The dispersion of a passive contaminant by turbulence is not only intrinsically interesting, but also is fundamental to practical heat- and mass-transfer problems. The mechanism of turbulent dispersion from the wall and effects of the molecular Prandtl number are examined. Transport properties for scalar dispersion from an instantaneous line source at the wall of a parallel-plate channel in which fully developed turbulent flow occurs, as well as from a continuous line source, are studied. Numerical experiments using a direct numerical simulation (DNS) of the flow, combined with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the flow field, are conducted. The method is applied for a wide range of molecular Prandtl (Pr) or Schmidt (Sc) number fluids (0.1⩽Pr⩽50,000, which corresponds to liquid metals, gases, refrigerants, lubricants and electrochemical fluids). Results from the DNS/LST method are compared with experimental data, which are available for low Pr or Sc fluids. The turbulent diffusivity and turbulent Prandtl number for the plume that results from a line source are calculated, and an appropriate scaling is proposed. Finally, a model is developed for the prediction of the mean temperature or mean concentration profile as a function of Pr or Sc.