  The classical scaling theory of turbulent parallel flow provides a framework
for the description of the mean velocity profile through two functions of one
variable, traditionally named law of the wall and law of the wake, and a
universal logarithmic law characterized by von K\'arm\'an's constant. Despite
its widespread adoption in research and in teaching, discrepancies between this
theory and both experiments and numerical simulations have been repeatedly
observed in the literature. Recently we have shown that in the logarithmic
layer such discrepancies can be physically interpreted and analytically
accounted for through an equally universal correction caused by the pressure
gradient. This finding opens the way to a likewise improvement in the
description of the law of the wall and of the law of the wake, an analytical
interpolation of either of which is often useful for practical applications.
