Mean temperature and concentration profiles in turbulent internal flows

We derive explicit formulas for the mean profiles of temperature or concentration of a diffusing substance (modeled as passive scalars) in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. The derivation leverages on the observed universality of the inner-layer thermal eddy diffusivity with respect to Reynolds and Prandtl number variations and across different flows, and on universality of the passive scalar defect in the core flow. Matching of the inner- and outer-layer expression yields a smooth compound passive scalar profile. We find excellent agreement of the analytical profile with data from direct numerical simulations of pipe and channel flows under various forcing conditions, and over a wide range of Reynolds and Prandtl numbers.