1. Introduction

Fluid flows are rarely without dispersed buoyant elements, ranging from rising bubbles to settling droplets and heavy particles. Among these dispersed buoyant bodies, the bubbles are incredibly effective at transporting gases, nutrients and heat – from aeration tanks where bubbly plumes help mix reactants, to the upper ocean where the bubbles redistribute dissolved gases and nutrients (Lohse Reference Lohse2018; Ni Reference Ni2024; Legendre & Zenit Reference Legendre and Zenit2025). As they ascend through the liquid, their wake-induced periodic zig-zags and spiralling paths stir up the liquid neighbourhood (Magnaudet & Eames Reference Magnaudet and Eames2000; Ern et al. Reference Ern, Risso, Fabre and Magnaudet2012; Mathai et al. Reference Mathai, Prakash, Brons, Sun and Lohse2015; Loisy & Naso Reference Loisy and Naso2017). When a swarm of such bubbles rise, what ensues is vigorous, turbulent mixing of the liquid (Risso Reference Risso2017; Lohse Reference Lohse2018; Mathai, Lohse & Sun Reference Mathai, Lohse and Sun2020; Ni Reference Ni2024; Legendre & Zenit Reference Legendre and Zenit2025), often referred to as bubble-induced turbulence (BIT) or pseudo-turbulence (Mazzitelli & Lohse Reference Mazzitelli and Lohse2009; Innocenti et al. Reference Innocenti, Jaccod, Popinet and Chibbaro2021; Pandey, Mitra & Perlekar Reference Pandey, Mitra and Perlekar2023). Bubble-induced turbulence is remarkably efficient at transporting momentum, heat and dissolved species, often rivalling even the mixing rates achieved in classical turbulence at a comparable energy input (Alméras et al. Reference Alméras, Risso, Roig, Cazin, Plais and Augier2015; Alméras et al. Reference Alméras, Mathai, Sun and Lohse2019; Wang, Mathai & Sun Reference Wang, Mathai and Sun2019).

Transport and mixing by turbulent fields is typically quantified from an Eulerian viewpoint, and given a long enough time approach Fickian behaviour (Taylor Reference Taylor1915; Richardson Reference Richardson1926; Batchelor Reference Batchelor1949). However, a Lagrangian perspective – of following fluid parcels and particles – can help us see further into the mechanisms that underlie this emergent diffusive law. For fluid tracers in isotropic turbulence, Taylor (Reference Taylor1922) showed that the mean-squared displacement (MSD) is given by

(1.1) \begin{equation} \sigma ^{2}(\unicode{x1D6E5} _\tau x) \approx \begin{cases} \sigma ^{2}(u_{\!f})\,\tau ^{2}, & \text{for } \tau \ll T_{L,u_{\!f}}\; \text{(ballistic)},\\[6pt] 2\,\sigma ^{2}(u_{\!f})\,T_{L,u_{\!f}}\,\tau , & \text{for } \tau \gg T_{L,u_{\!f}}\; \text{(diffusive)}, \end{cases} \end{equation}

where $\sigma ^{2}(\unicode{x1D6E5} _\tau x) \equiv \big \langle \bigl (\unicode{x1D6E5} _{\tau }x - \langle \unicode{x1D6E5} _{\tau }x\rangle \bigr )^{2}\big \rangle$ . Here, $\unicode{x1D6E5} _{\tau }x$ is the displacement along $x$ direction during a time lag $\tau$ , $\langle \,\boldsymbol{\cdot }\,\rangle$ denotes an average over trajectories and $T_{L,u_{\!f}}$ is the Lagrangian integral time scale of the flow. Further, $\boldsymbol{u}_{\!f}$ is the fluid velocity, and the subscript $f$ denotes fluid (liquid) phase quantities, while a subscript $b$ will be reserved for bubble quantities to be discussed later. In essence, for fluid turbulence the correlated movements persist for a finite interval of time, $\tau \ll T_{L,u_{\!f}}$ ; however, for longer interval spans comparable to $T_{L,u_{\!f}}$ , the correlations reduce exponentially until giving rise to a ballistic-to-diffusive transition for $\tau \gg T_{L,u_{\!f}}$ , with a turbulent diffusion coefficient $\approx 2\,\sigma ^{2}(u_{\!f})\,T_{L,u_{\!f}}$ .

Taylor’s analysis provided an elegant, explicit link between Lagrangian dispersion and Eulerian concentration fields. (Note that the discussion here concerns turbulent dispersion (Taylor Reference Taylor1922), which is distinct from the Taylor–Aris dispersion in pipe or channel flows (Taylor Reference Taylor1953)). One could go a step further and ask how the turbulence disperses a vertically drifting particle. For heavy particles that sink through atmospheric turbulence, Csanady (Reference Csanady1963) provided us with the answer: a reduced diffusivity, owing to the effect of crossing trajectories (Yudine Reference Yudine1959; Sabban & van Hout Reference Sabban and van Hout2011). In contrast, the dispersion of buoyant bubbles has remained largely unexplored. For isolated millimetric bubbles in nearly homogeneous and isotropic turbulence (HIT), Mathai et al. (Reference Mathai, Huisman, Sun, Lohse and Bourgoin2018) provided the first measurements, revealing a ballistic-to-diffusive transition occurring at a much earlier time compared with the liquid. But how does a rising bubble disperse in still liquid? And for a bubble within a bubble swarm (BIT), what kind of dispersion behaviour should we expect? These are precisely what Huang et al. (Reference Huang, Hessenkemper, Tan, Ni, Sommer, Bragg and Ma2025) have set forth to study, by exploring the dispersion of an isolated bubble in quiescent fluid and then comparing it with the dispersion of bubbles within BIT (figure 1 a).

Figure 1. Trajectories of rising bubbles in three different experimental configurations: (a) a bubble rising in quiescent liquid (left) and in a bubble swarm (right) at $\alpha = 1.2\,\%$ . (b) An isolated bubble rising through nearly homogeneous isotropic turbulence. The trajectories are coloured by bubble velocity magnitude in (a). Figures and data in (a) and (b) were adapted from Mathai et al. (Reference Mathai, Huisman, Sun, Lohse and Bourgoin2018) and Huang et al. (Reference Huang, Hessenkemper, Tan, Ni, Sommer, Bragg and Ma2025), respectively. Here La refers to the larger bubbles and $\boldsymbol {u}_b$ is the instantaneous rise velocity of the bubble.

Figure 2. Mean-squared displacement in the horizontal direction for bubbles in HIT (a) and that of bubbles and liquid (tracers) in BIT (b), as a function of time lag, $\tau$ . In (a) the MSD is normalised by the standard deviation of the bubble velocity, which collapses the short-time ballistic part. Figures adapted from Mathai et al. (Reference Mathai, Huisman, Sun, Lohse and Bourgoin2018) and Huang et al. (Reference Huang, Hessenkemper, Tan, Ni, Sommer, Bragg and Ma2025). Here Sm refers to the smaller bubbles and $\textit{Re}_\lambda$ is the Taylor Reynolds number.