Law of the wall and law of the wake in parallel turbulent flow

The classical scaling theory of Prandtl, von Kármán and Millikan, based upon the distinction in a wall layer and a defect layer, describes the mean velocity profile through two functions of one variable, after Coles traditionally named law of the wall and law of the wake. In the overlap of the two layers, the law of the wall reduces to the universal logarithmic law characterized by von Kármán's constant. Discrepancies between the logarithmic law and both experiments and numerical simulations have been repeatedly observed in the literature; despite its widespread adoption in research and in teaching serious doubts ensued about its precise form and universality, leading to the formulation of alternate theories and hindering ongoing experimental efforts to measure von K'arm'an's constant. By comparing different geometries of pipe, plane-channel and plane-Couette flow, we have recently shown that such discrepancies can be physically interpreted, and analytically accounted for, through a proper account of the wake component. In an asymptotic expansion of the logarithmic layer the wake component reduces to a universal higher-order correction proportional to the pressure gradient.