Motivated by anomalous entrainment behaviour in cumulus clouds, Bhat et al. (Exp. Fluids, vol. 7, 1989, pp. 99–102) pioneered a laboratory experiment to study turbulent jets subjected to a volumetric heating away from the momentum source. The study concluded that the use of a constant entrainment coefficient was insufficient for the flow, and that the results did not confirm the analysis of Hunt (Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes, 1994, pp. 309–334, Kluwer Academic), which suggested that an increase in relative turbulent transport of streamwise momentum could lead to a decrease in entrainment. The present paper re-evaluates theoretical aspects of both studies, and includes a decomposition of the factors contributing to entrainment. The reworked analysis is then used to examine three-dimensional numerical simulations of turbulent jets with off-source heating. The data are consistent with previous work, but give deeper insight not easily obtainable through experiment. Specifically, direct measurement of flux integrals shows that previous inference from experimental measurements of centreline velocity and profile widths under the assumption of self-similarity can lead to underestimation of the mass flux by over 50 % in some cases. Radial profiles of temperature, radial velocity and turbulent correlations show significant departures from self-similarity. The flux measurements show that there is actually an increase in the entrainment coefficient with heating, and that it is locally enhanced by positive forcing and decreased by an increase in turbulent transport of streamwise momentum, thereby confirming the essence of the original proposal of Hunt.