Three-dimensional phase-field simulations of the effect of convection on free dendritic growth

Three-dimensional free dendritic growth of a pure material into an undercooled melt in the presence of fluid flow is investigated numerically using the phase-field method. Such computations are made possible by solving the Navier-Stokes equations for the flow and the energy equation for the heat transport on a grid that is twice as coarse as the grid for the phase-field equation. The effect of the flow on the upstream growing dendrite tip velocity and radius of curvature is investigated as a function of the imposed flow velocity, undercooling, crystalline anisotropy, and Prandtl number. The results are compared to available theories of dendritic growth with and without convection. The predicted growth Péclet numbers as a function of the flow Péclet number are in reasonable agreement with the theoretical predictions. The dendrite tip selection parameter is essentially independent of the flow velocity within the range studied, which is also in accordance with theory. The three-dimensional dendrite tip shape is found to be well fitted by the same universal scaling relation as without flow.