In this paper, the unsteady 2-dimensional turbulent flow model for thermal stratification in a pressurizer surge line of PWR plant is proposed to numerically investigate the heat transfer and flow characteristics. The turbulence model is adapted to the low Reynolds number K-$varepsilon$ model (Davidson model). The dimensionless governing equations are solved by using the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The results are compared with simulated experimental results of TEMR Test. The time-dependent temperature profiles in the fluid and pipe nil are shown with the thermal stratification occurring in the horizontal section of the pipe. The corresponding thermal stresses are also presented. The numerical result for thermal stratification by the outsurge during heatup operation of PWR shows that the maximum dimensionless temperature difference is about 0.83 between hot and cold sections of pipe well and the maximum thermal stress is calculated about 322MPa at the dimensionless time 28.5 under given conditions.