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).
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High-resolution direct simulation of deep water breaking waves: transition to turbulence, bubbles and droplets production
We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number ( Re , wave inertia relative to viscous effects) and Bond number ( Bo , wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at high Re and Bo , and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulent Re up to a limiting value $Re_\lambda \sim 100$ , consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deike et al. ( J. Fluid Mech. , vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale, $r_H$ . We confirm two regimes in the bubble size distribution, $N(r/r_H)\propto (r/r_H)^{-10/3}$ for $r>r_H$ , and $\propto (r/r_H)^{-3/2}$ for $r
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- Award ID(s):
- 1849762
- NSF-PAR ID:
- 10414842
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 942
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2022.330
