A numerical simulation of the filling stage along the thickness direction is proposed by combining the free-surface boundary condition with the relevant governing equations. The mathematical model is based on the equations of continuity, momentum, and energy along with the inelastic power-law model and relevant boundary conditions. Because of the significant implications for microstructure development in the product, the fountain effect at the advancing free surface is explicitly taken into consideration in the simulation. The model yields data on free-surface shape and velocity, pressure, temperature, and shear stress distributions within the mold cavity. The rearrangement of the velocity and temperature profiles in the vicinity of the melt front is considered in detail.