Numerical computation has been performed to investigate temperature and velocity distributions for different stages of the Kyropoulos sapphire single crystal-growth process. The finite-element method is employed to solve the governing equations with proper boundary conditions. In the power history considered here, a vortex appears in the melt during growth, and its strength decreases as the input power is reduced. Isotherms in the melt are distorted by flow motion. The crystal-melt interface is always convex towards the melt and in early stages the convexity increases as the input power decreases. When the crystal-melt interface is close to the bottom of the crucible, this interface is flat near the apex because of reduction in growth rate near the upper region caused by input heat from the bottom of the crucible. Therefore, convexity of the crystal-melt interface decreases the input power decreases. The crystal shape predicted by the present simulation is similar to that of crystals grown in the industry. (C) 2010 Elsevier B.V. All rights reserved.