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An enthalpy-based hybrid lattice-Boltzmann method for modelling solid–liquid phase transition in the presence of convective transport

  • SUMAN CHAKRABORTY (a1) and DIPANKAR CHATTERJEE (a1)

Abstract

An extended lattice Boltzmann model is developed for simulating the convection–diffusion phenomena associated with solid–liquid phase transition processes. Macroscopic hydrodynamic variables are obtained through the solution of an evolution equation of a single-particle density distribution function, whereas, the macroscopic temperature field is obtained by solving auxiliary scalar transport equations. The novelty of the present methodology lies in the formulation of an enthalpy-based approach for phase-change modelling within a lattice-Boltzmann framework, in a thermodynamically consistent manner. Thermofluidic aspects of phase transition are handled by means of a modified enthalpy–porosity formulation, in conjunction with an appropriate enthalpy-updating closure scheme. Lattice-Boltzmann simulations of melting of pure gallium in a rectangular enclosure, Rayleigh–Bénard convection in the presence of directional solidification in a top-cooled cavity, and crystal growth during solidification of an undercooled melt agree well with the numerical and experimental results available in the literature, and provide substantial evidence regarding the upscaled computational economy provided by the present methodology.

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Askar, H. G. 1987 The front tracking scheme for the one-dimensional freezing problem. Intl J. Numer.Meth. Engng 24, 859869.
Barrios, G., Rechtman, R., Rojas, J. & Tovar, R. 2005 The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall. J. Fluid Mech. 522, 91100.
Bartoloni, A., Battista, C., Cabasino, S. et al. 1993 LBE simulations of Rayleigh–Bénard convection on the APE100 parallel processor. Intl J. Mod. Phys. C 4, 9931006.
Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A. & Tong, X. 1999 Modeling melt convection in phase-field simulations of solidification. J. Comput. Phys. 154, 468496.
Bennon, W. D. & Incropera, F. P. 1987 A continuum model for momentum, heat and species transport in binary solid–liquid phase-change systems. I. Model formulation. Intl J. Heat Mass Transfer 30, 21612170.
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in charged and neutral one-component system. Phys. Rev. 94, 511525.
Brent, A. D., Voller, V. R. & Reid, K. 1988 Enthalpy-porosity technique for modeling convection–diffusion phase change: application to the melting of a pure metal. Numer. Heat Transfer 13, 297318.
Chakraborty, S. & Dutta, P. 2001 A generalized formulation for evaluation of latent heat functions in enthalpy-based macroscopic models for convection–diffusion phase change process. Metall. Mat. Trans. B 32, 562564.
Chatterjee, D. & Chakraborty, S. 2005 An enthalpy based lattice Boltzmann model for diffusion dominated solid–liquid phase transformation. Phys. Lett. A 341, 320330.
Chatterjee, D. & Chakraborty, S. 2006 A hybrid lattice Boltzmann model for solid–liquid phase transition in presence of fluid flow. Phys. Lett. A 351, 359367.
Chatterjee, D. & Chakraborty, S. 2007 An enthalpy-source based lattice Boltzmann model for conduction dominated phase change of pure substances. Int. J. Therm. Sci. (accepted).
Chen, H., Chen, S. & Matthaeus, W. H. 1992 Recovery of the Navier–Stokes equations through a lattice gas Boltzmann equation method. Phys. Rev. A 45, R53395342.
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329364.
Chen, S., Chen, H., Martnez, D. & Matthaeus, W. 1991 Lattice Boltzmann model for simulation of magnetohydrodynamics. Phys. Rev. Lett. 67, 37763779.
Chen, Y., Ohashi, H. & Akiyama, M. 1995 Heat transfer in lattice BGK modeled fluid. J. Stat. Phys. 81, 7185.
Dantzig, J. A. 1989 Modeling liquid–solid phase changes with melt convection. Intl J. Numer. Meth. Engng 28, 17691785.
De Fabritiis, G., Mancini, A., Mansutti, D. & Succi, S. 1998 Mesoscopic models of liquid/solid phase transitions. Intl J. Mod. Phys. C 9, 14051415.
Frisch, U., Hasslacher, B. & Pomeau, Y. 1986 Lattice-gas automata for the Navier–Stokes equation. Phys. Rev. Lett. 56, 15051508.
Gau, C. & Viskanta, R. 1986 Melting and solidification of a pure metal on a vertical wall. J. Heat Transfer 108, 174181.
Guo, Z., Zheng, C. & Shi, B. 2002 Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308-1046308-6.
Harrowell, P. R. & Oxtoby, D. W. 1987 On the interaction between order and a moving interface: dynamical disordering and anisotropic growth rates. J. Chem. Phys. 86, 29322942.
He, X., Luo, L.-S. & Dembo, M. 1997 Some progress in lattice Boltzmann method: enhancement of Reynolds number in simulations. Physica A 239, 276285.
Jeong, J. H., Goldenfeld, N. & Dantzig, J. A. 2001 Phase field model for three-dimensional dendritic growth with fluid flow. Phys. Rev. E 64, 041602 (114).
Kendon, V. M., Cates, M. E., Pagonabarraga, I., Desplat, J.-C. & Bladon, P. 2001 Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study. J. Fluid Mech. 440, 147203.
Khachaturyan, A. G. 1996 Long-range order parameter in field model of solidification. Phil. Mag. A 74, 314.
Kim, Y-T., Provatas, N., Goldenfeld, N. & Dantzig, J. A. 1999 Universal dynamics of phase-field models for dendritic growth. Phys. Rev. E 59, R2546R2549.
Kumar, P., Chakraborty, S., Srinivasan, K. & Dutta, P. 2002 Rayleigh–Bénard convection during solidification of a eutectic solution cooled from the top. Metall. Trans. B 33, 605612.
Lallemand, P. & Luo, L.-S. 2003 Hybrid finite-difference thermal Lattice Boltzmann equation. Intl J. Mod. Phys. B 17, 4147.
Lan, C. W., Hsu, C. M., Liu, C. C. & Chang, Y. C. 2002 Adaptive phase field simulation of dendritic growth in a forced flow at various supercoolings. Phys. Rev. E 65, 061601(111).
Martys, N. S., Shan, X. & Chen, H. 1998 Evaluation of the external force term in the discrete Boltzmann equation. Phys. Rev. E 58, 68556857.
Melchionna, S. & Succi, S. 2004 Electrorheology in nanopores via lattice Boltzmann simulation. J. Chem. Phys. 120, 44924497.
Mezrhab, A., Bouzidi, M. & Lallemand, P. 2004 Hybrid lattice-Boltzmann finite-difference simulation of convective flows. Computers Fluids 33, 623641.
Mikheev, L. V. & Chernov, A. A. 1991 Mobility of a diffuse simple crystal melt interface. J. Cryst. Growth. 112, 591596.
Miller, W. 2001 The lattice Boltzmann method: a new tool for numerical simulation of the interaction of growth kinetics and melt flow. J. Cryst. Growth 230, 263269.
Miller, W. & Schroder, W. 2001 Numerical modeling at the IKZ: an overview and outlook. J. Cryst. Growth 230, 19.
Miller, W. & Succi, S. 2002 A Lattice Boltzmann model for anisotropic crystal growth from melt. J. Stat. Phys. 107, 173186.
Miller, W., Succi, S. & Manutti, D. 2001 Lattice Boltzmann model for anisotropic liquid–solid phase transition. Phys. Rev. Lett. 86, 35783581.
Miller, W., Rasin, I. & Pimentel, F. 2004 Growth kinetics and melt convection. J. Cryst. Growth 266, 283288.
Pal, D., Bhattacharya, J., Dutta, P. & Chakraborty, S. 2006 An enthalpy model for simulation of dendritic growth. Numer. Heat Transfer B 50, 5978
Palle, N. & Dantzig, J. A. 1996 An adaptive mesh refinement scheme for solidification problems. Mettal. Trans. 27A, 707717.
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere/McGraw-Hill.
Qian, Y., D'Humieres, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479484.
Rubinsky, B. & Cravahlo, E. G. 1981 A finite element method for the solution of one-dimensional phase change problems. Intl J. Heat Mass Transfer 24, 19871989.
Sankaranarayanan, K., Shan, X., Kevrekidis, I. G. & Sundaresan, S. 2002 Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method. J. Fluid Mech. 452, 6196.
Sasikumar, R. & Sreenivasan, R. 1994 Two-dimensional simulation of dendrite morphology. Acta Metall. Mater. 42, 23812386.
Shan, X. 1997 Simulation of Rayleigh–Bénard convection using a Lattice Boltzmann model. Phys. Rev. E 55, 27802788.
Shan, X. & Chen, H. 1993 Lattice Boltzmann model for simulating flows with multiple phase and components. Phys. Rev. E 47, 18151819.
Shan, X. & He, X. 1998 Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80, 6568.
Shin, Y. H. & Hong, C. P. 2002 Modeling of dendritic growth with convection using a modified cellular automaton model with a diffuse interface. ISIJ Intl 42, 359367.
Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.
Tong, X., Beckermann, C., Karma, A. & Li, Q. 2001 Phase-field simulations of dendritic crystal growth in a forced flow. Phys. Rev. E 63, 061601-(116).
Voller, V. & Cross, M. 1981 Accurate solutions of moving boundary problems using the enthalpy method. Intl J. Heat Mass Transfer 24, 545556.
Voller, V. & Cross, M. 1983 An explicit numerical method to track a moving phase change front. Intl J. Heat Mass Transfer 26, 147150.
Voller, V. R. & Prakash, C. 1987 A fixed grid numerical modeling methodology for convection–diffusion mushy region phase change problems. Intl J. Heat Mass Transfer 30, 17091719.
Voller, V., Swaminathan, C. R. & Thomas, B. G. 1990 Fixed grid techniques for phase change problems: a review. Intl J. Numer. Methods Engng 30, 875898.
Weaver, J. A. & Viskanta, R. 1986 Freezing of liquid saturated porous media. Intl J. Heat Mass Transfer 33, 27212734.
Zhang, R. & Chen, H. 2003 Lattice Boltzmann method for simulations of liquid–vapor thermal flows. Phys. Rev. E 67, 066711(16).
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