Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-18T22:29:06.625Z Has data issue: false hasContentIssue false

Three-dimensional dynamics of a pair of deformable bubbles rising initially in line. Part 2. Highly inertial regimes

Published online by Cambridge University Press:  06 June 2022

Jie Zhang
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an, PR China
Ming-Jiu Ni*
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an, PR China School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing, PR China
Jacques Magnaudet*
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
Email addresses for correspondence:,
Email addresses for correspondence:,


The buoyancy-driven dynamics of a pair of gas bubbles released in line is investigated numerically, focusing on highly inertial conditions under which isolated bubbles follow non-straight paths. In an early stage, the second bubble always drifts out of the wake of the leading one. Then, depending on the ratios of the buoyancy, viscous and capillary forces which define the Galilei ($Ga$) and Bond ($Bo$) numbers of the system, five distinct regimes specific to such conditions are identified, in which the two bubbles may rise independently or continue to interact and possibly collide in the end. In the former case, they usually perform large-amplitude planar zigzags within the same plane or within two distinct planes, depending on the oblateness of the leading bubble. However, for large enough $Ga$ and low enough $Bo$, they follow nearly vertical paths with small-amplitude erratic horizontal deviations. Increasing $Bo$ makes the wake-induced attraction toward the leading bubble stronger, forcing the two bubbles to realign vertically one or more times along their ascent. During such sequences, wake vortices may hit the trailing bubble, deflecting its path and, depending on the case, promoting or hindering further possibilities of interaction. In some regimes, varying the initial distance separating the two bubbles modifies their lateral separation beyond the initial stage. Similarly, minute initial angular deviations favour the selection of a single vertical plane of rise common to both bubbles. These changes may dramatically affect the fate of the tandem as, depending on $Bo$, they promote or prevent future vertical realignments.

JFM Papers
© The Author(s), 2022. 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.)



Adoua, R., Legendre, D. & Magnaudet, J. 2009 Reversal of the lift force on an oblate bubble in a weakly viscous linear shear flow. J. Fluid Mech. 628, 2341.CrossRefGoogle Scholar
Alméras, E., Risso, F., Roig, V., Cazin, S., Plais, C. & Augier, F. 2015 Mixing by bubble-induced turbulence. J. Fluid Mech. 776, 458474.CrossRefGoogle Scholar
Auguste, F., Magnaudet, J. & Fabre, D. 2013 Falling styles of disks. J. Fluid Mech. 719, 388405.CrossRefGoogle Scholar
Auton, T.R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199218.CrossRefGoogle Scholar
Bhaga, D. & Weber, M.E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 6185.CrossRefGoogle Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7, 12651274.CrossRefGoogle Scholar
Brücker, C. 1999 Structure and dynamics of the wake of bubbles and its relevance for bubble interaction. Phys. Fluids 11, 17811796.CrossRefGoogle Scholar
Bunner, B. & Tryggvason, G. 2002 Dynamics of homogeneous bubbly flows. Part 1. Rise velocity and microstructure of the bubbles. J. Fluid Mech. 466, 1752.CrossRefGoogle Scholar
Bunner, B. & Tryggvason, G. 2003 Effect of bubble deformation on the properties of bubbly flows. J. Fluid Mech. 495, 77118.CrossRefGoogle Scholar
Cano-Lozano, J.C., Martinez-Bazan, C., Magnaudet, J. & Tchoufag, J. 2016 Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability. Phys. Rev. Fluids 1, 053604.CrossRefGoogle Scholar
Chesters, A.K. & Hofman, G 1982 Bubble coalescence in pure liquids. Appl. Sci. Res. 38, 353361.CrossRefGoogle Scholar
Duineveld, P.C. 1998 Bouncing and coalescence of bubble pairs rising at high Reynolds number in pure water or aqueous surfactant solutions. Appl. Sci. Res. 58, 409439.CrossRefGoogle Scholar
Esmaeeli, A. & Tryggvason, G. 1999 Direct numerical simulations of bubbly flows. Part 2. Moderate Reynolds number arrays. J. Fluid Mech. 385, 325358.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Filella, A., Ern, P. & Roig, V. 2020 Interaction of two oscillating bubbles rising in a thin-gap cell: vertical entrainment and interaction with vortices. J. Fluid Mech. 888, A13.CrossRefGoogle Scholar
Fortes, A.F., Joseph, D.D. & Lundgren, T.S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467483.CrossRefGoogle Scholar
Gvozdic, B., Alméras, E., Mathai, V., Zhu, X., van Gils, D.P.M., Verzicco, R., Huisman, S.G., Sun, C. & Lohse, D. 2018 Experimental investigation of heat transport in homogeneous bubbly flow. J. Fluid Mech. 845, 226244.CrossRefGoogle Scholar
Hallez, Y. & Legendre, D. 2011 Interaction between two spherical bubbles rising in a viscous liquid. J. Fluid Mech. 673, 406431.CrossRefGoogle Scholar
Innocenti, A., Jaccod, A., Popinet, S. & Chibbaro, S. 2021 Direct numerical simulation of bubble-induced turbulence. J. Fluid Mech. 918, A23.CrossRefGoogle Scholar
Joseph, D.D., Fortes, A., Lundgren, T.S. & Singh, P. 1986 Nonlinear mechanics of fluidization of spheres, cylinders and disks in water. In Advances in Multiphase Flow and Related Problems (ed. G. Papanicolaou), pp. 101–122. SIAM.Google Scholar
Kong, G., Mirsandi, H., Buist, K.A., Peters, E.A.J.F., Baltussen, M.W. & Kuipers, J.A.M. 2019 Hydrodynamic interaction of bubbles rising side-by-side in viscous liquids. Exp. Fluids 60, 155.CrossRefGoogle Scholar
Kusuno, H. & Sanada, T. 2021 Wake-induced lateral migration of approaching bubbles. Intl J. Multiphase Flow 139, 103639.CrossRefGoogle Scholar
Kusuno, H., Yamamoto, H. & Sanada, T. 2019 Lift force acting on a pair of clean bubbles rising in-line. Phys. Fluids 31, 072105.CrossRefGoogle Scholar
Loisy, A., Naso, A. & Spelt, P.D.M. 2017 Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction. J. Fluid Mech. 816, 94141.CrossRefGoogle Scholar
Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311337.CrossRefGoogle Scholar
Moore, D.W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749766.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 Path instability of a rising bubble. Phys. Rev. Lett. 88, 014502.CrossRefGoogle ScholarPubMed
Mougin, G. & Magnaudet, J. 2006 Wake-induced forces and torques on a zigzagging/spiralling bubble. J. Fluid Mech. 567, 185194.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 58385866.CrossRefGoogle Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J. Comput. Phys. 302, 336358.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Risso, F. & Ellingsen, K. 2002 Velocity fluctuations in a homogeneous dilute dispersion of high-Reynolds-number rising bubbles. J. Fluid Mech. 453, 395410.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Sanada, T., Sato, A., Shirota, M. & Watanabe, M. 2009 Motion and coalescence of a pair of bubbles rising side by side. Chem. Engng Sci. 64, 26592671.CrossRefGoogle Scholar
Sangani, A.S. & Didwania, A.K. 1993 Dynamic simulations of flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 250, 307337.CrossRefGoogle Scholar
Smereka, P. 1993 On the motion of bubbles in a periodic box. J. Fluid Mech. 254, 79112.CrossRefGoogle Scholar
Stewart, C.W. 1995 Bubble interaction in low-viscosity liquids. Intl J. Multiphase Flow 21, 10371046.CrossRefGoogle Scholar
Tchoufag, J., Fabre, D. & Magnaudet, J. 2014 a Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders. J. Fluid Mech. 740, 278311.CrossRefGoogle Scholar
Tchoufag, J., Fabre, D. & Magnaudet, J. 2015 Weakly nonlinear model with exact coefficients for the fluttering and spiraling motion of buoyancy-driven bodies. Phys. Rev. Lett. 115, 114501.CrossRefGoogle ScholarPubMed
Tchoufag, J., Magnaudet, J. & Fabre, D. 2014 b Linear instability of the path of a freely rising spheroidal bubble. J. Fluid Mech. 751, R4.CrossRefGoogle Scholar
Yin, X. & Koch, D.L. 2008 Velocity fluctuations and hydrodynamic diffusion in finite-Reynolds-number sedimenting suspensions. Phys. Fluids 20, 043305.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Zenit, R. & Magnaudet, J. 2008 Path instability of rising spheroidal air bubbles: a shape-controlled process. Phys. Fluids 20, 061702.CrossRefGoogle Scholar
Zhang, J., Ni, M. & Magnaudet, J. 2021 Three-dimensional dynamics of a pair of deformable bubbles rising initially in line. Part 1. Moderately inertial regimes. J. Fluid Mech. 920, A16.CrossRefGoogle Scholar