We present axisymmetric numerical simulations of the turbulent flow of a liquid metal driven by a low-frequency, low-induction rotating magnetic field, whose magnetic Taylor number is in the range (1.2−46)T a²_cr^D with T a²_cr^D given by Grants and Gerbeth [I. Grants, G. Gerbeth. J. Fluid Mech., 431:407-426, 2001]. The computations were performed for cylindrical container of aspect ratio (diameter / height) R equals to unity. The axisymmetric numerical simulations (ANS) can accurately reproduce the experimental data available. The prediction of a low Re formulation of the k − ω turbulence model with modified boundary condition for ω on the wall agree well with the ANS data provided for T a < 1.2 · 10⁵ (E > 2 · 10⁻⁴). The numerical simulations showed that the interior core of the liquid rotates faster than the periphery with increase of the Lorentz force.