Numerical simulation of dendritic solidification with convection: Three-dimensional flow

A numerical method for the simulation of the effect of melt flow on the three-dimensional growth of a dendrite is described. The method is an extension of the technique for two-dimensional flow described in Al-Rawahi and Tryggvason [J. Comput. Phys. 180 (2002) 471] and is based on the explicit tracking of connected marker points that describe the liquid-solid interface. An explicit projection method is used to solve the energy and the Navier-Stokes equations on a regular stationary grid and the solidified region is represented by setting the velocities in the solid phase to zero. The latent heat released during solidification is calculated using the normal temperature gradient near the interface. The method is validated by a comparison with an exact solution for a Stefan problem and a grid refinement study. The simulations show that the speed of a dendrite arm growing into the flow is increased due to an increase in the temperature gradient on the upstream side and the formation of side branches is promoted, as in two-dimensions. The effect of the flow on the growth of dendrite arms growing in the downstream direction is smaller than in two-dimensions, due to a smaller wake.