The phenomenon of the free-surface vortex forming over a draining intake is well known, together with its detrimental effects. While analytical solutions have been helpful in clarifying some features of the phenomenon, no extensions have been readily provided in solving instances of practical importance. Therefore, efficient anti-vortex devices have been traditionally conceived by means of physical model studies. However, a numerical simulation of the whole flow field would be nowadays desirable. The proposed numerical solution of the flow field is based on an axial-symmetric finite volume model, which solves the incompressible Navier-Stokes equations on irregular geometries. Boundary conditions include both the Dirichlet and the Neumann type. The mesh is staggered. The numerical scheme is a semi-implicit one, where the terms controlling the diffusion and those controlling the pressure field are discretized implicitly, while the convective terms are approximated via an Euler-Lagrange approach. The discrete version of the continuity equation becomes, by a substitution, a system having the pressure values as the only unknowns. The solution proceeds via an iterative scheme, which solves a symmetric and semi-positive-definite system for the pressure, by a standard pre-conditioned conjugate gradient method. The discrete velocity field at each iteration can then be explicitly obtained. The numerical solution has been verified by the laboratory experimental data obtained by Daggett and Keulegan (1974). This comparison demonstrated that the proposed numerical model is capable of predicting the whole steady flow field. Of special value is the comparison with the radial velocity distribution, which has a typical jet-like profile along the vertical direction. According to the most recent experimental evidence, it seems that the very onset of the vortex can be traced to this special feature of the radial velocity profile.