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transport and the pressure/velocity gradient correlations,
while
D
K
is the molecular diffusion. The turbulent
pseudo-dissipation, given by the correlation of the products
of the fluctuating velocity gradients, is denoted by
K
.
The turbulent dissipation plays as important a physical role
as does production. The readers should note a few
modifications in relation to equation [2.20]. The correct
definitions for molecular diffusion and dissipation are,
respectively
ε
⎛
⎝
⎞
⎠
∂
∂
∂
u
i
x
k
+
∂
u
k
∂
D
*
= ν
u
k
⎜
⎟
x
i
∂
x
i
and
⎛
⎝
⎞
⎠
= ν
∂
u
k
∂
∂
u
i
x
k
+
∂
u
k
∂
ε
*
⎜
⎟
x
i
∂
x
i
ε
*
, combined with incompressibility,
give rise to the terms
D
K
and
The sum of
D
*
with
ε
K
in equation [2.29].
Figure 2.2 shows the distribution, in inner scales, of the
different terms in equation [2.29] in a low Reynolds number
channel flow. These results were obtained by Mansour
et al.
[MAN 88] using direct numerical simulation (DNS). As
we can see, the production and dissipation are indeed the
dominant terms. The production reaches its maximum
va
lue
when
y
+
=
12
, in the same place as the maximum of
uu
. In
particular, we can see that at
y
+
>
30
, the dissipation and
production are at equilibrium.
6
6 We will see in Chapter 4 that equality between production and
dissipation can only truly occur if the Reynolds number is high.
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