Direct numerical simulations of bubble-mediated gas transfer and dissolution in quiescent and turbulent flows

Abstract

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 Farsoiyaet 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 Theofanouset al.(Intl J. Heat Mass Transfer, vol. 19, issue 6, 1976, pp. 613–624).