We investigate the transport of a passive scalar in a fully developed turbulent axisymmetric jet at a Reynolds number of $\mathit{Re}=4815$ using data from direct numerical simulation. In particular, we simulate the response of the concentration field to an instantaneous variation of the scalar flux at the source. To analyse the time evolution of this statistically unsteady process we take an ensemble average over 16 independent simulations. We find that the evolution of $C_{m}(z,t)$, the radial integral of the ensemble-averaged concentration, is a self-similar process, with the front position and spread both scaling as $\sqrt{t}$. The longitudinal mixing of $C_{m}$ is shown to be primarily caused by shear-flow dispersion. Using the approach developed by Craske & van Reeuwijk (J. Fluid Mech., vol. 763, 2014, pp. 538–566), the classical theory for shear-flow dispersion is applied to turbulent jets to obtain a closure that couples the integral scalar flux to the integral concentration $C_{m}$. Model predictions using the dispersion closure are in good agreement with the simulation data. Application of the dispersion closure to a two-dimensional jet results in an integral transport equation that is fully consistent with that of Landel et al. (J. Fluid Mech., vol. 711, 2012, pp. 212–258).
Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan-Majda model and Taylor-Aris Dispersion
