The significance of coherent flow structures for the turbulent mixing ...

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L
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§
§
:
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;
õ
:
I
I
I
W
§
§
(
ï
Æ
[
;
õ
:
:
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5.1 The statistical description of turbulence
§JI
¾ and (
§J
¾ Q1 or Outward interaction (5.9)
§J
¾ and (
¾ Q2 or Ejection (5.10)
¾ and (
¾ Q3 or Inward interaction (5.11)
The complexity of equation (5.8) can be further reduced by assuming that the production of
turbulence, due to the conversion of the kinetic energy of the mean motion into turbulent
fluctuations, is in equilibrium with the dissipation due to the molecular stresses:
¾ and (
§JI
¾ Q4 or Sweep (5.12)
w
(5.13)
As the turbulent shearing stress §÷
component
expression
Ø ; ¾
;
( becomes zero at the wall the following
í ÕK;
Ù÷
( Ø7L
w (5.14)
M N ÕO;
can be used to determine the integration constant L
w. With the friction velocity defined by
from equation (5.13)
§QP
ØSR
wà[÷ and the assumption that ]÷
( ØV¾ holds, the following relation can be deduced
§QP õ
Ø
§EP
(5.15)
In the literature this so called law-of-the-wall is usually written in non dimensional wall-units
õUT and : T defined by
§QP õ
: T Ø and
§EP
(5.16)
õVT†Ø
It is clear that the law-of-the-wall is only approximately valid in the vicinity of the wall (: T
È §÷ (
PXW
Ê ( §QP §QP
é‚à
: :
) where is small relative to the molecular stresses. This flow domain is usually called
viscous sub-layer of the boundary layer [90]. Another important relation can be deduced from
equation (5.13) by assuming that the molecular stresses are small relative to the turbulent ones.
In this case
holds and the velocity profile becomes independent of the Reynolds
number provided the velocities and spatial dimensions are normalised with and . This
can only depend on and .
is clear from the analysis of the dimensions because õ-à ?
Thus it follows from õÝà ?
§QP
§QP
:1Y
àZ: with the von Kármán constant [
"0]
(5.17)
õ T Ø
\: T
This so called log-law is valid : T for
¾ÁÌÛ ). The region between : T
(:à¿Ü
until the effect of intermittency becomes important
¾
and : T ¾ on the other hand, where the molecular
È
and turbulent stresses are of equal order, is called buffer-layer.
For a deeper understanding of the turbulent motion two more equations need to be considered.
The conversion of the kinetic energy of the mean motion given by equation (5.15) and
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