The weak axisymmetric fountain that results from the injection of a dense fluid upwards into a large container of homogeneous fluid of lower density is studied numerically. Using a time-accurate finite volume code, the behaviour of fountains with both a uniform and a parabolic profile of the discharge velocity at the source has been investigated. The evolution of the transient fountain flow has been analysed and two distinct stages have been identified. The time series of the passage of the fountain front has been presented and the initial, temporary and final characteristic fountain heights have been determined and scaled with the Froude number at the source. At steady state, the final fountain height and the fountain width are found to be the height and horizontal length scales which provide the full parameterization of the flow in the fountain core. The vertical velocity and temperature on the symmetry axis have been scaled with the height scale and an explicit correlation is also obtained for the former. The radial distributions of both the vertical and horizontal velocities in the zone of self-similarity in the fountain core at steady state have been scaled with the two length scales and empirical correlations have been obtained.