Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 310

is most likely negligible at the start
of the regeneration process because the streamwise length
scale of the mother structure is significantly large with
100

The dependency ∂∂ x

, although this is only one order of magnitude
greater than the wall-normal length scale linked to the
structure

+

∝

. From this, we conclude that the secondary

+

∝ 10

vorticity layer engendered by the mother structure

ω

xS

( )

, where the subscript
0 refers to the wall. In addition, with the hypothesis whereby
the local velocity field is independent of x in the zones near
to the quasi-streamwise structures, it is possible to show
that the components v and w are essentially governed by
the near -wall vorticity

takes the form

ω > 0

ω∂ ∝ wy ∂ < 0

x xS

0

( )

. 15 Indeed, if , then

ω

uyzt ,;

x

. Also, if we typically consider the
viscous sublayer, or to a certain extent the lower buffer
sublayer, then 16

and

ω∂ =− uy ∂ u ∝− ω y

z z

, and therefore

ω∂∂ ∂∂ ∂∂ = wy vz wy − ≈

x

. The vertical velocity component results from the
continuity

wy

∝ ω

equation, which is thus written as

. When we combine this with the previous
relations, we find

∂∂ vy =− ∂ ∂ wz

( )

. The main problem
thrown up by the regeneration mechanism is the creation of
a dependency on x , without which the terms of production
by stretching, tilting or twisting in equation [5.48] are
canceled out, and the secondary vorticity is doomed to diffuse
and dissipate.

2

v =− 12 y ∂ω ∂ z

x

15 These arguments are identical to those given in [JIM 91] for the near-
wall zone, but can, to a certain extent, be generalized to apply to the field
near to the quasi-streamwise structure.
16 The length scale in the spanwise direction is typically the distance of
spacing of the streaks, which is 100 inner variables and one order of
magnitude greater than the length scale in direction y .