The relation between the Reynolds stress gradient and the vorticity fluxes in a turbulent shear flow has been known since the time of G. I. Taylor. With recent advances in scientific computing, this question has received a renewed attention. In this work, we present results from a well-resolved direct numerical simulation of an axisymmetric turbulent jet and plume. The simulation reproduces the self-preserving features of the two flows reported in the literature and satisfies the identity relating the Reynolds stress derivative to the vorticity fluxes, to a reasonably good degree, establishing the veracity of the simulation. The axial derivative term in the identity is shown to be negligibly small, particularly when the jet/plume is in a self-preserving state. The radial profiles of the two vorticity fluxes for the jet and plume are nearly identical, suggesting a similarity of the underlying structure. The significant result is that the vorticity flux term involving the radial vorticity and azimuthal velocity is non-zero in the core of the jet as well as plume, while it is zero in the outer region of these flows. Furthermore, this term, in the core of the flow, is equal in magnitude (but opposite in sign) to the second flux term relating azimuthal vorticity and radial velocity. Coherent vorticity in the outer region of the jet/plume is in the form of hairpin vortices whereas in the core of these flows it shows a more complex shape.