Power law and log law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers

The open equations of a turbulent boundary layer subjected to a pressure gradient analysed for classical two layers (inner wall and outer wake), while matched in the overlap region of MAX through the Millikan-Kolmogorov hypothesis leads to an open functional equation, and its classical solution for the velocity distribution is the log. region. It is shown here that the same open functional equation also predicts a power law velocity distribution and a power law skin friction in the overlap region. The uniformly valid solution for the composite wall power law and wake velocity profile is obtained. The connection between the power law and the classical log. law solutions of the open functional equation is analyzed. At large Reynolds number, the power law solutions reduce to the classical log. law solutions, and the equivalence predicts a certain relationship between the constants in power and log. laws. The results are compared with the experimental data.