Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 84

coordinate axes, in which case the expression of the
dissipation is

⎡ ⎤

2

2 2

⎛ ⎞ ⎠

⎛ ⎝ ⎞ ⎛ ⎝ ⎞

= ν ∂

u k
∂

∂ u k

∂

∂ u ∂ v ∂ w

∂

⎢ ⎥

*

ε K hom

x i =

2 ν + +

+

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎥

x i ∂ x ⎠ ∂ y z ⎠

⎝

⎣ ⎦

[2.41]

⎡ ⎤ 2 2

2 2 2 2

⎛ ⎞ ⎛ ⎞

⎛ ⎞ ⎠ ⎛ ⎝ ⎞ ⎛ ⎞ ⎠ ⎛ ⎝ ⎞

∂ u ∂ v ∂ u ∂ w ∂ v ∂ w

∂

⎢ ⎥

+ ν + + + + +

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

∂ y ⎝ ∂ x ∂ z ⎠ ⎝ ∂ x ∂ z ⎠ y

⎝ ⎠ ⎝ ⎠

⎣ ⎦

It is easy to show that the homogeneous enstrophy is
directly linked to

( )

*

[2.42] ωω = νε

i i K hom

hom

Using DNS in a channel with a low Reynolds number,
[ANT 91] showed that local homogeneity is an acceptable
hypothesis in wall turbulence for the case of dissipation, and
that ≈ ε K ho * constitutes a good approximation. In light of

equation [2.51], the ratio

ε *

i + /

* + is near to 1 throughout

ω ω ε

i K

the channel (Figure 2.9)

*

ε *

Figure 2.9. Distribution of a) and b) ε in a turbulent channel flow

Re=180
(solid line) and 360 (dotted line) according to [ANT 91].
The ratio of the enstrophy to the dissipation in inner variables is near to
1 c). This figure is adapted from [ANT 91]

K hom

when

τ