General Points - Transport and Coherent Structures in Wall Turbulence - page 30

viscosity, govern the mechanism of turbulent transport in
both cases. Coles 8 proposed a velocity distribution that is
valid both in the logarithmic sublayer and the outer layer, by
adding a function known as a “wake function”, to the
distribution [1.25]:

⎛ ⎞

1 ln Π y

()

Uy yB W
κ

++

=

+

+ + , Π [1.43]

⎜ ⎟

c

κ Λ

⎝ ⎠

0

where the argument of the wake function W is the distance
from the wall, rendered dimensionless by the external scale

Λ (the half-height h of a channel or the thickness of a

δ

0

boundary layer) and is a constant that depends slightly

on the Reynolds number in a canonic boundary layer without
pressure gradient and varies between 0.54 and 0.55. The
empirical function

Π

W , proposed by Coles, is

( )

21
1 in

η − π

()

[1.44]

W

η =+

2

where depends (slightly) on the

Reynolds number in a boundary layer without pressure
gradient and typically varies between 0.54 and 0.55.

η =Λ y . The constant Π

0

Let us look again at the distribution [1.28] in outer scales.
Generally speaking, this distribution depends not only on the
variable η =Λ= y y δ , but also on the Reynolds number,

0

+

expressed as δδν = u

τ

. Thus, relation [1.28] is expressed in

the general form:

()

UUy g
u τ

−

( )

[1.45]

∞ +

= η δ ,

8 These results were published in the first edition of the Journal of Fluid
Mechanics (vol. 1, p. 191, 1956), a renowned journal in this field.