Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 64

Indeed, the production and advection terms disappear in
the ab sence of mean velocity, and all the terms such as
∂∂

()

are zero for reasons of homogeneity. For
homogeneous, isotropic turbulence, u i u j

x i

3 . The = u k u k δ

ij

transport equation is then

∂ u i u j

∂

2
3 ε

*

δ ij [2.11] =−

t

where the isotropic dissipation rate is x j . By

rearranging equations [2.10] and [2.11], we find a formal
relation

*

ε = ∂ u i ∂ x j ∂ u i ∂

⎡ ⎤

⎛ ⎞

∂ u

i u

∂ u ∂ u

1
ρ

i + ∂

u i ∂ u i

2
3 ε

2
3 ε

j

⎢

j j

⎥

* *

δ ij [2.12]

⎜ ⎟

= p

−

2 ν

l +

δ ij

−

⎜

∂ t ⎢ ∂ x ∂ x ∂ x ∂ x

⎝ ⎠

j l

⎣ ⎦

The term in square brackets in equation [2.12] is clearly
responsible for the return to isotropy. Let us introduce the
modified tensor of the pressure/velocity gradient

correlations, defined by

⎡ ⎤

⎛ ∂ u ⎞ ∂ u

1 ∂ u ∂ u 2

⎢ ⎥

** j j *

[2.13]

επ = p +

i

− 2 ν

i

+ εδ

⎜ ⎟

⎜ ⎟

ij ij

⎢ ρ∂ ∂ x x ∂∂ x x 3 ⎥

⎝ ⎠

⎣

i j l l

⎦

Let us also introduce the anisotropy tensor, which we will
return to later on in this chapter:

uu

K δ

i j ij

b =− [2.14]

ij

2 3

In this relation, K u i u i 2 is the kinetic energy. Note that

=

0 for homogeneous isotropic turbulence. Let us also
introduce a time scale

b ij ≡

*

ε

dt * dt =

2 K