Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 65

By combining equations [2.12], [2.13] and [2.14], we are
able to write

⎛ ⎞

π i *
2 b ij −

db ij
dt *

[2.15]

⎜ ⎟

=− 2 b ij − 1

⎜ ⎟

⎝ ⎠

* , the
tensor b ij returns to isotropy. The simplest way of modeling
π i * is then to consider that

Suppose that under the influence of the term π

ij

b ij , in which case the term in
parentheses on the right of equation [2.15] is a constant C
and the tensor b i * becomes isotropic with long times, because
b i *

π i * ∝

e − Ct *

. The constant C must be positive,
and therefore the constant C R in

0 when t *

→∞

= →

2 C R b ij must be
greater than 1. To simplify this further, suppose that the
dissipation is isotropic, with

π i * ∝ b ij =−

∂ u

∂ u 2

3 ε

j

*

δ ij

i

2 ν

l =

∂ x ∂ x

l

By adding the condition of linear return to isotropy

π i * C R b ij and rearranging equation [2.13], we find ∝ b ij =

⎛ ⎞

⎛ ⎝ ⎞ ⎠

∂ u j u i u j

K

2 δ

1
ρ x i + ∂

u i

C R ∂

u i ∂ u i * b ij ij

3

⎜ ⎟

[2.16] p ⎜ ⎟

=−

2 C R ε =− −

∂ ∂ x j ∂ x l ∂ x l

⎝ ⎠

Equation [2.16] is the original form of the model of return
to isotropy suggested by Rotta [ROT 51].

Choi and Lumley [CHO 01] hypothesized that in
homogeneous turbulence, the tensor of return to isotropy π i *

defined by equation [2.13] is symmetrical and depends solely
on the anisotropy tensor b ij and the Reynolds number. Using
the invariant-based approach, we thus obtain the expression

*

a 0 b i 0 a 1 b i 1 a 2 b i 2

[2.17]

π = + +

ij