A multiple-grid-time-step lattice Boltzmann method for transport phenomena with dissimilar time scales: Application in dendritic solidification

In this paper, a multiple-grid-time-step model was developed for lattice Boltzmann (LB) simulations when combined transport equations need to be solved, so that the time step and grid size can be selected independently for each transport domain. The proposed multiple-grid-time-step technique was applied for modeling dendritic solidification of metallic alloys which involves a combination of transport phenomena occurring over different time and length scales. Momentum, solute and heat transfer equations with dissimilar diffusive properties were solved to model growth of a single dendrite. When dendrite growth is simulated under diffusive and convective transport, significant computational saving is achieved by applying the multiple-grid-time-step technique. When the heat transport equation is added, a high Lewis number requires the multiple-grid-time-step model in order to achieve a stable solution of the LB equations. In this case, it is shown that either a smaller time step or a larger grid size for the heat transfer domain compared to the solute transfer domain can stabilize the solution. The computing time is discussed for various cases. It is shown how the proposed model can significantly improve the computational efficiency when multiple transport equations with dissimilar properties need to be solved using the LB method.