Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation

Abstract

Due to distinctions of the heat and mass transport in two (2D) and three (3D) dimensions, the morphology and growth of numerically simulated dendrites are usually different. In order to quantitatively address such differences, the free dendritic growth of a binary alloy from undercooled melt under forced flow is simulated using the phase-field (PF) method. The vector-valued approach has been employed to solve the equations of fluid dynamics to expedite simulations. The classical Ananth-Gill solutions are used to validate the results of PF simulations, and good agreements between the simulated growth Péclet numbers of the upstream tip and analytical predictions are achieved. The tip velocity and radius of the steady-state upstream tip, and the solute profile ahead of the tip solid-liquid (S-L) interface, are compared quantitatively between 2D and 3D simulations. In 3D, since the rejected solute can be rapidly transported away from the S-L interface by liquid flow, the solute concentration at the liquid side of the S-L interface is lower, but its gradient is higher, and the thickness of the solute boundary layer is smaller, thus leading to a higher tip velocity and smaller tip radius. However, the ratios of the tip velocity and tip radius between 2D and 3D are not confined in a certain range but varies with the supersaturation and liquid inflow velocity. Nevertheless, the differences are relieved with increasing the supersaturation or the inflow strength. Quantitative distinctions can be found from the ratios of these characteristics and the tip stability parameter in 2D versus 3D simulations, which follow a power law of supersaturation or growth Péclet number.