A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solidliquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes.

This paper is devoted to the study of convergence orders of several numerical methods that are implemented in the TrioCFD code dedicated to the simulation of turbulent flows and heat transfer in nuclear engineering applications. The spatial discretization is based on Finite Difference-Volume or Finite Element-Volume methods. A projection method is applied to update the velocity and the pressure. The time scheme can be either explicit or implicit, and hexahedral or tetrahedral meshes can be used for simulations. In this paper, the test cases are relative to steady Stokes problems, steady and unsteady Navier-Stokes problems, and finally the well-known lid-driven cavity flow case. The latter proposes several comparisons between our simulations and numerical data already published in the literature, while the other cases yield the values of convergence orders by using the analytical solutions. The accuracy of the results obtained with TrioCFD differs according to the types of mesh used for simulations, the viscosity values or the source terms in the equations.