In this paper, the mathematical model and numerical simulations of the molten steel flow by the submerged entry nozzle and the filling process of the continuous casting mould cavity are presented. In the mathematical model, the temperature fields were obtained by solving the energy equation, while the velocity fields were calculated by solving the momentum equations and the continuity equation. These equations contain the turbulent viscosity which is found by solving two additional transport equations for the turbulent kinetic energy and its rate of dissipation. In the numerical simulations, coupling of the thermal and fluid flow phenomena by changes in the thermophysical parameters of alloy depending on the temperature has been taken into consideration. This problem (2D) was solved by using the finite element method. Numerical simulations of filling the continuous casting mould cavity were performed for two variants of liquid metal pouring. The effect of the cases of pouring the continuous casting mould on the velocity fields and the solid phase growth kinetics in the process of filling the continuous casting mould was evaluated as these magnitudes have an influence on the high quality of the continuous cast steel slab.