General Points - Transport and Coherent Structures in Wall Turbulence - page 29

equation [1.39] gives us the following logarithmic
distribution:

1 ln

()

f y

+

= y

+

+ B [1.40]

κ

and the outer distribution:

1 ln

()

g η =− η + B [1.41]

ext

κ

The constant

ex B is directly linked to B by the relation:

1 ln

ext BU hB =−

+ +

− [1.42]

κ

The logarithmic distribution can be interpreted as the
consequence of two distinct zones, where the scales are,
respectively, linked to the inner and the outer flow. This
argument, which is commonly called the Izakson-Millikan-
von Mises overlap, constitutes an indirect proof of the
existence of the logarithmic layer. The underlying hypothesis
is that the intermediary zone is independent of the Reynolds
number, i.e. the function f accepts a local similarity, it is

yu

⎛ ⎞

not in the form f

τ

, Re . Panton [PAN 07] proceeded

⎜ ⎟

ν

⎝ ⎠

differently to reach a similar conclusion. He used the
asymptotic Poincaré series in the outer and inner regions
and rigorously overlaped them to create the composite
profiles of mean velocity and the Reynolds shear stress. He
clearly showed that the logarithmic law is the internal
asymptote of the distribution in outer scales, and the
external asymptote of the profiles in wall units.

The structure of the external layer is greatly similar to
that of a wake. The structures with large scales, which
depend essentially on the inertia, and depend little on the