The article is dedicated to numerically simulating the formation of vortex stabilization of an electric arc and to analyzing the gas flow behavior. The three-dimensional model of transient laminar flow is designed for the actual geometry of the plasma torch. Reynolds numbers are presented for varying values of gas flow consumption. The formation time of a steady flow is determined by creating an automatic start algorithm of the plasma torch. The three-dimensional model allows to obtain the distribution of velocities needed for setting the boundary conditions of a two-dimensional axisymmetric task. The general goal of the study is creating a mathematical model of a plasma torch with vortex stabilization of the electric arc.