Approximate calculation of Reynolds analogy for turbulent boundary layer with pressure gradient.

Abstract

An approximate method is developed to calculate the Reynolds analogy factor for a turbulent boundary layer with pressure gradient. The inverse of the Reynolds analogy factor is found to equal the integral across the boundary layer of the product of the slope of the nondimensional velocity profile and the ratio of the nondimensional energy transfer to the nonmensional shear. This result requires a Prandtl number of unity. Approximations for the shear stress, energy transfer, and velocity distribution across the boundary layer make the Reynolds analogy factor depend on a nondimensional pressure gradient parameter and on a velocity profile shape parameter. An adverse pressure gradient is predicted to increase and a favorable pressure gradient to decrease the Reynolds analogy factor. Comparison with experimental data indicates this result to be qualitatively correct, but more data are needed for a quantitative test. Also obtained is a generalization of Crocco's relation between velocity and total enthalpy for a nonadiabatic surface without pressure gradient. As the pressure gradient departs from zero, the relation between the nondimensional total enthalpy and the nondimensional velocity varies widely from Crocco's relation.