Numerical modeling was performed to investigate the developing turbulent flow and heat transfer characteristics of water near the critical point in a curved pipe. The renormalization group (RNG) kappa-epsilon model was used to account for the turbulent flow and heat transfer in the curved pipe at a constant wall temperature with or without buoyancy force effect. A control volume finite element method (CVFEM) was used to solve the three-dimensional full elliptic governing equations for the problem numerically. Due to the great variation in physical properties of water near the critical point, the turbulent flow heat transfer can be significantly altered compared with the pure forced convection in the curved pipe. This study explored the influence of the near-critical pressure and wall temperature on the development of fluid flow and heat transfer along the pipe. The numerical results for forced convective flow and heat transfer were compared with experiments available in the literature. Based on the results of this research, the velocity, temperature, heat transfer coefficient, friction factor distribution, and effective viscosity are presented graphically and analyzed.