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Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction

  • Aurore Loisy (a1), Aurore Naso (a1) and Peter D. M. Spelt (a1)


Various expressions have been proposed previously for the rise velocity of gas bubbles in homogeneous steady bubbly flows, generally a monotonically decreasing function of the bubble volume fraction. For suspensions of freely moving bubbles, some of these are of the form expected for ordered arrays of bubbles, and vice versa, as they do not reduce to the behaviour expected theoretically in the dilute limit. The microstructure of weakly inhomogeneous bubbly flows not being known generally, the effect of microstructure is an important consideration. We revisit this problem here for bubbly flows at small to moderate Reynolds number values for deformable bubbles, using direct numerical simulation and analysis. For ordered suspensions, the rise velocity is demonstrated not to be monotonically decreasing with volume fraction due to cooperative wake interactions. The fore-and-aft asymmetry of an isolated ellipsoidal bubble is reversed upon increasing the volume fraction, and the bubble aspect ratio approaches unity. Recent work on rising bubble pairs is used to explain most of these results; the present work therefore forms a platform of extending the former to suspensions of many bubbles. We adopt this new strategy also to support the existence of the oblique rise of ordered suspensions, the possibility of which is also demonstrated analytically. Finally, we demonstrate that most of the trends observed in ordered systems also appear in freely evolving suspensions. These similarities are supported by prior experimental measurements and attributed to the fact that free bubbles keep the same neighbours for extended periods of time.



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Auton, T. R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199218.
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245268.
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 6185.
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100, 335354.
Brenner, H. & Cox, R. G. 1963 The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17, 561595.
Bunner, B & Tryggvason, G. 2002a Dynamics of homogeneous bubbly flows. Part 1. Rise velocity and microstructure of the bubbles. J. Fluid Mech. 466, 1752.
Bunner, B. & Tryggvason, G. 2002b Dynamics of homogeneous bubbly flows. Part 2. Velocity fluctuations. J. Fluid Mech. 466, 5384.
Bunner, B. & Tryggvason, G. 2003 Effect of bubble deformation on the properties of bubbly flows. J. Fluid Mech. 495, 77118.
Cartellier, A., Andreotti, M. & Sechet, P. 2009 Induced agitation in homogeneous bubbly flows at moderate particle Reynolds number. Phys. Rev. E 80 (6), 065301.
Cartellier, A. & Rivière, N. 2001 Bubble-induced agitation and microstructure in uniform bubbly flows at small to moderate particle Reynolds numbers. Phys. Fluids 13 (8), 21652181.
Chorin, A 1968 Numerical solution of the Navier–Stokes equations. Maths Comput. 22, 745762.
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic Press.
Colombet, D., Legendre, D., Risso, F., Cockx, A. & Guiraud, P. 2015 Dynamics and mass transfer of rising bubbles in a homogenous swarm at large gas volume fraction. J. Fluid Mech. 763, 254285.
Davis, R. H. & Acrivos, A 1985 Sedimentation of noncolloidal particles at low Reynolds numbers. Annu. Rev. Fluid Mech. 17, 91118.
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97121.
Esmaeeli, A. & Tryggvason, G. 1998 Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays. J. Fluid Mech. 377, 313345.
Esmaeeli, A. & Tryggvason, G. 1999 Direct numerical simulations of bubbly flows. Part 2. Moderate Reynolds number arrays. J. Fluid Mech. 385, 325358.
Esmaeeli, A. & Tryggvason, G. 2005 A direct numerical simulation study of the buoyant rise of bubbles at O (100) Reynolds number. Phys. Fluids 17, 093303.
Fabre, D., Tchoufag, J. & Magnaudet, J. 2012 The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 2436.
Garnier, C., Lance, M. & Marié, J.-L. 2002 Measurement of local flow characteristics in buoyancy-driven bubbly flow at high void fraction. Exp. Therm. Fluid Sci. 26, 811815.
Gillissen, J. J. J., Sundaresan, S. & Van Den Akker, H. E. A. 2011 A lattice Boltzmann study on the drag force in bubble swarms. J. Fluid Mech. 679, 101121.
Glendinning, A. B. & Russel, W. B. 1982 A pairwise additive description of sedimentation and diffusion in concentrated suspensions of hard spheres. J. Colloid Interface Sci. 89, 124143.
Gottlieb, S. & Shu, C.-W. 1998 Total variation diminishing Runge–Kutta schemes. Maths Comput. 67, 7385.
Grace, J. R. 1973 Shapes and velocities of bubbles rising in infinite liquids. Trans. Inst. Chem. Engrs 51, 116120.
Guazzelli, E. & Hinch, J. 2011 Fluctuations and instability in sedimentation. Annu. Rev. Fluid Mech. 43, 97116.
Hadamard, J. 1911 Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide visqueux. C. R. Acad. Sci. Paris 152, 17351738.
Hallez, Y. & Legendre, D. 2011 Interaction between two spherical bubbles rising in a viscous liquid. J. Fluid Mech. 673, 406431.
Harper, J. F. 1970 On bubbles rising in line at large Reynolds numbers. J. Fluid Mech. 41, 751758.
Harper, J. F. 1997 Bubbles rising in line: why is the first approximation so bad? J. Fluid Mech. 351, 289300.
Harten, A., Engquist, B., Osher, S. & Chakravarthy, S. R. 1987 Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231303.
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213241.
Hua, J., Stene, J. F. & Lin, P. 2008 Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method. J. Comput. Phys. 227, 33583382.
Ishii, M. & Zuber, N. 1979 Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J. 25, 843855.
Jenny, M., Dusek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201239.
Jiang, G.-S. & Shu, C.-W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202228.
Katz, J. & Meneveau, C. 1996 Wake-induced relative motion of bubbles rising in line. Intl J. Multiphase Flow 22 (2), 239258.
Keh, H. J. & Tseng, Y. K. 1992 Slow motion of multiple droplets in arbitrary three-dimensional configurations. AIChE J. 38, 18811904.
Koch, D. L. 1993 Hydrodynamic diffusion in dilute sedimenting suspensions at moderate Reynolds numbers. Phys. Fluids A 5, 1141.
Kushch, V. I., Sangani, A. S., Spelt, P. D. M. & Koch, D. L. 2002 Finite Weber number motion of bubbles through a nearly inviscid liquid. J. Fluid Mech. 460, 241280.
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81126.
Legendre, D., Magnaudet, J. & Mougin, G. 2003 Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid. J. Fluid Mech. 497, 133166.
Loisy, A.2016 Direct numerical simulation of bubbly flows: coupling with scalar transport and turbulence. PhD thesis, Université Claude Bernard Lyon 1.
Loth, E. 2008 Quasi-steady shape and drag of deformable bubbles and drops. Intl J. Multiphase Flow 34, 523546.
Martinez-Mercado, J., Palacios-Morales, C. A. & Zenit, R. 2007 Measurement of pseudoturbulence intensity in monodispersed bubbly liquids for 10 < Re < 500. Phys. Fluids 19, 103302.
Mei, R., Klausner, J. F. & Lawrence, C. J. 1994 A note on the history force on a spherical bubble at finite Reynolds number. Phys. Fluids 6, 418420.
Meland, R., Gran, I. R., Olsen, R. & Munkejord, S. T. 2007 Reduction of parasitic currents in level-set calculations with a consistent discretization of the surface-tension force for the CSF model. In 16th Australasian Fluid Mechanics Conference (AFMC) (ed. Jacobs, P. et al. ), pp. 862865. The University of Queensland.
Merle, A., Legendre, D. & Magnaudet, J. 2005 Forces on a high-Reynolds-number spherical bubble in a turbulent flow. J. Fluid Mech. 532, 5362.
Moore, D. W. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113130.
Naso, A. & Prosperetti, A. 2010 The interaction between a solid particle and a turbulent flow. New J. Phys. 12, 033040.
Osher, S. & Sethian, J. A. 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 1249.
Phillips, R. J., Brady, J. F. & Bossis, G. 1988 Hydrodynamic transport properties of hardsphere dispersions. I. Suspensions of freely mobile particles. Phys. Fluids 31, 34623472.
Prosperetti, A. & Tryggvason, G. 2007 Computational Methods for Multiphase Flow. Cambridge University Press.
Ray, B. & Prosperetti, A. 2014 On skirted drops in an immiscible liquid. Chem. Engng Sci. 108, 213222.
Riboux, G., Risso, F. & Legendre, D. 2010 Experimental characterization of the agitation generated by bubbles rising at high Reynolds number. J. Fluid Mech. 643, 509539.
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidisation: Part I. Trans. Inst. Chem. Engrs 32, 3553.
Roghair, I., Lau, Y. M., Deen, N. G., Slagter, H. M., Baltussen, M. W., Van Sint Annaland, M. & Kuipers, J. A. M. 2011 On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers. Chem. Engng Sci. 66, 32043211.
Russo, G. & Smereka, P. 2000 A remark on computing distance functions. J. Comput. Phys. 163, 5167.
Rybczynski, W. 1911 Uber die fortschreitende Bewegung einer flussigen Kugel in einem zahen Medium. Bull. Intl Acad. Sci. Cracovie A 1, 4046.
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 1935.
Sabelnikov, V., Ovsyannikov, A. Y. & Gorokhovski, M. 2014 Modified level set equation and its numerical assessment. J. Comput. Phys. 278, 130.
Salih, A. & Ghosh Moulic, S. 2009 Some numerical studies of interface advection properties of level set method. Sadhana 34, 271298.
Sangani, A. S. 1987 Sedimentation in ordered emulsions of drops at low Reynolds numbers. Z. Angew. Math. Phys. 38, 542556.
Sangani, A. S. & Acrivos, A. 1983 Creeping flow through cubic arrays of spherical bubbles. Intl J. Multiphase Flow 9, 181185.
Sankaranarayanan, K., Shan, X., Kevrekidis, I. G. & Sundaresan, S. 2002 Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method. J. Fluid Mech. 452, 6196.
Sankaranarayanan, K. & Sundaresan, S. 2002 Lift force in bubbly suspensions. Chem. Engng Sci. 57, 35213542.
Spelt, P. D. M. 2006 Shear flow past two-dimensional droplets pinned or moving on an adhering channel wall at moderate Reynolds numbers: a numerical study. J. Fluid Mech. 561, 439.
Spelt, P. D. M. & Sangani, A. S. 1998 Properties and averaged equations for flows of bubbly liquids. Appl. Sci. Res. 58, 337386.
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146159.
Sussman, M. & Uto, S. 1998 A computational study of the spreading of oil underneath a sheet of ice. UCLA Computational and Applied Mathematics Report 98.
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 466476.
Theodoropoulos, C., Sankaranarayanan, K., Sundaresan, S. & Kevrekidis, I. G. 2004 Coarse bifurcation studies of bubble flow lattice Boltzmann simulations. Chem. Engng Sci. 59, 23572362.
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.
Veldhuis, C. H. J. & Biesheuvel, A. 2007 An experimental study of the regimes of motion of spheres falling or ascending freely in a Newtonian fluid. Intl J. Multiphase Flow 33, 10741087.
Wacholder, E. 1973 Sedimentation in a dilute emulsion. Chem. Engng Sci. 28, 14471453.
Yin, X. & Koch, D. L. 2008 Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains. Phys. Fluids 20, 103304.
Yuan, H. & Prosperetti, A. 1994 On the in-line motion of two spherical bubbles in a viscous fluid. J. Fluid Mech. 278, 325.
Zenit, R., Koch, D. L. & Sangani, A. S. 2001 Measurements of the average properties of a suspension of bubbles rising in a vertical channel. J. Fluid Mech. 429, 307342.
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Loisy et al. supplementary movie
Top view of the bubble motion at a volume fraction of 3.8 % for case E1 (8 bubbles in the cell).

 Video (17.6 MB)
17.6 MB

Loisy et al. supplementary movie
Top view of the bubble motion at a volume fraction of 0.24 % for case E1 (8 bubbles in the cell).

 Video (443 KB)
443 KB


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