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Linear stability analysis of Taylor bubble motion in downward flowing liquids in vertical tubes

Published online by Cambridge University Press:  27 April 2022

H.A. Abubakar  and
O.K. Matar Open the ORCID record for O.K. Matar [Opens in a new window]
H.A. Abubakar
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK Department of Chemical Engineering, Ahmadu Bello University, Zaria 810107, Nigeria
O.K. Matar*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

Taylor bubbles are a feature of the slug flow regime in gas–liquid flows in vertical pipes. Their dynamics exhibits a number of transitions such as symmetry breaking in the bubble shape and wake when rising in downward flowing and stagnant liquids, respectively, as well as breakup in sufficiently turbulent environments. Motivated by the need to examine the stability of a Taylor bubble in liquids, a systematic numerical study of a steadily moving Taylor bubble in stagnant and flowing liquids is carried out, characterised by the dimensionless inverse viscosity $( N_f )$, Eötvös $( Eo )$ and Froude numbers $( U_m )$, the latter being based on the centreline liquid velocity, using a Galerkin finite-element method. A boundary-fitted domain is used to examine the dependence of the steady bubble shape on a wide range of $N_f$, $Eo$ and $U_m$. Our analysis of the bubble nose and bottom curvatures shows that the intervals $Eo = [ 20,30 )$ and $N_f=[60,80 )$ are the limits below which surface tension and viscosity, respectively, have a strong influence on the bubble shape. In the interval $Eo = (60,100 ]$, all bubble features studied are weakly dependent on surface tension. A linear stability analysis of the axisymmetric base states shows that there exist regions of $(N_f,Eo,U_m)$ space within which the bubble is unstable and assumes an asymmetric shape. To elucidate the mechanisms underlying the instability, an energy budget analysis is carried out which reveals that perturbation growth is driven by the bubble pressure for $Eo \geq 100$, and by the tangential interfacial stress for $Eo < 100$. Examples of the asymmetric bubble shapes and their associated flow fields are also provided near the onset of instability for a wide range of $N_f$, $Eo$ and $U_m$.

JFM classification

Drops and Bubbles: Bubble dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 941 , 25 June 2022 , A2
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Abubakar, H.A. 2019 Taylor bubble rise in circular tubes: steady-states and linear stability analysis. PhD thesis, Imperial College London.Google Scholar
Anjos, G., Mangiavacchi, N., Borhani, N. & Thome, J.R. 2014 3D ALE finite-element method for two-phase flows with phase change. Heat Transfer Engng 35 (5), 537–547.CrossRefGoogle Scholar
Araújo, J.D.P., Miranda, J.M., Pinto, A.M.F.R. & Campos, J.B.L.M. 2012 Wide-ranging survey on the laminar flow of individual Taylor bubbles rising through stagnant Newtonian liquids. Intl J. Multiphase Flow 43, 131–148.CrossRefGoogle Scholar
Bae, S.H. & Kim, D.H. 2007 Computational study of the axial instability of rimming flow using Arnoldi method. Intl J. Numer. Meth. Fluids 53, 691–711.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bendiksen, K. 1985 On the motion of long bubbles in vertical tubes. Intl J. Multiphase Flow 11, 797–812.CrossRefGoogle Scholar
Boomkamp, P.A.M. & Miesen, R.H.M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.CrossRefGoogle Scholar
Brown, R.A.S. 1965 The mechanics of large gas bubbles in tubes. I. Bubble velocities in stagnant liquids. Can. J. Chem. Engng 43, 217–223.CrossRefGoogle Scholar
Bugg, J.D., Mack, K. & Rezkallah, K.S. 1998 A numerical model of Taylor bubbles rising through stagnant liquids in vertical tubes. Intl J. Multiphase Flow 24, 271–281.CrossRefGoogle Scholar
Bugg, J.D. & Saad, G.A. 2002 The velocity field around a Taylor bubble rising in a stagnant viscous fluid: numerical and experimental results. Intl J. Multiphase Flow 28, 791–803.CrossRefGoogle Scholar
Campos, J.B.L.M. & Guedes de Carvalho, J.R.F. 1988 An experimental study of the wake of gas slugs rising in liquids. J. Fluid Mech. 196, 27–37.CrossRefGoogle Scholar
Capponi, A., James, M.R. & Lane, S.J. 2016 Gas slug ascent in a stratified magma: implications of flow organisation and instability for Strombolian eruption dynamics. Earth Planet. Sci. Lett. 435, 159–170.CrossRefGoogle Scholar
Carvalho, M.S. & Scriven, L.E. 1999 Three-dimensional stability analysis of free surface flows: application to forward deformable roll coating. J. Comput. Phys. 151, 534–562.CrossRefGoogle Scholar
Christodoulou, C.N. & Scriven, L.E. 1988 Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3 (4), 355–406.CrossRefGoogle Scholar
Clift, R., Grace, J.R. & Weber, M.E. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
Collins, R., De Moraes, F., Davidson, J. & Harrison, D. 1978 The motion of a large gas bubble rising through liquid flowing in a tube. J. Fluid Mech. 89, 497–514.CrossRefGoogle Scholar
Cuvelier, C. & Schulkes, R.M.S.M. 1990 Some numerical methods for the computation of capillary free boundaries governed by the Navier–Stokes equations. SIAM Rev. 32 (3), 355–423.CrossRefGoogle Scholar
Davies, R.M. & Taylor, G. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200, 375–390.Google Scholar
Dumitrescu, D.T. 1943 Strömung an einer Luftblase im senkrechten Rohr. Z. Angew. Math. Mech. 23 (3), 139–149.CrossRefGoogle Scholar
Fabre, J. 2016 A long bubble rising in still liquid in a vertical channel: a plane inviscid solution. J. Fluid Mech. 794, R4.CrossRefGoogle Scholar
Fabre, J. & Figueroa-Espinoza, B. 2014 Taylor bubble rising in a vertical pipe against laminar or turbulent downward flow: symmetric to asymmetric shape transition. J. Fluid Mech. 755, 485–502.CrossRefGoogle Scholar
Fabre, J. & Liné, A. 1992 Modeling of two-phase slug flow. Annu. Rev. Fluid Mech. 24, 21–46.CrossRefGoogle Scholar
Feng, J.Q. 2008 Buoyancy-driven motion of a gas bubble through viscous liquid in a round tube. J. Fluid Mech. 609, 377–410.CrossRefGoogle Scholar
Fershtman, A., Babin, V., Barnea, D. & Shemer, L. 2017 On shapes and motion of an elongated bubble in downward liquid pipe flow. Phys. Fluids 29, 112103.CrossRefGoogle Scholar
Figueroa-Espinoza, B. & Fabre, J. 2011 Taylor bubble moving in a flowing liquid in vertical channel: transition from symmetric to asymmetric shape. J. Fluid Mech. 679, 432–454.CrossRefGoogle Scholar
Fraggedakis, D., Pavlidis, M., Dimakopoulos, Y. & Tsamopoulos, J. 2016 On the velocity discontinuity at critical volume of a bubble rising in a viscoelastic fluid. J. Fluid Mech. 789, 310–346.CrossRefGoogle Scholar
Funada, T., Joseph, D., Maehara, T. & Yamashita, S. 2005 Ellipsoidal model of the rise of a Taylor bubble in a round tube. Intl J. Multiphase Flow 31, 473–491.CrossRefGoogle Scholar
Ganesan, S. & Tobiska, L. 2008 An accurate finite element scheme with moving meshes for computing 3D-axisymmetric interface flows. Intl J. Multiphase Flow 57, 119–138.Google Scholar
Goldsmith, H.L. & Mason, S.G. 1962 The movement of single large bubbles in closed vertical tubes. J. Fluid Mech. 14, 42–58.CrossRefGoogle Scholar
Griffith, P. & Wallis, G.B. 1961 Two phase slug flow. Trans. ASME J. Heat Transfer 83, 7–320.CrossRefGoogle Scholar
Ha Ngoc, H. & Fabre, J. 2006 A boundary element method for calculating the shape and velocity of two-dimensional long bubble in stagnant and flowing liquid. Engng Anal. Bound. Elem. 30, 539–552.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Math. 20 (3-4), 251–265.CrossRefGoogle Scholar
Hooper, A.P. & Boyd, W.G.C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528.CrossRefGoogle Scholar
Hu, H.H & Joseph, D.D. 1989 Lubricated pipelining: stability of core-annular flow. Part 2. J. Fluid Mech. 205, 359–396.CrossRefGoogle Scholar
Johnson, A.A. & Tezduyar, T.E. 1994 Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput. Meth. Appl. Mech. Engng 119, 73–94.CrossRefGoogle Scholar
Kang, C.W., Quan, S.P. & Lou, J. 2010 Numerical study of a Taylor bubble rising in stagnant liquids. Phys. Rev. E 81, 1539–3755.CrossRefGoogle Scholar
Kruyt, N.P., Cuvelier, C., Segal, A. & Van Der Zanden, J. 1988 A total linearization method for solving viscous free boundary flow problems by the finite-element method. Intl J. Numer. Meth. Fluids 8, 351–363.CrossRefGoogle Scholar
Lehoucq, R.B., Sorensen, D.C. & Yang, C. 1997 ARPACK user's guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM.Google Scholar
Lizarraga-Garcia, E., Buongiorno, J., Al-Safran, E. & Lakehal, D. 2017 A broadly applicable unified closure relation for Taylor bubble rise velocity in pipes with stagnant liquid. Intl J. Multiphase Flow 89, 345–358.CrossRefGoogle Scholar
Llewellin, E.W., Del Bello, E., Taddeucci, J., Scarlato, P. & Lane, S.J. 2012 The thickness of the falling film of liquid around a Taylor bubble. Proc. R. Soc. A 468, 1041–1064.CrossRefGoogle Scholar
Lu, X. & Prosperetti, A. 2006 Axial stability of Taylor bubbles. J. Fluid Mech. 568, 173–192.CrossRefGoogle Scholar
Lu, X. & Prosperetti, A. 2009 A numerical study of Taylor bubbles. Ind. Engng Chem. Res. 48, 242–252.CrossRefGoogle Scholar
Mao, Z.S. & Dukler, A.E. 1989 An experimental study of gas-liquid slug flow. Exp. Fluids 8, 169–1821.CrossRefGoogle Scholar
Mao, Z.S. & Dukler, A.E. 1990 The motion of Taylor bubbles in vertical tubes. I. A numerical simulation for the shape and rise velocity of Taylor bubbles in stagnant and flowing liquids. J. Comput. Phys. 91, 2055–2064.CrossRefGoogle Scholar
Mao, Z.S. & Dukler, A.E. 1991 The motion of Taylor bubbles in vertical tubes. II. Experimental data and simulations for laminar and turbulent flow. Chem. Engng Sci. 46, 132–160.CrossRefGoogle Scholar
Martin, C.S. 1976 Vertically downward two-phase slug flow. Trans. ASME J. Fluids Engng 98, 715–722.CrossRefGoogle Scholar
Maxworthy, T. 1967 A note on the existence of wakes behind large, rising bubbles. J. Fluid Mech. 27, 367–368.CrossRefGoogle Scholar
Miller, C.A. & Scriven, L.E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.CrossRefGoogle Scholar
Moissis, R. & Griffith, P. 1962 Entrance effect in a two-phase slug flow. Trans. ASME J. Heat Transfer 84, 29–38.CrossRefGoogle Scholar
Nickens, H. & Yannitel, D. 1987 The effect of surface tension and viscosity on the rise velocity of a large gas bubble in a closed vertical liquid-filled tube. Intl J. Multiphase Flow 13, 57–69.CrossRefGoogle Scholar
Nicklin, D., Wilkes, J. & Davidson, J. 1962 Two-phase flow in vertical tubes. Trans. Inst. Chem. Engrs 40, 61–68.Google Scholar
Nigmatulin, T.R. 2001 Surface of a Taylor bubble in vertical cylindrical flows. Dokl. Phys. 46, 803–805.CrossRefGoogle Scholar
Nogueira, S., Riethmuller, M.L., Campos, J.B.L.M. & Pinto, A.M.F.R. 2006 a Flow in the nose region and annular film around a Taylor bubble rising through vertical columns of stagnant and flowing Newtonian liquids. Chem. Engng Sci. 61, 845–857.CrossRefGoogle Scholar
Nogueira, S., Riethmuller, M.L., Campos, J.B.L.M. & Pinto, A.M.F.R. 2006 b Flow patterns in the wake of a Taylor bubble rising through vertical columns of stagnant and flowing Newtonian liquids: an experimental study. Chem. Engng Sci. 61, 7199–7212.CrossRefGoogle Scholar
Ó Náraigh, L., Spelt, P.D.M., Matar, O.K. & Zaki, T.A. 2011 Interfacial instability in turbulent flow over a liquid film in a channel. Intl J. Multiphase Flow 37, 812–830.CrossRefGoogle Scholar
Pering, T.D. & McGonigle, A.J.S. 2018 Combining spherical-cap and Taylor bubble fluid dynamics with plume measurements to characterize basaltic degassing. Geosciences 8 (2), 42.CrossRefGoogle Scholar
Pinto, A.M.F.R., Coelho Pinheiro, M.N. & Campos, J.B.L.M. 1998 Coalescence of two gas slugs rising in a co-current flowing liquid in vertical tubes. Chem. Engng Sci. 53 (16), 2973–2983.CrossRefGoogle Scholar
Polonsky, S., Shemer, L. & Barnea, D. 1999 The relation between the Taylor bubble motion and the velocity field ahead of it. Intl J. Multiphase Flow 25, 957–975.CrossRefGoogle Scholar
Pringle, C.C.T., Ambrose, S., Azzopardi, B.J. & Rust, A.C. 2015 The existence and behaviour of large diameter Taylor bubbles. Intl J. Multiphase Flow 72, 318–323.CrossRefGoogle Scholar
Prosperetti, A. 1980 Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid. J. Méc. 19 (1), 149–181.Google Scholar
Rana, B.K., Das, A.K. & Das, P.K. 2015 Mechanism of bursting Taylor bubbles at free surfaces. Langmuir 31, 9870–9881.CrossRefGoogle ScholarPubMed
Sahu, K.C., Ding, H., Valluri, P. & Matar, O.K. 2009 Linear stability analysis and numerical simulation of miscible two-layer channel flow. Phys. Fluids 21, 042104.CrossRefGoogle Scholar
Sahu, K.C., Valluri, P., Spelt, P.D.M. & Matar, O.K. 2007 Linear instability of pressure-driven channel flow of a Newtonian and Herschel-Bulkley fluid. Phys. Fluids 19, 122101.CrossRefGoogle Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flows with viscosity stratification. J. Fluid Mech. 592, 23–49.CrossRefGoogle Scholar
Slikkeveer, P.J. & Van Loohuizen, E.P. 1996 An implicit surface tension algorithm for Picard solvers of surface-tension-dominated free and moving boundary problems. Intl J. Numer. Meth. Fluids 22, 851–865.3.0.CO;2-R>CrossRefGoogle Scholar
Taha, T. & Cui, Z.F. 2002 Hydrodynamic analysis of upward slug flow in tubular membranes. Desalination 145, 179–182.CrossRefGoogle Scholar
Taha, T. & Cui, Z.F. 2006 CFD modeling of slug flow in vertical tubes. Chem. Engng Sci. 61, 676–687.CrossRefGoogle Scholar
Tsamopoulos, J., Dimakopoulos, Y., Chatzidai, N., Karapetsas, G. & Pavlidis, M. 2008 Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions of bubble entrapment. J. Fluid Mech. 601, 123–164.CrossRefGoogle Scholar
Viana, F., Pardo, R., Yánez, R., Trallero, J.L. & Joseph, D.D. 2003 Universal correlation for the rise velocity of long gas bubbles in round pipes. J. Fluid Mech. 494, 379–398.CrossRefGoogle Scholar
White, E.T. & Beardmore, R.H. 1962 The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes. Chem. Engng Sci. 17, 351–361.CrossRefGoogle Scholar
Yu, Y.E., Magnini, M., Zhu, L., Shim, S. & Stone, H.A. 2021 Non-unique bubble dynamics in a vertical capillary with an external flow. J. Fluid Mech. 911, A34–1–19.CrossRefGoogle Scholar
Zhou, W. & Dusek, J. 2017 Marginal stability curve of a deformable bubble. Intl J. Multiphase Flow 89, 218–227.CrossRefGoogle Scholar
Zukoski, E.E. 1966 Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech. 25, 821–837.CrossRefGoogle Scholar
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