Lattice Boltzmann scheme for crystal growth in external flows

Abstract

A composite phase-field lattice-Boltzmann scheme is used to simulate dendritic growth from a supercooled melt, allowing for heat transport by both diffusion and convection. The phase-transition part of the problem is modelled by the phase-field approach of Karma and Rappel, whereas the flow of the liquid is computed by the lattice-Boltzmann-BGK (LBGK, referring to Bhatnagar, Gross, and Krook) method into which interactions with solid and thermal convection are incorporated. For simplicity, we have so far restricted ourselves to the symmetric model. Heat transport is simulated via the multicomponent LBGK method. Depending on the level of anisotropy and undercooling, dendrites or doublons are obtained in our simulations. Crystal growth in a shear flow is considered for different flow velocities and undercoolings. Doublons turn out to be robust against the perturbation imposed by a shear flow and display interesting dynamic behavior, quite different from that of dendrites. In addition, the influence of a parallel flow on the operating state of the tip of dendrites is studied. To complement information from selection theories such as the one presented by Bouissou and Pelcé, we measure selected growth characteristics of dendrites as a function of a flow imposed parallel to the growth direction, for intermediate undercoolings. The observed dependencies are compatible with power law behavior, if the undercooling is not too high. It is shown that for sufficiently large flow velocities, an oncoming flow can lead to tip oscillations of the dendrite and, consequently, to the generation of coherent side branches.

Influence of external flows on crystal growth: Numerical investigation

“…Clearly, more numerical or experimental data are needed to both check the existing theories and guide further theoretical development. The purpose of this article is to provide first elements of these data using our combined phase-field lattice-Boltzmann approach [22,23]. A successful selection theory on the microscale (constructed on the basis of these and similar data) would then yield useful information for more applied work on macrosegregation and related questions.…”

“…A first attempt to extend selection theory to situations with a flow was made by Bouissou and Pelcé [40] and there were a number of less rigorous approaches to the problem as well (references are given in [23]). However, while one experiment seems to support this theory [41], another one contradicts it [42].…”

“…This approach was pioneered by Miller et al [19,20,21], and a slightly different variant, the advantages of which will be discussed in Sec. II, was developed by ourselves [22,23].…”

Phase-field-lattice Boltzmann study for lamellar eutectic growth in a natural convection melt

“…Encouragingly, a robust parallel-adaptive mesh refinement algorithm (Para-AMR) has been developed to efficiently solve the phase field equations including the dendritic [9] and eutectic transitions [10] . The lattice Boltzmann method (LBM) has also emerged with great potential to numerically solve the energy, momentum and mass transport problems by a relaxation to a local equilibrium [11][12][13] . Therefore, it seems reasonable to couple the LBM into eutectic phase-field models to retrieve the interaction of eutectic microstructure and flows.…”

“…The lattice Boltzmann method (LBM) has also emerged with great potential to numerically solve the energy, momentum and mass transport problems by a relaxation to a local equilibrium [11][12][13] . Therefore, it seems reasonable to couple the LBM into eutectic phase-field models to retrieve the interaction of eutectic microstructure and flows.…”