A boundary element method for calculating the shape and velocity of two-dimensional long bubble in stagnant and flowing liquid. (English) Zbl 1195.76282

Summary: A numerical method based on a boundary element method (BEM) has been developed for computing the velocity and the shape of long bubbles moving steadily in stagnant and flowing liquid in 2D case: plane and axisymmetrical. The flow is assumed to be inviscid and incompressible. The method consists in solving simultaneously a Poisson equation characterizing the flow and an equation for bubble shape in the form of a functional-differential equation resulting from both Bernoulli equation and the jump conditions at the interface. The Poisson equation is solved by a BEM with an iterative loop for nonlinear source term while the system of nonlinear algebraic equations obtained by discretizing the equation on the interface is solved by the Powell’s hybrid algorithm. The bubble shape and velocity are obtained as a part of the solution. The problem of multiple solutions is investigated numerically and the maximum velocity criterion is used for selecting the physical solution. The results obtained by the simulation are in good agreement with the experimental and numerical results of previous studies.


76M15 Boundary element methods applied to problems in fluid mechanics
76T10 Liquid-gas two-phase flows, bubbly flows
Full Text: DOI


[1] Davies, R. M.; Taylor, G., The mechanics of large bubbles rising through extended liquids and through liquids in tubes, Proc R Soc Ser A, 200, 375-390 (1950)
[2] Benjamin, T. B., Gravity currents and related phenomena, J Fluid Mech, 31, 209-248 (1968) · Zbl 0169.28503
[3] Zukoski, E. E., Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes, J Fluid Mech, 25, 821-837 (1968)
[5] Maneri, C.; Zuber, N., An experimental study of plane bubbles rising at inclination, Int J Multiphase Flow, 1, 623-645 (1974)
[6] Harmathy, T. Z., Velocity of large drops and bubbles in media of infinite or restricted extent, AIChE J, 6, 281-288 (1960)
[7] Nicklin, J.; Wilkes, J. O.; Davidson, J. F., Two phase flow in vertical tubes, Trans Inst Chem Eng, 40, 61-68 (1962)
[8] White, E. T.; Beardmore, R. H., The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes, Chem Eng Sci, 17, 351-361 (1962)
[9] Griffith, P., The prediction of low-quality boiling voids (1963), [ASME Paper No. 63-HT-20]
[10] Collins, R., A simple model of the plane gas bubble in a finite liquid, J Fluid Mech, 22, 763-771 (1965) · Zbl 0131.23601
[11] Bendiksen, K. H., An experimental investigation of the motion of long bubbles in inclined tubes, Int J Multiphase Flow, 10, 467-483 (1984)
[12] Hasan, R.; Kabir, C. S., Predicting multiphase flow behavior in a deviated well, SPE 15449, 61st annual technical conference, New Orleans. LA 5-8 October (1986)
[13] Wallis, G. B., One dimensional two-phase flow (1969), MacGraw-Hill: MacGraw-Hill New York
[14] Dumitrescu, D. T., Strömung an einer Luftblase im Senkrechten Rohr, Z Angew Math Mech, 23, 139 (1943) · JFM 67.0352.03
[15] Collins, R.; De, F. F.; Davidson, J. F.; Harrison, D., The motion of large gas bubble rising through liquid flowing in a tube, J Fluid Mech, 89, 497-514 (1978)
[16] Bendiksen, K. H., On the motion of long bubbles in vertical tubes, Int J Multiphase Flow, 11, 797-812 (1985) · Zbl 0649.76060
[17] Nickens, H. V.; Yannitell, D. W., The effects of surface tension and viscosity on the rise velocity of a large gas bubble in close, vertical liquid-filled tube, Int J Multiphase Flow, 13, 57-69 (1987)
[18] Birkhoff, G.; Carter, D., Rising plane bubbles, J Math Mech, 6, 769-779 (1957) · Zbl 0080.18403
[19] Garabedian, P. R., On steady-state bubbles generated by Taylor instability, Proc R Soc A, 241, 423-431 (1957) · Zbl 0078.16602
[20] Vanden-Broëck, J. M., Bubbles rising in a tube and jets falling from a nozzle, Phys Fluids, 27, 1090-1093 (1984) · Zbl 0577.76095
[21] Vanden-Broëck, J. M., Rising bubble in two-dimensional tube with surface tension, Phys Fluids, 27, 2604-2607 (1984) · Zbl 0599.76122
[22] Couët, B.; Strumolo, G. S., The effects of surface tension and tube inclination on a two-dimensional rising bubble, J Fluid Mech, 184, 1-14 (1987)
[24] Mao, Z.-S.; Dukler, A. E., 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, 132-160 (1990) · Zbl 0711.76102
[25] Mao, Z.-S.; Dukler, A. E., The motion of Taylor bubbles in vertical tubes II. Experimental data and simulations for laminar and turbulent flow, Chem Eng Sci, 46, 2055-2064 (1991)
[26] Bugg, J. D.; Mack, K.; Rezkallah, K. S., A numerical model of Taylor bubbles rising though stagnant liquids in vertical tubes, Int J Multiphase Flow, 24, 271-281 (1998) · Zbl 1121.76429
[27] Benkenida, A., Développement et validation d’une méthode de simulation d’écoulement diphasique sans reconstruction d’interface, Application à la dynamique des bulles de Taylor (1999), Thèse de doctorat, Institut National Polytechnique: Thèse de doctorat, Institut National Polytechnique Toulouse, France
[28] Blake, J. R.; Taib, B. B.; Doherty, G., Transient cavities near boundaries. Part 1. Rigid boundaries, J Fluid Mech, 170, 479-497 (1986) · Zbl 0606.76050
[29] Chakrabarti, R.; Harris, P. J.; Verma, A., On the interaction of an explosion bubble with a fixed rigid structure, Int J Numer Methods Fluids, 29, 389-396 (1999) · Zbl 0938.76060
[30] Harris, P. J., An investigation into the use of the boundary integral method to model the motion of a single gas or vapour bubble in a liquid, Eng Anal Bound Elem, 28, 325-332 (2004) · Zbl 1078.76048
[31] Heister, S. D., Boundary element methods for two-fluid free surface flows, Eng Anal Bound Elem, 19, 309-317 (1997)
[32] Kamiya, N.; Nakayama, K., Prediction of free surface of die swell using the boundary element method, Comput Struct, 46, 387-395 (1993) · Zbl 0767.76032
[33] Yoon, S. S.; Heister, S. D., A fully non-linear model for atomization of high-speed jets, Eng Anal Bound Elem, 28, 345-357 (2004) · Zbl 1130.76322
[34] Lundgren, T. T.; Mansour, N. N., Oscillations of drops in zero gravity with weak viscous effects, J Fluid Mech, 194, 479-510 (1988) · Zbl 0645.76110
[36] Lamb, H., Hydrodynamics (1932), Cambridge University Press: Cambridge University Press Cambridge · JFM 26.0868.02
[37] Batchelor, G. K., An introduction to fluid dynamics (1967), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0152.44402
[38] Hawthorne, W. R., (Sovran, G., Fluid mechanics of internal flow (1967), Elsevier: Elsevier Amsterdam)
[39] Vied, I. A.; Bajalieva, S. S., A boundary problem for a system of second degree integral-differential equations of Volter type, Investigations on integral-differential equations, vol. 13 (1980), Ilim Press: Ilim Press Frunze, [in Russian]
[40] Brebbia, C. A., The Boundary element method for engineers (1978), Pentech Press: Pentech Press London · Zbl 0414.65060
[41] Powell, M. J.-D., A hybrid method for nonlinear algebraic equations, (Rabinowitz, P., Numerical methods for nonlinear algebraic Equation (1970), Gordon and Breach: Gordon and Breach London) · Zbl 0277.65028
[42] Brown, R. A.S., The mechanics of large gas bubbles in tube. I. Bubble velocities in stagnant liquid, Can J Chem Eng, 43, 217-223 (1965)
[43] DeJesus, J. D.; Ahmad, W.; Kawaji, M., Experimental study of flow structure in vertical slug flow, Proceedings of second int. conf. multiphase flow, Kyoto, 3-7 April (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.