Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 316

. The wall-normal vorticity layer becomes
approximately streamwise a little later on. It behaves like a
Burgers vortex, subjected to ux

δ ∝ l ς t

x ω

and v , where the

= β =− β y

1 d

dt

δ 1

strain rate is β =− = . The 2D spatiotemporal

δ t

development which governs the regeneration of ω is then

x

2

∂ω ∂ω ω ∂ ω

y

x x x x

[5.54]

− = + ν

∂ t t ∂ y t ∂ y

2

This equation can have a self-similar solution, of the form

( )

ω ∝ t

− 1/ 2

exp − 3 y

2

4 ν t

x

The viscosity governs the thickness at long times for

δ

and

which . The behaviors for short times

δ ∝ 2 ν t δ ∝ l ξ t

x ω

( )

1/3

* 2 2

long times δ ∝ 2 ν t coincide at time t ∝ l ω 4 νξ . The

x

vorticity reaches its peak at tt =

*

and the thickness

( )

1/ 3

reaches its minimum.

δ

*

∝ 2 ν t

*

= 4 l ν ξ

x ω

Jiménez [JIM 94] assumes that the origin of the wall-
normal vorticity layer is the instability of the streaks.

These are represented schematically as two coflowing plane
jets (high- and low-velocity streaks), at a distance

ω

y

apart.
The most unstable linear mode of a jet is sinuous 18 with a
wavelength 2.5 times greater than the thickness of the jet
2

λ

is half of the most unstable
wavelength, i.e. combining

. The thickness

λ

l ω

. Each streak engenders
two vorticity layers of opposite signs. These layers roll up
into counter-rotating vortices, if they are sufficiently intense.

l ω

= 58 λ

18 However, it should be noted that the streaks are mainly linearly stable
in the inner layer. We will return to look at this point in detail later on in
this chapter.