×

zbMATH — the first resource for mathematics

A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. (English) Zbl 1087.76089
Summary: A stable discretization of the lattice Boltzmann equation (LBE) for non-ideal gases is presented for simulation of incompressible two-phase flows having high density and viscosity ratios. The stiffness of the discretized forcing terms in LBE for non-ideal gases is known to trigger severe numerical instability and restrict practical application of the LBE method. Use of a proper pressure updating scheme is also crucial to the stability of the LBE method because of non-negligible pressure variation across the phase interface. To deal with these issues, we propose a stable discretization scheme, which assumes the low Mach number approximation, and utilizes the stress and potential forms of the surface tension force, the incompressible transformation, and the consistent discretization of the intermolecular forcing terms. The proposed stable discretization scheme is applied to simulate 1-D advection equation with a source term, a stationary droplet, droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying Reynolds numbers. The numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the Laplace’s law. The time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law.

MSC:
76M28 Particle methods and lattice-gas methods
76T99 Multiphase and multicomponent flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L.M. Pismen, Nonlocal diffuse interface theory of thin films and the moving contact line, Phys. Rev. E, 64, 2001, 021693-1
[2] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. rev. fluid mech., 30, 329, (1998) · Zbl 1398.76180
[3] Gunstensen, A.K.; Rothman, D.H.; Zaleski, S.; Zanetti, G., A discrete Boltzmann equation model for non-ideal gases, Phys. rev. A, 43, 4320, (1991)
[4] Shan, X.; Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. rev. E, 47, 1815, (1993)
[5] Shan, X.; Chen, H., Simulation of nonideal gases and liquid-gas phase-transition by the lattice Boltzmann-equation, Phys. rev. E, 49, 2941, (1994)
[6] Swift, M.R.; Osborn, W.R.; Yeomans, J.M., Lattice Boltzmann simulation of nonideal fluids, Phys. rev. lett., 75, 830, (1995)
[7] Swift, M.R.; Orlandini, E.; Osborn, W.R.; Yeomans, J.M., Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Phys. rev. E, 54, 5041, (1996)
[8] Martys, N.S.; Chen, H., Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method, Phys. rev. E, 53, 743, (1996)
[9] He, X.; Chen, S.; Zhang, R., A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability, J. comput. phys., 152, 642, (1999) · Zbl 0954.76076
[10] He, X.; Zhang, R.; Chen, S.; Doolen, G.D., On the three-dimensional Rayleigh-Taylor instability, Phys. fluids, 11, 1143, (1999) · Zbl 1147.76410
[11] Zhang, R.; He, X.; Chen, S., Interface and surface tension in incompressible lattice Boltzmann multiphase model, Comput. phys. commun., 129, 121, (2000) · Zbl 0990.76073
[12] Zhang, R.; He, X.; Doolen, G.; Chen, S., Surface tension effects on two-dimensional two-phase Kelvin-Helmholtz instabilities, Adv. water resour., 24, 461, (2001)
[13] Nie, X.; Doolen, G.D.; Chen, S., Lattice-Boltzmann simulations of fluid flows in MEMS, J. stat. phys., 107, 279, (2002) · Zbl 1007.82007
[14] Lim, C.Y.; Shu, C.; Niu, X.D.; Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Phys. fluids, 14, 2299, (2002) · Zbl 1185.76227
[15] Succi, S., Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis, Phys. rev. lett., 89, 064502, (2003)
[16] T. Lee, C.-L. Lin, Rarefaction and compressibility effects of the lattice Boltzmann equation method in a gas microchannel, in review, 2004
[17] Anderson, D.M.; McFadden, G.B.; Wheeler, A.A., Diffuse-interface methods in fluid mechanics, Annu. rev. fluid mech., 30, 139, (1998) · Zbl 1398.76051
[18] Jamet, D.; Lebaigue, O.; Coutris, N.; Delhaye, J.M., The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change, J. comput. phys., 169, 624, (2001) · Zbl 1047.76098
[19] Lee, T.; Lin, C.-L., A pressure evolution lattice Boltzmann equation method for two-phase flow with phase change, Phys. rev. E, 67, 056703, (2003)
[20] He, X.; Shan, X.; Doolen, G.D., A discrete Boltzmann equation model for non-ideal gases, Phys. rev. E, 57, R13, (1998)
[21] He, X.; Luo, L., A discrete Boltzmann equation model for non-ideal gases, Phys. rev. E, 55, R6333, (1997)
[22] Holdych, D.J.; Rovas, D.; Georgiadis, J.G.; Buckius, R.O., An improved hydrodynamics formulation for multiphase flow lattice-Boltzmann models, Int.J. mod. phys. C, 9, 1393, (1998)
[23] Nourgaliev, R.R.; Dinh, T.N.; Theofanous, T.G.; Joseph, D., The lattice Boltzmann equation method: theoretical interpretation, numerics and implications, Int. J. multiphase flow, 29, 117, (2003) · Zbl 1136.76594
[24] Majda, A.; Sethian, J., The derivation and numerical solution of the equations for zero Mach number combustion, Combust. sci. technol., 42, 185, (1985)
[25] Day, M.S.; Bell, J.B., Numerical simulation of laminar reacting flows with complex chemistry, Combust. theory modelling, 4, 4, 535, (2000) · Zbl 0970.76065
[26] Chen, Y.; Teng, S.L.; Shukuwa, S.; Ohashi, H., Lattice Boltzmann simulation of two-phase fluid flows, Int. J. mod. phys. C, 9, 1383, (1998)
[27] Hirsch, C., Numerical computation of internal and external flows, vol. II, (1990), John Wiley and Sons New York · Zbl 0742.76001
[28] Lee, T.; Lin, C.-L., An Eulerian description of the streaming process in the lattice Boltzmann equation, J. comput. phys., 185, 445, (2003) · Zbl 1047.76106
[29] Inamuro, T.; Ogata, T.; Ogino, F., Lattice Boltzmann simulation of bubble flows, computational science-ICCS2003 LNCS 2657, (2003), Springer-Verlag Berlin, 1015
[30] Inamuro, T.; Ogata, T.; Tajima, S.; Konishi, N., A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. comput. phys., 198, 628, (2004) · Zbl 1116.76415
[31] Sankaranarayanan, K.; Shan, X.; Kevrekidis, I.G.; Sundaresan, S., Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method, J. fluid mech., 452, 61, (2002) · Zbl 1059.76070
[32] Qian, Y.-H.; Chen, S.-Y., Dissipative and dispersive behavior of lattice-based models for hydrodynamics, Phys. rev. E, 61, 2712, (2000)
[33] Nadiga, B.T.; Zaleski, S., Investigations of a two-phase fluid model, Euro. J. mech. B: fluids, 15, 885, (1996) · Zbl 0886.76097
[34] Hsieh, D.-Y.; Wang, X.-P., Phase transition in van der Waals fluid, SIAM J. appl. math., 57, 4, 871, (1997) · Zbl 0896.76091
[35] D. Jacgmin, An energy approach to the continuum surface method, in: 34th Aerospace Sciences Meeting, Reno, NV 96, 1996, p. 0858
[36] Filippova, O.; Hänel, D., A novel lattice BGK approach for low Mach number combustion, J. comput. phys., 158, 139, (2000) · Zbl 0963.76072
[37] Rowlinson, J.S.; Widom, B., Molecular theory of capillary, (1989), Oxford Press Oxford
[38] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc.R. soc. lond. A, 454, 2617, (1998) · Zbl 0927.76007
[39] Angelopoulos, A.D.; Paunov, V.N.; Burganos, V.N.; Payatakes, A.C., Lattice Boltzmann simulation of nonideal vapor-liquid flow in porous media, Phys. rev. E, 57, 3, 3237, (1998)
[40] Luo, L., Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. rev. E, 62, 4982, (2000)
[41] Palmer, B.J.; Rector, D.R., Lattice-Boltzmann algorithm for simulating thermal two-phase flow, Phys. rev. E, 61, 5295, (2000)
[42] Lee, T.; Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, J. comput. phys., 171, 336, (2001) · Zbl 1017.76043
[43] M. Dai, J.B. Perot, D.P. Schmidt, Heat transfer within deforming droplets, in: Proceedings of ASME, Internal Combustion Engine Division, September, New Orleans, 2002
[44] Lamb, H., Hydrodynamics, (1932), Dover New York · JFM 26.0868.02
[45] Josserand, C.; Zaleski, S., Droplet splashing on a thin liquid film, Phys. fluids, 15, 1650, (2003) · Zbl 1186.76263
[46] Weiss, D.A.; Yarin, A.L., Single drop impact ont liquid films: neck distortion, jetting, tiny bubble entrainment, and crown formation, J. fluid mech., 385, 229, (1999) · Zbl 0931.76011
[47] Kim, H.-Y.; Feng, Z.; Chun, J.-H., Instability of a liquid jet emerging from a droplet upon collision with a solid surface, Phys. fluids, 12, 531, (2000) · Zbl 1149.76433
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.