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Extensional kinematics governs turbulent bubble breakup

Published online by Cambridge University Press:  14 October 2026

Leonel Beckedorff Open the ORCID record for Leonel Beckedorff [Opens in a new window]  and
Alfredo Soldati Open the ORCID record for Alfredo Soldati [Opens in a new window]
Leonel Beckedorff
Affiliation:
Institute of Fluid Mechanics and Heat Transfer , TU Wien, 1060 Wien, Austria
Alfredo Soldati*
Affiliation:
Institute of Fluid Mechanics and Heat Transfer , TU Wien, 1060 Wien, Austria Polytechnic Department, University of Udine , 33100 Udine, Italy
*
Corresponding author: Alfredo Soldati, alfredo.soldati@tuwien.ac.at
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Abstract

Content of image described in text.

Bubble breakup in turbulence is driven by velocity increments acting on the bubble interface. Traditionally, these instantaneous increments are replaced by the inertial-range second-order moment scaling, but whether this averaged description captures the kinematics leading to breakup remains unclear. Here, we experimentally investigate how bubble deformation preceding breakup relates to carrier-phase statistics. The experiments are conducted in nearly homogeneous isotropic turbulence with $\textit{Re}_\lambda =635$. Five high-speed cameras are used to reconstruct both the three-dimensional bubble topology and the trajectories of flow tracers in the carrier phase. A two-point description of bubble deformation is used to enable a direct comparison with particle pair dispersion theory and velocity increments. We find that the pre-breakup deformation stage can be described as a ballistic-dispersion problem, mirroring the short-time behaviour of particle pairs. For bubbles with $\textit{We} = \mathcal{O}(1)$, corresponding here to sizes close to the Hinze scale, the deformation is biased towards extensional events in which positive longitudinal velocity increments separate the daughter-associated subvolumes. Using a multifractal representation of the carrier-phase longitudinal-increment probability density function, conditioned by a capillary threshold, our results suggest that the measured deformation increments are consistent with stronger-than-average extensional increments, rather than with the inertial-range estimate.

JFM classification

Drops and Bubbles: Breakup/coalescence Multiphase and Particle-laden Flows: Multiphase flow Drops and Bubbles: Bubble dynamics

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JFM Rapids
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Journal of Fluid Mechanics , Volume 1044 , 10 October 2026 , R3
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
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© The Author(s), 2026. Published by Cambridge University Press

1. Introduction

In turbulence, bubbles undergo a cascade-like fragmentation process in which large bubbles break into smaller ones until capillary stresses balance turbulent stresses (Chan et al. Reference Chan, Johnson and Moin2021). Together with coalescence, breakup therefore controls the bubble size distribution, which, in turn, determines the total interfacial area and directly impacts transport phenomena such as mass and heat transfer (Liao & Lucas Reference Liao and Lucas2009; Soligo, Roccon & Soldati Reference Soligo, Roccon and Soldati2019; Ni Reference Ni2024). The classical basis for describing the breakup of bubbles and droplets in turbulence was established by the seminal works of Kolmogorov (Reference Kolmogorov1949) and Hinze (Reference Hinze1955), commonly known as the Kolmogorov–Hinze framework. Although originally developed for neutrally buoyant droplets, the framework is commonly applied to bubbles, despite differences in density and viscosity ratios. In this picture, a bubble of diameter upper D $D$ is expected to break if the dynamic pressure difference induced by the surrounding flow, rho delta u squared $\rho \delta u^2$ , surpasses the surface tension stresses, sigma divided by upper D $\sigma /D$ . This balance defines the Weber number, italic We equals rho left parenthesis delta u right parenthesis squared divided by left parenthesis sigma divided by upper D right parenthesis $\textit{We} = {\rho (\delta u)^2}/(\sigma /D)$ , where rho $\rho$ is the carrier-phase density, sigma $\sigma$ is the interfacial tension and delta u $\delta u$ is the characteristic velocity increment. The latter is particularly important for fragmentation, as the instantaneous action of turbulence on a deforming bubble is set by local velocity increments acting on the interface (Lalanne et al. Reference Lalanne, Masbernat and Risso2019).

Within the Kolmogorov–Hinze framework, this velocity increment at the bubble scale is usually replaced by the inertial-range second-order moment estimate, left parenthesis delta u right parenthesis squared almost equals upper C 2 left parenthesis left angle bracket epsilon right angle bracket upper D right parenthesis Superscript 2 divided by 3 $(\delta u)^2 \approx C_2(\langle \epsilon \rangle D)^{2/3}$ , where left angle bracket epsilon right angle bracket $\langle \epsilon \rangle$ is the mean energy dissipation rate and upper C 2 $C_2$ is the Kolmogorov constant. This simplification, considered in many breakup models (Prince & Blanch Reference Prince and Blanch1990; Martínez-Bazán et al. Reference Martínez-Bazán, Montañés and Lasheras1999; Lehr et al. Reference Lehr, Millies and Mewes2002), leads to a breakup criterion based on mean turbulent statistics. This closure is useful for estimating a characteristic breakup scale, but it averages out the instantaneous and directional nature of the velocity increments and does not specify how these increments act on the deforming interface. Several studies have moved beyond this mean-field closure by relating droplet or bubble deformation to local flow quantities, interface kinematics, or time-resolved shape evolution (Perlekar et al. Reference Perlekar, Biferale, Sbragaglia, Srivastava and Toschi2012; Qi et al. Reference Qi, Tan, Corbitt, Urbanik, Salibindla and Ni2022, Reference Qi, Xu, Tan, Zhong, Wu and Ni2024; Xu et al. Reference Xu, Qi, Zhong, Tan, Wu and Ni2024). However, a kinematic description that connects local turbulent velocity increments to the bubble deformation leading to breakup remains missing.

As discussed by Qi et al. (Reference Qi, Xu, Tan, Zhong, Wu and Ni2024), bubble deformation is typically quantified using global measures such as interfacial area, aspect ratio or spherical-harmonic mode amplitudes (Stone et al. Reference Stone, Bentley and Leal1986; Ravelet et al. Reference Ravelet, Colin and Risso2011; Vejražka et al. Reference Vejražka, Zedníková and Stanovský2018; Masuk et al. Reference Masuk, Salibindla and Ni2021; Perrard et al. Reference Perrard, Rivière, Mostert and Deike2021; Rivière et al. Reference Rivière, Abahri and Perrard2024). Some of these studies describe turbulence-induced deformation more generally, whereas others focus on the last stages preceding breakup. While these quantities are useful for describing the state of the interface, they are not easily related to turbulent velocity increments. Recently, Xu et al. (Reference Xu, Qi, Zhong, Tan, Wu and Ni2024) moved in this direction by analysing the velocity of the most stretched tip of deforming bubbles and comparing its probability density function (p.d.f.) to those of Eulerian velocity increments. Here, we adopt similar ideas, but focus specifically on breakup events. Instead of the most stretched tip of the bubble, we use a two-point description of the deformation: the parent-bubble volume is partitioned into two subvolumes associated with the daughter bubbles to be formed, and the motion of these subvolumes’ centroids is tracked up to the breakup instant. This two-point construction serves as an analogue to particle pair dispersion theory, also providing a kinematic link to carrier-phase velocity increments. Testing this interpretation requires time-resolved three-dimensional (3-D) measurements of the deforming bubble shape and the surrounding turbulent velocity field. The experimental methods used to obtain these measurements are described next.

2. Methods

2.1. Experimental set-up

The experiments, involving air bubbles in water, were conducted in the jet-stirred turbulent water tank illustrated in figure 1(a). Turbulence was generated within the octagonal tank by two opposing jet arrays, each consisting of 12 jets that continuously injected water at 8 m s Superscript negative 1 $\text{m s}^{-1}$ . The turbulence was nearly homogeneous and isotropic in the central region of the test section, where the measurement volume was located. The measurement volume was approximately 4 times 5 times 4 $4\times 5\times 4$ cm cubed $^3$ . Millimetre-sized air bubbles were injected below this central region through the bottom part of the tank. Further details of the experimental facility can be found from Beckedorff et al. (Reference Beckedorff, Caridi and Soldati2025).

Figure 1.

Experimental set-up and measurement procedure. (a) Schematic of the jet-stirred turbulent water tank. (b) Arrangement of the five high-speed cameras and (c) photograph of the apparatus. Each camera has an opposing LED panel, not shown here. (d) Example of raw backlight images and corresponding segmented particle and bubble images. (e) Reconstructed bubble together with surrounding tracer-particle trajectories.

A diagram of an experimental setup for studying bubble fragmentation in a turbulent water tank.
Figure 1. Long description

Panel A: A schematic diagram of a jet-stirred turbulent water tank. The tank has inflow and outflow points, with jet flow and bubble injection mechanisms. Panel B: A diagram showing the arrangement of five high-speed cameras around the water tank. The cameras are labeled Cam 1 through Cam 5 and are positioned to capture different views of the tank. Panel C: A photograph of the experimental apparatus, showing the water tank and the five high-speed cameras positioned around it. Each camera is labeled accordingly. Panel D: Examples of raw backlight images and corresponding segmented particle and bubble images. The raw images show particles and bubbles, while the segmented images highlight the particles and bubbles separately. Panel E: A reconstructed bubble with surrounding tracer-particle trajectories, illustrating the movement and interaction of particles around the bubble.

Flow images were recorded using five high-speed cameras (Phantom VEO 340L at 2000 f.p.s.) and a backlight illumination system. The camera arrangement is displayed in figures 1(b) and 1(c). The illumination system consisted of five white LED panels placed opposite to each camera (not displayed in figure 1), casting the shadows of bubbles and flow tracers (60 mu ${\unicode{x03BC}}$ m polyamide particles) onto the cameras’ imaging planes. The raw images were preprocessed and segmented into particle images and binarised bubble images; an example is shown in figure 1(d). The segmented images were then used for two purposes: (i) 3-D reconstruction of the bubble topology and (ii) carrier-phase measurements using 3-D particle tracking velocimetry (3-D-PTV). Bubble reconstruction used the visual Hull (VH) technique (Baumgart Reference Baumgart1974; Laurentini Reference Laurentini1994; Masuk et al. Reference Masuk, Salibindla and Ni2019) with a voxel size of 0.1 cubed $0.1^3$ mm cubed $^3$ – equivalent to less than 1 percent sign $1\,\%$ of the average bubble volume. For the carrier-phase measurements, we obtained particle trajectories using the shake-the-box (STB) technique (Schanz et al. Reference Schanz, Gesemann and Schröder2016). Between 8000 and 12 000 particles were successfully tracked at each time step, corresponding to an average particle concentration of 0.125 particles mm– cubed $^3$ . Figure 1(e) shows an example of a reconstructed bubble together with the surrounding flow trajectories. Image acquisition, segmentation and STB processing were carried out using LaVision DaVis 10.2.1. The velocity data obtained from STB were then used to characterise the turbulence in the measurement volume.

Figure 2.

Turbulence characterisation and experimental scales. (a) Eulerian second-order structure function, upper S 2 Superscript upper E $S_2^E$ , and longitudinal component, upper D Subscript italic LL $D_{\textit{LL}}$ , with inertial-range scaling shown for reference. (b) Estimates of the mean energy dissipation rate from second- and third-order structure functions; the shaded region indicates the estimated extent of the inertial range. (c) Summary of the relevant experimental length and time scales. The grey histogram indicates the bubble size distribution.

Three graphs depict turbulence characterization and experimental scales.

We conducted the experiments at italic Re Subscript lamda Baseline equals 635 $\textit{Re}_\lambda =635$ , where the Taylor-microscale Reynolds number is defined as italic Re Subscript lamda Baseline equals u Superscript prime 2 Baseline StartRoot 15 divided by left parenthesis nu left angle bracket epsilon right angle bracket right parenthesis EndRoot $\textit{Re}_\lambda = u^{\prime 2}\sqrt {15/(\nu \langle \epsilon \rangle )}$ . Here, nu $\nu$ is the kinematic viscosity, u prime equals left bracket left parenthesis u Subscript x Superscript prime 2 Baseline plus u Subscript y Superscript prime 2 Baseline plus u Subscript z Superscript prime 2 Baseline right parenthesis divided by 3 right bracket Superscript 1 divided by 2 Baseline equals 0.28 $u^\prime =[(u_x^{\prime 2}+u_y^{\prime 2}+u_z^{\prime 2})/3]^{1/2}=0.28$ m s–1 is the root-mean-square velocity fluctuation and left angle bracket epsilon right angle bracket equals 0.29 $\langle \epsilon \rangle = 0.29$ m squared $^2$ s– cubed $^{3}$ is the mean energy dissipation rate. We estimated left angle bracket epsilon right angle bracket $\langle \epsilon \rangle$ from the Eulerian second-order structure function, upper S 2 Superscript upper E Baseline equals left angle bracket StartAbsoluteValue bold italic u left parenthesis bold italic x plus r comma t right parenthesis minus bold italic u left parenthesis bold italic x comma t right parenthesis EndAbsoluteValue squared right angle bracket $S^E_2 = \langle |\boldsymbol{u}(\boldsymbol{x}+r,t) - \boldsymbol{u}(\boldsymbol{x},t)|^2\rangle$ , computed from velocity increments between particle pairs separated by a distance r $r$ . Figure 2(a) presents upper S 2 Superscript upper E $S^E_2$ and its longitudinal component ( upper D Subscript italic LL $D_{\textit{LL}}$ ) as functions of r $r$ . The dashed line indicates the slope of the inertial-range scaling law, upper S 2 Superscript upper E Baseline equals left parenthesis 11 divided by 3 right parenthesis upper C 2 left parenthesis r left angle bracket epsilon right angle bracket right parenthesis Superscript 2 divided by 3 $S^E_2 = (11/3)C_2(r\langle \epsilon \rangle )^{2/3}$ , where the Kolmogorov constant is upper C 2 equals 2.13 $C_2 = 2.13$ . Figure 2(b) shows the estimation of left angle bracket epsilon right angle bracket $\langle \epsilon \rangle$ based on the second- and third-order structure functions. The shaded area in figure 2(b) indicates the inertial range, which spans from 1.5 $1.5$ to 9.8 $9.8$ mm ( 40 less than r divided by eta less than 260 $40 \lt r/\eta \lt 260$ ). We considered the inertial range as the region where the compensated third-order longitudinal structure function, minus upper D Subscript italic LLL divided by left parenthesis r left angle bracket epsilon right angle bracket right parenthesis $-D_{\textit{LLL}}/(r\langle \epsilon \rangle )$ , deviates by less than 10 percent sign $10\,\%$ from 4 divided by 5 $4/5$ , where upper D Subscript italic LLL Baseline equals left angle bracket delta u Subscript upper L Superscript 3 Baseline right angle bracket $D_{\textit{LLL}} = \langle \delta u_L^3\rangle$ . Based on left angle bracket epsilon right angle bracket $\langle \epsilon \rangle$ , the Kolmogorov length and time scales were eta equals left parenthesis nu cubed divided by left angle bracket epsilon right angle bracket right parenthesis Superscript 1 divided by 4 Baseline equals 38 $\eta = (\nu ^3/\langle \epsilon \rangle )^{1/4}=38$ mu ${\unicode{x03BC}}$ m and tau Subscript eta Baseline equals left parenthesis nu divided by left angle bracket epsilon right angle bracket right parenthesis Superscript 1 divided by 2 Baseline equals 1.7 $\tau _\eta = (\nu /\langle \epsilon \rangle )^{1/2} = 1.7$ ms. The integral length scale, estimated from the longitudinal velocity autocorrelation, was upper L equals 4.1 $L = 4.1$ cm, giving an integral time scale upper T Subscript upper L Baseline almost equals upper L divided by u Superscript prime Baseline equals 146 $T_L \approx L/u^\prime = 146$ ms. Figure 2(c) summarises the relevant time and length scales from the experiments. We note that the VH voxel size is larger than eta $\eta$ . This does not limit the present analysis, because the statistics considered here are evaluated at inertial-range scales comparable to the bubble size, rather than from small-scale interfacial quantities such as local curvature.

The dataset comprised 124 bubble breakup events. The parent-bubble equivalent diameters, upper D $D$ , ranged from 2.2 to 8.8 mm ( 55 less than upper D divided by eta less than 245 $55 \lt D/\eta \lt 245$ ) with a mean of 6.3 $6.3$ mm. The diameters fell within the inertial range; the bubble size distribution is illustrated in figure 2(c). The Weber number, italic We equals rho upper C 2 left angle bracket epsilon right angle bracket Superscript 2 divided by 3 Baseline upper D Superscript 5 divided by 3 Baseline divided by sigma $\textit{We}=\rho C_2 \langle \epsilon \rangle ^{2/3} D^{5/3}/\sigma$ , ranged from 0.4 to 6.2, with a mean value of italic We equals 2.85 $\textit{We}=2.85$ . The quantities upper D divided by eta $D/\eta$ and italic We $\textit{We}$ were computed using the dissipation rate estimated for each individual case. Breakup events were morphologically diverse: in our dataset, most events produced unequal daughter bubbles, consistent with the U-shaped daughter-size distributions measured by Rodríguez-Rodríguez et al. (Reference Rodríguez-Rodríguez, Martínez-Bazán and Montañés2003) and the predominance of uneven breakups reported by Qi et al. (Reference Qi, Masuk and Ni2020). The smallest daughter bubble detected had a diameter of approximately 0.8 mm, close to the practical reconstruction limit imposed by the VH voxel size.

Figure 3.

(a) Reconstructed breakup event of a bubble with diameter upper D equals 4.3 $D=4.3$ mm. The two daughter-associated subvolumes, upper Omega 1 $\varOmega _1$ and upper Omega 2 $\varOmega _2$ , are shown together with their centroids, bold italic upper X 1 $\boldsymbol{X}_1$ and bold italic upper X 2 $\boldsymbol{X}_2$ ; red and grey lines indicate the corresponding centroid trajectories. (b) Evolution of the separation r left parenthesis t right parenthesis equals StartAbsoluteValue bold italic upper X 2 left parenthesis t right parenthesis minus bold italic upper X 1 left parenthesis t right parenthesis EndAbsoluteValue $r(t)=|\boldsymbol{X}_2(t)-\boldsymbol{X}_1(t)|$ , coloured by the velocity-increment magnitude delta u equals StartAbsoluteValue bold italic u 2 left parenthesis t right parenthesis minus bold italic u 1 left parenthesis t right parenthesis EndAbsoluteValue $\delta u=|\boldsymbol{u}_2(t)-\boldsymbol{u}_1(t)|$ . Black and white markers indicate the onset of the inertial deformation stage and breakup, respectively. The inset shows the corresponding alignment angle theta $\theta$ between bold italic r $\boldsymbol{r}$ and delta bold italic u $\delta \boldsymbol{u}$ .

Two graphs depict the deformation and breakup of a bubble in a turbulent flow.
Figure 3. Long description

Panel A: A 3-D plot shows the trajectories of two subvolumes of a deforming bubble. The subvolumes are labeled V1 and V2, with their centroids marked as X1(t) and X2(t). The trajectories are indicated by red and grey lines. The distance between the centroids, r(t), is also shown. Panel B: A line graph displays the evolution of the separation distance r(t) over time t, colored by the velocity-increment magnitude delta u. The graph includes markers for the onset of the inertial deformation stage and the breakup point. The inset shows the alignment angle theta between the separation distance and another reference direction.

2.2. Bubble subvolume definition and tracking

We analysed the bubble deformation by partitioning the parent-bubble volume into two main subvolumes and tracking the relative separation of their centroids. An example of a reconstructed breakup event is shown in figure 3(a), where the grey and red lines represent the trajectories associated with the two subvolumes. These subvolumes, denoted by upper Omega 1 $\varOmega _1$ and upper Omega 2 $\varOmega _2$ , were defined recursively backward in time based on the volumes of the two largest daughter bubbles formed after the breakup, whose volumes are upper V 1 $V_1$ and upper V 2 $V_2$ .

The subvolume identification was initialised at the last frame in which the parent bubble remained connected. The centroids of the two largest daughter bubbles in the subsequent frame were mapped onto the voxelised parent-bubble volume and used as reference points, from which the geodesic distances d 1 left parenthesis bold italic x right parenthesis $d_1(\boldsymbol{x})$ and d 2 left parenthesis bold italic x right parenthesis $d_2(\boldsymbol{x})$ to every voxel bold italic x $\boldsymbol{x}$ within the parent bubble were computed. Voxels were then assigned according to their relative proximity to these reference points and partitioned into upper Omega 1 $\varOmega _1$ and upper Omega 2 $\varOmega _2$ based on the daughter-volume ratio, with upper Omega 1 $\varOmega _1$ containing the fraction upper V 1 divided by left parenthesis upper V 1 plus upper V 2 right parenthesis $V_1/(V_1+V_2)$ of voxels and upper Omega 2 $\varOmega _2$ defined by the complement. In non-binary breakup events, upper Omega 2 $\varOmega _2$ included a small residual volume associated with minor fragments. However, the influence of this residual was small, as non-binary breakups accounted for only 12 events in our dataset, with an average residual volume of 1.3 percent sign $1.3\,\%$ . The centroids of upper Omega 1 $\varOmega _1$ and upper Omega 2 $\varOmega _2$ were then used as reference points in the preceding frame and the same voxel-assignment procedure was applied recursively backward in time to define the evolution of the two daughter-associated subvolumes. This procedure is not intended as an exact material reconstruction, but rather as a temporally coherent partition of the parent bubble near breakup for kinematic analysis. Furthermore, the two-point construction is breakup-conditioned: the daughter bubbles are first identified after breakup, and their measured volumes and centroids are then used to define a deformation coordinate over the preceding connected frames. The approach is therefore not used to predict the breakup event or to determine the daughter bubbles before they are observed.

After defining the two subvolumes, upper Omega 1 left parenthesis t right parenthesis $\varOmega _1(t)$ and upper Omega 2 left parenthesis t right parenthesis $\varOmega _2(t)$ , we computed their centroids, bold italic upper X 1 left parenthesis t right parenthesis $\boldsymbol{X}_1(t)$ and bold italic upper X 2 left parenthesis t right parenthesis $\boldsymbol{X}_2(t)$ . The centroid trajectories were filtered using a second-order Savitzky–Golay filter with a 7-frame window (window length almost equals 2 tau Subscript eta $\approx 2\tau _\eta$ ), from which we obtained the relative separation bold italic r left parenthesis t right parenthesis equals bold italic upper X 2 left parenthesis t right parenthesis minus bold italic upper X 1 left parenthesis t right parenthesis $\boldsymbol{r}(t) = \boldsymbol{X}_2(t) - \boldsymbol{X}_1(t)$ and the corresponding velocity increment delta bold italic u left parenthesis t right parenthesis equals bold italic u 2 left parenthesis t right parenthesis minus bold italic u 1 left parenthesis t right parenthesis $\delta \boldsymbol{u}(t) = \boldsymbol{u}_2(t)-\boldsymbol{u}_1(t)$ . The centroid velocities, bold italic u 1 left parenthesis t right parenthesis $\boldsymbol{u}_1(t)$ and bold italic u 2 left parenthesis t right parenthesis $\boldsymbol{u}_2(t)$ , were computed by differentiating the fitted second-order polynomials from the Savitzky–Golay procedure. Note that these velocities are integral observables of the parent-bubble deformation. By construction, delta bold italic u equals normal d bold italic r divided by normal d t $\delta \boldsymbol{u} = \mathrm{d}\boldsymbol{r}/\mathrm{d}t$ and its projection onto the deformation axis gives the ‘stretching rate’ of the parent bubble, delta bold italic u bold dot bold italic r divided by r equals normal d r divided by normal d t $\delta \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{r}/r = \mathrm{d}r/\mathrm{d}t$ . These velocities should not be interpreted as pointwise gas-phase or carrier-phase velocities. Figure 3(a) illustrates the trajectories bold italic upper X Subscript i Baseline left parenthesis t right parenthesis $\boldsymbol{X}_i(t)$ , the subvolumes upper Omega Subscript i $\varOmega _i$ , the daughter-bubble volumes upper V Subscript i $V_i$ and the relative separation bold italic r left parenthesis t right parenthesis $\boldsymbol{r}(t)$ , while figure 3(b) presents the evolution of the separation magnitude, r equals StartAbsoluteValue bold italic r EndAbsoluteValue $r = |\boldsymbol{r}|$ , coloured by the velocity increment magnitude, delta u equals StartAbsoluteValue delta bold italic u EndAbsoluteValue $\delta u = |\delta \boldsymbol{u}|$ , for the same breakup event. The inset of figure 3(b) reports theta $\theta$ , the angle between the separation vector and the velocity increment, defined as theta identical to arc cosine left bracket delta bold italic u bold dot bold italic r divided by left parenthesis r delta u right parenthesis right bracket $\theta \equiv \arccos [\delta \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{r} / (r\, \delta u)]$ (Shnapp et al. Reference Shnapp, Brizzolara, Neamtu-Halic, Gambino and Holzner2023); this quantity is analysed in § 3.

This two-point description of the deformation is initially evaluated using an analogy with particle pair dispersion theory (Richardson Reference Richardson1926; Batchelor Reference Batchelor1950; Bourgoin et al. Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006). Building on this analogy, we analysed the deformation using the mean squared separation, left angle bracket StartAbsoluteValue bold italic r left parenthesis t right parenthesis minus bold italic r 0 EndAbsoluteValue squared right angle bracket $\langle |\boldsymbol{r}(t) - \boldsymbol{r}_0|^2\rangle$ , relative to an initial separation bold italic r 0 $\boldsymbol{r}_0$ . Since we focus on the kinematics preceding breakup, bold italic r 0 $\boldsymbol{r}_0$ was defined at the onset of the inertial deformation stage, identified for each event as the start of the last sustained increase in the separation r left parenthesis t right parenthesis $r(t)$ before breakup. The onset corresponding to r 0 $r_0$ was identified with a standard change-point method (Lavielle Reference Lavielle2005; Killick et al. Reference Killick, Fearnhead and Eckley2012). The separation r left parenthesis t right parenthesis $r(t)$ was first smoothed with a centred moving-median filter of width 3 tau Subscript eta $3\tau _\eta$ . The smoothed signal was then approximated by two straight lines from the start of the recording to the instant of maximum separation. The onset was taken as the time for which this two-line description fits best in the least-squares sense: the first line captures the waiting stage and the second the sustained growth towards breakup. The onset was further required to lie at least 3 tau Subscript eta $3\tau _\eta$ from breakup, a margin comparable to the capillary time scale estimated in § 3. The inertial deformation stage is highlighted in figure 3(b), where the black- and white-filled markers denote the deformation onset ( r 0 $r_0$ ) and breakup.

We emphasise that this two-point approach describes the geometric evolution of the parent-bubble deformation; it is not intended as a material reconstruction, nor as a measurement of the gas velocity inside the bubble. The approach is most reliable during the stage where the two daughter-associated subvolumes define a coherent deformation axis. During the inertial deformation stage, the separation vector was mostly aligned with the bubble semi-major axis, indicating that bold italic r $\boldsymbol{r}$ captures the dominant elongation direction. At earlier times, farther from breakup, the connection between the reconstructed subvolumes and the eventual daughter bubbles becomes less reliable. For this reason, the main breakup-conditioned statistics are evaluated over the inertial deformation stage. Here, ‘breakup-conditioned’ means that only bubbles for which breakup is directly observed are included and that the statistics are evaluated over the corresponding pre-breakup deformation stage.

Lastly, we computed two-point statistics of tracer-particle pairs sampled over the same pre-breakup time interval, with initial separations in the range 10 eta less than or slanted equals r 0 less than or slanted equals 300 eta $10\eta \leqslant r_0 \leqslant 300\eta$ . An example of surrounding trajectories is shown in figure 1(e). We emphasise that the bubble-pair separation is defined kinematically from the centroids of the two daughter-associated subvolumes, whereas the carrier-phase separation is obtained from tracer-particle pairs. These carrier-phase statistics therefore provide a reference for comparison with the bubble deformation, rather than a material counterpart to the bubble-pair construction. Hereafter, superscripts b $b$ and f $f$ denote quantities associated with the bubble deformation and with the carrier phase, respectively. Quantities without superscript denote definitions common to both cases.

Figure 4.

(a) Normalised mean squared separation of deforming bubbles (blue markers) and particle pairs (red lines) as a function of t divided by t 0 $t/t_0$ . From light to dark red, the particle-pair initial separations are r 0 equals 10 eta $r_0 = 10\eta$ , 20 eta $20\eta$ , 30 eta $30\eta$ , 40 eta $40\eta$ , 50 eta $50\eta$ , 100 eta $100\eta$ , 200 eta $200\eta$ and 300 eta $300\eta$ . The inset shows the mean bubble separation with all events aligned at breakup, left parenthesis t minus t Subscript b Baseline right parenthesis divided by tau Subscript eta Baseline equals 0 $(t-t_b)/\tau _\eta =0$ . (b) Mean alignment angle theta $\theta$ between the separation vector and velocity increment for deforming bubbles and particle pairs. The inset shows the breakup-aligned evolution of left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b\rangle$ ; shaded regions indicate the average duration of the inertial deformation stage and the estimated capillary time.

Graphs depict the normalized mean squared separation and alignment angle of deforming bubbles and particle pairs.

3. Results

We first examine whether the two-point description of bubble deformation can be interpreted as a relative-dispersion problem. Figure 4(a) compares the mean squared separation of the bubble-subvolume centroids with that of particle pairs in the surrounding flow. The separation is normalised by the second-order structure function at the initial separation, upper S 2 left parenthesis r 0 right parenthesis equals left angle bracket StartAbsoluteValue delta bold italic u 0 EndAbsoluteValue squared right angle bracket $S_2(r_0) = \langle |\delta \boldsymbol{u}_0|^2 \rangle$ , while time is normalised by the corresponding eddy turnover time, t 0 equals r 0 Superscript 2 divided by 3 Baseline left angle bracket epsilon right angle bracket Superscript negative 1 divided by 3 $t_0 = r_0^{2/3}\langle \epsilon \rangle ^{-1/3}$ . Using the mean onset separation, left angle bracket r 0 right angle bracket equals 122 eta $\langle r_0\rangle =122\eta$ , this gives a representative value left angle bracket t 0 right angle bracket almost equals 42 $\langle t_0 \rangle \approx 42$ ms, although the normalisation is applied event by event. For each breakup event, time is measured from the identified onset of the inertial deformation stage, so that t equals 0 $t = 0$ corresponds to r equals r 0 $r=r_0$ . Ensemble averages are then computed in fixed, non-overlapping bins of normalised time. As different bubbles break at different instants, late-time bins contain progressively fewer events, so we truncate the curves when fewer than 25 % of the events remain. Because r 0 $r_0$ varies between breakup events, the normalisation cannot be based on a single value of upper S 2 left parenthesis r 0 right parenthesis $S_2(r_0)$ . We therefore group the events into five r 0 $r_0$ -ranges, each containing approximately 25 breakup events, and compute upper S 2 left parenthesis r 0 right parenthesis $S_2(r_0)$ per range. Each breakup event is then normalised using the corresponding range-averaged value. Throughout this paper, shaded areas and error bars indicate 95 % bootstrap confidence intervals (Efron & Tibshirani Reference Efron and Tibshirani1994).

We observe that bubble deformation exhibits ballistic growth at early times, closely resembling the behaviour of particle pairs. The dispersion in this ballistic regime is governed by a persistent initial velocity increment, leading to the well-known scaling left angle bracket StartAbsoluteValue bold italic r left parenthesis t right parenthesis minus bold italic r 0 EndAbsoluteValue squared right angle bracket equals upper S 2 left parenthesis r 0 right parenthesis t squared $\langle |\boldsymbol{r}(t) - \boldsymbol{r}_0|^2 \rangle = S_2(r_0)t^2$ (Batchelor Reference Batchelor1950; Bourgoin Reference Bourgoin2015). In canonical pair dispersion, t divided by t 0 almost equals 1 $t/t_0 \approx 1$ marks the transition from the initial ballistic regime to subsequent dispersion regimes, whose nature depends on the initial separation r 0 $r_0$ and the initial velocity increment (Shnapp & Liberzon Reference Shnapp and Liberzon2018; Tan & Ni Reference Tan and Ni2022; Li et al. Reference Li, Wang, Qi and Coletti2024). Reference slopes for possible subsequent regimes are included in figure 4(a): the slope t cubed $t^3$ indicates the Richardson superdiffusive regime, while the t $t$ slope indicates the diffusive regime. Here, most bubbles break before reaching one eddy turnover time, making the transition to a second regime unclear and leaving ballistic dispersion as the dominant observed regime. The fact that most breakup events occur within t divided by t 0 less than 1 $t/t_0 \lt 1$ indicates that the pre-breakup deformation develops over a time shorter than (or comparable to) the turnover time of an eddy at the initial separation scale. This observation is consistent with classical breakage-time models that estimate the deformation time from an eddy-turnover time, such as that of Coulaloglou & Tavlarides (Reference Coulaloglou and Tavlarides1977).

While figure 4(a) indicates that bubbles deform following the same ballistic trend as particle pairs, defining r 0 $r_0$ at the onset of inertial deformation may select a subset of events with distinct dispersion properties. In particular, the conditioned bubble-deformation data likely probe flow regions that promote stronger separation than those sampled by unconditioned particle pairs. This tendency can be evaluated through the alignment angle between the velocity increment and the separation vector, theta identical to arc cosine left bracket delta bold italic u bold dot bold italic r divided by left parenthesis r delta u right parenthesis right bracket $\theta \equiv \arccos [\delta \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{r} / (r\, \delta u)]$ (Shnapp et al. Reference Shnapp, Brizzolara, Neamtu-Halic, Gambino and Holzner2023; Tan et al. Reference Tan, Zhong, Xu, Qi and Ni2025). Angles smaller than 90 Superscript ring $90 ^{\circ }$ indicate increasing separation, whereas angles larger than 90 Superscript ring $90 ^{\circ }$ indicate decreasing separation.

Figure 4(b) shows the evolution of the mean alignment angle for deforming bubbles and particle pairs. The particle-pair results (red lines) follow the behaviour reported by Shnapp et al. (Reference Shnapp, Brizzolara, Neamtu-Halic, Gambino and Holzner2023): left angle bracket theta Superscript f Baseline right angle bracket $\langle \theta ^f \rangle$ starts at 90 Superscript ring $90 ^{\circ }$ , indicating no preferential alignment, then decreases monotonically during the ballistic regime and approaches approximately 59.3 Superscript ring $59.3 ^{\circ }$ in the inertial range for t divided by t 0 greater than or equivalent to 0.5 $t/t_0 \gtrsim 0.5$ . For deforming bubbles, left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b \rangle$ remains mostly below 59.3 Superscript ring $59.3 ^{\circ }$ in the inertial deformation stage. This indicates that the breakup-conditioned deformation is dominated by an extensional relative motion, with delta bold italic u Superscript b $\delta \boldsymbol{u}^b$ preferentially aligned with bold italic r $\boldsymbol{r}$ . The inset of figure 4(b) presents the evolution of left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b \rangle$ with all events aligned in time by subtracting the breakup instant t Subscript b $t_b$ , so that breakup occurs at left parenthesis t minus t Subscript b Baseline right parenthesis divided by tau Subscript eta Baseline equals 0 $(t-t_b)/\tau _\eta = 0$ . The inset of figure 4(a) uses the same temporal reference for the mean separation. The red-shaded region indicates the average duration of the inertial deformation stage. The inset shows that left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b\rangle$ is close to 90 Superscript ring $90 ^{\circ }$ at early times, before the onset of the inertial deformation stage. This value should not be interpreted in the same way as the early-time limit of left angle bracket theta Superscript f Baseline right angle bracket $\langle \theta ^f\rangle$ for particle pairs. In the latter, left angle bracket theta Superscript f Baseline right angle bracket almost equals 90 Superscript ring $\langle \theta ^f\rangle \approx 90 ^{\circ }$ results from random sampling of particle pairs in an unconditioned ensemble. For the bubble deformation, instead, the two daughter-associated subvolumes remain constrained by the interface and surface tension before the inertial deformation stage, so left angle bracket theta Superscript b Baseline right angle bracket almost equals 90 Superscript ring $\langle \theta ^b\rangle \approx 90 ^{\circ }$ reflects a preferential perpendicular alignment between delta bold italic u Superscript b $\delta \boldsymbol{u}^b$ and bold italic r $\boldsymbol{r}$ , not random sampling. Approximately 8 tau Subscript eta $8\tau _\eta$ before breakup, left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b\rangle$ decreases sharply to approximately 40 Superscript ring $40 ^{\circ }$ , indicating the onset of an extensional event in which the relative velocity becomes aligned with the deformation direction. This behaviour is consistent with the trend in the main panel.

The subsequent increase of left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b\rangle$ close to breakup likely reflects the rapid retraction observed in some events during pinch-off, where capillary effects are significant. A characteristic capillary time scale can be estimated as tau Subscript c Baseline equals StartRoot rho d cubed divided by sigma EndRoot divided by left parenthesis 2 StartRoot 3 EndRoot right parenthesis $\tau _c=\sqrt {\rho d^3/\sigma }/(2\sqrt {3})$ (Rivière et al. Reference Rivière, Ruth, Mostert, Deike and Perrard2022; Ruth et al. Reference Ruth, Aiyer, Rivière, Perrard and Deike2022; Qi et al. Reference Qi, Xu, Tan, Zhong, Wu and Ni2024), where d $d$ is the neck width at the onset of the capillary instability. Since d $d$ is not robustly available for all events, we approximate it by the mean diameter of the smaller daughter bubble, d almost equals 2.6 $d\approx 2.6$ mm, which gives tau Subscript c Baseline almost equals 2.65 tau Subscript eta $\tau _c\approx 2.65\tau _\eta$ . This estimate is shown by the green-shaded region in the inset and is comparable to the time interval over which left angle bracket theta Superscript b Baseline right angle bracket $\langle \theta ^b\rangle$ increases before breakup.

The preferential alignment observed in figure 4(b) suggests an enhanced longitudinal velocity increment. To quantify this effect, we compare the longitudinal velocity increment at scale r $r$ , denoted delta Subscript r Baseline u Subscript upper L $\delta _r u_L$ , with the inertial-range velocity scale, left parenthesis r left angle bracket epsilon right angle bracket right parenthesis Superscript 1 divided by 3 $(r \langle \epsilon \rangle )^{1/3}$ . Figure 5(a) shows the p.d.f. of this normalised longitudinal increment for bubble-deformation data collected over the inertial deformation stage (blue markers) and for particle pairs (red markers). Particle pairs are selected in a narrow separation range ( 120 eta less than or slanted equals r less than or slanted equals 124 eta $120\eta \leqslant r \leqslant 124\eta$ ) centred on r equals 122 eta $r=122\eta$ , which corresponds to the mean initial separation left angle bracket r 0 Superscript b Baseline right angle bracket $\langle r_0^b\rangle$ of the bubbles. The particle-pair distribution is slightly negatively skewed, consistent with the asymmetry associated with Kolmogorov’s 4 divided by 5 $4/5$ law. In comparison, the bubble-deformation distribution exhibits a clear shift from the delta Subscript r Baseline u Subscript upper L Superscript f $\delta _r u^f_L$ distribution. The bubble-deformation p.d.f. is based on a much smaller sample than the carrier-phase p.d.f. and should be interpreted only qualitatively. We therefore do not draw conclusions from higher-order moments, tail shape or intermittency levels. The most robust feature of the bubble-deformation p.d.f. is the positive shift of its mean value, consistent with the alignment trend in figure 4(b). This indicates that, conditioned on breakup, the pre-breakup deformation preferentially samples extensional increments. We therefore interpret left angle bracket delta Subscript r Baseline u Subscript upper L Superscript b Baseline vertical bar normal b normal r normal e normal a normal k normal u normal p right angle bracket $\langle \delta _r u_L^b \mid \mathrm{breakup}\rangle$ as a breakup-conditioned mean extensional increment and ask whether its magnitude is compatible with the positive tail of the carrier-phase distribution once a capillary threshold is imposed.

Figure 5.

(a) P.d.f. of the normalised longitudinal velocity increment, delta Subscript r Baseline u Subscript upper L divided by left parenthesis r left angle bracket epsilon right angle bracket right parenthesis Superscript 1 divided by 3 $\delta _r u_L/(r\langle \epsilon \rangle )^{1/3}$ , for bubble-deformation data during the inertial deformation stage (blue), particle pairs with r equals 122 eta plus or minus 2 eta $r=122\eta \pm 2\eta$ (red) and the Chevillard model (Chevillard et al. Reference Chevillard, Castaing, Lévêque and Arneodo2006) (black). The grey region indicates positive increments satisfying delta Subscript r Baseline u Subscript upper L Superscript f Baseline greater than or slanted equals u Subscript c $\delta _r u_L^f\geqslant u_c$ . (b) Normalised mean bubble-deformation increments (blue markers) compared with the conditioned estimate from (3.2) (red line) and the inertial-range estimate (dash-dotted line). The inset shows the conditioned estimate as a function of upper D divided by upper D Subscript upper H $D/D_H$ ; the red-shaded region marks italic We greater than 10 $\textit{We}\gt 10$ .

Two graphs depict data on bubble deformation and particle pairs.

To examine this interpretation, we define a local longitudinal Weber number based on carrier-phase velocity increments, italic We Subscript upper L Baseline equals rho left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis squared divided by left parenthesis sigma divided by upper D right parenthesis $\textit{We}_L = \rho (\delta _r u^f_L) ^2/(\sigma /D)$ . Assuming that only extensional increments can increase the separation between the daughter-associated subvolumes, the threshold must be applied to the signed velocity increments. If breakup requires italic We Subscript upper L $\textit{We}_L$ to exceed a critical value, say italic We Subscript upper L Baseline greater than or slanted equals italic We Subscript c Baseline equals script upper O left parenthesis 1 right parenthesis $\textit{We}_L \geqslant \textit{We}_c = \mathcal{O}(1)$ , the corresponding threshold is

(3.1) delta Subscript r Baseline u Subscript upper L Superscript f Baseline greater than or slanted equals u Subscript c Baseline equals StartRoot StartFraction italic We Subscript c Baseline sigma Over rho upper D EndFraction EndRoot period \begin{equation} \delta _r u^f_L \geqslant u_c = \sqrt {\frac {\textit{We}_c\sigma }{\rho D}}. \end{equation}

This conditioning is illustrated in figure 5(a): the grey region indicates the positive carrier-phase longitudinal increments that satisfy the threshold in 3.1. We then ask whether the conditional mean over this region, left angle bracket delta Subscript r Baseline u Subscript upper L Superscript f Baseline vertical bar delta Subscript r Baseline u Subscript upper L Superscript f Baseline greater than or slanted equals u Subscript c Baseline right angle bracket $\langle \delta _r u^f_L\mid \delta _ru^f_L \geqslant u_{c}\rangle$ , provides a reasonable estimate of the measured breakup-conditioned bubble increment, left angle bracket delta Subscript r Baseline u Subscript upper L Superscript b Baseline vertical bar normal b normal r normal e normal a normal k normal u normal p right angle bracket $\langle \delta _ru_L^b \mid \mathrm{breakup}\rangle$ . Estimating this conditional mean directly from a finite set of coupled bubble-flow measurements would require sufficient carrier-phase samples to resolve the positive tail of the p.d.f. over several decades of probability, which is impractical here. Instead, we use a model to provide a smooth representation of the carrier-phase longitudinal-increment p.d.f., namely script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis $\mathcal{P}_{\delta _r u_L}(\delta _r u^f_L)$ . Specifically, we adopt the unified multifractal description proposed by Chevillard et al. (Reference Chevillard, Castaing, Lévêque and Arneodo2006) to estimate script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis $\mathcal{P}_{\delta _r u_L}(\delta _r u^f_L)$ . We refer the reader to Chevillard et al. (Reference Chevillard, Castaing, Lévêque and Arneodo2006) for details of the model; the parameters are set to script upper R Superscript asterisk Baseline equals 52 $\mathcal{R}^*=52$ , c 1 equals 0.37 $c_1=0.37$ and c 2 equals 0.025 $c_2=0.025$ , as defined by the authors. The solid line in figure 5(a) displays the resulting script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis $\mathcal{P}_{\delta _r u_L}(\delta _r u^f_L)$ for r equals 122 eta $r = 122\eta$ . The model provides a close approximation to our measured distributions of delta Subscript r Baseline u Subscript upper L Superscript f $\delta _r u^f_L$ in the range 20 eta less than or slanted equals r less than or slanted equals 300 eta $20\eta \leqslant r \leqslant 300\eta$ (not shown here). We then use this p.d.f. to evaluate the conditional mean over positive events selected by the local Weber criterion. The mean velocity increment conditioned on this criterion at scale r $r$ is computed as

(3.2) left angle bracket delta Subscript r Baseline u Subscript upper L Superscript f Baseline vertical bar delta Subscript r Baseline u Subscript upper L Superscript f Baseline greater than or slanted equals u Subscript c Baseline right angle bracket equals StartFraction integral Subscript u Subscript c Baseline Superscript normal infinity Baseline delta Subscript r Baseline u Subscript upper L Superscript f Baseline script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Subscript Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis d left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis Over integral Subscript u Subscript c Baseline Superscript normal infinity Baseline script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Subscript Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis d left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis EndFraction period \begin{equation} \langle \delta _r u^f_L \mid \delta _r u^f_L \geqslant u_{c} \rangle = \frac {\int _{u_{c}}^\infty \delta _r u^f_L \mathcal{P}_{\delta _r u_L}(\delta _r u^f_L) d(\delta _r u^f_L)} {\int _{u_{c}}^\infty \mathcal{P}_{\delta _r u_L}(\delta _r u^f_L) d(\delta _r u^f_L)}. \end{equation}

In summary, this characteristic velocity increment can be estimated in three steps: (i) for a bubble of diameter upper D $D$ in a flow of known rho $\rho$ , sigma $\sigma$ and left angle bracket epsilon right angle bracket $\langle \epsilon \rangle$ , and a specified italic We Subscript c $\textit{We}_c$ , compute the capillary threshold u Subscript c $u_c$ from (3.1); (ii) estimate the carrier-phase longitudinal-increment p.d.f. at scale r almost equals upper D $r \approx D$ ; (iii) compute the characteristic velocity increment as the mean of this p.d.f. above u Subscript c $u_c$ , from (3.2). Any representation that reproduces the carrier-phase increment statistics at the relevant scales can be used for the p.d.f. in step (ii), including a directly measured distribution, when available. Here, we chose the Chevillard et al. (Reference Chevillard, Castaing, Lévêque and Arneodo2006) model because it provides a closed-form representation of the increment p.d.f., with universal constants taken from the original publication (no parameters were fitted to our data).

Figure 5(b) compares the longitudinal velocity increment of deforming bubbles (blue markers) with the conditioned estimate from (3.2) (red line). The markers are obtained by sorting the dataset into three upper D divided by eta $D/\eta$ bins, each containing approximately 41 $41$ events, and averaging delta Subscript r Baseline u Subscript upper L Superscript b $\delta _r u_L^b$ over the inertial deformation stage. Each marker is placed at the corresponding bin-averaged diameter, left angle bracket upper D divided by eta right angle bracket $\langle D/\eta \rangle$ . Equation (3.2) is evaluated using r equals left angle bracket r Superscript b Baseline right angle bracket equals 0.97 upper D $r=\langle r^b \rangle = 0.97D$ , where left angle bracket r Superscript b Baseline right angle bracket $\langle r^b \rangle$ is the mean separation of bubbles during the inertial deformation stage. For comparison, the unconditioned inertial-range velocity scale, delta u Subscript upper L Baseline almost equals StartRoot upper C 2 EndRoot left parenthesis left angle bracket epsilon right angle bracket upper D right parenthesis Superscript 1 divided by 3 $\delta u_L \approx \sqrt {C_2}(\langle \epsilon \rangle D)^{1/3}$ , is shown by the dash-dotted line. All velocity increments are normalised by left parenthesis left angle bracket epsilon right angle bracket upper D right parenthesis Superscript 1 divided by 3 $(\langle \epsilon \rangle D)^{1/3}$ . The conditioned estimate obtained with italic We Subscript c Baseline equals upper C 2 left parenthesis 0.725 right parenthesis Superscript 5 divided by 3 $\textit{We}_c = C_2 (0.725)^{5/3}$ , corresponding to Hinze’s classical criterion (Hinze Reference Hinze1955), agrees reasonably well with the measured magnitude of the bubble deformation increment and captures its decrease with increasing upper D $D$ . By contrast, the unconditioned inertial-range estimate based on the mean energy dissipation rate underpredicts the measured increments and does not reproduce their dependence on upper D $D$ . This difference is consistent with the fact that, for the present bubble sizes, capillary stresses are not negligible: breakup therefore preferentially samples stronger-than-average extensional increments from the carrier-phase distribution.

For the bubble breakup cascade, similarly to the turbulence cascade, an intermediate bubble-size subrange may exist in which turbulent velocity fluctuations dominate over capillary stresses (Chan et al. Reference Chan, Johnson and Moin2021). This range is expected for upper D Subscript upper H Baseline much less than upper D much less than upper L $D_H \ll D \ll L$ , where upper D Subscript upper H $D_H$ is the Hinze scale, upper D Subscript upper H Baseline equals left bracket italic We Subscript c Baseline sigma divided by left parenthesis rho upper C 2 left angle bracket epsilon right angle bracket Superscript 2 divided by 3 Baseline right parenthesis right bracket Superscript 3 divided by 5 $D_H=[\textit{We}_c\sigma /(\rho C_2\langle \epsilon \rangle ^{2/3})]^{3/5}$ . The bubble sizes in the present experiments are of the same order as upper D Subscript upper H $D_H$ , so the data do not lie deeply within this intermediate bubble-size subrange. In this regime, capillary stresses remain important and strongly affect the threshold u Subscript c $u_c$ , and therefore the conditioning in (3.2). This is illustrated in the inset of figure 5(b), where upper D $D$ is normalised by upper D Subscript upper H $D_H$ . As upper D divided by upper D Subscript upper H $D/D_H$ increases, the threshold decreases and left angle bracket delta Subscript r Baseline u Subscript upper L Superscript f Baseline vertical bar delta Subscript r Baseline u Subscript upper L Superscript f Baseline greater than or slanted equals u Subscript c Baseline right angle bracket $\langle \delta _r u^f_L \mid \delta _r u^f_L \geqslant u_{c} \rangle$ approaches a finite value, indicating that capillary effects become progressively less restrictive. The red-shaded region marks italic We greater than 10 $\textit{We} \gt 10$ , computed from upper D divided by upper D Subscript upper H Baseline equals left parenthesis upper W e divided by italic We Subscript c Baseline right parenthesis Superscript 3 divided by 5 $D/D_H = (We/\textit{We}_c)^{3/5}$ . At sufficiently large upper D divided by upper D Subscript upper H $D/D_H$ , the inertial-range estimate is expected to become an adequate description of the characteristic velocity increment, as predicted by the Kolmogorov–Hinze framework. At smaller upper D divided by upper D Subscript upper H $D/D_H$ , however, stronger positive deviations from the mean inertial-range scaling are required to overcome capillary stresses, so a closure based on the conditioned carrier-phase p.d.f., script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L Baseline left parenthesis delta Subscript r Baseline u Subscript upper L Superscript f Baseline right parenthesis $\mathcal{P}_{\delta _r u_L}(\delta _r u_L^f)$ , is more appropriate than the unconditioned estimate. We note that the conditioned estimate in the inset is not expected to converge exactly to StartRoot upper C 2 EndRoot $\sqrt {C_2}$ at large upper D divided by upper D Subscript upper H $D/D_H$ . The dash-dotted line is based on the simplified estimate delta Subscript upper D Baseline u Subscript upper L Baseline almost equals StartRoot upper C 2 EndRoot left parenthesis left angle bracket epsilon right angle bracket upper D right parenthesis Superscript 1 divided by 3 $\delta _D u_L\approx \sqrt {C_2}(\langle \epsilon \rangle D)^{1/3}$ , whereas the corresponding second-order moment scaling is left angle bracket left parenthesis delta Subscript upper D Baseline u Subscript upper L Baseline right parenthesis squared right angle bracket equals upper C 2 left parenthesis left angle bracket epsilon right angle bracket upper D right parenthesis Superscript 2 divided by 3 $\langle (\delta _D u_L)^2\rangle =C_2(\langle \epsilon \rangle D)^{2/3}$ . We use the simplified estimate only as a reference velocity scale, because the present dataset is not large enough to evaluate a comparable second-order moment for the bubble-deformation increments. Finally, we emphasise that the present experiments cover a limited range of upper D divided by upper D Subscript upper H $D/D_H$ . Further measurements over a broader range of bubble sizes and italic We $\textit{We}$ are needed to test whether (3.2) remains valid in other regimes.

4. Discussion

The role of the velocity increment can be further understood through the K62 refined similarity hypothesis (Kolmogorov Reference Kolmogorov1962; Oboukhov Reference Oboukhov1962). The longitudinal velocity increment may be written as delta Subscript r Baseline u Subscript upper L Superscript f Baseline equals upper V Subscript upper L Baseline left parenthesis epsilon Subscript r Baseline r right parenthesis Superscript 1 divided by 3 $\delta _r u^f_L = V_L (\epsilon _r r)^{1/3}$ (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997), where upper V Subscript upper L $V_L$ is a dimensionless stochastic variable and epsilon Subscript r $\epsilon _r$ is the local average of the energy dissipation rate over a volume of size r cubed $r^3$ (Iyer et al. Reference Iyer, Sreenivasan and Yeung2017). Since epsilon Subscript r Baseline greater than 0 $\epsilon _r\gt 0$ , the sign of delta Subscript r Baseline u Subscript upper L Superscript f $\delta _r u^f_L$ is entirely determined by upper V Subscript upper L $V_L$ . Thus, large local dissipation can increase the magnitude of the forcing, but it does not guarantee breakup. The local velocity increment must also be sufficiently extensional to separate the daughter-associated subvolumes. This distinction suggests that deformation-conditioned events are better described by the distribution of local longitudinal increments, which contains both the fluctuations of epsilon Subscript r $\epsilon _r$ and the signed contribution of upper V Subscript upper L $V_L$ . Connecting breakup with intermittency remains a direction for future work. For bubbles well below the Hinze scale, the threshold u Subscript c $u_c$ lies deep in the positive tail of the longitudinal-increment distribution, so the breakup frequency should become sensitive to intermittency corrections to the tail of script upper P Subscript delta Sub Subscript r Subscript u Sub Subscript upper L $\mathcal{P}_{\delta _r u_L}$ .

In conclusion, bubble breakup in turbulence can be characterised as a ballistic relative-dispersion process, similar to the short-time separation of tracer-particle pairs. This analogy, however, does not imply that breakup samples ordinary turbulent motion. With italic We equals script upper O left parenthesis 1 right parenthesis $\textit{We} = \mathcal{O}(1)$ , breakup is biased towards extensional events in which local velocity increments are sufficiently strong to separate the daughter-associated subvolumes. In this sense, the Kolmogorov–Hinze framework correctly identifies velocity increments as central to turbulent breakup; however, our results suggest that breakup models for bubbles near or below the Hinze scale should account for the signed probability distribution of longitudinal velocity increments and its conditioning by capillary stresses, rather than relying solely on the inertial-range second-order moment estimate.

Acknowledgements

The authors acknowledge TU Wien for financial support through its Open Access Funding Programme. We thank F. Sortini, V. Giurgiu and O.M.H. Rodriguez for inspiring discussions and suggestions.

Funding

The funding from the Faculty of Mechanical and Industrial Engineering, TU Wien, Sonderprojekt No. A230719_UP_E322 is gratefully acknowledged. A.S. gratefully acknowledges financial support from the European Union-NextGenerationEU PNRR M4.C2.1.1 - PRIN 2022, ‘The fluid dynamics of interfaces: mesoscale models for bubbles, droplets, and membranes and their coupling to large scale flows’ 2022R9B2MW - G53C24000810001.

Declaration of interests

The authors report no conflict of interest.

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Figure 0

Figure 1. Figure 1 long description.Experimental set-up and measurement procedure. (a) Schematic of the jet-stirred turbulent water tank. (b) Arrangement of the five high-speed cameras and (c) photograph of the apparatus. Each camera has an opposing LED panel, not shown here. (d) Example of raw backlight images and corresponding segmented particle and bubble images. (e) Reconstructed bubble together with surrounding tracer-particle trajectories.

Figure 1

Figure 2. Turbulence characterisation and experimental scales. (a) Eulerian second-order structure function, S2E$S_2^E$, and longitudinal component, DLL$D_{\textit{LL}}$, with inertial-range scaling shown for reference. (b) Estimates of the mean energy dissipation rate from second- and third-order structure functions; the shaded region indicates the estimated extent of the inertial range. (c) Summary of the relevant experimental length and time scales. The grey histogram indicates the bubble size distribution.

Figure 2

Figure 3. Figure 3 long description.(a) Reconstructed breakup event of a bubble with diameter D=4.3$D=4.3$ mm. The two daughter-associated subvolumes, Ω1$\varOmega _1$ and Ω2$\varOmega _2$, are shown together with their centroids, X1$\boldsymbol{X}_1$ and X2$\boldsymbol{X}_2$; red and grey lines indicate the corresponding centroid trajectories. (b) Evolution of the separation r(t)=|X2(t)−X1(t)|$r(t)=|\boldsymbol{X}_2(t)-\boldsymbol{X}_1(t)|$, coloured by the velocity-increment magnitude δu=|u2(t)−u1(t)|$\delta u=|\boldsymbol{u}_2(t)-\boldsymbol{u}_1(t)|$. Black and white markers indicate the onset of the inertial deformation stage and breakup, respectively. The inset shows the corresponding alignment angle θ$\theta$ between r$\boldsymbol{r}$ and δu$\delta \boldsymbol{u}$.

Figure 3

Figure 4. (a) Normalised mean squared separation of deforming bubbles (blue markers) and particle pairs (red lines) as a function of t/t0$t/t_0$. From light to dark red, the particle-pair initial separations are r0=10η$r_0 = 10\eta$, 20η$20\eta$, 30η$30\eta$, 40η$40\eta$, 50η$50\eta$, 100η$100\eta$, 200η$200\eta$ and 300η$300\eta$. The inset shows the mean bubble separation with all events aligned at breakup, (t−tb)/τη=0$(t-t_b)/\tau _\eta =0$. (b) Mean alignment angle θ$\theta$ between the separation vector and velocity increment for deforming bubbles and particle pairs. The inset shows the breakup-aligned evolution of ⟨θb⟩$\langle \theta ^b\rangle$; shaded regions indicate the average duration of the inertial deformation stage and the estimated capillary time.

Figure 4

Figure 5. (a) P.d.f. of the normalised longitudinal velocity increment, δruL/(r⟨ϵ⟩)1/3$\delta _r u_L/(r\langle \epsilon \rangle )^{1/3}$, for bubble-deformation data during the inertial deformation stage (blue), particle pairs with r=122η±2η$r=122\eta \pm 2\eta$ (red) and the Chevillard model (Chevillard et al.2006) (black). The grey region indicates positive increments satisfying δruLf⩾uc$\delta _r u_L^f\geqslant u_c$. (b) Normalised mean bubble-deformation increments (blue markers) compared with the conditioned estimate from (3.2) (red line) and the inertial-range estimate (dash-dotted line). The inset shows the conditioned estimate as a function of D/DH$D/D_H$; the red-shaded region marks We>10$\textit{We}\gt 10$.