Inner Region of Smooth Pipes and Open Channels

The eddy viscosity function, proposed in this note, is characterized by a damping coefficient, Γ_o, the asymptotic value for a large Reynolds number. It is related to the Reynolds stress in the near-wall region. As the Reynolds number decreases in pipes and subcritical open-channel flow, the velocity profile is progressively displaced from the universal log-linear relation, which is accounted for by increasing values of the damping coefficient. The inverse relation between the Reynolds number and the damping coefficient follows from an analytical solution of the velocity profile, which is composed of both a viscous and a turbulent component. The inverse relation, derived from the logarithmic gradient of the viscous component, also yields the minimum Reynolds number for completely turbulent flow. Furthermore, it provides the basis for the correlation of various characteristics in the laminar-turbulent transition and of heat and mass transfer coef-ficients. All of these relations, which are singular functions of the asymptotic value of the damping coefficient, Γ_o, support the cubic, rather than the quartic, variation of Reynolds stress and eddy viscosity in the near-wall region.