Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 96

whose length scale is such that y does not depend

explicitly on the viscosity (on the one hand), or on the
position y at the wall (on the other). We can, therefore, write

Λ 0 >>

( )

E pp k x Λ 0

( )

[2.54] = f 1 k x Λ 0

2 u 4

ρ

τ

for the zone with small wavenumber k x shown in Figure
2.17, which clearly accepts an outer scale law.

In the opposite direction, the range of high wavenumber is
under the influence of the structures which are locally
isotropic and depend on the Kolmogorov velocity scale
v Ko = ν ()

( )

1/ 4

1/4 and Kolmogorov length scale

3

.

η = νε

Ko

Hence, the scaling law in this particular spectral zone is
( )

Ek η

( )

pp x Ko

[2.55]

= fk η

3 x Ko

24

ρν

Ko

Production reaches a state of equilibrium with dissipation
in the constant stress sublayer, which is written as

ε *

3

[2.56] = u κ y

τ

where it must be remembered that is the von Kárman

constant. Consequently, the Kolmogorov scales are

κ

1/4

⎛ ⎝ ⎞ ⎠

3

ν u

τ

v Ko = ⎜ ⎟

κ y

[2.57]

1/ 4

⎛ ⎝ ⎞

3 y

κν

η Ko =

⎜ ⎟

3

u

⎠ τ

The structures whose length scale is approximately y
come into play in the range of intermediary wavelengths.
Hence

( )

Eky

( )

pp x

= f ky [2.58]

2 x

24

ρ u τ