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Understanding the complex physical phenomena involved in the continuous casting process simulation requires continuous and complementary research to improve mathematical modeling. The paper presents a mathematical model and numerical simulations of the molten steel flow by the submerged entry nozzle and the filling process of the continuous casting mold cavity. 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 the thermophysical parameters of alloy depending on the temperature has been taken into consideration. The finite element method was used to solve this problem. Numerical simulations of filling the continuous casting mold cavity were performed for different variants of liquid metal pouring. The effect of the cases of pouring the continuous casting mold on the velocity fields and the solid phase growth kinetics in the process of filling the continuous casting mold was evaluated. It turns out that the appropriate selection of the submerged entry nozzle is very important for the failure-free operation of the continuous casting machine and obtaining a high-quality slab, which was the purpose of this work.