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Lattice Boltzmann model for melting with natural convection

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Lattice Boltzmann model for melting with natural convectionIntroductionReview of pure substance meltingLattice Boltzmann (LB) modelThermal lattice BoltzmannPhase change with lattice BoltzmannResults and discussionConduction meltingConvection meltingConclusionsAcknowledgementAcknowledgmentsReferencesLattice Boltzmann model for melting with natural convectionChristian Hubera,*, Andrea Parmigianib, Bastien Chopardb, Michael Mangac, Olivier BachmanndaDepartment of Earth and Planetary Science, University of California – Berkeley, 307 McCone Hall 4767, Berkeley, CA 94720-4767, USAbComputer Science Department, University of Geneva, 24, Rue du Général Dufour, 1211 Geneva 4, SwitzerlandcDepartment of Earth and Planetary Science, University of California – Berkeley, 177 McCone Hall 4767, Berkeley, CA 94720-4767, USAdDepartment of Earth and Space Science, University of Washington, Johnson Hall 070, Seattle WA 98195-1310, USAarticle infoArticle history:Received 17 October 2007Received in revised form 1 May 2008Accepted 5 May 2008Available online 20 June 2008Keywords:Lattice BoltzmannHeat transferMeltingConvectionabstractWe develop a lattice Boltzmann method to couple thermal convection and pure-substance melting. Thetransition from conduction-dominated heat transfer to fully-developed convection is analyzed and scal-ing laws and previous numerical results are reproduced by our numerical method. We also investigatethe limit in which thermal inertia (high Stefan number) cannot be neglected. We use our results to extendthe scaling relations obtained at low Stefan number and establish the correlation between the meltingfront propagation and the Stefan number for fully-developed convection. We conclude by showing thatthe model presented here is particularly well-suited to study convection melting in geometrically com-plex media with many applications in geosciences.Ó 2008 Elsevier Inc. All rights reserved.1. IntroductionMelting caused by natural convection occurs in many settings,from large-scale phenomena in geosciences to small-scale indus-trial processes during alloy solidification and crystal growth. Mod-els of convection melting need to account for thermally-drivenflow coupled with a moving interface where latent heat is eitherabsorbed (melting) or released (solidification). The interplay be-tween the fluid flow and the moving boundary leads to a complexdynamical behavior, as the position of the solid liquid interface be-comes one of the unknowns of the problem (Jany and Bejan, 1988).The variables of interest are the melting front position and theNusselt number, which describe, respectively, the evolution ofthe geometry of the system and the heat transfer. The movingboundary problem usually requires complex numerical schemessuch as front tracking methods (Bertrand et al., 1999), adaptativegrid methods (Mencinger, 2004), level set methods (Tan and Zab-aras, 2006), phase field approaches (Boettinger et al., 2002) or vol-ume-of-fluid methods (Hirt and Nichols, 1981).Natural convection melting has been investigated by numerousexperimental (Bénard et al., 2006; Wolff and Viskanta, 1987; Donget al., 1991; Hirata et al., 1993; Wang et al., 1999), theoretical(Viskanta, 1982; Viskanta, 1985; Jany and Bejan, 1988; Zhangand Bejan, 1989) and numerical studies (Webb and Viskanta,1986; Bertrand et al., 1999; Mencinger, 2004; Usmani et al.,1992; Chatterjee and Chakraborty, 2005; Javierre et al., 2006).The development of appropriate scaling laws (e.g. Jany and Bejan,1988) and powerful computational methods have significantly im-proved the understanding of the convection melting processes.Heat transfer correlations for the Nusselt number have been devel-oped, but the range of values for the key dimensionless groups(Rayleigh, Prandtl and Stefan numbers) over which the correlationshave been tested remains limited.The lattice Boltzmann method, developed over the last two dec-ades, provides a powerful alternative approach for studying con-vection involving phase changes. Compared with classiccomputational fluid dynamics methods, it offers two significantadvantages. First, no-slip boundary conditions in complex geome-tries can be implemented through simple local rules (Chopard andDroz, 1998). Second, the main part of the algorithm is purely local,making the LB method straightforward to parallelize even thoughnot necessarily more efficient than for example the ‘‘continuum” fi-nite difference method (Nourgaliev et al., 2000, 2003).In this study we develop a Lattice Boltzmann (LB) method tomodel pure substance conduction and convection melting. Westart by introducing the mathematical description of the problemtogether with previously established scaling laws (Jany and Bejan,1988) for the convection case. We then introduce the LB method,with extensions for the thermal model (using a multiple distribu-tion approach) and the phase transition (using a modified versionof Jiaung et al. (2001) algorithm). In Section 4, we compare the re-sults of our model with analytical solutions (for the conductioncase) and with the scaling laws obtained by Jany and Bejan0142-727X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.ijheatfluidflow.2008.05.002* Corresponding author.E-mail addresses: [email protected] (C. Huber), [email protected] (A. Parmigiani), [email protected] (B. Chopard), [email protected] (M. Manga), [email protected] (O. Bachmann).International Journal of Heat and Fluid Flow 29 (2008) 1469–1480Contents lists available at ScienceDirectInternational Journal of Heat and Fluid Flowjournal homepage: www.elsevier.com/locate/ijhff(1988) for the convection problem. We extend their scaling laws tohigh Stefan numbers (i.e. where thermal inertia is non-negligible)for the Nusselt number and the position of the melting front. Final-ly, we show that the model developed here is able to handle non-idealized problems with complex geometries such as porous mediawith no additional complications.2. Review of pure substance meltingThe problem of half space conduction melting with homoge-neous, isotropic thermal diffusivity, has been solved analyticallyin 1860 by Neumann. Heat transfer in the liquid is given byoTot¼jr2T; ð1Þwhere T is temperature, and j the thermal diffusivity. Nomencla-ture is summarized in Table 1. At the melt-solid boundary, whenthe solid is kept at the melting temperature, the energy balance re-quires


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