General Points - Transport and Coherent Structures in Wall Turbulence - page 42

1.11. Vorticity

Chapter 5 of [TAR 11a] and [TAR 11b] is entirely devoted
to the characteristics of the vorticity field near to the wall.
We will briefly summarize certain aspects here because the
generation of near-wall coherent structures is closely linked
to the dynamics of the vorticity, and in particular to its
streamwise component ω . The vorticity is the rotational

x

G

G G of the velocity field. The transport equations for

are thus obtained by applying the rotational to the
momentum balance equations (which eliminates
the pressure terms). The transport equation for the
instantaneous vorticity field thus obtained is

ω =∇∧ u ω

i

( )

D Ω+ ω

2

∂ ∂

( ) ( )

( )

i i

=Ω + ω Uu + + ν Ω + ω

j j i i i i

[1.55]

Dt ∂ x ∂ x ∂ x

j l l

=+ PD

ω ω

i i

where the choice of indicial notation enables us to make use
of the Einstein conventio n of summation over the repeated
indices. The component

G G

G G

() ()

⎡ ⎤ corresponds to

Ω=∇∧ x Ux

⎣ ⎦

i

i

G GG represents
the fluctuating component. The left-hand term in equation
[1.55] contains the inertial terms. The first term on the
right-hand side represents the terms of vorticity production

( ) ( )

the average vorticity and ⎡ ⎤

ω x t , =∇∧ u x t ,

⎣ ⎦

i

i

, which are of crucial importance in the dynamics of wall
turbulence, and which we will discuss in detail later on. The
last term in equation [1.55] represents the diffusion

P ω

i

of

D ω

i

G G . The mean vorticity in a fully developed
turbulent channel flow, homogeneous in the streamwise and

() ( )

Ωω x + xt ,

i i