2.1. Experimental set-up and conditions

The experimental set-up, shown in figure 2, consists of a polycarbonate tank with inner dimensions $47 \times 47 \times 61 \,\text{cm}^3$ , similar to the one used by Néel & Deike (Reference Néel and Deike2021), which is filled to a height of $35\,\text{cm}$ with a solution of artificial seawater (made of deionised (DI) water and artificial sea salt, ASTM D1141-98, Lake Products Company LLC). The water height is chosen so that the waves/fluctations on the water surface remain small and the surface bubbles can be clearly imaged. Bubbles are continuously generated as compressed air flows through the needles positioned at the bottom of the tank, arranged in a square with eight needles extending horizontally from each side. The air flow rate through the needles is controlled at 100 cm $^3$ min⁻¹ per needle, so that the resulting plume of rising bubbles is fairly monodisperse.

Figure 2.

Sketch of the experimental set-up, with measurements of bubbles, drops and dry aerosol particles. Bubbles of nearly identical sizes are generated by compressed air flowing through needles at the bottom of the tank. When two underwater pumps are turned on (as illustrated in the sketch), the injected bubbles are broken up into many smaller bubbles as they rise into the region of turbulence generated by the water jets from the pumps (bubbles not drawn to scale). The bubbles are imaged from the side view to measure the rising bulk bubbles and from the top view to capture the bubbles at the surface. Large fields of view (black) and small fields of view (orange) are drawn approximately to scale and permit the measurement of bubble radii from $30\,\unicode{x03BC} \mathrm{m}$ to $5\,\mathrm{mm}$ . Liquid drops ejected into the air by bursting bubbles are measured through an inline holographic set-up (measuring liquid drop radii from $10\,\unicode{x03BC} \mathrm{m}$ to $500\,\unicode{x03BC} \mathrm{m}$ ), while dry aerosol particles are measured by extracting and drying the liquid drops and analysing them using an optical particle sizer (OPS) and scanning mobility particle sizer (SMPS, composed of a differential mobility analyzer (DMA) and a condensation particle counter (CPC)) (for dry particle radii from $20\,\mathrm{nm}$ to $5\,\unicode{x03BC} \mathrm{m}$ ). A line of filtered, compressed air is connected to the lid to flush the air above the water surface before the start of each run. Relative humidity and both air and water temperature are monitored throughout the experiments.

The artificial seawater solution is prepared by mixing DI water with ASTM D1141-98 artificial sea salt, which contains the same proportions of elements found in natural seawater in amounts greater than $0.0004\,\%$ by weight (Lake Products Company LLC). Solutions discussed in this paper were prepared with a salinity of 36 g L⁻¹ (35 g kg⁻¹) to mimic the composition of ocean water. The solution has liquid viscosity $\unicode{x03BC}= 1.001 \,\text{mPa s}$ , liquid density $\rho= 1025 \,\text{kg}\,\text{m}^{-3}$ and static surface tension $\gamma = 52 \,\text{mN m}^{-1}$ . The low static surface tension indicates that the salt mixture likely includes some surfactant. At the beginning and end of each day of experiments, the solution was characterised by measuring the surface tension isotherm with a Langmuir trough (KSV NIMA, model KN 1003). This measurement was used to confirm that the surface properties of the solution remained consistent over multiple days of experiments.

Two pumps (Rule 20 DA 800 GPH Bilge Pump) are mounted $6\,\text{cm}$ above the needle tips, pointed towards each other from opposite corners. When the pumps are turned on, they produce water jets that meet in the center of the tank, creating a region of turbulent flow that causes the rising bubbles to fragment into many smaller bubbles, with turbulence similar to the set-up described in Ruth et al. (Reference Ruth, Aiyer, Rivière, Perrard and Deike2022). Experiments can be performed with or without forced turbulence, where the turbulence has two effects: agitate the flow and break the bubbles, leading to a broad-banded size distribution. Note that bubble imaging occurs above the region of active breakup.

Three experimental parameters were varied to create different initial bubble size distributions for the different cases run in this experiment: turning the underwater pumps off or on to create either a narrow- or broad-banded bulk bubble distribution, changing the needle size/bubble production method, and reducing the number of needles (which effectively reduces the injected air volume per unit time because the flow rate per needle is kept constant). An aquarium bubbler is also used to generate a plume of submillimetric bubbles centered around $250\,\unicode{x03BC} \mathrm{m}$ .

Figure 1 illustrates the wide range of surface bubble populations explored in this paper, showing representative images of the various conditions detailed in table 1. The surface bubble distributions display marked differences in terms of both sizes and clustering, which will control the emitted drop size distribution. There are also significant differences in the number of bubbles present at the surface in each condition, which are associated with very different numbers of produced drops (see Appendix A for the total number of drops compared with the number of bubbles for each case). Because of the variety in size, number, and clustering of surface bubbles for various conditions, it is necessary to characterise the surface properly through many measurements, across all scales, before attempting to link the bursting surface bubbles to the drops they produce.

Table 1.

Experimental parameters. The bulk bubble size distribution across cases was varied by changing between quiescent and turbulent flow underwater (by turning the underwater pumps off or on, causing the bubbles to break up in the turbulent case), by changing the injection bubble size (using two different needle sizes or an aquarium bubbler) and by halving the volume (using 16 instead of 32 needles with the same air flow rate per needle). Case names describe the range of radii over which a significant number of bubbles are concentrated (narrow, broad) and the initial injection bubble size before any breakup in turbulence ( $2\,\mathrm{mm}$ , $3\,\mathrm{mm}$ , $250\,\unicode{x03BC} \mathrm{m}$ ). The broad- $2\,\mathrm{mm}$ -half volume case is characterised by bubbles arriving at the surface isolated from each other, bursting individually instead of clustered as the smaller number of needles leads to both a reduced volume and increased needle spacing. Here $R_{b, inj}$ represents the typical injection bubble size by the needles or bubbler; $n_{R_b\lt 2\,\textrm{mm}}$ shows the number of bubbles (per unit volume) smaller than radius $2\,\mathrm{mm}$ , which reinforces the designation of certain cases as narrow- or broad-banded. We provide the total air flow rate through the needles for each case, but we caution that this flow rate value should not be used to approximate the bubble population in our configuration, as it is primarily related to the volume contained in the largest bubbles and does not capture the variations in the bubble size distributions (broad and narrow configurations) necessary when linking bursting bubbles to emitted drops. Bubble size distributions for each case are shown throughout §§ 2.2.4, 4.1 and 4.2. Representative surface bubble images for each case are shown in figure 1.

In order to capture the large range of length scales present in the system – spanning bubble radii of $30{-}5000\,\unicode{x03BC} \mathrm{m}$ and drop sizes of $0.05{-}500\,\unicode{x03BC} \mathrm{m}$ – we make multiple measurements of bulk bubbles, surface bubbles, drops and dry particles. Bulk bubbles are imaged in both a large and small field of view by cameras positioned above the region of active breakup. Surface bubbles are also captured in two fields of view by cameras positioned perpendicular to the water surface, where the bubbles cluster near the center of the tank before bursting and ejecting drops into the air above the water surface, which is enclosed in the tank with an acrylic lid.

Figure 3.

Experimental protocol for data acquisition illustrated by typical time series of temperature, relative humidity, bubbles, drops and dry particles through a single run on a representative case (broad- $2\,\text{mm}$ in table 1). Sample images are shown with bubble detection overlaid in red (c–e). Scale bars (shown as white lines on images) represent $1\,\text{cm}$ for the bulk and surface large fields of view (c, right image ine) and $1\,\text{mm}$ for the small fields of view (d, left image in e). Panel (a) illustrates the steps in the experimental protocol, showing first flushing to obtain clean air, followed by equilibration during which the relative humidity reaches a value of about $90\,\%$ , followed by a steady state. Air and water temperature are also shown in (a). The dry aerosol particle counts are shown in (b), with the SMPS particle count close to 0 (less than 10 particles per cm $^3$ ) at the end of the flushing stage, and reaching a steady state during the acquisition stage. Drop measurements from the holography are shown in (b) by a concentration $\rho _d$ (number per measurement volume), with the rolling mean and standard deviation plotted for each holographic measurement. For the bubble measurements in panels (c–e), the lines represent the rolling mean of the count of bubbles detected in each frame, with shading for the rolling standard deviation. All rolling quantities were computed using a window size equal to $20\,\%$ of the total number of frames in each dataset. Each measurement shows some fluctuations around a rolling mean that remains relatively steady throughout the measurement period.

Drops are sized using a combination of an inline holographic system positioned just above the water surface (Erinin et al. Reference Erinin, Wang, Liu, Towle, Liu and Duncan2019, Reference Erinin, Néel, Mazzatenta, Duncan and Deike2023), an optical particle sizer (OPS from TSI, Inc., OPS 3330), and a scanning mobility particle sizer (SMPS from TSI, Inc., which consists of an advanced aerosol neutralizer, model 3088; an electrostatic classifier, model 3082; a differential mobility analyzer, model 3081; and a condensation particle counter (CPC), model 3752). The SMPS operates with an aerosol flow rate of 0.3 L min⁻¹ and sheath flow rate of 2.4 L min⁻¹. The OPS and SMPS are connected through the lid and pull air with suspended drops out of the tank, where they are dried (TSI diffusion dryer, model 3062) and then measured. Relative humidity and air temperature are monitored $25\,\text{cm}$ above the calm water surface throughout all experimental runs, along with the temperature of the solution.

Table 2.

Measurement details for imaging of bulk bubbles, surface bubbles, and drops (holography and shadow) in the present experiment. The shadow imaging drop measurement was performed for certain cases (narrow- $2\,\mathrm{mm}$ , narrow- $3\,\mathrm{mm}$ , narrow- $250\,\unicode{x03BC} \mathrm{m}$ and broad- $2\,\mathrm{mm}$ ) to allow for the measurement of drops very close to the surface. A range is specified for the depth of field of the bulk measurement resulting from the size-dependent depth-of-field correction, and this range corresponds to the depth of field of bubble radii in the range $R_b= [150, 3000]\,\unicode{x03BC} \mathrm{m}$ (see details in § 2.2.3 and figure 6 of Erinin et al. (Reference Erinin, Néel, Mazzatenta, Duncan and Deike2023)).

2.2. Experimental protocol and statistically steady-state measurements of bubbles, drops and particles

We present the systematic experimental protocol used to characterize drop production by bubble bursting in the statistically steady-state bubbling tank. Figure 3 illustrates the time series of the various measurements for a typical case (broad- $2\,\text{mm}$ : broad-banded distribution of bubbles created by underwater turbulence that breaks the bubbles into many smaller sizes).

Once the artificial seawater solution is in the tank, a lid is placed on top to seal the system. A constant stream of filtered, compressed air is then flowed through the air above the water surface to flush aerosols from the system. The background aerosols escape through a single port in the lid, which is left open throughout the run to maintain equilibrium within the system. The flushing continues until the CPC component of the SMPS measures a particle count below 10 particles cm⁻ $^3$ , as shown in figure 3(b), which can be compared with an ambient particle count of approximately 1000 particles cm⁻ $^3$ . This ensures that the sea salt aerosols we sample in the air are nearly all emitted by bubble bursting. The air through the top of the tank is then turned off, and the system is left to equilibrate for 40 mins. The CPC particle counts from the pre-run period are shown in figure 3(b). We confirm that the particle count and relative humidity level (figure 3 a) have reached a steady state before proceeding with the run. Relative humidity, air temperature and water temperature are tracked throughout the run using a Thorlabs sensor (TSP01, TSP-TH) mounted to the tank lid. Values are comparable to realistic ocean conditions and remain steady throughout the run (temperatures vary by approximately 1 °C throughout the measurement period).

After the system reaches a steady state of particle counts, the SMPS, OPS, and holographic measurements begin simultaneously (shown in terms of the drop concentration in figure 3 b). The SMPS and OPS sample continuously throughout the entire 50 min run. Three independent holographic movies are recorded during the run, with approximately 15 mins of saving time required between each measurement. For the bubbles (figure 3 c,d,e), the large field-of-view surface measurement is made at the start of the run, followed by simultaneous measurements of bulk bubbles for both fields of view. Surface and bulk bubbles are recorded separately so that the backlighting for each measurement does not interfere with the other. Finally, the small field-of-view surface measurement is made at the end of the run. Compared with the large surface measurement, the small surface images provide a significantly more zoomed-in view that requires more light than the large surface view, so the two measurements are made separately. Details of each measurement are given below and summarised in table 2. Representative sample images for each bubble measurement are also shown in figure 3(c–e) with detection overlaid. Some fluctuations are present in the number of bubbles and drops with time, but the magnitude of the fluctuations is fairly consistent throughout the run. Therefore, we consider the system to be running in a statistically steady state.

2.2.4. Typical bubble and drop size distributions

The systematic measurements described above for a wide range of scales allow us to extract full size distributions for both drops and bubbles and probe different conditions in a systematic way (conditions summarised in table 1, with sample surface images in figure 1). We illustrate the characterisation of the size distribution for two representative cases in figure 4, the broad- $2\,\text{mm}$ and narrow- $2\,\text{mm}$ cases.

Figure 4.

Size distributions of bulk bubbles, $N_b(R_b)$ (a,d), surface bubbles, $N_s(R_b)$ (b,e) and drops, $N_d(r_d)$ (c,f) for broad-banded (a–c) and narrow-banded (d–f) distributions of rising bubbles (see cases broad- $2\,\text{mm}$ and narrow- $2\,\text{mm}$ in table 1). Different symbols indicate the different fields of view (bubble measurements) or the different measurement techniques (drop/particle measurements). The drop size distribution is shown in terms of the liquid drop radius, $r_d$ . All distributions are interpolated over the logarithmically-spaced bins, with the interpolated distribution plotted over the data points.

Using bubble sizes and counts extracted from the images through the methods described in the previous sections, we plot the size distributions of the detected bulk and surface bubbles, $N_b(R_b)$ and $N_s(R_b)$ , shown in figure 4, normalised by the bin size and by the measurement volume or area, respectively. For both bubble size distributions, data from the large and small field-of-view measurements are represented by different symbols and nicely overlap. Note that the bubble radius $R_b$ represents the volumetric bubble radius for both the bulk and surface measurements.

For the drop size distribution, all measurements are expressed as $N_d(r_d)$ , the number of drops per unit bin size and unit measurement volume. Data from the SMPS, OPS, holographic imaging, and shadow imaging overlap well, and the combined measurements result in a single distribution that characterizes drops spanning $50\,\text{nm}$ to $500\,\unicode{x03BC} \mathrm{m}$ . For the broad-banded case, we observe a continuum of bubble radii spanning $R_b= [30, 5000]\,\unicode{x03BC} \mathrm{m}$ in both the bulk and the surface (figure 4 a,b). We see a corresponding continuum of drop radii spanning $r_d= [0.05, 500]\,\unicode{x03BC} \mathrm{m} $ (figure 4 c). In the narrow-banded case, we see a narrow peak of bubble sizes around the injection bubble radius, $R_b= 2.3 \,\text{mm}$ (figure 4 d,e). We refer to this case as narrow-banded, or nearly monodisperse, because the majority of bubbles are found in this peak around 2 mm. Some amount of smaller bubbles are also present for this narrow-banded case (formed as bubbles pinch off from the needles), and there are also smaller drops, shown in the drop size distribution below $r_d \sim 200 \,\unicode{x03BC} \mathrm{m}$ (figure 4 f). The bulk and surface distributions show similar trends in both conditions, suggesting limited coalescence (which is expected for artificial seawater).

2.3. Summary of the experimental analysis

Having obtained the full size distributions of bubbles and drops, we now analyse the link between the different measurements to systematically quantify the attribution of drops to bursting surface bubbles. In § 3 we link the measured bulk bubble size distribution (which integrates to a number per unit volume) with the surface bubble size distribution (which integrates to a number per unit area) using a simple flux argument involving the bubble rise velocity and the bubble lifetime at the surface. We then analyse the link between bursting bubbles and drops in § 4. We consider a direct attribution of measured drops to measured bubbles in § 4.1, and in § 4.2 we predict a drop size distribution from the measured bubbles using existing scaling laws developed for individual bubble bursting.

4.1. Direct attribution between measured bubbles and drops

We first show the drop size distributions resulting from various bubble size distributions and discuss how different sizes of drops may be attributed to different sizes of bursting bubbles. Figure 6 shows the size distributions of bursting bubbles, $N_b(R_b)$ (a,d,g), and drops, $N_d(r_d)$ (b,e,h), as well as the ratio of the number of drops produced per bubble $\eta _d$ (c,f,i), in chosen size ranges.

Figure 6.

Size distributions of bursting bubbles, $N_b(R_b)$ (a,d,g), and drops, $N_d(r_d)$ (b,e,h), in terms of the liquid drop radius, $r_d$ , for various case comparisons. The bubble size distributions presented vary significantly, including broad-banded cases and three narrow-banded cases peaking at radii of $2\,\mathrm{mm}$ , $3\,\mathrm{mm}$ and $250\,\unicode{x03BC} \mathrm{m}$ , with a corresponding variety of drop size distributions. To calculate the drop production efficiency presented in the rightmost column (c,f,i), the bubble and drop size distributions are first integrated in the shaded or outlined size buckets, where the link between the bubble and drop sizes approximates the link suggested by scaling laws for individual bubble bursting. The resulting number of drops in each size range is then divided by the number of bubbles in the corresponding shaded size range (outlined for the film flap range) to obtain an average number of drops produced per bubble in each bubble size bucket (production efficiency $\eta _d$ , shown as circles within the jet drop size range corresponding to the blue shaded regions and diamonds for the assumed film flap drop size range corresponding to the orange boxes). Shaded grey lines show the number of drops per bubble predicted by the individual bubble bursting scalings for jet (solid line) and film flapping (dotted line) production mechanisms (tested in § 4.2; see table 3 for details). Cases shown are (a–c) narrow- $2\,\mathrm{mm}$ versus broad- $2\,\mathrm{mm}$ , (d–f) narrow- $3\,\mathrm{mm}$ versus narrow- $250\,\unicode{x03BC} \mathrm{m}$ and (g–i) broad- $2\,\mathrm{mm}$ versus broad- $2\,\mathrm{mm}$ -half volume. Further details of each case are found in table 1, and the uncertainty associated with the production efficiency values shown in (c,f,i) is discussed in Appendix B.

Figure 6(a–c) shows the two sample cases described in § 2.2.4 (figures 4 and 5, narrow- and broad- $2\,\mathrm{mm}$ injection cases). Both cases use the same needles and flow rate, but in the broad case turbulent flow is created underwater. As described above, turbulence induces bubble breakup and creates a broad-banded bubble size distribution with a continuum of bubbles spanning radii of $30\,\unicode{x03BC} \mathrm{m}$ to almost $5\,\mathrm{mm}$ . Without turbulence, the distribution has a peak around $R_b= 2 \,\text{mm}$ with a lower number of smaller bubbles. The two cases (broad- and narrow-banded) have a relatively comparable number of bubbles around $2\,\mathrm{mm}$ , near the needle injection size. For the submillimetric bubbles, there is a large difference of about two orders of magnitude in the number of bubbles (because without turbulence, the only small bubbles present are formed by pinch-off at the needles, see § 2). Moving to the drop size distribution, we observe a corresponding difference of similar magnitude between the drop size distributions for nearly the entire range of sizes measured in the experiment, until the number of drops becomes more comparable between cases for $r_d \gt 400 \,\unicode{x03BC} \mathrm{m}$ . This suggests that the drops we measure below radius $400 \,\unicode{x03BC} \mathrm{m}$ are produced by bubbles with radii smaller than approximately $2 \,\text{mm}$ (shaded in colour).

We therefore focus on the submillimetric portion of the bubble size distribution (shaded in colour) and consider the link between the measured submillimetric bubbles and produced drops. We first test a direct attribution of drops to bubbles, dividing the number of measured drops by the number of measured bubbles in discrete size bins (shaded/outlined with the corresponding colours).

Size buckets are chosen to approximate the individual bubble size scaling laws for jet and film flapping production mechanisms (discussed in more detail in § 4.2 and Berny et al. Reference Berny, Popinet, Séon and Deike2021; Deike et al. Reference Deike, Reichl and Paulot2022; Jiang et al. Reference Jiang, Rotily, Villermaux and Wang2022). The corresponding bubble and drop ranges are shown in shaded/outlined colours in figure 6, where the overlap in bubble sizes represents the fact that bubbles can produce both jet and film flap drops in parallel processes. For the film flap scaling, we test the link between bubbles of radii $R_b= [70, 1200] \,\unicode{x03BC} \mathrm{m}$ and drops of radii $r_d\lt 0.5\,\unicode{x03BC} \mathrm{m}$ (outlined/shaded in orange). For the jet drop scaling, we consider the ability of bubbles in the range $R_b= [30, 2000] \,\unicode{x03BC} \mathrm{m}$ to produce drops of $r_d \gt 0.5 \,\unicode{x03BC} \mathrm{m}$ (shaded in blue). The total regions tested for jet drop production are approximately evenly divided into three smaller log-spaced size buckets for both bubbles and drops, and the buckets are linked in size order.

The distributions are then integrated within each assigned size bucket to calculate a total number of bubbles or drops (per measurement volume) within that size range, given by $\rho _b = \int _{R_{b1}}^{R_{b2}} N_b(R_b) \,dR_b$ for bubbles within the size range $[R_{b1},R_{b2}]$ and $\rho _d = \int _{r_{d1}}^{r_{d2}} N_d(r_d) \,dr_d$ for drops in the size range $[r_{d1},r_{d2}]$ . By dividing the number of drops in a size bucket by the number of bubbles in the corresponding size bucket, we calculate a drop production efficiency (the number of drops produced per bubble) for each pair of size buckets, defined as $\eta _d = \rho _d/\rho _b$ .

Resulting drop production efficiencies are shown in the right column of figure 6(c,f,i), calculated from the shaded ranges (jet drop, circles) or outlined ranges (film flap drop, diamonds) of bubble and drop size buckets for the two cases. Note that the values of the efficiencies would shift somewhat if we chose to split the drop size distribution at a value other than $r_d=0.5\,\unicode{x03BC} \mathrm{m}$ when testing the attribution to the assumed bubble size ranges. The cutoff radius at $r_d= 0.5 \,\unicode{x03BC} \mathrm{m}$ was chosen because it falls between the largest predicted film drop and smallest predicted jet drop for the range of bubble sizes present in the experiment.

For the broad-banded case in figure 6(c), we observe approximately 50 drops/bubble in the assumed film flap range (black diamond) and approximately $1.0{-}1.8$ drops/bubble across the jet drop size buckets (black circles). For the narrow-banded case, we observe a high efficiency of around 100 drops/bubble in the film flap size range (blue diamond), while the small jet drop efficiency is around 9 and the larger jet drop efficiency is below unity (blue circles).

Table 3.

Equations and parameters used to calculate the drop size distribution $N_d(r_d)$ from input bubble size distribution $N_b(R_b)$ , following (with adaptation) the scaling laws developed for individual bubble bursting. Equations used for the mean size $\langle r_{d} \rangle (R_b)$ and number $n_d(R_b)$ of emitted drops are shown, as well as the radius bounds [ $R_{b1}$ , $R_{b2}$ ] and the order of the gamma distribution (assumed to represent the distribution of drop sizes $p(r_d/\langle r_d \rangle , R_b)$ ) required to compute $N_d(r_d)$ as described in (4.1). Film flap: size scaling based on the data from Jiang et al. (Reference Jiang, Rotily, Villermaux and Wang2022), discussed in Deike et al. (Reference Deike, Reichl and Paulot2022). The drop number is taken as a single value of 40 drops/bubble, based on Jiang et al. (Reference Jiang, Rotily, Villermaux and Wang2022). Jet (G-C)/Jet (B-R): uses the formulation for the first drop size given by Gañán-Calvo (Reference Gañán-Calvo2017) and Blanco–Rodríguez & Gordillo (Reference Blanco–Rodríguez and Gordillo2020), respectively. The drop number scaling from Berny et al. (Reference Berny, Popinet, Séon and Deike2021) is used with a modified prefactor suggested by Wang et al. (Reference Wang2017). The Laplace number $La = R_b/l_{\unicode{x03BC} }$ controls jet drop ejection, comparing the radius of the bursting bubble, $R_b$ , to the visco-capillary length, $l_{\unicode{x03BC} } = \unicode{x03BC} ^2/\rho \gamma$ , using the values for physical parameters given in § 2.1. Here $La_* = 550$ is taken for the critical Laplace number for jet drop formation in the G-C formulation (Walls et al. Reference Walls, Henaux and Bird2015; Berny et al. Reference Berny, Deike, Séon and Popinet2020). Note that the B-R expression provided here is valid for $Bo = \rho gR_b^2/\gamma \leq 0.05$ and $La \gt 1111$ , which corresponds here to bubbles in the radius range $R_b= [20, 500]\,\unicode{x03BC} \mathrm{m}$ , so this formulation is applied only to our measured bubbles of radii smaller than $500\,\unicode{x03BC} \mathrm{m}$ . The G-C scaling is applied to bubbles of radii larger than $30\,\unicode{x03BC} \mathrm{m}$ and would diverge for smaller radii between $10{-}20\,\unicode{x03BC} \mathrm{m}$ . Film centrifuge: scalings developed by Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012), with size and number of drops determined by the bubble size and film thickness $h_b$ , where $h_b$ scales with ${R_b}^2$ / $l_c$ and spans $[0.2, 30]\,\unicode{x03BC} \mathrm{m}$ for bubbles with $R_b \gt 0.4l_c$ (capillary length $l_c = \sqrt {\gamma /\rho g} = 2.3\,\text{mm}$ for the artificial seawater solution). Sensitivity of the predicted drop size distributions to different choices of $R_{b1}$ , $R_{b2}$ and the order of the gamma distribution are discussed in Appendix B.

These numbers can be compared with the efficiencies discussed in the literature. The light grey lines in the same plot represent the drop number given by the individual bubble bursting scaling laws used throughout the paper for jet drops (solid, between $0.6{-}3$ drops produced per bubble (Wang et al. Reference Wang2017)) and film flap drops (dotted, 40 drops per bubble (Jiang et al. Reference Jiang, Rotily, Villermaux and Wang2022)). Detailed equations for the scalings shown here, based on past work in the literature, are provided in table 3 and described in § 4.2. Comparing the calculated efficiencies for the two sample cases to the approximate individual bursting scaling laws, we see reasonable agreement for the broad-banded case. For the nearly monodisperse case, significant differences are observed and various interpretations can be proposed. The higher efficiency of film flap drops could be related to the proposed bubble pinch-off effect/small aerosol formation in the bulk discussed in the recent work of Jiang et al. (Reference Jiang, Rotily, Villermaux and Wang2024), with this effect potentially smeared out or not occurring in the presence of turbulence (note that, for the narrow- $2 \,\text{mm}$ case, high-speed movies confirm collisions at the needle tip/during bubble pinch-off as discussed in Jiang et al. (Reference Jiang, Rotily, Villermaux and Wang2024); these collisions at the needle tips are not observed in the narrow- $3 \,\text{mm}$ case). The higher jet drop efficiency for the small bubbles, which closely match direct numerical simulations for similar conditions (Berny et al. Reference Berny, Popinet, Séon and Deike2021), could be due to the fact that either the low surface density of bubbles or a calmer free surface are necessary to obtain such efficiency. Larger jet drop low production efficiency might be related to a lack of statistical convergence in measuring these drops with the holographic system. Sources of uncertainty involved in the calculation of the production efficiency values presented in figure 6 are discussed in Appendix B.

To further probe the link between collective bursting bubbles and ejected drops, we perform the same direct attribution analysis for several initial bubble configurations and analyse the corresponding change in the drop size distribution and associated uncertainties. We look next at figure 6(d–f) to analyse how injecting bubbles larger or smaller than $2 \,\text{mm}$ , with a nearly monodisperse size distribution, affects the drop production. For the injection of larger bubbles (panel d), the peak of the size distribution is closer to a radius of approximately $3.5 \,\text{mm}$ (compared with the previous case of approximately $2.3 \,\text{mm}$ bubbles). The small bubble case features a wider peak of primarily submillimetric bubbles centered near $250 \,\unicode{x03BC} \mathrm{m}$ (see details in table 1 and surface images in figure 1).

From the size distributions, it is immediately evident that many more submicron droplets are produced (figure 6 e) when more small submillimetric bubbles are present (figure 6 d). We observe an approximately 3 order-of-magnitude difference between the size distributions of the small bubble case and the large bubble case, where very few submicron drops are produced. As shown by the large bubble case, when we measure very few submillimetric bubbles, we measure very few submicron drops. This demonstrates that bubbles predominantly larger than $2 \,\text{mm}$ do not produce a significant amount of submicron aerosols in this configuration.

Comparing to individual bursting scalings, in figure 6(f) we see a reasonable efficiency of approximately 60 drops/bubble for both cases for the proposed film flap range (green and orange diamonds). For the two largest jet drop bins in figure 6(f), the efficiencies for bubbles are low (green and orange circles; efficiencies below 0.1 associated with a very small number of drops are not shown on the plot), with similar values to those discussed above for the $2 \,\text{mm}$ narrow-banded case (figure 6 c, blue circles). For the smallest jet drop bin, the efficiencies are high, at around 20 drops/bubble. We note that the overall efficiencies are consistent across all the cases despite the large differences in the bubble size distributions (see the plot of the number of drops versus the number of bubbles in Appendix A), and it is only once we assume a link between the bubbles sizes and drop sizes for this attribution that differences in efficiencies appear for the narrow-banded cases compared with the broad-banded cases. The different efficiency values found in the narrow-banded cases imply that some subtleties of the bubble–drop link may be blurred across neighbouring size buckets for the continuum of bubble and drop sizes present in the broad-banded cases, which seem to match the single bubble scaling laws well.

The lack of large drops for two of the narrow-banded cases (peaking at radii of $2 \,\text{mm}$ and $3 \,\text{mm}$ ) can be explained by either missing drops due to the small field of view or by a physical collective effect where neighbours may somehow limit production efficiency in these nearly monodisperse conditions. To take a step in investigating a potential collective effect that may modulate the bubble to drop size link or affect overall production efficiencies, we look at the broad-banded cases and compare cases with 32 versus 16 bubbling needles (‘broad- $2 \,\text{mm}$ ’ and ‘broad- $2 \,\text{mm}$ -half volume’ – the cases have the same flow rate per needle, so we effectively halve the injected volume of bubbles). Aside from decreasing the injected volume, decreasing the number of needles allows the bubbles to arrive at the surface spaced apart from each other, so that most bubbles burst individually instead of clustered, as visualised in figure 1(f). Looking at the size distributions for these two cases, shown in figure 6(g–i), we observe that a smaller number of bursting bubbles (panel g) results in a drop size distribution that is very similar to that of the clustered case, especially for the submicron drops (panel h). Looking at the efficiencies (panel i), the 16 needle case has a higher efficiency than the 32 needle case across all bubble sizes, indicating that a collective effect could potentially be limiting the drop production efficiency.

Through the attribution of measured drops to measured bubbles analysed in this section, we have observed that the broad-banded cases show an overall agreement between our calculated bursting efficiencies and those found from the single bubble scaling laws for film flap and jet drop production, with a possible reduction of efficiency for the collective bursting (figure 6 g–i). The narrow-banded cases demonstrate that submicron drops are produced by bubbles with radii smaller than approximately $1 \,\text{mm}$ (figure 6 d–f), and that nearly monodisperse clusters may display a reduced efficiency in jetting for bubbles with radii above approximately $100 \,\unicode{x03BC} \mathrm{m}$ and an increased efficiency for the smaller bubbles (figure 6 c,f).

4.2. Using individual bubble scaling laws to predict drops produced by measured bubbles

Now we investigate the distributions of drops predicted by the individual bubble bursting scaling laws – which provide the number and size of drops for a given bubble – combined with the measured bubble size distributions, and we then compare the predicted drop size distributions to our measured drop size distributions. We use the results to further evaluate how well the individual bursting relations describe collective bursting results across all cases. To predict the drops produced by the collection of bursting bubbles, we consider the approach from Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012), initially proposed for production of film drops via a centrifuge mechanism and extended to jet drops (Berny et al. Reference Berny, Popinet, Séon and Deike2021) and film flap drops (Deike et al. Reference Deike, Reichl and Paulot2022). For a specific drop production process, the total drop size distribution $N_d(r_d)$ produced by an ensemble of bubbles is obtained by integrating individual bursting scalings over the size distribution of bursting bubbles $N_b(R_b)$ in the size range [ $R_{b1}$ , $R_{b2}$ ]:

(4.1) \begin{equation} N_d(r_d) = \int _{R_{b1}}^{R_{b2}} \frac {N_b(R_b)n_{d}(R_b)}{\langle r_d \rangle (R_b)} p\left (\frac {r_d}{\langle r_d \rangle }, R_b\right ) \,{\rm d}R_b.\end{equation}

Here $N_b(R_b)$ is the measured bubble bursting size distribution, $\langle r_d \rangle (R_b)$ is the mean drop size produced by a bursting bubble of size $R_b$ , $n_{d}(R_b)$ is the number of drops produced by a bubble bursting of size $R_b$ , and $p(r_d/\langle r_d \rangle , R_b)$ is the probability density function of drop sizes produced by a bubble of size $R_b$ , assumed to be a gamma distribution of known order (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Berny et al. Reference Berny, Popinet, Séon and Deike2021; Deike et al. Reference Deike, Reichl and Paulot2022). Sensitivity of the resulting drop size distributions to the order of the gamma distribution and to the chosen radius bounds ( $R_{b1}$ and $R_{b2}$ ) are discussed in Appendix B.

We perform the integration, using the measured bubble size distribution, for each individual drop production mechanism, as summarised in table 3. Scalings for the number and size of drops produced by the bursting of single bubbles have been developed for three main modes of drop production: film centrifuge, film flap and jet drop production. The integration for each mechanism over many bubble sizes is similar to the analysis performed by Deike et al. (Reference Deike, Reichl and Paulot2022) (using assumed/modelled bubble size distributions) and Néel et al. (Reference Néel, Erinin and Deike2022) (measuring only larger scales of drops and bubbles), as well as for individual mechanisms in Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012) (film centrifuge drops, assumed/modelled bubble size distribution), Blanco–Rodríguez & Gordillo (Reference Blanco–Rodríguez and Gordillo2020) (jet drops), and Gañán-Calvo (Reference Gañán-Calvo2023) (extended jet drops using assumed/modelled bubble size distributions).

Figure 7 shows the result of the integration in (4.1) for each production mechanism, compared with the measured drop size distribution for the $2\,\text{ mm}$ bubble size cases (broad-banded and narrow-banded). The scalings and coefficients used for each drop production mechanism are given in table 3 and depicted graphically in figure 6.

Figure 7.

Measured bubble ( $N_b(R_b)$ ) and liquid drop ( $N_d(r_d)$ ) size distributions for the $2\,\mathrm{mm}$ injected bubble cases with (a) broad-banded and (b) narrow-banded distributions. Overlaid coloured lines represent the predicted drop size distribution from each production mechanism for individual bubble bursting (see details in table 3), integrated over many bubble sizes using the measured bubble size distribution. Bubble size ranges used in the integration for each mechanism are labelled in the same colour on the measured bubble size distributions. Note that the jet formulation from Blanco–Rodriguez & Gordillo (Reference Blanco–Rodríguez and Gordillo2020) (B-R) is only applied in the range $R_b= [30,500]\,\unicode{x03BC} \mathrm{m}$ .

From figure 7, we can discuss how well scaling laws for each drop production mechanism describe the experimental drop size distribution, beginning with the broad-banded case presented in the top row.

We start with jet drop production, where drops are created by fragmentation of the jet formed by the focusing of capillary waves into the collapsed bubble cavity. For bubbles of sizes between $30\,\unicode{x03BC} \mathrm{m}$ and the capillary length ( $l_c\sim 2.3\,\text{ mm}$ for the artificial seawater solution), which are able to produce jet drops (Walls et al. Reference Walls, Henaux and Bird2015; Berny et al. Reference Berny, Deike, Séon and Popinet2020), we apply the theoretical formulation developed by Gañán-Calvo (Reference Gañán-Calvo2017) for the resulting drop size. Because there is no theoretical prediction for the number of jet drops produced by the bursting bubble, we consider the scaling proposed by Berny et al. (Reference Berny, Popinet, Séon and Deike2021) for the number of drops (coherent with experimental data from Spiel Reference Spiel1997b ; Ghabache et al. Reference Ghabache, Antkowiak, Josserand and Séon2014; Ghabache & Séon Reference Ghabache and Séon2016) with a modified prefactor. The modified prefactor means that bubbles of radii $R_b = [30, 2300]\,\unicode{x03BC} \mathrm{m}$ are predicted to produce $n_d= [3, 0.6] $ jet drops per bubble, which is lower than the $n_d= [11, 3]$ drops from the Berny et al. (Reference Berny, Popinet, Séon and Deike2021) fit to various data for individual bubble bursting, but is in agreement with the collective bursting experiment from Wang et al. (Reference Wang2017) (see figure 1 from their paper).

For the broad-banded case, shown in figure 7(a) (bubbles on the left, drops on the right), the jet drop scaling laws (blue lines) match the measured drop size distribution (in black) very well for drops of radii approximately $1$ to $500 \,\unicode{x03BC} \mathrm{m}$ . Because the magnitude of the distributions lines up well using the $0.6{-}3$ jet drops produced per bubble across the radius range (where smaller bubbles produce more drops), the average number of jet drops produced by many collective bursting events is likely lower than the $3{-}11$ drops that a single bubble is physically able to produce (across the same radius range). Each bursting event is not necessarily as efficient as every other event. In the collective bursting set-up, some bubbles may be disturbed by the rising or bursting of their neighbours or by water fluctuations that could affect the cavity collapse and jet formation, which would reduce the overall drop production efficiency. The jet drop size formulation developed in Blanco–Rodríguez & Gordillo (Reference Blanco–Rodríguez and Gordillo2020), based on capillary wave selection, was also tested (blue dotted line). Since it yields very similar results over the range of bubble radii we measure where the scaling is valid (as the drop sizes as a function of bubble sizes were derived using the same experimental and numerical data in both formulations), we use the equation from Gañán-Calvo (Reference Gañán-Calvo2017) for the rest of the analysis.

Next we consider the mode of film drop production proposed in Jiang et al. (Reference Jiang, Rotily, Villermaux and Wang2022), where they report submicron drops produced by bursting bubbles in the approximate radius range $R_b= [70, 1000]\,\unicode{x03BC} \mathrm{m}$ , which they attribute to a film flapping mechanism. As shown in figure 7, the size scaling and order of magnitude associated with the film flap drops (orange) match well with the measured distribution (black) for the broad-banded case.

Finally, for film drop production through a centrifuge mechanism, Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012) describe the fragmentation of ligaments formed as the cap of a bubble with radius $R_b \gt 0.4l_c$ retracts. We observe that the predicted distribution (green) does not describe the data well, falling below the measured distribution for the broad-banded case. The film centrifuge scaling leads to significantly fewer drops than the jet drop counterpart (in the size range where both mechanisms are applicable) due to the fact that there are fewer larger bubbles (above the capillary length) where the centrifuge mechanism is effective. Note that the film centrifuge scaling does not predict submicron drops.

Looking now at the narrow-banded case presented in figure 6(b), we observe that the film flap results again match the peak and magnitude relatively well (orange line), only slightly below the experimental data (black line). The jet scalings (blue) approximate the correct drop magnitude at their peak, around $1\,\unicode{x03BC} \mathrm{m}$ , but overpredict some larger drops. While this overprediction could be related to limits in measuring larger drops in the experiment, we also note that if a truncated gamma distribution is used, such that $p(r_d/\langle r_d \rangle , R_b)= 0$ for bubbles of radii $R_b \gt 2 \,\text{mm}$ , the resulting predicted drop size distribution describes the large drops well (shown in Appendix B for the broad- $2\,\mathrm{mm}$ and narrow- $2\,\mathrm{mm}$ cases). The film centrifuge mechanism again does not describe the data, this time overpredicting the number of drops. This significant overprediction may be related to fact that the centrifuge mechanism can frequently eject drops horizontally, reducing their measured efficiency as these drops likely fall back to the water without being transported upwards into the measurement region (Néel & Deike Reference Néel and Deike2022; Shaw & Deike Reference Shaw and Deike2024).

From the two sample cases, we find that the experimental drop size distributions can be predicted well from knowledge of the experimental bubble size distribution using scaling laws for film flap drop and jet drop production for individual bubble bursting, with some differences between data and model for jet drops above $r_d=10\,\unicode{x03BC} \mathrm{m}$ . The film centrifuge mechanism does not appear to be statistically relevant and is ignored in the rest of the analysis. In the next section we explore how well this framework describes spray generation for the other bubble size distributions.

Figure 8.

Bursting bubble ( $N_b(R_b)$ ) and liquid drop ( $N_d(r_d)$ ) size distributions for the cases: (a) broad- $3\,\mathrm{mm}$ , (b) broad- $2\,\mathrm{mm}$ -half volume, (c) narrow- $3\,\mathrm{mm}$ and (d) narrow- $250\,\unicode{x03BC} \mathrm{m}$ , with representative surface images inlaid for each case. The predicted drop size distributions from individual bubble bursting scaling laws are plotted in colour for film flapping (orange) and jet (blue) drop production (see details in table 3), with the bounds of the associated portions of the bubble size distribution shown with dashed lines of the same colour.

4.3. Jet and film drop individual bubble bursting scaling laws for all cases

Figure 8 compares the measured and predicted drop size distributions for the remaining cases, using the same set of parameters described in table 3. As introduced in the previous section, the individual bubble scaling laws match the data very well for the broad-banded cases, shown in figure 8(a) (bubbles on the left, drops on the right) for bubbles of original injection size $3\,\mathrm{mm}$ , which are again characterised by a continuum of bubble sizes after breakup in turbulence. The magnitude of the distributions again match well with the data (black) using the $0.6{-}3$ drop per bubble assumption for the jet drops (blue), as discussed in the previous section, and the film flap (orange) magnitude also lines up well with the data. In this case, the framework works well across all scales with the two modes of production – with the jet drop mechanism dominating drop production above $1\,\unicode{x03BC} \mathrm{m}$ and the proposed film flap mechanism producing submicron drops.

Figure 8(b) shows the distributions for the case with 16 bubbling needles, where bubbles are spaced out on the surface and burst individually. While the jet drop scaling still matches well, the film flap prediction seems to miss some measured drops. The slight underprediction for this case – compared with the better match for submicron drops in figure 7(a) – may indicate a higher efficiency for the individual bursting case due to a potential collective effect limiting drop production. For the cases with narrower distributions, shown in figure 8(c,d), the measured and predicted distributions match well in the size range of film flap drops but are less coherent for the region described by jet drop production. In the large bubble case (panel c), again fewer large jet drops are measured than predicted. For the small bubbles, the prediction overestimates drops around radius $r_d=10\,\unicode{x03BC} \mathrm{m}$ . As discussed in § 4.1 for the direct attribution, the narrow-banded cases reveal that the assumed size link between bubbles and drops is not entirely accurate in this collective bursting set-up when large aggregates of bubbles of nearly the same size are present. Indeed, it seems that the predicted jet drop distribution for the cases in figure 8(c,d) may be missing some drops below radius $r_d=1\,\unicode{x03BC} \mathrm{m}$ , which can be seen from the gap between the predicted and measured distributions where the film and jet curves overlap. Néel & Deike (Reference Néel and Deike2021) and Néel et al. (Reference Néel and Deike2022) observed a similar shift/broadening in the peak size of ejected drops for collective bursting of millimetric bubbles in surfactant solutions of various concentrations, which remains unexplained. These inconsistencies may be explained by a surfactant or collective effect and require further investigation. We note also that the predicted jet drop production is sensitive to the chosen critical Laplace number ( $La^*$ , the drop ejection threshold), shifting the predicted drop size distribution to somewhat smaller sizes if this value is increased.

Overall, the drop size distribution produced by clustered bursting bubbles in artificial seawater can be reasonably well described from knowledge of the bubble size distribution, especially for the broad-banded distributions. By integrating individual bubble bursting scaling laws for size, number and distribution of ejected drops for all measured bubble sizes, we arrive at a reasonable estimate of the drop size distribution. Further study is required to explain the details of deviations in large jet drop production and spray generation by denser rafts of small bubbles, and to more precisely connect the two production mechanisms around their respective cutoffs around $1\, \unicode{x03BC} \mathrm{m}$ (large size for film drops and small size for jet drops), with possible collective effects.