions in a quiescent flow assumed in the models.

We calculated the probability density function (PDF) of the bubble widths $d_{x,max}$ at departure, as well as the PDF of ${\it\delta}t$ , for the departure events obtained for all active sites at a given $T_{b}-T_{on}$ and cavity separation $l$ . Since $d_{x,max}$ was smaller than the cavity separation $l$ for all cases, its PDF was calculated over the range 0 to $l$ . The bin size for the $d_{x,max}$ PDF equalled the pixel size of the images (0.016 mm). The bin size for the PDF of ${\it\delta}t$ was different for each PDF since the range between maximal and minimal values of ${\it\delta}t$ varied for each $l$ and superheat value.

In figure 11(a) the PDFs of $d_{x,max}$ for $l=0.60$  mm at different superheat values are shown. Each of them could be fitted well by a Gaussian function. In figure 11(b) the PDFs of ${\it\delta}t$ at different superheat values, also fitted by Gaussian functions, are shown. At small ${\it\delta}t$ there are significant deviations from these fits. Similarly, Gaussian distributions were fitted to the PDFs of both $d_{x,max}$ and ${\it\delta}t$ obtained for wafers with $l=1.0$ and 2.0 mm.

Figure 11.

(a) PDF of bubble widths at departure $d_{x,max}$ . (b) PDF of the time difference between consecutive bubble departures ${\it\delta}t$ for various bottom-plate superheat values $T_{b}-T_{on}$ (see labels) and $l=0.60$  mm. Each PDF was fitted by a Gaussian distribution.

At the lowest superheat value $T_{b}-T_{on}=3.74$  K, the PDFs of $d_{x,max}$ and ${\it\delta}t$ for the $l=0.60$  mm wafer showed two peaks, indicating that some active nucleation sites behaved differently from others: some sites nucleated smaller bubbles at a higher rate, while others nucleated bubbles that reached approximately the same $d_{x,max}\simeq 0.51$  mm observed for larger superheat values. The widths of the small departing bubbles and the uncertainty associated with them were of the same order. For such small bubbles the departure frequency $f$ was so large that the corresponding peak in the PDF of ${\it\delta}t$ was an order of magnitude larger than the peak located at ${\it\delta}t\simeq 6$  s (see figure 13 a below) associated with the larger bubbles. In order to include the results from cavities nucleating larger bubbles at $T_{b}-T_{on}=3.74$  K, only departure diameters larger than 0.1 mm were considered for figure 11(a) and values for ${\it\delta}t>3$  s were used for figure 11(b). A similar behaviour was found in the measurements for the $l=1.0$  mm wafer at $T_{b}-T_{on}=4.28$  K; in this case we neglected $d_{x,max}$ values smaller than 0.25 mm (which represented less than 15 % of the 3408 departure events considered) and fitted a Gaussian distribution to the PDF of $d_{x,max}>0.25$  mm.

Means and standard deviations were computed from the Gaussian distributions such as those in figure 11. In figure 12 the mean departure diameter found for all cavity separations is plotted as a function of both superheat and $Ja$ . The vertical bars indicate the corresponding standard deviation of the distributions. One can see that for $l=0.60$  mm the bubble departure diameter remained nearly constant at $d_{x,max}\simeq 0.5$  mm in the entire superheat range. The results for $l=1.0$  mm show a weak dependance of $d_{x,max}$ on superheat; $d_{x,max}$ decreased with decreasing superheat for $T_{b}-T_{on}<8$  K. The results for $l=2.0$  mm are comparable with those obtained for $l=1.0$  mm at $T_{b}-T_{on}\simeq 10$ and 8 K; but at the largest superheat the bubbles detached from the surface with the largest $d_{x,max}$ .

Figure 12.

The mean values of the Gaussian fits to the maximal bubble width at departure $d_{x,max}$ as a function of the bottom plate superheat $T_{b}-T_{on}$ and the Jakob number $Ja$ . The vertical bars indicate the standard deviations of the distributions. Cavity separation indicated in the label.

The time-averaged values $\langle {\it\delta}t\rangle$ of ${\it\delta}t$ are plotted on a double logarithmic scale as a function of the bottom-plate superheat and $Ja$ in figure 13(a) for all cavity separations $l$ . Within our resolution they were independent of $l$ . All data points shown (except that for $l=1.0$  mm at lowest superheat) were fitted by the exponential function $\langle {\it\delta}t\rangle ={\it\tau}_{0}\exp (-(T_{b}-T_{on})/C)$ . The fit gave ${\it\tau}_{0}=31.0$  s and $C=2.12$  K (dashed line in the figure).

Figure 13.

(a) The mean values of the Gaussian fits to the time difference between consecutive bubble departures ${\it\delta}t$ as a function of the bottom-plate superheat $T_{b}-T_{on}$ and the Jakob number $Ja$ for different cavity separations $l$ . A fit of the exponential function $\langle {\it\delta}t\rangle ={\it\tau}_{0}\exp (-(T_{b}-T_{on})/C)$ to the data gave ${\it\tau}_{0}=31.0$ and $C=2.12$  K (dashed line). (b) The standard deviations divided by the corresponding mean value ${\it\sigma}_{{\it\delta}t}/\langle {\it\delta}t\rangle$ as a function of $T_{b}-T_{on}$ and $Ja$ . Cavity separation specified in label.

The bubble growth rate and departure frequency greatly increased with increasing superheat and $Ja$ , reflecting the larger rate at which heat was supplied to an active nucleation site and to the surrounding superheated liquid. A higher heat flux at the bottom plate did not strongly affect the volume at departure, suggesting that the heat flux into the bubbles remained below a certain critical value (Og̃uz & Prosperetti Reference Og̃uz and Prosperetti1993) in the studied superheat range.

The ratio of the standard deviation ${\it\sigma}_{{\it\delta}t}$ to the corresponding $\langle {\it\delta}t\rangle$ is plotted in figure 13(b) as a function of both superheat and $Ja$ . As $\langle {\it\delta}t\rangle$ increased for decreasing superheat values, it was also subjected to increasingly larger fluctuations around the mean. The result was that ${\it\sigma}_{{\it\delta}t}/\langle {\it\delta}t\rangle$ reached values up to 60 % at the lowest superheats. The fluctuations around the mean were stronger at the larger separations between active sites; that is, nucleating bubbles with closer neighbouring active sites departed the surface in a more regular manner in time than bubbles departing from wafers with larger cavity separation.

4.2.2 Bubble latent heat content

The heat per unit time $Q_{2ph}$ for two-phase flow was enhanced with respect to that for one-phase flow $Q_{1ph}$ in the entire superheat (and thus $Ja$ ) range. The enhancement $Q_{2ph}-Q_{1ph}$ had three main contributions due to different mechanisms; here we quantify the contribution of the latent heat content of bubbles growing on the heated surface. Figure 14 shows, on a log-linear scale, $Q_{2ph}-Q_{1ph}$ divided by the corresponding number of active sites $N_{a}$ , i.e. the heat per unit time enhancement due to a single active site, as a function of superheat and $Ja$ (solid symbols) measured for both $l=0.60$ and $1.0$  mm wafers. Over the entire superheat range depicted, $(Q_{2ph}-Q_{1ph})/N_{a}$ was larger for the wafer with a larger cavity separation, and for a given cavity spacing $l$ it remained approximately constant for $T_{b}-T_{on}\gtrsim 5$  K and then declined slightly for decreasing superheat values smaller than about 5 K.

Figure 14.

Left vertical axis: the enhancement $Q_{2ph}-Q_{1ph}$ of the heat per unit time due to the bubbles, divided by the number of active sites $N_{a}$ (solid symbols) on a logarithmic scale as a function of the bottom-plate superheat $T_{b}-T_{on}$ and the Jakob number $Ja$ . Right vertical axis: the ratio of the latent heat per unit time $Q_{bub}N_{a}$ to the enhancement $Q_{2ph}-Q_{1ph}$ (open symbols) as a function of $T_{b}-T_{on}$ and $Ja$ . The vertical error bars represent the uncertainty range. Labels indicate the cavity separation for each vertical axis.

Assuming that the entire heat absorbed by a bubble with volume $V$ was used to generate vapour at the saturation density and pressure (partial pressure of the vapour was 86 kPa), we estimated the latent heat content $H$ of a single bubble. Knowing the superheat dependence of the mean bubble departure frequency $1/\langle {\it\delta}t\rangle$ allowed us to compute the heat per unit time associated to the departing vapour bubbles of a single nucleation site:

We note that there was a considerable amount of dissolved air in the liquid (see § 3.4) and that the bubble volume at departure was probably a mixture of vapour and air; however, the concentration of each inside the bubble is unknown. Therefore, $Q_{bub}$ was an upper bound to the actual latent heat absorbed by the growing bubbles from a single site. The contribution of bubble latent heat to the enhancement $Q_{bub}N_{a}/(Q_{2ph}-Q_{1ph})$ is plotted as a function of superheat and $Ja$ (open symbols) in figure 14 (right vertical axis) for both $l=0.60$ and $1.0$  mm. The vertical bars show the uncertainty range by taking into account the uncertainty of $V$ and ${\it\sigma}_{{\it\delta}t}$ . For both $l$ values, the ratio $Q_{bub}N_{a}/(Q_{2ph}-Q_{1ph})$ increased with superheat, mostly due to the exponential decrease of $\langle {\it\delta}t\rangle$ . For $l=0.60$  mm, at $T_{b}-T_{on}\simeq 4$  K the contribution $Q_{bub}N_{a}/(Q_{2ph}-Q_{1ph})\simeq 0$ and it reached a maximum value of 25 % at the largest superheat. The two points obtained for $l=1.0$  mm were smaller than $Q_{bub}N_{a}/(Q_{2ph}-Q_{1ph})$ obtained for $l=0.60$  mm. Considering that $Q_{2ph}-Q_{1ph}$ showed little dependence on $l$ (Narezo Guzman et al. Reference Narezo Guzman, Xie, Chen, Fernandez Rivas, Sun, Lohse and Ahlers2015), that $d_{x,max}$ and $V$ were slightly larger for the $l=1.0$  mm case (for $T_{b}-T_{on}\gtrsim 8$  K) and that $\langle {\it\delta}t\rangle$ did not vary from one cavity separation to another, the smaller $Q_{bub}N_{a}/(Q_{2ph}-Q_{1ph})$ calculated for $l=1.0$  mm was a consequence of the smaller number of active sites  $N_{a}$ .

The increasing relative contribution of bubble latent heat implied a smaller relative contribution from other mechanisms such as micro-convection, transient conduction, enhanced buoyancy and the advection of heat by the bubbles. However, the absolute contribution of the effective buoyancy to the total heat flux enhancement increased with superheat, similar as reported by Lakkaraju et al. (Reference Lakkaraju, Stevens, Oresta, Verzicco, Lohse and Prosperetti2013), as the number of bubbles per unit time rising across the flow increased and their volume either remained constant or increased with superheat.