We present a new DNS data set for a statistically axisymmetric turbulent jet, plume and forced plume in a domain of size 40r$\mathrm{<mrow\; data-mjx-texclass="ORD">0</mrow>}$ x 40r$\mathrm{<mrow\; data-mjx-texclass="ORD">0</mrow>}$ x 60r$\mathrm{<mrow\; data-mjx-texclass="ORD">0</mrow>}$, where r$\mathrm{<mrow\; data-mjx-texclass="ORD">0</mrow>}$ is the source diameter. The data set supports the validity of the Priestley and Ball entrainment model in unstratified environments (excluding the region near the source), which is corroborated further by the Wang and Law and Ezzamel et al. experimental data sets, the latter being corrected for a small but influential co-flow that affected the statistics. We show that the second-order turbulence statistics in the core region of the jet and the plume are practically indistinguishable, although there are significant differences near the plume edge. The DNS data indicates that the turbulent Prandtl number is about 0.7 for both jets and plumes. For plumes, this value is a result of the difference in the ratio of the radial turbulent transport of radial momentum and buoyancy. For jets however, the value originates from a different spread of the buoyancy and velocity profiles, in spite of the fact that the ratio of radial turbulent transport terms is approximately unity. The DNS data does not show any evidence of similarity drift associated with gradual variations in the ratio of buoyancy profile to velocity profile widths.