Coalescence of bubbles in a high Reynolds number confined swarm

J. Fluid Mech. (2022), vol.944, A13, doi:10.1017/jfm.2022.492 Coalescence of bubbles in a high Reynolds number confined swarm J. Ruiz-Rus1,†,P.Ern 1,V.Roig 1and C. Martínez-Bazán2,3 1Institut de Mécanique des Fluides de Toulouse, Université de Toulouse – CNRS, France 2Departamento de Mecánica de Estructuras e Ingeniería Hidráulica, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain 3Andalusian Institute for Earth System Research, University of Granada, Avda. del Mediterráneo s/n, 18006 Granada, Spain (Received 9 January 2022; revised 15 May 2022; accepted 30 May 2022) We investigate experimentally the coalescence cascade process for a confined swarm of deformable bubbles immersed in a bidimensional vertical cell filled with water. For different gas volume fractions, air bubbles of size D0larger than the cell thickness are injected at the bottom of the cell. The bubbles swarms transformation is explored using high-speed visualizations. The time evolution of each bubble in the swarm is determined using a specifically developed algorithm, enabling bubble tracking and coalescence detection. We determine the evolution of the bubble size distribution downstream from the injection point, and show that the stages of the coalescence cascade are characterized by the diameter, DV90, representative of the largest bubbles. The collision frequency of pairs of bubbles of sizes Dkand Dk,h(Dk,Dk), and their coalescence efficiency, λ,are obtained from the experiments. The efficiency is nearly constant, independently of the bubble sizes and of the gas volume fraction. Concerning collision frequency, our results reveal the existence of two different coalescence regimes depending on the capability of the bubbles to deform. Models describing h(Dk,Dk)for both regimes are provided. They take into account the specific response of the bubble pair, which depends on the reduced diameter Dp=2DkDk/(Dk+Dk), to the global swarm-induced agitation governed by DV90 and the gas volume fraction. In the first regime, occurring for smaller Dp, bubbles are brought together by agitation and rapidly coalesce, while for sufficiently large Dp,both bubbles are able to deform and spend more time adapting mutually their shapes before coalescing. Key words: breakup/coalescence, gas/liquid flow, bubble dynamics †Email address for correspondence: [email protected] © The Author(s), 2022. Published by Cambridge University Press. 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 in any medium, provided the original work is properly cited. 944 A13-1 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 1. Introduction The dynamics of gas bubbles in liquids drives a wide variety of operations in the chemical process industry, mineral processing and food industry, among many other examples. It also leads the mass exchange between ocean and atmosphere and the generation of aerosols (Deane & Stokes 2002). Breakup and coalescence of bubbles is commonly present in most applications, and equipment is designed based on the understanding of the gas–liquid interaction phenomena and on the bubble’s behaviour. Thus, a profound knowledge of interaction between bubbles as well as between the gas and the liquid phases is crucial to understand the processes and optimize their performance. Bubble fragmentation has been extensively investigated under different flow configurations, and different models have been proposed (Tsouris & Tavlarides 1994; Martínez-Bazán, Montañés & Lasheras 1999a; Wang, Wang & Jin 2003; Qi, Masuk & Ni 2020, among many others);however, there is still a large amount of scientific effort devoted to this topic. Coalescence is also essential to describe the dynamics and evolution of a population of bubbles. It is usually defined as a three-step process involving three different mechanisms (Prince & Blanch 1990; Chesters 1991). The first step consists of bringing close together the bubbles involved in the process, typically two of them. This step is controlled by the external liquid flow that induces the bubbles motion and causes them to collide. Once the bubbles are in contact, in order to coalesce, it is necessary to drain the thin liquid film separating them. The last stage initiates when the film becomes thin enough so that inter-molecular forces, such as van der Waals ones for pure fluids, become dominant, breaking the liquid film and thus making the bubbles coalesce. Considerable effort has been also devoted to better understand the underlying physics that characterizes the growth of the neck which forms just after a hole appears on the drained liquid film between coalescing bubbles or drops (Eggers, Lister & Stone 1999; Paulsen et al. 2014; Anthony et al. 2017;MorenoSotoet al. 2018). It should be noted that the last stage of the drainage is very fast compared with the previous ones. Due to marked contrasts of coalescence efficiency observed when physico-chemical properties of the fluids vary, most of the coalescence studies concentrate on the drainage of the liquid film once the bubbles are sufficiently close (Marrucci 1969; Chesters & Hofman 1982; Oolman & Blanch 1986; Zhang & Thoroddsen 2008; Ghosh 2009; Huisman, Ern & Roig 2012). This analysis of coalescence as a three-step process has been fruitful to build several global models combining knowledge obtained from different studies. It remains that, in a given flow configuration, steps come one after another without real discontinuity, thus,the ratios of their respective lifetimes may vary depending on their ambiguous definition. In this sense, it can thus be interesting to analyse the coalescence process as a whole, using phenomenological models that avoid the complexity of describing in detail these stages. In the present work we focus on the global process, analysing the hydrodynamics controlling the bubble coalescence in a high Reynolds number confined bubble swarm. In order to explain precisely the aim of our study we first present the modelling formalism that we adopt and the closure laws that we discuss. The evolution of sizes of a population of bubbles is often modelled by means of a Boltzmann-type partial integro-differential conservation equation (Williams 1985), ∂n ∂t+∇·(¯un)+∂(Rn) ∂v =˙ Qr+˙ Qd,(1.1) where n(v, x,t)dvdxis the probable number of bubbles with volume in the range dv about vin the spatial range dxabout xat time t,¯uis the mean velocity of bubbles of volume vat location xat time t,˙ Qrand ˙ Qdare the birth and death rates of change of the number of bubbles due to breakup and coalescence, and Ris the rate of change of 944 A13-2 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm the volume vof a bubble which, for flows with no thermal effects, may be due to mass dissolution. In the present work we do not include thermal effects and dissolution can be neglected (R=0) since the dissolution times are much larger than the characteristics residence time of bubbles for the bubble sizes considered. Equation (1.1) may depend on space and time if the problem is non-homogeneous and unsteady. A dependence on the velocities of the bubbles can also be introduced if, for a given size, possible velocities are distributed in a large range where coalescence and breakup dominant mechanisms may vary. For simpler presentation, we do not incorporate this effect in the following equations, as the flow regime that we consider does not require this supplementary complexity to be represented. When neglecting changes of volume due to a thermodynamical phase change, taking into account breakup as well as coalescence, this equation writes as (Coulaloglou &Tavlarides1977; Martínez-Bazán 1999; Marchisio & Fox 2013) ∂n(v, x,t) ∂t+∇·[n(v, x,t)¯u(v, x,t)]=˙ Qc+˙ Qb,(1.2) where ˙ Qcrepresents the sink or source terms of n(v, x,t)due to coalescence and ˙ Qbdue to breakup. Thus, the equation that determines the transport and the evolution of n(v, x,t) is the Liouville–Boltzmann’s equation. It is a generalization of Smoluchowski’s equation established for particle coagulation (Smoluchowski 1917), usually called the population balance equation (PBE) (Williams 1985). Moments of order 0 and 1 of n(v, x,t)are respectively the total number of bubbles per unit volume, N∞(x,t), whatever their sizes, and the volume fraction of the dispersed phase α(x,t)(Ramkrishna 2000; Marchisio & Fox 2013), N∞(x,t)=∞ 0 n(v, x,t)dv, (1.3) α(x,t)=∞ 0 vn(v, x,t)dv. (1.4) The coalescence and breakup rates in (1.2) read as ˙ Qc(v, x,t)=1 2v 0 λ(v−v,v)h(v−v,v)n(v −v,x,t)n(v,x,t)dv−gc(v)n(v, x,t), (1.5) and ˙ Qb(v, x,t)=∞ v f(v,v)m(v)gb(v)n(v,x,t)dv−gb(v)n(v, x,t), (1.6) where h(v, v)is the collision frequency between bubbles of volumes vand v;λ(v, v)is the collision efficiency between bubbles of volumes vand v;gc(v) is the coalescence rate of bubbles of volume vwith any other bubble; gb(v) is the breakup or fragmentation frequency of bubbles of volume v;m(v) is the number of bubbles resulting from the fragmentation of bubbles of volume v;andf(v,v) is the bubble size distribution of daughter bubbles resulting from the fragmentation of a mother bubble of volume v. In (1.5)the first integral term on the right-hand side is a source term and the second one a sink term, both due to coalescence. Similarly, in (1.6) source and sink terms due to fragmentation also contribute to the evolution of the population of bubbles. As a complement to (1.5), the coalescence rate of bubbles of volume vwith any other bubble is 944 A13-3 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán defined by gc(v) =∞ 0 λ(v, v)h(v, v)n(v,x,t)dv.(1.7) Several closure laws and models for each of these terms have been proposed in the past. Their validity is most often limited to given hydrodynamical regimes of breakup and coalescence enforced by turbulent agitation or by mean shear flow. Sometimes they also include the influence of physico-chemical properties of the liquid or of the interface. Alarge amount of information on the adopted models can be found in references such as Coulaloglou & Tavlarides (1977), Prince & Blanch (1990), Chesters (1991)and Martínez-Bazán et al. (2010) or in literature reviews such as Kolev (1993), Lasheras et al. (2002), Liao & Lucas (2009,2010). However, the present work is focused on the coalescence processes of bubbles rising in liquid initially at rest, leaving breakage out of its scope. Thus, without including bubble breakup, (1.2) reduces to ∂n(v, x,t) ∂t+∇·[n(v, x,t)¯u(v, x,t)] =1 2v 0 λ(v −v,v)h(v −v,v)n(v −v,x,t)n(v,x,t)dv −∞ 0 λ(v, v)h(v, v)n(v, x,t)n(v,x,t)dv.(1.8) In the literature there are two types of models that have been proposed for h(v, v),or for the coalescence time, including phenomenological and physical models. The physical models are mainly focused on the description of the drainage process of the liquid film separating two bubbles when they get close enough (Chesters & Hofman 1982; Oolman & Blanch 1986; Ghosh 2009; Huisman et al. 2012). They are thus limited to coalescence in stagnant liquids, unperturbed by the bubbles in motion. In contrast, the phenomenological models are introduced for moving bubbles and are based on models of collision of molecules applied in physical gas dynamics (Coulaloglou & Tavlarides 1977; Sovova 1981; Prince & Blanch 1990; Tsouris & Tavlarides 1994). In these models, the coalescence rate of bubbles of volume vwith any other bubble, defined by (1.7), is commonly reduced to the product between a collision frequency times a coalescence efficiency. The objective of this work is to determine λ(v, v)and h(v, v)from bubble coalescence experiments performed in a high-Reynolds-number swarm of bubbles injected at the bottom of a planar vertical thin-gap cell filled with liquid at rest, and to analyse their contribution to gc(v). This confined configuration favours observation of coalescence. Starting from injection, the bubbles are greater than the gap thickness, favouring their interaction since the bubbles cannot escape out of the plane. Thus, coalescence is enhanced, being the coalescence rate larger than in the three-dimensional configuration (Lundin & McCready 2009). In the regime explored here, there is no dewetting of the liquid films between the bubbles and the walls, and bubbles move at large Bond and Archimedes (or Reynolds) numbers. Thus, the cascade of sizes generated by coalescence is expected to create a complex self-induced gravity-driven agitation in the swarm. Indeed, for isolated bubbles, it is already known that bubbles whose sizes vary in a range similar to the one that we observe, exhibit contrasted oscillating paths and shapes (Kelley & Wu 1997;Roiget al. 2012; Filella, Ern & Roig 2015; Piedra, Ramos & Herrera 2015; Hashida, Hayashi & Tomiyama 2019;Pavlovet al. 2021). In the present work we do not study the statistical properties of bubble agitation, but it has to be kept in mind that the self-induced agitation 944 A13-4 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm results from wake interactions, as already discussed in a homogeneous swarm of confined bubbles where coalescence was inhibited (Bouche et al. 2012,2014). It is worth pointing out that this flow configuration finds promising applications in chemical engineering since it is expected to be an alternative reactor of intermediate size that takes advantage of the confinement to enhance mass transfer, as in monolith reactors, and of the intense bubble-induced agitation to develop satisfactory in-plane mixing (Roudet et al. 2017; Alméras et al. 2018). Some recent applications have been developed concerning light-activated reactions or cultivation of micro-algae in photo-reactors that need narrow geometries due to light absorption and attenuation, while keeping efficient mixing and mass transfer requirements (Oelgemoller 2016;Pruvostet al. 2017; Thobie et al. 2017). The paper is organized as follows. The experimental facility and the techniques developed to describe the evolution of the population of bubbles are presented in §2.A careful examination of the performances of the bubble tracking algorithm is also presented in this section. The statistical properties of the gas flow and their evolution with the bubble population, for different values of the void fraction at injection, are reported in §3.In particular, in § 3.1 we will introduce a parameter to properly characterize the evolution of bubble sizes. Then, the results of the rate of change of the population of bubbles and the measurements of the bubble collision frequency are summarized and discussed in § 3.2. The model associated with the collision frequency is introduced in § 4.Finally ,§5is devoted to conclusions. 2. Experimental facility and techniques The experimental facility used to characterize the evolution of the bubble size distribution in a high Reynolds number confined bubble swarm is presented in § 2.1. This particular flow configuration allowed a direct analysis of the whole bubble population using the shadowgraphy technique, since the planar motion prevented bubble overlap in the recording plane. In addition to the description of the operating conditions and of the shadowgraphy technique used, an overview of the image processing algorithm developed to detect, classify and track the bubbles in the swarm is given in Appendix A. 2.1. Experimental facility and operating conditions The confined bubble swarm was generated within a quasi-bidimensional vertical cell filled with distilled water at ambient temperature, with the top section open to atmospheric pressure (Bouche et al. 2012,2014). The cell consists of two parallel glass plates (800 mm high and 400 mm wide) separated by a thin gap of width w=1mm(figure 1a). Unlike in previous works related to confined bubble swarms (Bouche et al. 2012,2013, 2014; Alméras et al. 2016,2018), no electrolyte was added to the liquid so that bubble coalescence was not inhibited. In addition, the distilled water was regularly renewed to prevent interface contamination. Air bubbles of uniform size were periodically injected from the bottom of the cell through an array of 16 capillary tubes of 0.6 mm inner diameter and 0.8 mm outer diameter (figure 1a). The tubes were equally distributed along the bottom of the cell and connected to a controlled pressure air feeding chamber. The pressure drop along the air injection tubes was sufficiently large to ensure a constant air flow rate along the tubes (Gordillo, Sevilla & Martínez-Bazán 2007). The bubble detachment frequency, fb, was accurately selected firstly setting a certain pressure in the air feeding chamber, pg, and, secondly, controlling the air flow rate through each tube using individual valves. 944 A13-5 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán (a) (b) 20 mm w = 1 mm Backlight 800 mm Gas injector Camera g 400 mm Uz Figure 1. (a) Sketch of the experimental facility, showing the field of view of size 358.40 mm ×179.20 mm for one of the camera recording positions. The zoomed area schematizes the lateral view of the cell with a bubble flattened between the side walls. (b) Example of image taken in one of the three different vertical positions of the camera. This frequency was checked for each injector using a stroboscopic light before running each experiment. The volume of the injected bubbles ensured that their sizes were always larger than the width of the gap, w. Therefore, the bubbles were flattened between the cell walls, forming a thin liquid film between the bubble interface and the wall (see zoomed area in figure 1a). In this configuration, independently of the bubble size distribution of the swarm, no dewetting at the glass plates was observed and the bubbles degrees of freedom were bounded. The flow above the bubbles goes around sideways, and does not enter the thin liquid films at rest as in Pavlov et al. (2021). Therefore, these thin films are not expected to play any role in the bubble coalescence processes described in the present work. Hence, a two-dimensional description of the motion can be adopted as in Roig et al. (2012)and Filella et al. (2015). In that sense, every bubble in the swarm is described by an equivalent diameter which is defined as D=(4Ab/π)1/2,with Abthe projected area of the bubble on the cell plane. The injected gas volume fraction, α0=ΣiAi bw/(LxLzw), was determined from the total volume occupied by all the bubbles in a measuring window a few millimetres above the capillary tubes (W1infigure 2), where Lz=50.83 mm and Lx=328.67 mm are respectively the height and the width of the window. In the current experiments, α0was varied from 2.4%to6.7 % by adjusting the bubble generation frequency. Table 1 shows the experimental conditions of the four experimental sets considered in the present work, including the mean equivalent diameter of the injected bubbles, D0. The variations in the sizes of the bubbles generated by each tube was negligible and a monodispersed bubble swarm was initially formed, as in Bouche et al. (2012,2014). Additionally, the total air flow rate was estimated as Qg=4πD02wfb, showing a linear increase with α0. The bubble swarm at the bottom of the cell is characterized by α0and by the non-dimensional parameters governing the motion of an isolated bubble of equivalent diameter at injection, D0. These include the Archimedes number, Ar0=√gD0D0/ν,the confinement ratio, δ0=w/D0, and the Bond number Bo0=ρgD2 0/σ, where gis the 944 A13-6 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm 514.58 W15 W14 W13 W12 W11 W10 W9 W8 W7 W6 W5 W4 W3 W2 W1 Lx Lz 463.75 412.92 324.58 273.75 z (mm) 222.92 142.58 91.75 40.92 0 Figure 2. General view of the three recording positions for α0=3.2 %. The 15 measuring windows used for the spatial discretization are superimposed on the images. The height of each measuring window is Lz= 50.83 mm while its width almost comprises the whole transverse spanwise of the cell, Lx=328.67 mm. The vertical axis denotes the position of the middle point of some of the measuring windows. gravity, νand ρthe liquid kinematic viscosity and density, wthe thickness of the cell and σthe surface tension. Under the conditions reported here, these parameters lie in the following ranges: 630 ≤Ar0≤850, 0.24 ≤δ0≤0.29, and 1.79 ≤Bo0≤2.11. For confined bubbles of equivalent diameter D≥D0, the mean rise velocity of an isolated bubble can be estimated by (Filella et al. 2015) Ub≃0.75(w/D)1/6gD,(2.1) 944 A13-7 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán α0D0pgfbQg (%) (mm) (bar) (s−1)(m 3s−1)×106 Set 1 2.4 3.65 ±0.20 0.6 7 1.18 ±0.13 Set 2 3.2 3.68 ±0.22 0.7 9 1.53 ±0.19 Set 3 4.9 3.85 ±0.28 0.8 13.5 2.51 ±0.38 Set 4 6.7 3.96 ±0.23 0.9 18 3.54 ±0.42 Table 1. Injection conditions of the four experimental sets considered in the present work: α0,gasvolume fraction at the bottom of the cell; D0, mean equivalent diameter of the bubbles injected; pg, pressure at the air feeding chamber; fb, bubble generation frequency; Qg, air flow rate. which in dimensionless form can be expressed as, Re ≃0.75δ1/6Ar,(2.2) where Re =UbD/ν and Ar =√gDD/ν are the Reynolds and Archimedes number of a bubble of size D. The bubble Reynolds number, Re0=UbD0/ν, can then be defined, and ranges here from 380 to 500. The gap Reynolds number, Re0δ02, can then also be introduced to assess the inertial regime, as it varies between 28 and 32. Thus, the flow can be considered to be dominated by inertia (Bush & Eames 1998) for all bubble sizes involved in the swarm. Bubbles at injection initially behave as isolated bubbles exhibiting oscillatory paths coupled to their unsteady wakes that generate periodic vortex shedding. Their in-plane projected shape is deformed and can be considered as an ellipse of moderate eccentricity (Roig et al. 2012; Filella et al. 2015). For further discussion, general ideas can be retained. First, the velocity disturbances induced by bubbles in the liquid are mainly parallel to the plates, except in the close vicinity of the bubbles. Then, the order of magnitude of the liquid velocity in the bubble’s wake is √gD. The wake is nevertheless strongly attenuated by shear stress at the walls within a characteristic time scale proportional to the viscous one, τν=w2/(4ν). Therefore, in the swarm the agitation in the liquid results from direct interactions of localized random flow disturbances of various sizes as in the homogeneous swarm studied by Bouche et al. (2014). The swarm of bubbles generated was recorded using shadowgraphy in a measurement region that spanned almost the entire horizontal width of the cell (figure 1). To that aim, the cell was illuminated from behind with an uniform, constant and diffused white light perpendicular to the cell plane (figure 1a). Placing the light source at one side of the cell, a camera (Photron APX) equipped with a 85 mm lens was used to take images of 210 levels of grey and of size 1024 ×512 pixels, with an exposure time of 1/2000 s, from the other side. In order to analyse the evolution of the population of bubbles as they rise, while maintaining the desired resolution, the backlight and the camera were placed at three different positions (figure 2). Transverse homogeneity of the flow was observed. Therefore, the downstream evolution of the bubble swarm was described by a statistical analysis of the bubble population characteristic parameters averaged over the horizontal width of the cell. In order to avoid possible errors due to border effects, the analysis was performed in a recording region, placed in the middle of each image, which consisted of a rectangle of size 328.67 mm ×152.49 mm with a pixel-size resolution of 350 μm(figure 1b). Moreover, to increase the spatial resolution of the measurements, the recording region was divided into five windows of horizontal length Lx=328.67 mm and vertical length Lz=50.83 mm, with 50 % of overlapping, as indicated in figure 2. Two types of measurements were performed, depending on the image acquisition frequency. First, a series of experiments 944 A13-8 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm taking video images of the swarm at 250f.p.s. were conducted. This recording rate was high enough to track the bubbles as they rose along the field of view. Thus, bubble collisions could be detected and tracked to determine whether the colliding bubbles coalesced, generating new larger bubbles, or eventually separated, which allowed us to determine the bubble collision frequency, h,aswellastheefficiency,λ. Measurements also allowed us to detect breakup events, giving birth to smaller daughter bubbles, although this phenomenon was rare in this study. For this analysis, high-speed movies of 25 s duration were recorded at the three positions. The total duration of the recordings included between 20 000 and 75000 bubble records per position, depending on the injection conditions and on the measuring location. In addition to the experiments recorded at high frequency, at each recording position, sets of around 3000 uncorrelated images were recorded at a frame rate of 1/2f.p.s. to ensure that the bubbles in one image were not recorded in the following one. Thus, the total number of bubbles detected at each position varied between 25 000 and 250 000, depending on the injected air flow rate and on the recording position. This ensured a statistically converged and unbiased measurement of parameters of the bubble population that can be obtained from low frequency experiments such as the volume probability density function. Satisfactory comparison of the information that could be obtained from both types of measurements indicated that the statistical parameters extracted only from the high-frequency records were indeed robust and meaningful. Details of the digital image processing methods specifically developed in this work to detect and classify the bubbles in the swarm are given in §A.1. In addition, the techniques designed to track the bubbles and detect the collisions, as well as the coalescence and breakup events are described in §A.2. Additional information can be found in Ruiz-Rus (2019). 2.2. Performance of the bubble tracking algorithm (BTA) The results obtained with the bubble tracking algorithm (BTA) described in Appendix A, consist of the record of the bubbles along the field of view for each position. The information stored for each bubble includes the projected area (bubble volume), the centroid location, the bubble velocity components, as well as the bubble lifespan and the types of birth and death events. In addition, family trees are established for each newly generated bubble, including the parents in a birth from coalescence, or the mother and the sibling in a birth from breakage. The performance of the BTA algorithm can be estimated, first, from the fact that more than 99 % of the detected bubbles can be tracked, the remaining ones being associated to specific events with simultaneous coalescence and breakup in agglomerates of bubbles. As an example, figure 3 shows a set of bubble trajectories for each injection condition at the first recording position. In this figure, regardless of where the bubble trajectories begin, they have been displaced to the same origin, with xoand zobeing the initial positions of the bubbles. It can be observed that the horizontal dispersion of the bubbles increases with the injected air volume fraction and with the vertical position, showing the effects of the liquid velocities induced by the wakes of the population of bubbles. The bubble lifespan is typically larger for the lowest values of α0(figure 3a,b), since the number of coalescence events is still low at this recording position. The trajectories show the characteristic path oscillations described by Roig et al. (2012) for isolated bubbles, indicating the weak effect of the hydrodynamic interactions at these low void fractions. However, the degree of coalescence substantially increases with α0, resulting in much shorter trajectories (figure 3c,d). 944 A13-9 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 025 8 1.0 0.8 0.6 0.4 0.2 0 (a)(b) 7 6 5 4 3 2 1 0 z/D0DV90/D0 DV90/D0 50 75 100 125 150 1 2 3 4 5 6 7 8 N∞ * U ¯z/N∞ * U ¯z(0) α0 = 6.7 % α0 = 4.9 % α0 = 3.2 % α0 = 2.4 % Figure 8. (a) Evolution of DV90 with the downstream location, both normalized with the corresponding bubble injection diameter D0, for the different injection conditions. Note that DV90/D0remains unchanged until the coalescence process starts, leading to larger bubbles. (b) Evolution of the flux of bubbles normalized with that at the first measuring window, N∗ ∞¯ Uz(0), as a function of DV90/D0. corroborating that the population of bubbles follow similar coalescence cascade processes and that DV90 is the proper variable to describe the evolution of the population of bubbles of the different swarms. In that sense, DV90/D0can be seen as a variable similar to the dimensionless time, t/τ , proposed by Smoluchowski (1917) to describe the Brownian coagulation of particles within nearly monodispersed systems (see Chandrasekhar 1943; Friedlander 1977, for reports of this work in English), being τthe characteristic coagulation or decay time, usually called half-life of the population. So, in the following, DV90 will be considered as the characteristic parameter to determine the evolution of the swarm. This will allow us to consider the coalescence frequency as an unknown that varies with the self-induced evolution of the population, which includes the influence of the a priori unknown velocities of each size, as well as the α0dependence. As already indicated above, the aim of the present work is to experimentally determine the coalescence frequency of a pair of bubbles of sizes Dand D,respectively, in the bubble swarm. To be able to do this, large enough size ranges have to be defined to ensure that the number of bubbles included in each range is sufficiently high to have an adequate number of colliding bubbles and, thus, obtain statistically converged results. For this purpose, we discretized the population of bubbles obtained from the experiments in different size classes, denoted as class 0, 1, 2 and so on. Every class is represented by a diameter Dk and includes bubble sizes in the range Dk−Δk/2≤D≤Dk+Δk/2, within a size bin of width Δk. The diameter Dkrepresents the middle size of the bin and corresponds to integer values of the injection bubble volume, accounting for volume-conservative coalescence events. Thus, the initial class corresponds to the injection bubbles, D0. The following class is associated to the coalescence of two bubbles of size D0,beingD2 1=2D2 0. The rest of the classes, Dk, are defined as the diameter of the bubble formed from the coalescence of two bubbles belonging to the two preceding classes, D2 k=D2 k−1+D2 k−2. Therefore, the different classes represent added volumes of the injection bubbles resulting in the following diameters: D1=√2D0,D2=√3D0,D3=√5D0,D4=√8D0,D5=√13D0, D6=√21D0,D7=√34D0and D8=√55D0. The different sizes of the bins, Δk,were chosen looking for substantial but smooth variations of the bubble characteristics. Such definition of bubble classes and their corresponding limits allowed us to have an amount of bubbles sufficiently large in each class to observe enough collision events among bubbles 944 A13-16 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm Set 1 Set 2 Set 3 Set 4 α0(%) 2.4 3.2 4.9 6.7 kD k±Δk/2 (mm) 0()3.65 ±0.27 3.68 ±0.27 3.85 ±0.32 3.96 ±0.32 1( )5.16 ±0.37 5.20 ±0.38 5.45 ±0.43 5.60 ±0.44 2( )6.32 ±0.78 6.37 ±0.78 6.67 ±0.78 6.86 ±0.81 3( )8.16 ±1.06 8.23 ±1.08 8.62 ±1.10 8.85 ±1.18 4( )—10.40 ±1.10 10.90 ±1.20 11.20 ±1.20 5( )—13.26 ±1.77 13.90 ±1.80 14.27 ±1.87 6( ♦)— —17.67 ±2.00 18.14 ±2.10 7( )— — 22.48 ±2.81 23.08 ±2.85 8()— — —29.36 ±3.43 Table 2. Description of the bubble size classes defined for each experimental set. Here, kdenotes the bubble class indicating the symbols used to represent them, Dkis the mean diameter describing the class and Δkthe width of the size bin containing the class. of different classes, ensuring a good statistical convergence of the data. Details of the bubble classes defined for the different injection conditions are listed in table 2. Considering the bubble classes defined in table 2, the number of bubbles of a certain class per unit volume, Nk, is obtained by integrating (1.3) between the size limits of the corresponding bin, Dk−Δk/2andDk+Δk/2. The evolution of the fraction of bubbles of each class, Nk/N∞, is represented in figure 9, as a function of DV90/D0for α0=6.7%, showing the contribution of the amount of bubbles of each class to the whole population in every stage of the coalescence process. Note that, as shown in the inset of figure 9,the fraction of injection bubbles (class k=0) is always larger than those of the other classes, indicating that there is still a considerable amount of bubbles of size D0remaining even far from the injection point. This fact reveals the appreciable presence of these small bubbles even at stages in which the coalescence cascade has accounted nearly 82events. After a rapid decrease during the early stages of the evolution, the fraction of injection bubbles almost stabilizes around half of the total number of bubbles. However, the evolution of the fraction of all the other classes remarkably differs from that of the initial bubbles. In fact, a certain class of bubbles does not begin to form until a pair of smaller bubbles, whose sum of volumes is equal to the volume of the forming bubble, coalesce. First, the number of bubbles of classes k>0 increases due to coalescence of smaller bubbles. Afterwards, when the number of bubbles of a given class becomes sufficiently large, they start to coalesce forming larger bubbles. Eventually, when the number of coalescing bubbles of a class kis larger than the number of bubbles that are generated from the coalescence of smaller ones, Nk/N∞decreases. Indeed, the evolution of the concentration of a certain bubble class within the swarm is driven by the balance between the coalescence rate of smaller bubbles that form bubbles of this class, and their rate of coalescence with all the others, as is clearly established by (1.5). The coalescence rate in the balance equation (1.8)is determined from the collision frequencies of pairs of bubbles, h(Dk,Dk), whose experimental values will be obtained in §3.2 andmodelledin§4. 3.2. Determination and analysis of the bubble pair collision frequency, h(Dk,Dk) Considering the discretization in bubble classes given in table 2, the number of bubbles per unit volume of a given class kthat die per unit time due to coalescence represents the 944 A13-17 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 0.30 1.0 0.8 0.6 0.4 0.25 0.20 0.15 0.10 Nk/N∞ 0.05 123 DV90/D0 456 2468 78 0 Figure 9. Evolution of the fraction of bubbles of each class with DV90/D0for α0=6.7 %. Similar values are obtained for the other injection conditions. The fraction of bubbles belonging to the injection class is displayed in the inset for clarity. The symbols represent the different bubble classes according to table 2. coalescence death rate, commonly expressed as ˙ Dec(Dk)=−∞ 0 λ(Dk,Dk)h(Dk,Dk)n(Dk)n(Dk)dDk=−gc(Dk)n(Dk). (3.2) In (3.2), h(Dk,Dk)is the collision frequency of bubbles of class Dkwith bubbles of size Dkand has units of m3s−1,andn(Dk)is the number density of bubbles with units of m−3, thus,˙ Dec(Dk)has units of m−3s−1. Note that, since λ(Dk,Dk)=λ∞is constant in the present case, h(Dk,Dk)can be treated indistinctly as the collision or the coalescence frequency of the pair of bubbles. Having this in mind, h(Dk,Dk)was obtained from the experiments performed at high acquisition rates, applying the bubble tracking and coalescence detection algorithm described in Appendix A,as h(Dk,Dk)=Γkk NkNk.(3.3) In (3.3), Γkkis the number of bubbles of class kcolliding with bubbles of class kin the measuring window, of volume Lz×Lx×w(see figure 2), per unit time, and Nkand Nk are respectively the number of bubbles of class kand kaccounted in the volume of the measuring window. Thus, the rate of loss of bubbles of class k,˙ Dec(Dk), corresponds to the frequency at which effective collisions of bubbles of size Dkwith the rest of the bubbles take place, which, assuming that λ(Dk,Dk)=λ∞is constant (see figure 4), reduces to ˙ Dec(Dk)=λ∞Σ∞ k=0Γkk.Note that, considering all the bubbles of the distribution which collide per unit time, Γ∞=Σ∞ k=0Σ∞ k=0Γkk, the total number of coalescence events in expression (2.4), can also be obtained as γ∞/T=λ∞Γ∞/2. It is worth indicating that the models for hhave been commonly derived by making an analogy with the kinetic theory of gases (Vincenti & Kruger 1965). These models (see, e.g. the review in Liao & Lucas 2010) generally consider has the volume swept per unit time by the colliding bubbles. It is usually referred to as the collision kernel, or the coalescence kernel if the efficiency is also included (Marchisio & Fox 2013). However, taking into account that his symmetric, resulting in h(Dk,Dk)=h(Dk,Dk), in the present work it will be referred to as the bubble pair collision frequency. 944 A13-18 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm 1.2 1.0 (i) (ii) (iii) 0.8 0.6 0.4 h(Dk, Dk) (m3 s–1) 0.2 4 6 8 10 12 14 16 18 20 0 (×10–5) Dk (mm) Figure 10. Experimental measurements of the bubble pair collision frequency, h(Dk,Dk), obtained for the injection condition α0=4.9 % at a stage of the evolution of the swarm where DV90 =18.25 mm (DV90/D0= 4.74). The symbols represent the different bubble classes according to table 2. Solid symbols represent pairs of bubbles colliding in the first regime, while hollow ones denote collisions within the second regime. The series corresponding to Dk=6.67 mm and Dk=10.90 mm are not plotted for clarity. The points indicated by arrows correspond to the cases shown in figure 13. The evolution of h(Dk,Dk)is described hereafter for the experimental case corresponding to α0=4.9 %, as a representative example. Figure 10 shows the results of the bubble pair collision frequency for various bubble pairs at a stage of the evolution of the swarm where DV90 =18.25 mm (DV90/D0=4.74). The adopted representation displays the evolution of h(Dk,Dk)with Dkfor the different classes of bubbles (constant values of Dk). We should remember that, according to figure 8, since the bubble size distributions evolve in a similar way for all values of α0,DV90/D0can be used as a parameter to describe the coalescence cascade process. Considering the results corresponding to the smallest bubble size present in the distribution, Dk=3.85 mm (), a monotonous increment of the collision frequency is observed when the size of the other colliding bubble, Dk, increases. This behaviour is also initially observed for larger bubbles, i.e. larger values of Dk, up to a certain value of Dkbeyond which the frequency begins to decrease. Taking into account the interchangeability of Dkand Dk, it can be stated that the slope of the curve h(Dk,Dk)−Dkincreases with Dk. This monotonically increasing behaviour of the bubble pair collision frequency, characterized by the fact that at least one of the bubbles involved in the collision is relatively small, will be referred to as the first regime (represented by solid symbols in figure 10). In this regime, as soon as the bubbles collide, they coalesce or (in a very few cases) bounce back, but their surfaces do not deform during a certain time until they coalesce. On the other hand, as observed for Dk≥8.62 mm (,,), the evolution of the bubble pair collision frequency shows a local maximum at certain values of Dk, which decreases as Dkincreases. After this maximum a different behaviour appears, which defines a second collision/coalescence regime (represented by hollow symbols in figure 10), where the two involved bubbles are large enough to be able to deform during the coalescence process. In this regime, once the bubbles get in touch, their interfaces deform and flatten forming a thin liquid film between the two bubbles, which mainly drains sideways (Huisman et al. 2012;Pavlovet al. 2021). The time spent on the 944 A13-19 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 25 (a)(b)(c)(d) (e)(f)(g)(h) 20 15 Dk (mm) 10 5 0 25 20 15 Dk (mm) 10 5 0 25 1.2 1.0 0.8 0.6 0.4 0.2 0 20 15 Dk (mm) 10 5 0 25 20 15 Dk (mm) 10 5 0 25 20 15 Dk (mm) 10525 20 15 Dk (mm) 105025 20 15 Dk (mm) 105025 20 15 Dk (mm) 1050 25201510525 2015105025201510502520151050 (×10–5) h(Dk, Dk) (m3 s–1) 1.2 1.0 0.8 0.6 0.4 0.2 0 h(Dk, Dk) Nk (1/s) Figure 11. Contour plots of: (a–d)h(Dk,Dk)and (e–h) the product h(Dk,Dk)Nkfor α0=4.9 % at four different instants of the bubble coalescence cascade process, characterized by DV90 (indicated by dashed lines in each plot). Results are shown for (a,e)DV90 =10.93 mm (DV90/D0=2.84); (b,f)DV90 =13.96 mm (DV90/D0=3.62); (c,g)DV90 =18.25 mm (DV90/D0=4.74) and (d,h)DV90 =20.79 mm (DV90/D0= 5.40). The solid black lines in (a–d) indicate constant values of the reduced diameter Dp. bubbles surface deformation and on the liquid film drainage, before coalescence, increases the time during which the bubbles interact, leading to a reduction of the corresponding collision/coalescence frequency. Figure 11(a–d)shows the contours of h(Dk,Dk)at four different instants of the coalescence cascade process corresponding to α0=4.9 %. In this case, bubbles of diameter D0=3.85mm are initially injected and, as they coalesce while they rise, a population of bubbles of increasing diameters is generated. The values of DV90 of the bubble size distributions corresponding to panels (a–d), indicated with horizontal and vertical dashed lines, are 10.93 mm, 13.96 mm, 18.25 mm and 20.79 mm, respectively (DV90/D0=2.84; 3.62; 4.74 and 5.40). The plots clearly show that h(Dk,Dk)is a symmetric function that increases with DV90. In fact, it can be observed that the coalescence frequencies between two small or two large bubbles are small compared with the collision frequency between bubbles of intermediate sizes. Note,for instance, that the maximum coalescence frequency in figure 11(d) falls in a fringe corresponding to the coalescence of bubbles of Dk=8mm (Dk=20 mm) with bubbles of Dk=20 mm (Dk=8 mm), or to the coalescence between two bubbles of diameter Dk=Dk≈12 mm. Interestingly, a close inspection of the contour plots indicates that the colour levels nearly follow lines of constant values of Dp=1 21 Dk+1 Dk−1 =2DkDk Dk+Dk,(3.4) drawn with solid lines in figure 11(a–d). The diameter Dprepresents the diameter of a bubble whose radius of curvature is equal to the mean radius of curvature of the pair of interacting bubbles, and will be further called the reduced diameter. This diameter has been commonly used to describe the deformation of interacting bubbles and in models of drops/bubbles coalescence (Chesters & Hofman 1982;Kampet al. 2001; Neitzel & Dell’Aversana 2002). In addition to h(Dk,Dk)shown in figure 11(a–d), figure 11(e–h) 944 A13-20 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm displays the bubble pair collision frequency weighted by the number of bubbles of size Dk. This quantity represents the frequency of collision of a bubble of size Dkwith bubbles of size Dkand obviously depends on the number of bubbles of size Dkin the distribution, n(Dk). It represents the contribution of bubbles of class kto the coalescence frequency of bubbles of class k,gc(Dk), as established in (1.7). It can be seen that h(Dk,Dk)Nk, obtained experimentally as Γkk/Nkis no longer symmetric and it depends on the bubbles size distribution, having the maximum values for Dk=D0in our particular case, since the number of bubbles of size D0is larger than the number of bubbles of other sizes, as shown in figure 9. A description of the bubble pair collision frequency, h(Dk,Dk), is therefore sought after in terms of the reduced diameter, Dp. The experimental values of h(Dk,Dk)displayed in figure 10 are represented as a function of the corresponding Dpin figure 12(a). They fall on a single curve, represented by the thick solid line, which corresponds to the averaged bubble pair collision frequency along lines of constant Dpin figure 11(c). This curve clearly indicates the existence of the two different collision regimes commented above, properly distinguished now as a function of Dp. Initially, in the first regime (solid symbols), h(Dp)increases with Dpuntil it reaches a maximum value beyond which it begins to decrease with Dp(hollow symbols). In this figure, as in figure 10, the cases indicated by arrows correspond to the time series of experimental images shown in figure 13. Equivalent results are found for different values of DV90, as shown in figure 12(a) (different lines) for the instants of the coalescence cascade presented in figure 11(a–d). This result corroborates that Dpproperly captures the dependence of the collision frequency on the bubble sizes. The values of the reduced diameter at which the maximum of h(Dp)occurs, i.e. the change from the first to the second collision regime, denoted as ˜ Dp, are displayed in figure 12(b) for the different experimental cases tested. For the smallest value of the injected volume fraction, α0=2.4 %, the entire coalescence cascade took place in the first regime, without transitioning to the second one, and no data are represented in this case. It can be observed that ˜ Dpincreases with the concentration of bubbles and with DV90, following a power law given by ˜ Dp/w∝(αDV90/w)1/2as it will be commented later on in § 4. In order to better illustrate the main characteristics of the two regimes mentioned above, different coalescence events are shown in the time series of snapshots displayed in figure 13. The time intervals between images are the same for the four series. The black dots inside the bubbles denote the instantaneous position of their centroids, while the coloured ones indicate their previous positions, describing the bubbles trajectories. The cases shown are representative of the collisions taking place during the evolution of the swarm. The first three series (figure 13a–c) have been selected for the same injection condition and stage of the swarm (α0=4.9% and DV90/D0=4.74), and correspond to the points indicated with arrows in figures 10 and 12. The processes involve pairs of bubbles where the diameter of one of them is Dk=13.90 mm (), whose trajectory is represented with red dots in figure 13. The other bubbles that form the pairs have different diameters Dk, as indicated in figure 10, and their trajectories are represented by blue dots in the corresponding panels of figure 13. On the other hand, figure 13(d) represents a coalescence event in a swarm also at DV90/D0≈4.74, where the pair of bubbles are similar to those in figure 13(c), but at a higher void fraction, α0=6.7%. As specifically indicated in the figure, coalescence events belonging to both regimes have been represented. The processes shown in figure 13(a,b) represent typical collisions taking place in the first regime. It can be seen that in both cases the coalescence happens as soon as the two bubbles collide, around t=−20 ms in both series. In these cases, the 944 A13-21 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 2 00.2 0.6 1.0 1.4 1.8 2.2 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (×10–5) 46810 Dp (mm) αDV90/w α0 = 6.7 % α0 = 4.9 % α0 = 3.2 % D ˜p/w h(Dk, Dk) (m3 s–1) 12 (i) (ii) (iii) 2 1 14 16 18 (a)(b) Figure 12. (a) Bubble pair collision frequency, h(Dk,Dk), as a function of the reduced diameter, Dp.The symbols represent the experimental results shown in figure 10. Collisions taking place in the first regime are depicted with solid symbols, while hollow ones are used to represent collisions in the second regime. As in figure 10, the cases labelled as (i), (ii) and (iii), respectively, correspond to the series (a–c)infigure 13. Lines represent the averaged value of h(Dk,Dk)along isolines of Dpin figure 11 for DV90/D0=5.40 (thick dashed line); 4.74 (thick solid line); 3.62 (thick dashed-dotted line) and 2.84 (thick dotted line). (b) Dependence of ˜ Dp/wwith (αDV90/w)for the experimental cases tested. The solid line indicates that ˜ Dp/w∝(αDV90/w)1/2, according to (4.10). Here αis the local gas volume fraction. bubbles do not significantly change their shape when they interact and rapidly contact at a point before coalescing. It could be said that they meet and kiss each other. This type of coalescence was reported for unconfined configurations by Howarth (1964) and later on modelled as the ratio between the interfacial energy and the energy of collision (Sovova 1981; Tsouris & Tavlarides 1994). However, the collision shown in figure 13(c) is markedly different and belongs to the second regime. In this case, when the bubbles get close enough (after t=−100ms) they deform and flatten, with the upper bubble surrounding the lower one. They move together for a while before being able to open a hole in the liquid film that separates them, and coalesce. During this period, the lower bubble, Dk(red dots), decelerates before contacting the upper one, Dk(blue dots), using its kinetic energy to deform the interface (Chesters 1991;Kampet al. 2001) and to increase the pressure in the liquid film that forms between the two bubbles (Duineveld 1998). Thus, unlike in figure 13(a,b), the bubbles meet and dance for awhile before kissing in figure 13(c). The dancing time increases with the size of the bubbles since large bubbles are able to deform more easily, what makes h(Dk,Dk)decrease in the second regime. It is worth noting that in the earlier stages of the cascade process, i.e. for low values of DV90,theswarm is formed by relatively small bubbles which collide exclusively in the first regime. The time the bubbles take to deform and adapt their shapes before coalescing not only depends on their sizes but also on the liquid velocity fluctuations, induced by the motion of the bubbles in the swarm. Larger velocity fluctuations favour the destabilization of the liquid film separating the contacting bubbles and, thus, their coalescence. To illustrate this effect, figure 13(d) shows a pair of bubbles similar to those in figure 13(c) in a bubble swarm also characterized by DV90 ≈18.25 mm (DV90/D0≈4.74), but with a higher concentration of bubbles, α0=6.7 %. In this case, although the interacting bubbles are similar to those displayed in panel (c), the coalescing time is shorter because the liquid fluctuation velocities are larger for α0=6.7% than for α0=4.9 %. In fact, it can be observed in figure 13 that at t=−160 ms the distance between the centroids of the two bubbles is larger in (d)thanin(c). Therefore, it can be asserted that the global motion of the swarm 944 A13-22 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm Second regime First regime 10 mm 0 ms t = −160 ms −100 ms −60 ms −20 ms (a) (b) (c) (d) Figure 13. Images showing the time evolution of representative cases of the collision process at a stage of the evolution of the swarm where DV90 =18.25 mm (DV90/D0=4.74) for α0=4.9 %. They correspond to the cases denoted by (i), (ii) and (iii) in figures 10 and 12(a), with (a)Dp=7.86 mm, (b)Dp=9.64 mm, (c)Dp=15.55 mm. In (d)the bubbles belong to classes k=5andk=6(table 2)withDp=15.87 mm and for α0=6.7 %. The instantaneous location of the centroids of the bubbles are indicated with black dots. The position of the centroids in previous frames describing the trajectories of the bubbles are represented by sequences of coloured dots (only one out of three instants are plotted for clarity). The time to coalescence in each snapshot is indicated at the bottom of the figure. 944 A13-23 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán –5 –5.0 –2.5 0 2.5 5.0 7.5 –5.0 –2.5 0 2.5 5.0 7.5 0 (x – xo)/D0 (z – zo)/D0 510 –5 0 (x – xo)/D0 5 –5.0 –2.5 0 2.5 5.0 7.5 –5 0 (x – xo)/D0 5 –5.0 –2.5 0 2.5 5.0 7.5 –10 –5 (x – xo)/D0 0 (a)(b)(c)(d) Figure 14. Effect of bubble deformation in the first collision regime, illustrated by different characteristic interaction events of a small bubble of size Dkwith a larger one of size Dk, placed at the origin of coordinates, for (a)Dp=8.33 mm, (b)Dp=9.68 mm, (c)Dp=11.59 mm and (d)Dp=12.11 mm. The bubble swarm corresponds to DV90 =23.17 mm (DV90/D0=6.02) and α0=4.9 %. Both coordinates have been normalized by the corresponding bubble injection diameter D0. The arrows indicate the direction of the relative motion of Dk. is driven by buoyancy and, thus, governed mainly by the evolving bubble size distribution, characterized by DV90, and the concentration of bubbles, α0(see discussion of figure 6b in § 3.1). Furthermore, the bubble collision rate increases with the liquid perturbation velocity which increases as the bubbles coalesce and get larger, i.e. as DV90 increases. The role played by hydrodynamic interactions on the coalescence process is also illustrated in figure 14, now considering different events where the smallest bubble is above the largest one. In the figure, typical situations of bubbles interacting in the first regime are presented for different values of Dp, in a bubble swarm corresponding to α0=4.9% and DV90 =23.17 mm (DV90/D0=6.02). The relative location of the smallest bubble of the pair, Dk, is plotted as it interacts with a larger bubble, Dk. The origin of the system of coordinates (xo,zo) corresponds to the centroid of bubble Dkat each instant. The temporal evolution of the relative position of the centroid of bubble Dk, between the represented stages, is indicated by dots. Bubble pairs for increasing values of Dp(and of h(Dk,Dk)) are presented in figure 14(a–c). For a nearly constant value of Dk(largest bubble), the size Dk(smallest bubble) is progressively increased from left to right. In figure 14(a), illustrating the interaction of a small bubble with a big one, the small bubble barely deforms, and is ejected away from the larger one due to the overpressure established around its stagnation point. Consequently, this case is not considered as a collision in our analysis (see § A.1) and the collision frequency of this type of bubble is low. As Dkincreases (figure 14b,c), in addition to increasing the possible length of interaction with bubble Dk, the capability of Dkto be deformed also increases, favouring the collision between both bubbles. Besides the hydrodynamic mechanisms governing the response of bubble Dkto the flow around the top of bubble Dk, additional bubble deformation effects have been also observed when the two bubbles are sufficiently large (figure 14d). In this kind of collision, the liquid velocity field brings the bubbles sufficiently close so that bubble Dkdeforms towards the low pressure area of the wake of bubble Dk. Indeed, these final instants of the interaction process, based on the wake entrainment mechanism, lead to bubble collision when the interacting bubbles are large enough to deform (Miyahara, Tsuchiya & Fan 1991). In contrast, smaller bubbles that barely deform and respond faster to the liquid velocity fluctuations can eventually avoid the collision (Filella, Ern & Roig 2020). For figure 13(a,b), the discussion was focused primarily on the short time interaction, as well as on the wake effects of the bubbles once they were carried out close enough by the liquid motion. In contrast, the configurations shown in figure 14(a–d)focuses on 944 A13-24 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm the response of Dkto the hydrodynamic effects caused by its interaction with bubble Dk. They illustrate the influence of the bubble sizes, and of their capability to be deformed, on the increase of hwith Dpin the first regime. The cases shown in figures 13 and 14 are representative of the collision phenomenology observed in the coalescence cascade process and contribute equally to the bubble pair collision frequency obtained from the experiments. 4. Formulation of the coalescence model To model the rate of change of the population of bubbles due to coalescence, ˙ Qc,in(1.2), it is necessary to implement models for the integral kernels involved in the coalescence rate in (1.5), which consider the interaction between bubbles of different sizes, namely Dkand Dk, respectively. In this sense, it has to be taken into account that bubble coalescence includes a first approaching step, starting at distances typically larger than the bubble sizes, driven by the local transport mechanisms. Subsequently, a second step, characterized by the short-distance bubble-bubble interaction, takes place (Marchisio & Fox 2013). The whole process is commonly modelled considering both stages separately, the first accounting for the bubble collision frequency h(Dk,Dk)and the second for the coalescence efficiency term λ(Dk,Dk). The efficiency is usually expressed comparing the time required to drain and disrupt the liquid film formed between the two bubbles when they collide, td, with the time the bubbles remain in contact under the influence of the external flow, i.e. the residence time ¯ tc. As mentioned above, in the two-dimensional confined configuration of interest here, the coalescence efficiency λhas been defined as the fraction of pairs of colliding bubbles which end up coalescing. In this case, once the bubbles collide, they cannot move away in the direction perpendicular to the walls and, in most of the cases, eventually coalesce. Thus, the coalescence efficiency is high and nearly constant, being thus independent of the size of the colliding bubbles and of the external flow conditions, such as DV90 or α0 (figure 4). Consequently, the bubble collision frequency h(Dk,Dk)will be considered as the bubble coalescence frequency since both functions behave in the same way and, in the following, we will focus on developing a model for h(Dk,Dk), which jointly accounts for the local transport phenomena and for the effects derived from the bubbles deformations and their hydrodynamics interaction when they approach. In general, h(Dk,Dk)has been modelled as a characteristic relative velocity between the two colliding bubbles, ¯ur(Dk,Dk), multiplied by a characteristic cross-sectional collision area Sc(Dk,Dk), which usually depends on the size of the colliding bubbles, h(Dk,Dk)∼Sc¯ur.(4.1) In particular, Coulaloglou & Tavlarides (1977) proposed an expression for h(Dk,Dk) based on the collision of molecules in kinetic theory of gases, given by h(Dk,Dk)∼π 4(Dk+Dk)2u2(Dk)+u2(Dk)1/2 ,(4.2) where u(Dk)and u(Dk)are the root mean square of the velocity fluctuations of bubbles of sizes Dkand Dk, respectively. The relative velocity between the two bubbles can be established by different mechanisms, depending on the liquid field where the bubbles are immersed, i.e. turbulent fluctuations of the carrier fluid, size-dependent differences in the bubble rising velocities, wake entrainment or shear-layer induced velocity differences, among others. However, in the present flow where the liquid velocity is not externally imposed, the mean motion of the swarm and its agitation are driven by a gravity effect 944 A13-25 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán V. Roig https://orcid.org/0000-0002-9315-0467; C. Martínez-Bazán https://orcid.org/0000-0003-2023-4691. Appendix A A.1. Image processing methods for bubble detection and classification Since even the smallest bubbles in the swarm are already larger than the thickness of the cell, the surface of each bubble is mainly perpendicular to the direction of the light, showing up in the images as a region of connected pixels with a grey level similar to that of the liquid background, enclosed by a thin dark stripe representing the air–water interface not aligned with the cell plane (figure 19a). The thickness of the dark line that delimits the perimeter of the bubble is almost constant, regardless of its size, because the curvature of the bubble in a plane perpendicular to the field of view is constant (Bongiovanni, Dominguez & Chevaillier 2000). Detailed images of isolated bubbles within the same experimental facility and for the same range of sizes reached in this work can be found in Roig et al. (2012). Using this property, each image is analysed making use of a specifically developed image processing algorithm, followed by a bubble detection and classification method. The processing algorithm involves a first pre-processing step, followed by a second one where the image is binarized. Once the bubbles present in each image are detected, their centroid positions as well as their projected areas are measured. Figure 19 shows an example of the processing steps in one of the measuring windows. The first step of the process implies the improvement of the contrast of the original grey scale image (figure 19a). It involves the subtraction of a background reference image without bubbles and the normalization of the image brightness by correcting each value of the pixel intensity matrix. More detailed information regarding the brightness correction method can be found in Fu & Liu (2016). This brightness normalization reduces the uncertainties due to the variation of the illumination conditions and facilitates the next binarization step. Figure 19(b) shows an inversion of the resulting corrected image, where any physical noise (e.g. glass wall scratches and background noise) has been removed while the grey-level gradient between the bubbles edges and the background has been enhanced. Afterwards, the well-known Otsu’s method (Otsu 1979) is used in the second step as an automatic, robust, global binarization-threshold selection technique. After binarization, single bubbles can be detected as blobs of connected low-level pixels enclosed by unique edges of high-level pixels which are totally surrounded by background. Bubbles involved in a collision share the same edge of connected high-level pixels, however, there exists an independent blob of low-level pixels for each bubble. Unlike in other works which deal with bubble collisions or formation of clusters through a separation distance criteria (see Figueroa-Espinoza & Zenit 2005; Figueroa-Espinoza et al. 2018,for example), the contact of at least one edge pixel of each bubble is required in the present work to define a collision. This definition is crucial, since it determines the measurement of the collision rate and the efficiency of coalescence. Figure 19(c) shows detected single bubbles as filled objects and colliding bubbles as hollow ones. This procedure to detect the bubble collisions allows us to clearly distinguish between two independent bubbles involved in a collision (see the example highlighted by an arrow in figure 19)andthe newborn ones, just formed due to the coalescence of two colliding bubbles (boxed by dashed lines in figure 19). Once the bubbles have been detected and classified as single or in-collision bubbles, their instantaneous characteristics are determined. The bubble position is obtained as the centroid position of the in-side blob of low-level pixels. The projected area is defined as 944 A13-32 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm (a) (b) (c) Figure 19. Example of the image analysis algorithm showing the two-step binarization process. (a) Original grey scale image. (b) Inversion of the pre-processed image showing the brightness normalization and the improved grey-level gradient between the bubbles edges and the background. (c) Binarized image where the bubbles have been classified as single bubbles (filled objects) or as in-collision bubbles (hollow objects). A typical bubble collision is pointed by an arrow and a recently coalesced bubble is highlighted by the box with dashed frame. the area occupied by the pixels belonging to the in-side blob plus those belonging to the edge. A difficulty arises obtaining the area of the bubbles involved in collisions, since the pixels composing the edge of the agglomerate are shared among the colliding bubbles. To deal with this issue, the total area of the agglomerate edge is distributed among the bubbles according to the ratio between the number of pixels enclosing the inner perimeter of each bubble and those composing the outer perimeter of the agglomerate. Since the edge width remains constant independently of the size of the colliding bubbles, this simple procedure allows us to avoid any further computation devoted to the separation of the bubbles, as occurs in more complicated three-dimensional bubbly flows (see, e.g. Rueda Villegas et al. 2019). A.2. Bubble tracking and coalescence/breakage detection algorithm (BTA) The time evolution of each bubble in the swarm was obtained using a BTA specifically designed and developed for this work, which includes a coalescence/breakage detection algorithm. It consists of the detection of the bubbles in a frame, j, followed by the search and identification of the same bubbles in the previous one, j−1. For new bubbles, born in frame jeither due to coalescence or breakup, family trees are established between the daughter (in frame j) and the parents (in frame j−1). More precisely, the algorithm involves a first step where an image (frame j) is processed using the digital analysis described in § A.1. As a result, the bubbles in the frame are detected, obtaining the positions of their centroids as well as their projected areas. Afterwards, each bubble is classified as single or in-collision bubble. In addition to the bubbles, the detected collisions, defined as agglomerates of two or more bubbles in direct contact (see § A.1), are treated as independent entities and,thus, their characteristic parameters are also calculated, including the total number of bubbles in collision. In order to facilitate the search of corresponding objects in two consecutive frames, a bounding box 944 A13-33 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán 10 mm −108 ms −76 ms −48 ms −20 ms −8 ms −4 ms 0 ms 16 ms (a)(b)(c)(d) (e)(f)(g)(h) Figure 20. Characteristic sequence of the BTA performance showing the tracking process, superimposed on a region of the original grey scale images at different instants. The trajectories of the properly tracked bubbles are represented by a sequence of dots corresponding to the bubble centroid locations in the previous frames. (a–d) Examples of the correlation method applied to two different bubbles (red and blue, respectively), showing the positions of their centroids in the previous frame, j−1, (circle) lying inside the corresponding bounding box in frame j(dashed box). The black arrow in (b) indicates a new bubble entering the field of view. (e–f) Typical collision detected and tracked in two consecutive frames. The bounding box of the bubble agglomerate is shown with a dashed dark green rectangle in each frame and the bubbles involved are marked with coloured stars. ( f–g) Sequence of the end of a collision event due to bubble coalescence. The parent bubbles (coloured stars) give rise to a new bubble (green diamond). (h) The coalesced bubble is hereafter tracked as a single bubble (green circle). containing the target object is defined for each bubble or agglomerate of bubbles (colliding bubbles). Figure 20 shows the BTA performance superimposed on the original grey scale image at various instants, being the reference time, t=0, the frame where a coalescence event takes place (figure 20g). The trajectories of the bubbles are represented as a sequence of dots, which correspond to the locations of their centroids in the previous frames. The solid circles on each bubble denote the position of the centroid in the previous frame, j−1. The above mentioned bounding boxes are plotted in figure 20(a–d) for two different bubbles in red and blue, respectively, while those corresponding to a collision event are shown in figure 20(e,f) in green. Notice that the bounding box enclosing the collision becomes that of the newborn coalesced bubble in figure 20(g) since the collision event ends up with the coalescence of the two bubbles, as described in detail below. Once the bubbles in the images (frame j) have been detected, the key point of the tracking algorithm is to search for the corresponding ones in frame j−1. In that sense, every single and in-collision bubble must be related, at least, to one object in the previous frame. Moreover, every detected collision must be related to a previous collision or, at least, to two previous bubbles. The procedure works sequentially identifying objects, taking into account the continuity of the bubbles trajectories and the conservation of volume (projected areas) of the objects. Initially, only single bubbles in both frames are considered. Therefore, a single bubble in frame jis related to the single one in frame 944 A13-34 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press Coalescence of bubbles in a high Re confined swarm j−1 whose centroid falls inside the bounding box of the bubble in frame j.Giventhe experimental acquisition rate, this bounding-box criteria is highly effective, even for the smallest bubbles that are accelerated when they are trapped in the wake of larger ones (figure 20c,d). However, when bubbles of very different sizes get closer (without being in contact), more than a centroid detected in frame j−1 can be inside the bounding box of the larger bubble in frame j. To avoid possible errors, an additional criterion based on the conservation of volume is imposed (Rodríguez-Rodríguez, Martínez-Bazán & Montañés 2003). Therefore, the volume of the corresponding bubbles in both frames must be equal. Bubbles located near the bottom edge of the field of view in frame j, which cannot be associated with any object in frame j−1(see , e.g. the bubble highlighted by an arrow in figure 20b), are directly classified as new bubbles just entering the analysis region. Any other single bubble that cannot be related to a previous one is taken out for further analysis. The single bubbles properly tracked are stored in the data base and removed from both frames to facilitate the analysis of the colliding ones. The following step is devoted to the analysis of the agglomerates, which define collision events, detected in frame j. When several bubbles collide, the process can end with the bubbles coalescing or bouncing off each other. Thus, every collision detected in frame j, considered as a unique object, is analysed searching for the corresponding collision object in frame j−1, applying the algorithm described above to track individual bubbles. In addition to the volume of the agglomerate, the number of bubbles involved in the collision must also be conserved in both frames. If the corresponding collision is found in the previous frame, the bubbles that form the agglomerate (bubbles marked with coloured stars in figure 20e,f) are identified as well using the criteria used for single bubbles. On the other hand, if no corresponding collision is found in frame j−1, it is assumed that the collision detected in frame jis a new one occurring because several bubbles (usually two) have been brought together (figure 20e). In that case, the bubbles involved in the collision are analysed searching for the corresponding previous single bubbles leading to the collision. Once they have been processed, the agglomerates, as well as all the involved bubbles, are stored and no further action is performed with them in the current frame. At this point, the remaining bubbles in frame jare those emerging either from the breakup of a mother bubble in frame j−1orfromthedeath of a previous agglomerate. The latter gives rise to two different situations: (i) death by coalescence of the bubbles which form the agglomerate, creating a new larger bubble, or (ii) death by the separation of the involved bubbles, leading to different single bubbles in frame j. These two possibilities represent the basis of the collision efficiency concept. Situation (i) results in a efficient collision, giving rise to a coalescence event. On the other hand, (ii) indicates an inefficient collision, where the involved bubbles continue living without changes in the population. To determine this efficiency, it has to be pointed out that both situations respectively arise from a collision detected in frame j−1 which does not have a corresponding agglomerate in frame j. Therefore, a forward analysis, from frame j−1toframej,isappliedtothe remaining agglomerates in frame j−1. In this case, the correlation method used to track a single bubble is applied here for each bubble involved in the collision detected in frame j−1, searching for the corresponding bubble in frame j. Only the remaining bubbles in frame jwhose centroid falls inside the bounding box of the analysed agglomerate in frame j−1are considered. If the death of the collision is due to separation, any bubble forming the agglomerate in frame j−1 will be related to a corresponding single bubble in frame j, satisfying both the bubble bounding box as well as the bubble volume conservation criteria. However, if the collision event ends up in a coalescence, the bubble emerging from the coalescence is determined as the single bubble in frame jwhose centroid falls inside the bounding box of the analysed agglomerate in frame j−1, being its projected 944 A13-35 https://doi.org/10.1017/jfm.2022.492 Published online by Cambridge University Press J. Ruiz-Rus, P. Ern, V. Roig and C. Martínez-Bazán area equal to the sum of those of the parent bubbles. A typical coalescence event is shown in figure 20(f,g). The parent bubbles involved in the collision can be seen in frame j−1 (figure 20f), being their centroids indicated by coloured stars, while the newborn bubble is shown in frame j(figure 20g), with its centroid marked with a green diamond. From this point, the new bubble generated by coalescence is tracked as a single bubble (figure 20h). Finally, the daughter bubbles remaining in frame j, which are generated due to the breakup of a mother bubble in frame j−1, are identified through a backward–forward implementation of the correlation method, following the ideas proposed by Rodríguez-Rodríguez et al. (2003). For a potential daughter bubble in frame j, the corresponding mother bubble is searched in frame j−1 as the larger bubble whose bounding box includes the centroid of the analysed daughter one. Then, the second daughter bubble is additionally searched in frame jas that whose centroid falls inside the bounding box of the mother one and whose volume corresponds to the volume of the mother minus that of the sibling one. Therefore, both daughter bubbles in frame j are identified as new single bubbles appearing due to breakup, while the corresponding mother bubble in frame j−1isdefinedasdeath due to breakup. 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