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We describe the rising trajectory of bubbles in isotropic turbulence and quantify the slowdown of the mean rise velocity of bubbles with sizes within the inertial subrange. We perform direct numerical simulations of bubbles, for a wide range of turbulence intensity, bubble inertia and deformability, with systematic comparison with the corresponding quiescent case, with Reynolds number at the Taylor microscale from 38 to 77. Turbulent fluctuations randomise the rising trajectory and cause a reduction of the mean rise velocity$$\tilde {w}_b$$compared with the rise velocity in quiescent flow$$w_b$$. The decrease in mean rise velocity of bubbles$$\tilde {w}_b/w_b$$is shown to be primarily a function of the ratio of the turbulence intensity and the buoyancy forces, described by the Froude number$$Fr=u'/\sqrt {gd}$$, where$$u'$$is the root-mean-square velocity fluctuations,$$g$$is gravity and$$d$$is the bubble diameter. The bubble inertia, characterised by the ratio of inertial to viscous forces (Galileo number), and the bubble deformability, characterised by the ratio of buoyancy forces to surface tension (Bond number), modulate the rise trajectory and velocity in quiescent fluid. The slowdown of these bubbles in the inertial subrange is not due to preferential sampling, as is the case with sub-Kolmogorov bubbles. Instead, it is caused by the nonlinear drag–velocity relationship, where velocity fluctuations lead to an increased average drag. For$$Fr > 0.5$$, we confirm the scaling$$\tilde {w}_b / w_b \propto 1 / Fr$$, as proposed previously by Ruthet al.(J. Fluid Mech., vol. 924, 2021, p. A2), over a wide range of bubble inertia and deformability. 

We investigate drop break-up morphology, occurrence, time and size distribution, through large ensembles of high-fidelity direct-numerical simulations of drops in homogeneous isotropic turbulence, spanning a wide range of parameters in terms of the Weber number We, viscosity ratio between the drop and the carrier flow μr = μd/μl, where d is the drop diameter, and Reynolds (Re) number. For μr ≤ 20, we find a nearly constant critical We, while it increases with μr (and Re) when μr > 20, and the transition can be described in terms of a drop Reynolds number. The break-up time is delayed when μr increases and is a function of distance to criticality. The first break-up child-size distributions for μr ≤ 20 transition from M to U shape when the distance to criticality is increased. At high μr, the shape of the distribution is modified. The first break-up child-size distribution gives only limited information on the fragmentation dynamics, as the subsequent break-up sequence is controlled by the drop geometry and viscosity. At high We, a d−3/2 size distribution is observed for μr ≤ 20, which can be explained by capillary-driven processes, while for μr > 20, almost all drops formed by the fragmentation process are at the smallest scale, controlled by the diameter of the very extended filament, which exhibits a snake-like shape prior to break-up. 

We perform direct numerical simulations of a gas bubble dissolving in a surrounding liquid. The bubble volume is reduced due to dissolution of the gas, with the numerical implementation of an immersed boundary method, coupling the gas diffusion and the Navier–Stokes equations. The methods are validated against planar and spherical geometries’ analytical moving boundary problems, including the classic Epstein–Plesset problem. Considering a bubble rising in a quiescent liquid, we show that the mass transfer coefficient $$k_L$$ can be described by the classic Levich formula $$k_L = (2/\sqrt {{\rm \pi} })\sqrt {\mathscr {D}_l\,U(t)/d(t)}$$ , with $d(t)$ and $U(t)$ the time-varying bubble size and rise velocity, and $$\mathscr {D}_l$$ the gas diffusivity in the liquid. Next, we investigate the dissolution and gas transfer of a bubble in homogeneous and isotropic turbulence flow, extending Farsoiya et al. ( J. Fluid Mech. , vol. 920, 2021, A34). We show that with a bubble size initially within the turbulent inertial subrange, the mass transfer coefficient in turbulence $$k_L$$ is controlled by the smallest scales of the flow, the Kolmogorov $$\eta$$ and Batchelor $$\eta _B$$ microscales, and is independent of the bubble size. This leads to the non-dimensional transfer rate $${Sh}=k_L L^\star /\mathscr {D}_l$$ scaling as $${Sh}/{Sc}^{1/2} \propto {Re}^{3/4}$$ , where $${Re}$$ is the macroscale Reynolds number $${Re} = u_{rms}L^\star /\nu _l$$ , with $$u_{rms}$$ the velocity fluctuations, $L^*$ the integral length scale, $$\nu _l$$ the liquid viscosity, and $${Sc}=\nu _l/\mathscr {D}_l$$ the Schmidt number. This scaling can be expressed in terms of the turbulence dissipation rate $$\epsilon$$ as $${k_L}\propto {Sc}^{-1/2} (\epsilon \nu _l)^{1/4}$$ , in agreement with the model proposed by Lamont & Scott ( AIChE J. , vol. 16, issue 4, 1970, pp. 513–519) and corresponding to the high $Re$ regime from Theofanous et al. ( Intl J. Heat Mass Transfer , vol. 19, issue 6, 1976, pp. 613–624). 

Bubble-mediated gas exchange in turbulent flow is critical in bubble column chemical reactors as well as for ocean–atmosphere gas exchange related to air entrained by breaking waves. Understanding the transfer rate from a single bubble in turbulence at large Péclet numbers (defined as the ratio between the rate of advection and diffusion of gas) is important as it can be used for improving models on a larger scale. We characterize the mass transfer of dilute gases from a single bubble in a homogeneous isotropic turbulent flow in the limit of negligible bubble volume variations. We show that the mass transfer occurs within a thin diffusive boundary layer at the bubble–liquid interface, whose thickness decreases with an increase in turbulent Péclet number, $$\widetilde {{Pe}}$$ . We propose a suitable time scale $$\theta$$ for Higbie ( Trans. AIChE , vol. 31, 1935, pp. 365–389) penetration theory, $$\theta = d_0/\tilde {u}$$ , based on $$d_0$$ the bubble diameter and $$\tilde {u}$$ a characteristic turbulent velocity, here $$\tilde {u}=\sqrt {3}\,u_{{rms}}$$ , where $$u_{{rms}}$$ is the large-scale turbulence fluctuations. This leads to a non-dimensional transfer rate $${Sh} = 2(3)^{1/4}\sqrt {\widetilde {{Pe}}/{\rm \pi} }$$ from the bubble in the isotropic turbulent flow. The theoretical prediction is verified by direct numerical simulations of mass transfer of dilute gas from a bubble in homogeneous and isotropic turbulence, and very good agreement is observed as long as the thin boundary layer is properly resolved.