Numerical experiments using a direct numerical simulation (DNS) of turbulent flow between two parallel plates in conjunction with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the flow field are conducted for Prandtl or Schmidt numbers between 0.01 and 50,000. The LST methodology is used to generate mean temperature profiles as a function of the entry distance in the case of a step change in heat or mass flux at the walls of the channel. The heat transfer coefficient and the Nusselt number ratio, Nu(x)/Nu(x --> infinity), downstream from the step change in the wall flux are determined for the range of Pr or Sc fluids examined. Relations between the heat or mass transfer coefficient at the fully developed part of the channel and Pr or Sc are proposed for low and high Pr or Sc cases. Finally, unified correlations, which provide the heat or mass transfer coefficient for all Pr or Sc, in the Reynolds number range examined, are proposed. Also, the exponent of the asymptotic dependence of the eddy diffusivity close to the wall is obtained. (C) 2003 Elsevier Ltd. All rights reserved.