Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 253

The function

()

is a factor which remains to be defined.

γ r

( )

The contribution to vU U − made by the particles from

a b a

the vicinity of point y is

ry −

( )

() () ()

() ()

−

ty γ

rUr Uy p r r − Δ

y

*

The sum of the above relation at all positions r ranging
from the wall to the half-height h of the channel means we
can write

h

1

( ) ( )

() (

) ( ) () () ()

[4.58]

∫

vU U − − r y − γ rUr Uy p rdr −

a b a y

*

ty

0

Using

r
∫ s

dU
dy

() − U y () = ()

U r ds

[BER 90a] construct the simple relation

0

h

1 dU

( )

() ()

− ∫ vU U − rFrdr

ty y

[4.59]

()

a b a y

*

0

Combining this with equation [4.58] enables us to express
the function F y r

()

by

⎧

r

( ) () ()

∫

y − ssps s γ 0 ≤ ry ≤

⎪ ⎪

y

()

0

Fr = [4.60]

⎨ ⎪

y

h

( ) () ()

∫

sy sp sds − γ y ≤ rh ≤

⎪ ⎩

y

r

() () = 1 when

[BER 90a] determines F y r , supposing that γ r

y and another constant, in order to ensure that v a 0 in < = r

the channel, which necessitates