Asymptotic Analysis of Turbulent Channel Flow for Mixing Length Theory

Limit process expansion techniques are used to analyze fully developed turbulent incompressible flow in an infinitely long channel with plane parallel walls. Closure of the Reynolds time-averaged equations of motion is effected through the introduction of a model for the eddy viscosity that is a generalization and modification of the model of mixing length theory. Expansions are carried out in the limit of the turbulent Reynolds number (the ratio of the eddy viscosity to the molecular viscosity) going to infinity. To leading orders of approximation, it is determined that the flow divides into : a relatively thick turbulent centerline defect layer, where the Reynolds turbulent stress dominates the Newton laminar stress ; and a thin viscous wall layer, where the Reynolds and Newton stresses are of comparable magnitude. By proceeding to higher orders of approximation, however, it is determined that this two-layer formulation for the flow is not uniformly valid. For completeness, a very thin laminar wall layer, in which the Newton stress dominates the Reynolds stress, must be introduced immediately adjacent to the channel wall.