A reinterpretation of the turbulent Prandtl number

The eddy viscosity at any location in fully developed turbulent flow in a round pipe is shown by an exact analysis to be simply the ratio of the shear stress due to the time-averaged turbulent fluctuations in the velocity to that due to the molecular motion and thus independent of its heuristic diffusional origin. The eddy conductivity in fully developed turbulent convection is similarly shown to be simply the ratio of the corresponding contributions to the radial heat flux density. However, while providing a quantitative physical rationale for the eddy viscosity and the eddy conductivity, the analysis that leads to this insight at the same time eliminates their conceptual value except in a historical sense. On the other hand, the turbulent Prandtl number, as redefined directly in terms of the time-averaged fluctuations, remains an essential parametric variable. The local fraction of the radial heat flux density due to turbulence (the replacement for the eddy conductivity as a variable) appears, on the basis of both experimental evidence and asymptotic analyses, to be independent of geometry and the thermal boundary condition(s) and thereby a universal function only of the molecular Prandtl number and the local fraction of the shear stress due to the turbulence. It follows that the turbulent Prandtl number shares this restricted functionality. Although the functional and numerical behavior of the turbulent Prandtl number remains to this day incompletely defined either experimentally or theoretically, the rate of heat transfer, as represented by the Nusselt number, is fortuitously insensitive to these uncertainties.