  The problem of discerning key features of steady turbulent flow adjacent to a
wall has drawn the attention of some of the most noted fluid dynamicists of all
time. Standard examples of such features are found in the mean velocity
profiles of turbulent flow in channels, pipes or boundary layers. The aim of
this review article is to expound the essence of some elementary theoretical
efforts in this regard. Possibly the best known of them, and certainly the
simplest, is the argument (obtained independently) by Izakson (1937) and
Millikan (1939). They showed that if an inner scaling and an outer scaling for
the profile are valid near the wall and near the center of the flow (or the
edge of the boundary layer), respectively, and if there is an overlap region
where both scalings are valid, then the profile must be logarithmic in that
common region. That theoretical justification has been used and expanded upon
by innumerable authors for over 60 years, and at the present time is still
rightly enjoying popularity.
  Although background discussions of several related topics are included in the
present article, for example the classical ideas of Prandtl and von Karman, the
main foci will be on (i) a careful examination of the Izakson-Millikan
argument, together with a presentation of a better mathematical justification
for its conclusion; and (ii) a detailed clarification of a newer approach to
gaining theoretical understanding of the mean velocity and Reynolds stress
profiles based on the search for scaling patches.
  The two approaches share common goals, they are both heavily involved with
scaling concepts, and many results are similar, but the logical trains of
thought are entirely different. The first, as mentioned, dates back to the 30's
and the second was introduced in a series of recent papers by Fife, Wei and
Klewicki et al.
