| Three-dimensional dendritic solidification of a pure material is investigated numerically in the presence of fluid flow. A phase-field method is used to track the solid/liquid interface. The Navier-Stokes equations are solved using a modified version of a finite element CFD code, while the phase-field and energy equations are solved by finite difference schemes. The numerical algorithm is verified by solving numerous benchmark problems.; An effective double grid scheme is developed which allows for solving the flow and temperature fields on a coarser uniform grid, while the phase-field is solved on a finer uniform grid. Convergence studies performed on every case show that the single and double grid schemes yield very similar results.; For the cases studied, the selection of the tip velocity and tip radius of a dendrite in forced convection varies with flow velocity, material anisotropy, magnitude of undercooling and Prandtl number. Fluid flow significantly changes the dendrite morphology as well as the temperature field around the growing dendrite. Flow parallel to the growth direction greatly increases the upstream tip velocity, while flow perpendicular to the growth direction has almost no effect on the cross-stream tip growth velocity. The downstream dendrite tip grows much slower in the presence of fluid flow. For the upstream dendrite tip, it is found that both the actual tip radius and the parabolic fit tip radius decrease with increasing fluid flow velocity. However, when the interface thickness to diffusion length ratio is too large, the simulation results will be too far from convergence and the results provide misleading information.; Simulation results are also compared with available theories of dendritic growth assuming the tip to be a paraboloid of revolution. The relationship between the growth Peclet number and flow Peclet number, both based on the parabolic fit tip radius as the characteristic length, is in reasonable agreement with the theoretical predictions. Furthermore, the ratio of selection parameter without and with flow is compared with an available solvability theory that considers convection. Additional simulations are needed on the behavior of the selection parameter with fluid flow. |
