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A front-tracking method for the computations of multiphase flow. (English) Zbl 1047.76574
From the summary: Direct numerical simulations of multiphase flows, using a front-tracking method, are presented. The method is based on writing one set of governing equations for the whole computational domain, and treating the different phases as one fluid with variable material properties. Interfacial terms are accounted for by adding the appropriate sources as \(\delta\) functions at the boundary separating the phases. The unsteady Navier-Stokes equations are solved by a conventional finite volume method on a fixed, structured grid and the interface, or front, is tracked explicitly by connected marker points. Interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid. Advection of fluid properties such as density is done by following the motion of the front. The method has been implemented for fully three-dimensional flows, as well as for two-dimensional and axisymmetric ones.

76M12 Finite volume methods applied to problems in fluid mechanics
76T10 Liquid-gas two-phase flows, bubbly flows
Full Text: DOI
[1] Acrivos, A.; Jeffrey, D.J.; Sauille, D.A., Particle migration in suspension by thermocapillary or electrophoretic motion, J. fluid mech, 212, 95, (1990) · Zbl 0695.76050
[2] Adams, J., MUDPACK: multigrid FORTRAN software for the efficient solution of linear elliptic partial differential equations, Appl. math. comput, 34, 113, (1989) · Zbl 0685.65091
[3] Agresar, G.; Linderman, J.J.; Tryggvason, G.; Powell, K.G., An adaptive, Cartesian, front tracking method for the motion, deformation and adhesion of circulating cells, J. comput. phys, 43, 346, (1998) · Zbl 0935.76047
[4] Alexiades, V.; Solomon, A.D., Mathematical modeling of melting and freezing processes, 92, (1993)
[5] Al-Rawahi, N.Z.; Tryggvason, G., A numerical method for flow and solidification, Proceedings of the ASME FEDSM00 fluids engineering division summer meeting, Boston, MA, 11-15, (2000)
[6] Bayvel, L.; Orzechowski, Z., Liquid atomization, (1993)
[7] Beckermann, C.; Diepers, H.-J.; Steinbach, I.; Karma, A.; Tong, X., Modeling melt convection in phase-field simulations of solidification, J. comput. phys, 154, 468, (1999) · Zbl 0960.82015
[8] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys, 100, 335, (1992) · Zbl 0775.76110
[9] Bunner, B., Large scale simulations of bubbly flow, (2000)
[10] Bunner, B.; Tryggvason, G., Direct numerical simulations of three-dimensional bubbly flows, Phys. fluids, 11, 1967, (1999) · Zbl 1147.76342
[11] Bunner, B.; Tryggvason, G., An examination of the flow induced by buoyant bubbles, J. visualization, 2, 153, (1999)
[12] B. Bunner, and, G. Tryggvason, Dynamics of homogeneous bubbly flows. 1. Motion of the bubbles, submitted for publication. · Zbl 1042.76058
[13] B. Bunner, and, G. Tryggvason, Dynamics of homogeneous bubbly flows. 2. Turbulence of the liquid phase, submitted for publication. · Zbl 1152.76484
[14] B. Bunner, and, G. Tryggvason, Effect of bubble deformation on the stability and properties of bubbly flows, submitted for publication. · Zbl 1085.76067
[15] Che, J., Numerical simulations of complex multiphase flows: electrohydrodynamics and solidification of droplets, (1999)
[16] Chen, S.; Johnson, D.B.; Raad, P.E.; Fadda, D., The surface marker and micro cell method, Int. J. numer. meth. fluids, 25, 749, (1997) · Zbl 0896.76064
[17] Clift, R.; Grace, J.R.; Weber, M.E., Bubbles, drops, and particles, (1978)
[18] Cortez, R.; Minon, M., The blob projection method for immersed boundary problems, J. comp. phys, 161, 428, (2000) · Zbl 0962.74078
[19] Daly, B.J., Numerical study of the effect of surface tension on interface instability, Phys. fluids, 12, 1340, (1969) · Zbl 0177.56103
[20] Daly, B.J.; Pracht, W.E., Numerical study of density-current surges, Phys. fluids, 11, 15, (1968) · Zbl 0153.57103
[21] Dandy, D.S.; Leal, G.L., Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers, J. fluid mech, 208, 161, (1989)
[22] Eckert, E.R.G.; Drake, R.M., Analysis of heat and mass transfer, (1972) · Zbl 0247.76079
[23] Eggers, J., Theory of drop formation, Phys. fluids, 7, 941, (1995) · Zbl 1023.76523
[24] Ervin, E.A., Full numerical simulations of bubbles and drops in shear flow, (1993)
[25] Ervin, E.A.; Tryggvason, G., The rise of bubbles in a vertical shear flow, ASME J. fluid eng, 119, 443, (1997)
[26] Esmaeeli, A., Numerical simulations of bubbly flows, (1995)
[27] Esmaeeli, A.; Tryggvason, G., An inverse energy cascade in two-dimensional, low Reynolds number bubbly flows, J. fluid mech, 314, 315, (1996) · Zbl 0875.76636
[28] Esmaeeli, A.; Tryggvason, G., Direct numerical simulations of bubbly flows. part 1—low Reynolds number arrays, J. fluid mech, 377, 313, (1998) · Zbl 0934.76090
[29] Esmaeeli, A.; Tryggvason, G., Direct numerical simulations of bubbly flows. part 2—moderate Reynolds number arrays, J. fluid mech, 385, 325, (1999) · Zbl 0945.76087
[30] Fadlun, E.A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. comput. phys, 161, 35, (2000) · Zbl 0972.76073
[31] Farago, Z.; Chigier, N., Morphological classification of disintegration of round liquid jets in a coaxial air stream, Atom. sprays, 2, 137, (1992)
[32] Feng, J.; Hu, H.H.; Joseph, D.D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. 1. sedimation, J. fluid mech, 261, 95, (1994) · Zbl 0800.76114
[33] Feng, J.; Hu, H.H.; Joseph, D.D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. 2. Couette and poiseuilli flows, J. fluid mech, 277, 271, (1995) · Zbl 0876.76040
[34] Fortes, A.; Joseph, D.D.; Lundgren, T., Nonlinear mechanics of fluidization of beds of spherical particles, J. fluid mech, 177, 467, (1987)
[35] Fukai, J.; Shiiba, Y.; Yamamoto, T.; Miyatake, O.; Poulikakos, D.; Megaridis, C.M.; Zhao, Z., Wetting effects on the spreading of a liquid droplet colliding with a flat surface: experiment and modeling, Phys. fluids, 7, 236, (1995)
[36] Glicksman, M.E., Interaction of flows with the crystal-melt interface, Annu. rev. fluid mech, 18, 307, (1986)
[37] Glimm, J.; Grove, J.W.; Li, X.L.; Oh, W.; Sharp, D.H., A critical analysis of rayleigh – taylor growth rates, J. comput. phys, 169, 652, (2001) · Zbl 1011.76057
[38] Glowinski, R.; Pan, T.W.; Helsa, T.I.; Joseph, D.D.; Periaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys, 169, 363, (2001) · Zbl 1047.76097
[39] Göz, M.F.; Bunner, B.; Sommerfeld, M.; Tryggvason, G., On the unsteady dynamics of two-dimensional bubbles in a regular array, Proceedings of the ASME FEDSM00 fluids engineering division summer meeting, Boston, MA, 11-15, (2000)
[40] Gropp, W.; Lusk, E.; Skjellum, A., Using MPI: portable parallel programming with the message-passing interface, (1995)
[41] Han, J.; Tryggvason, G., Secondary breakup of liquid drops in axisymmetric geometry. I. constant acceleration, Phys. fluids, 11, 3650, (1999) · Zbl 1149.76400
[42] J. Han, and, G. Tryggvason, Secondary breakup of liquid drops in axisymmetric geometry. II. Impulsive acceleration, to appear. · Zbl 1184.76209
[43] Harper, J.F., The motion of bubbles and drops through liquids, Adv. appl. mech, 12, 59, (1972) · Zbl 0257.76094
[44] Harper, J.F., On bubbles with small immobile adsorbed films rising in liquids at low Reynolds numbers, J. fluid mech, 58, 539, (1973) · Zbl 0258.76074
[45] Homma, S.; Koga, J.; Matsumoto, S.; Tryggvason, G., Pinch-off dynamics of jet breakup in liquid – liquid systems, Proceedings of the ASME FEDSM00 fluids engineering division summer meeting, Boston, MA, 11-15, (2000)
[46] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., Boundary integral methods for multicomponent fluids and multiphase materials, J. comput. phys, 169, 302, (2001) · Zbl 1046.76029
[47] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., The long-time motion of vortex sheets with surface tension, Phys. fluids, 9, 1933, (1997) · Zbl 1185.76545
[48] Hu, H.H., Direct simulations of flows of solid – liquid mixtures, Int. J. multiphase flow, 22, 335, (1996) · Zbl 1135.76442
[49] Hu, H.H.; Patankar, N.A.; Zhu, M.Y., Direct numerical simulations of fluid – solid systems using the arbitrary lagrangian – eularian technique, J. comput. phys, 169, 427, (2001) · Zbl 1047.76571
[50] Jacqmin, D., Calculation of two-phase navier – stokes flows using phase-field modeling, J. comput. phys, 155, 96, (1999) · Zbl 0966.76060
[51] Jamet, D.; Lebaigue, O.; Coutris, N.; Delhaye, J.M., The second gradient method for the direct numerical simulations of liquid – vapor flows with phase-change, J. comput. phys, 169, 624, (2001) · Zbl 1047.76098
[52] Jan, Y.J., Computational studies of bubble dynamics, (1994)
[53] Y.-J. Jan and G. Tryggvason, Computational studies of contaminated bubbles in Proceedings of Symposium on Dynamics of Bubbles and Vortices near a Free Surface, edited by I. Sahin and G. TryggvasonASME, New York, 1991, AMD Vol. 119, pp. 46-54.
[54] Jiang, Y.J.; Umemura, A.; Law, C.K., An experimental investigation on the collision behavior of hydrocarbon droplets, J. fluid mech, 234, 171, (1992)
[55] Johnson, A.A.; Tezduyar, T.E., 3D simulation of fluid-particle interactions with the number of particles reaching 100, Comput. methods appl. mech. eng, 145, 301, (1997) · Zbl 0893.76043
[56] Juric, D., Computations of phase change, (1996), University of Michigan
[57] D. Juric, Direct numerical simulation of solidification microstructures affected by fluid flow, in Modeling of Casting, Welding and Advanced Solidification Processes VIII, edited by B. G. Thomas and C. Beckermann, TMS, 1998, pp. 605-612.
[58] Juric, D.; Shin, S.W., Direct computations of solidification with fluid flow, ASME FEDSM’00 numerical methods for multiphase flow, (2000)
[59] D. Juric, Interface stretching test, available at, http://www.me.gatech.edu/djuric.
[60] Juric, D.; Tryggvason, G., A front tracking method for dentritic solidification, J. comput. phys, 123, 127, (1996) · Zbl 0843.65093
[61] Juric, D.; Tryggvason, G., Computations of boiling flows, Int. J. multiphase flow, 24, 387, (1998) · Zbl 1121.76455
[62] Karma, A.; Rappel, W.J., Phase-field simulation of three-dimensional dendrites: Is microscopic solvability theory correct?, J. cryst. growth, 174, 54, (1997)
[63] Karma, A.; Rappel, W.-J., Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. rev. E, 57, 4323, (1998) · Zbl 1086.82558
[64] R. Kobayashi, Simulations of three-dimensional dendrites, in Pattern Formation in Complex Dissipative Systems, edited by S. KaiWorld Scientific, Singapore, 1992, pp. 121-128.
[65] Lafaurie, B.; Nardone, C.; Scardovelli, R.; Zaleski, S.; Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. comput. phys, 113, 134, (1994) · Zbl 0809.76064
[66] Ladd, A.J.C., Dynamical simulations of sedimenting spheres, Phys. fluids A, 5, 299, (1993)
[67] Lamb, H., Hydrodynamics, (1932) · JFM 26.0868.02
[68] Lee, R.C.; Nydahl, J.E., Numerical calculations of bubble growth in nucleate boiling from inception through departure, J. heat transfer, 111, 474, (1989)
[69] Lefebvre, A., Atomization and sprays, (1989)
[70] Leonard, B.P., A stable and accurate convection modelling procedure based on quadratic upstream interpolation, Comput. meth. appl. mech. eng, 19, 59, (1979) · Zbl 0423.76070
[71] Lock, N.; Jaeger, M.; Medale, M.; Occelli, R., Local mesh adaptation technique for front tracking problems, Int. J. numer. meth. fl, 28, 719, (1998) · Zbl 0932.76034
[72] Loth, E.; Taeibi-Rahni, M.; Tryggvason, G., Deformable bubbles in a free shear, Int. J. multiphase flow, 23, 977, (1997) · Zbl 1135.76483
[73] Mortazavi, S.; Tryggvason, G., A numerical study of the motion of drops in Poiseuille flow. 1. lateral migration of one drop, J. fluid mech, 411, 325, (2000) · Zbl 0964.76018
[74] Mortazavi, S., Computational investigation of particulated two-phase flows, (1995)
[75] Mullins, W.W.; Sekerka, R.F., Stability of a planar interface during solidification of a dilute binary alloy, J. appl. phys, 35, 444, (1964)
[76] Nas, S., Computational investigation of thermocapillary migration of bubbles and drops in zero gravity, (1995)
[77] S. Nas and G. Tryggvason, Computational investigation of the thermal migration of bubbles and drops, in AMD 174/FED 175 Fluid Mechanics Phenomena in Microgravity, edited by D. A. Siginer, R. L. Thompson, and L. M. Trefethen, Presented at the ASME 1993 Winter Annual Meeting ASME, New York, 1993, pp. 71-83.
[78] Nobari, M.R.H., Numerical simulations of drop collisions and coalescence, (1993)
[79] Nobari, M.R.; Tryggvason, G., Numerical simulations of three-dimensional drop collisions, Aiaa j, 34, 750, (1996)
[80] Nobari, M.R.; Jan, Y.-J.; Tryggvason, G., Head-on collision of drops—A numerical investigation, Phys. fluids, 8, 29, (1996) · Zbl 1023.76588
[81] Knoll, D.A.; Rider, W.J., A multigrid preconditioned newton – krylov method, SIAM J. sci. comput, 21, 691, (1998) · Zbl 0952.65102
[82] Oran, E.S.; Boris, J.P., Numerical simulation of reactive flow, (1987) · Zbl 0762.76098
[83] Osher, S.; Fedkiw, R.P., Level set methods: an overview and some recent results, J. comput. phys, 169, 463, (2001) · Zbl 0988.65093
[84] Patankar, S.V., Numerical heat transfer and fluid flow, (1980) · Zbl 0595.76001
[85] R. K. Patil, and, J. Prusa, Numerical solutions for asymptotic, diffusion controlled growth of a hemispherical bubble on an isothermally heated surface, in, Experimental/Numerical Heat Transfer in Combustion and Phase Change, edited by, M. F. Modest, T. W. Simon, and M. Ali, Ebadian, ASME, New York, 1991, HTD Vol, 170.
[86] Pozrikidis, C., Interfacial dynamics for Stokes flow, J. comput. phys, 169, 250, (2001) · Zbl 1046.76012
[87] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys, 25, 220, (1977) · Zbl 0403.76100
[88] C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, in SEB Symposium on Biological Fluid Dynamics, Leeds, England, July 5-8.
[89] Peskin, C.S.; Printz, B.F., Improved volume conservation in the computation of flows with immersed boundaries, J. comput. phys, 105, 33, (1993) · Zbl 0762.92011
[90] Plesset, M.S.; Zwick, S.A., The growth of vapor bubbles in superheated liquids, J. appl. phys, 25, 493, (1954) · Zbl 0056.20505
[91] Popinet, S.; Zaleski, S., A front-tracking algorithm for accurate representation of surface tension, Int. J. numer. meth. fluids, 30, 775, (1999) · Zbl 0940.76047
[92] K. Powell, Solution of the Euler equations on solution-adaptive Cartesian grids, in Computational Fluid Dynamics Reviews 1998, edited by M. Hafez and K. OshimaWorld Scientific, 1998, Vol. 1, pp. 65-92.
[93] Prosperetti, A.; Plesset, M.S., Vapor bubble growth in a superheated liquid, J. fluid mech, 85, 349, (1978)
[94] Prosperetti, A.; Plesset, M.S., The stability of an evaporating liquid surface, Phys. fluids, 27, 1590, (1984) · Zbl 0573.76092
[95] Qian, J.; Tryggvason, G.; Law, C.K., A front tracking method for the motion of premixed flames, J. comput. phys, 144, 52, (1998) · Zbl 1392.76050
[96] J. Qian, G. Tryggvason, and, C. K. Law, An experimental and computational study of bounching and deforming droplet collision, submitted for publication.
[97] Rayleigh, Lord, On the pressure developed in a liquid during the collapse of a spherical cavity, Philos. mag, 34, 94, (1917) · JFM 46.1274.01
[98] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys, 153, 509, (1999) · Zbl 0953.76069
[99] Ryskin, G.; Leal, L.G., 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, 19, (1984) · Zbl 0548.76032
[100] Sangani, A.S., Sedimentation in ordered emulsions of drops at low renolds number, J. appl. math. phys. ZAMP, 38, 542, (1988)
[101] Sangani, A.S.; Didwania, A.K., Dynamic simulations of flows of bubbly liquids at large Reynolds numbers, J. fluid mech, 250, 307, (1993) · Zbl 0773.76072
[102] Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Annu. rev. fluid mech, 31, 567, (1999)
[103] Schmidt, A., Computations of three dimensional dendrites with finite elements, J. comput. phys, 126, 293, (1996) · Zbl 0844.65096
[104] Sethian, J.A., Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. comput. phys, 169, 503, (2001) · Zbl 0988.65095
[105] Sethian, J.A.; Strain, J., Crystal growth and dendritic solidification, J. comput. phys, 98, 231, (1992) · Zbl 0752.65088
[106] Shopov, P.J.; Minev, P.D.; Bazhekov, I.B.; Zapryanov, Z.D., Interaction of a deformable bubble with a rigid wall at moderate Reynolds numbers, J. fluid mech, 219, 241, (1990)
[107] Smereka, P., On the motion of bubbles in a periodic box, J. fluid mech, 254, 79, (1993) · Zbl 0788.76085
[108] Song, M.; Tryggvason, G., The formation of a thick border on an initially stationary fluid sheet, Phys. fluids, 11, 2487, (1999) · Zbl 1149.76547
[109] Son, G.; Dhir, V.K., Numerical simulation of saturated film boiling on a horizontal surface, J. heat transfer, 119, 525, (1997)
[110] Son, G.; Dhir, V.K., Numerical simulation of film boiling near critical pressures with a level set method, J. heat transfer, 120, 183, (1998)
[111] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flows, J. comput. phys, 114, 146, (1994) · Zbl 0808.76077
[112] Sussman, M.; Almgren, A.S.; Bell, J.B.; Colella, P.; Howell, L.H.; Welcome, M.L., An adaptive level set approach for incompressible two-phase flows, J. comput. phys, 148, 81, (1999) · Zbl 0930.76068
[113] Taeibi-Rahni, M.; Loth, E.; Tryggvason, G., DNS simulations of large bubbles in mixing layer flow, Int. J. multiphase flow, 20, 1109, (1994) · Zbl 1134.76680
[114] Takagi, S.; Matsumoto, Y., Three-dimensional deformation of a rising bubble, Proceedings of the German-Japanese symposium on multiphase flow, 499, (1994)
[115] Tauber, W.; Tryggvason, G., Primary atomization of a jet, Proceedings of the ASME FEDSM00 fluids engineering division summer meeting, Boston, MA, 11-15, (2000)
[116] W. Tauber, S. O. Unverdi, and, G. Tryggvason, The nonlinear behavior of a sheared immiscible fluid interface, submitted for publication. · Zbl 1185.76364
[117] Tauber, W.; Tryggvason, G., Direct numerical simulations of primary breakup, Comput. fluid dyn, 9, (2000)
[118] T. Tezduyar, Large-scale fluid-particle interactions, http://www.arc.umn.edu/research/tezduyar/101sphere.html.
[119] Tonhardt, R.; Amberg, G., Phase-field simulations of dendritic growth in a shear flow, J. cryst. growth, 194, 406, (1998)
[120] Tryggvason, G.; Aref, H., Numerical experiments on Hele Shaw flow with a sharp interface, J. fluid mech, 136, 1, (1983) · Zbl 0525.76042
[121] Tryggvason, G., Numerical simulation of the rayleigh – taylor instability, J. comput phys, 75, 253, (1988) · Zbl 0638.76056
[122] Tryggvason, G.; Unverdi, S.O., Computations of three-dimensional Rayleigh-Taylor instability, Phys. fluids A, 2, 656, (1990)
[123] G. Tryggvason and S. O. Unverdi, The shear breakup of an immiscible fluid interface, in Fluid Dynamics at Interfaces, edited by W. Shyy and R. Narayanan, Cambridge Univ. Press, Cambridge, UK, 1999, pp. 142-155. · Zbl 0980.76027
[124] Unverdi, S.O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys, 100, 25, (1992) · Zbl 0758.76047
[125] Unverdi, S.O.; Tryggvason, G., Computations of multi-fluid flows, Physica D, 60, 70, (1992) · Zbl 0779.76101
[126] Udaykumar, H.S.; Kan, H.C.; Shyy, W.; Tran-Son-Tay, R., Multiphase dynamics in arbitrary geometries on fixed Cartesian grids, J. comput. phys, 137, 366, (1997) · Zbl 0898.76087
[127] Udaykumar, H.S.; Mittal, R.; Shyy, W., Computation of solid – liquid phase fronts in the sharp interface limit on fixed grids, J. comput. phys, 153, 535, (1999) · Zbl 0953.76071
[128] Ungar, L.H.; Brown, R.A., Cellular interface morphologies in directional solidification. IV. the formation of deep cells, Phys. rev. B, 31, 5931, (1985)
[129] Warren, J.A.; Boettinger, W.J., Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta metall, 43, 689, (1995)
[130] Welch, S.W.J.; Wilson, J., A volume of fluid based method for fluid flows with phase change, J. comput. phys, 160, 662, (2000) · Zbl 0962.76068
[131] Welch, S.W.J., Local simulation of two-phase flows including interface tracking with mass transfer, J. comput. phys, 121, 142, (1995) · Zbl 0839.76069
[132] Weatherburn, C.E., Differential geometry of three dimensions, I, (1927) · JFM 53.0658.08
[133] Wheeler, A.A.; Murray, B.T.; Schaefer, R.J., Computations of dendrites using a phase-field model, Physica D, 66, 243, (1993) · Zbl 0800.76491
[134] Wheeler, A.A.; Ahmad, N.A.; Boettinger, W.J.; Braun, R.J.; McFadden, G.B.; Murray, B.T., Recent development in phase-field models of solidification, Adv. space res, 16, 163, (1995)
[135] Wu, P.-K.; Miranda, R.F.; Faeth, G.M., Effects of initial flow conditions on primary breakup of nonturbulent and turbulent liquid jets, (1994)
[136] Yabe, T., Interface capturing and universal solution of solid, liquid and gas by CIP method, Proceedings of the high-performance computing of multi-phase flow, Tokyo, 18-19, (1997)
[137] Yabe, T.; Xiao, F.; Utsumi, T., The constrained interpolation profile (CIP) method for multiphase analysis, J. comput. phys, 169, 556, (2001) · Zbl 1047.76104
[138] Yang, Y.; Tryggvason, G., Dissipation of energy by finite amplitude surface waves, Comput. fluids, 27, 829, (1998) · Zbl 0964.76024
[139] Yu, P.-W.; Ceccio, S.L.; Tryggvason, G., The collapse of a cavitation bubble in shear flows—a numerical study, Phys. fluids, 7, 2608, (1995) · Zbl 1026.76564
[140] Zaleski, S.; Li, J.; Succi, S., 2-dimensional navier – stokes simulation of deformation and breakup of liquid patches, Phys. rev. lett, 75, 244, (1995)
[141] S. Zaleski, J. Li, R. Scardovelli, and G. Zanetti, Direct simulation of multiphase flows with density variations, Proceedings of IUTAM Symposium, Marseille, July 8-10, 1996, edited by F. Anselmet and L. FulachierKluwer Academic, Dordrecht, 1996.
[142] Deleted in proof.
[143] Esmaeeli, A.; Tryggvason, G., Direct numerical simulations of boiling flows, Proceedings of the fourth international conference on multiphase flows, new orleans, LA, (May 27-June 1, 2001)
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