General Points - Transport and Coherent Structures in Wall Turbulence - page 15

sustained. The length scale based on the friction velocity and
the viscosity is l

( )

constitutes the
internal scales. The quantity q , rendered dimensionless by
the internal scales, is denoted by q + , such as, e.g. the

= ν u . The couple

u τν

ν τ

+

+

velocity UUu
τ

= , the Reynolds stress

2

, the

− uu = − uu u τ

i j i j

+ 2 + 2

time in wall units.

The internal scale is constant in the case of a turbulent flow
in a channel, but it depends on the streamwise direction x in
a

t = tu

τ

ν or the frequency f = f u

τ

ν

turbulent boundary layer. Consequently, the

( )

()

adimensionalization by the wall scales only makes

sense for a given local position x in the latter case. Among
other things, this requires that the turbulent boundary layer
should be at equilibrium. The notion of turbulence at
equilibrium is an important one, and one that is sometimes
tricky to grasp. Interested readers can, for example, consult
[TOW 76], which is one of the classic works in this domain.

uxl
τ

,

ν

The external velocity scale is either the velocity U in the
center of the channel or its bulk velocity U , or the velocity
in the irrotational zone of a boundary layer U . In parallel,

∞

the external length scale is the half-height of the channel

h or the thickness of the turbulent boundary layer. The
external scales are universal, unlike the internal scales,
which are linked to the localized phenomenology, near to the
wall.

Λ

0

1.8. Eddy viscosity closures

Consider the terms:

()

∂ uv

∂

2

U 1 ∂ ⎛ ∂ U ⎞

ν − = μ − ρ uv

⎜ ⎟

2

∂ y ∂ y ρ ∂ y ∂ y

⎝ ⎠