Hostname: page-component-686fd747b7-qtbbk Total loading time: 0 Render date: 2025-02-13T11:21:23.667Z Has data issue: false hasContentIssue false
  • English
  • Français

Fundamental time scales of bubble fragmentation in homogeneous isotropic turbulence

Published online by Cambridge University Press:  03 May 2023

Declan B. Gaylo Open the ORCID record for Declan B. Gaylo [Opens in a new window] ,
Kelli Hendrickson Open the ORCID record for Kelli Hendrickson [Opens in a new window]  and
Dick K.P. Yue Open the ORCID record for Dick K.P. Yue [Opens in a new window]
Declan B. Gaylo
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Kelli Hendrickson
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Dick K.P. Yue*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: yue@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the fundamental time scales that characterise the statistics of fragmentation under homogeneous isotropic turbulence for air–water bubbly flows at moderate to large bubble Weber numbers, $We$. We elucidate three time scales: $\tau _r$, the characteristic age of bubbles when their subsequent statistics become stationary; $\tau _\ell$, the expected lifetime of a bubble before further fragmentation; and $\tau _c$, the expected time for the air within a bubble to reach the Hinze scale, radius $a_H$, through the fragmentation cascade. The time scale $\tau _\ell$ is important to the population balance equation (PBE), $\tau _r$ is critical to evaluating the applicability of the PBE no-hysteresis assumption, and $\tau _c$ provides the characteristic time for fragmentation cascades to equilibrate. By identifying a non-dimensionalised average speed $\bar {s}$ at which air moves through the cascade, we derive $\tau _c=C_\tau \varepsilon ^{-1/3} a^{2/3} (1-(a_{max}/a_H)^{-2/3})$, where $C_\tau =1/\bar {s}$ and $a_{max}$ is the largest bubble radius in the cascade. While $\bar {s}$ is a function of PBE fragmentation statistics, which depend on the measurement interval $T$, $\bar {s}$ itself is independent of $T$ for $\tau _r \ll T \ll \tau _c$. We verify the $T$-independence of $\bar {s}$ and its direct relationship to $\tau _c$ using Monte Carlo simulations. We perform direct numerical simulations (DNS) at moderate to large bubble Weber numbers, $We$, to measure fragmentation statistics over a range of $T$. We establish that non-stationary effects decay exponentially with $T$, independent of $We$, and provide $\tau _r=C_{r} \varepsilon ^{-1/3} a^{2/3}$ with $C_{r}\approx 0.11$. This gives $\tau _r\ll \tau _\ell$, validating the PBE no-hysteresis assumption. From DNS, we measure $\bar {s}$ and find that for large Weber numbers ($We>30$), $C_{\tau }\approx 9$. In addition to providing $\tau _c$, this obtains a new constraint on fragmentation models for PBE.

JFM classification

Drops and Bubbles: Breakup/coalescence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 962 , 10 May 2023 , A25
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Campbell, B. 2014 A mechanistic investigation of nonlinear interfacial instabilities leading to slug formation in multiphase flows. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Castro, A.M. & Carrica, P.M. 2013 Bubble size distribution prediction for large-scale ship flows: model evaluation and numerical issues. Intl J. Multiphase Flow 57, 131–150.CrossRefGoogle Scholar
Chan, W.H.R., Dodd, M.S., Johnson, P.L. & Moin, P. 2021 a Identifying and tracking bubbles and drops in simulations: a toolbox for obtaining sizes, lineages, and breakup and coalescence statistics. J. Comput. Phys. 432, 110156.CrossRefGoogle Scholar
Chan, W.H.R., Johnson, P.L. & Moin, P. 2021 b The turbulent bubble break-up cascade. Part 1. Theoretical developments. J. Fluid Mech. 912, A42.CrossRefGoogle Scholar
Chan, W.H.R., Johnson, P.L., Moin, P. & Urzay, J. 2021 c The turbulent bubble break-up cascade. Part 2. Numerical simulations of breaking waves. J. Fluid Mech. 912, A43.CrossRefGoogle Scholar
Deane, G.B. & Stokes, M.D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418, 839–844.CrossRefGoogle ScholarPubMed
Deike, L. 2022 Mass transfer at the ocean–atmosphere interface: the role of wave breaking, droplets, and bubbles. Annu. Rev. Fluid Mech. 54 (1), 191–224.CrossRefGoogle Scholar
Deike, L., Melville, W.K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Diemer, R.B. & Olson, J.H. 2002 A moment methodology for coagulation and breakage problems. Part 3. Generalized daughter distribution functions. Chem. Engng Sci. 57, 4187–4198.CrossRefGoogle Scholar
Garrett, C., Li, M. & Farmer, D. 2000 The connection between bubble size spectra and energy dissipation rates in the upper ocean. J. Phys. Oceanogr. 30 (9), 2163–2171.2.0.CO;2>CrossRefGoogle Scholar
Gaylo, D.B., Hendrickson, K. & Yue, D.K.P. 2021 Effects of power-law entrainment on bubble fragmentation cascades. J. Fluid Mech. 917, R1.CrossRefGoogle Scholar
Gaylo, D.B., Hendrickson, K. & Yue, D.K.P. 2022 An Eulerian label advection method for conservative volume-based tracking of bubbles/droplets. J. Comput. Phys. 470, 111560.CrossRefGoogle Scholar
Hendrickson, K., Weymouth, G.D. & Yue, D.K.P. 2020 Informed component label algorithm for robust identification of connected components with volume-of-fluid method. Comput. Fluids 197, 104373.CrossRefGoogle Scholar
Hinze, J.O. 1955 Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J. 1, 289–295.CrossRefGoogle Scholar
Liao, Y. & Lucas, D. 2009 A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Engng Sci. 64, 3389–3406.CrossRefGoogle Scholar
Lundgren, T.S. 2003 Linearly forced isotropic turbulence. Tech. Rep. Center for Turbulence Research, Stanford University.Google Scholar
Martínez-Bazán, C., Montañés, J.L. & Lasheras, J.C. 1999 a On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. J. Fluid Mech. 401, 157–182.CrossRefGoogle Scholar
Martínez-Bazán, C., Montañés, J.L. & Lasheras, J.C. 1999 b On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size pdf of the resulting daughter bubbles. J. Fluid Mech. 401, 183–207.CrossRefGoogle Scholar
Martínez-Bazán, C., Rodríguez-Rodríguez, J., Deane, G.B., Montaes, J.L. & Lasheras, J.C. 2010 Considerations on bubble fragmentation models. J. Fluid Mech. 661, 159–177.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.CrossRefGoogle Scholar
Popinet, S. 2018 Numerical models of surface tension. Annu. Rev. Fluid Mech. 50 (1), 49–75.CrossRefGoogle Scholar
Qi, Y., Mohammad Masuk, A.U. & Ni, R. 2020 Towards a model of bubble breakup in turbulence through experimental constraints. Intl J. Multiphase Flow 132, 103397.CrossRefGoogle Scholar
Qi, Y., Tan, S., Corbitt, N., Urbanik, C., Salibindla, A.K.R. & Ni, R. 2022 Fragmentation in turbulence by small eddies. Nat. Commun. 13, 469.CrossRefGoogle ScholarPubMed
Risso, F. & Fabre, J. 1998 Oscillations and breakup of a bubble immersed in a turbulent field. J. Fluid Mech. 372, 323–355.CrossRefGoogle Scholar
Rivière, A., Mostert, W., Perrard, S. & Deike, L. 2021 Sub-Hinze scale bubble production in turbulent bubble break-up. J. Fluid Mech. 917, A40.CrossRefGoogle Scholar
Rivière, A., Ruth, D.J., Mostert, W., Deike, L. & Perrard, S. 2022 Capillary driven fragmentation of large gas bubbles in turbulence. Phys. Rev. Fluids 7 (8), 083602.CrossRefGoogle Scholar
Rosales, C. & Meneveau, C. 2005 Linear forcing in numerical simulations of isotropic turbulence: physical space implementations and convergence properties. Phys. Fluids 17, 095106.CrossRefGoogle Scholar
Skartlien, R., Sollum, E. & Schumann, H. 2013 Droplet size distributions in turbulent emulsions: breakup criteria and surfactant effects from direct numerical simulations. J. Chem. Phys. 139, 174901.CrossRefGoogle ScholarPubMed
Solsvik, J., Maaß, S. & Jakobsen, H.A. 2016 Definition of the single drop breakup event. Ind. Engng Chem. Res 55, 2872–2882.CrossRefGoogle Scholar
Sporleder, F., Borka, Z., Solsvik, J. & Jakobsen, H.A. 2012 On the population balance equation. Rev. Chem. Engng 28, 149–169.Google Scholar
Tsouris, C. & Tavlarides, L.L. 1994 Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 40, 395–406.CrossRefGoogle Scholar
Valentas, K.J., Bilous, O. & Amundson, N.R. 1966 Analysis of breakage in dispersed phase systems. Ind. Engng Chem. Fundam. 5, 271–279.CrossRefGoogle Scholar
Vejražka, J., Zedníková, M. & Stanovský, P. 2018 Experiments on breakup of bubbles in a turbulent flow. AIChE J. 64, 740–757.CrossRefGoogle Scholar
Weymouth, G.D. & Yue, D.K.P. 2010 Conservative volume-of-fluid method for free-surface simulations on Cartesian-grids. J. Comput. Phys. 229 (8), 2853–2865.CrossRefGoogle Scholar
Yu, X., Hendrickson, K., Campbell, B.K. & Yue, D.K.P. 2019 Numerical investigation of shear-flow free-surface turbulence and air entrainment at large Froude and Weber numbers. J. Fluid Mech. 880, 209–238.CrossRefGoogle Scholar
Yu, X., Hendrickson, K. & Yue, D.K.P. 2020 Scale separation and dependence of entrainment bubble-size distribution in free-surface turbulence. J. Fluid Mech. 885, R2.CrossRefGoogle Scholar