Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 236

passive structures, in accordance with the terminology
introduced by Townsend [TOW 76]. The active structures
contribute to all the terms in the Reynolds tensor. Chapter 6
will be entirely dedicated to the effects of the large- (and very
large)-scale structures. In this section, however, we will
content ourselves with laying down a few preliminary
foundations.

Passive structures are irrotational. In fact, it is possible to
decompose the normal gradient of the Reynolds shear stress
in the following form [HIN 75]:

⎛ ⎞

∂ uv ∂

ωω

uu −− vv ww

[4.19]

−=−+ v w

⎜ ⎟

⎜ ⎟

z y

∂ y ∂ x 2

⎝ ⎠

simply by using the definitions or applying the more general
relation

uu

∂ ∂

j j

− uu = εω u −

ji k j k

∂ x ∂ x 2

j i

where the asymmetrical tensor ε ijk is zero if two of the

indices are arbitrarily identical, 1 if the indices are all

different, respectively, for even and odd numbers of
permutations. In addition, we have 13

ε ijk =±

∂ u

x

i

ωε =−

k ijk

∂

j

The velocity/vorticity correlations which appear in the
flux of the Reynolds shear stress in equation [4.19] can then
be interpreted as the rotational components (active) with the
decomposition

13 For example, see Chapter 5 of [TAR 11a] and [TAR 11b].