  Scaling of the Reynolds stresses has been sought by many researchers, since
it provides a template of universal dynamical patterns across a range of
Reynolds numbers. Various statistical and normalization schemes have been
attempted, but without complete or convincing similarity properties. Our prior
work on the transport processes in wall-bounded flows point toward
self-similarity in the gradient space, where the first and second derivatives
of the Reynolds stress components exhibit universal scaling across the entire
boundary layer. This scaling is extendable to compressible flows. Finally, a
universal, integral scaling for the mean velocity profiles is discovered and
presented.
