The local time-averaged velocity, the mixed-mean velocity, and the friction factor for fully turbulent flow between parallel plates and in round tubes and concentric circular annuli can be expressed in terms of integrals of the turbulent shear stress. The pressure distribution across a channel can similarly be expressed in terms of an integral of normal stresses. These formulations, which are simple and exact, can be integrated numerically using experimental data, computed values, or correlating equations for turbulent stresses. Their greatest merit, however, may arise from the insight they provide with respect to the contributions of the fluctuating components of the velocity. For example, for concentric circular annuli such a formulation identifies a difference between the locations of the maximum in the velocity and the zero in the total shear stress. This difference, which has been overlooked in most experimental and semitheoretical investigations, introduces an error of unknown but possibly significant magnitude into all of the results, ft also precludes the application of the mixing-length, eddy viscosity and k - ∈ models.
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