Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 285

a) b)

Figure 5.7. a) “Stabilizing” effect and b) “destabilizing”
effect caused by spanwise shear around an Oseen vortex
at an infinite distance from the walls

D ω

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

∂∂ v w ∂ u ∂ w ∂ v ∂ u

y

= + − + Ω −

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

z

D t ∂∂ x y ∂ z ∂ ∂ x y ∂

∂ω

y

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

[5.10]

2

∂ v

z

y

+ ν

∂ ∂ x ∂ x

l l

determining the regeneration of the wall-normal vorticity.
The component by tilting. The

formation of an HPV 5 proposed by Theodersen [THE 52] is
similar: the spanwise vorticity lines caused by a local
instability enter into a zone where the velocity is greater
than the near-wall velocity, and are quickly advected by the
mean flow. The legs of the structures thus formed are
subsequently stretched and intensified. The instantaneous
velocity profiles may locally become inflectional, as shown in
Figure 5.8. The angle of the structures far from the wall
reaches a maximum value of 45°, which corres pond s to the
maximum degree of stretching of the enstrophy

then transforms into ω ω

x

y

. Indeed,
the pred ominant production term in the transport equation
for

ωω

i i

, expressed in tensor form as

ωω

i i

2

1
2

D ωω ∂ U ∂ ω

, [5.11]

i i

= ωω

i

+ νω

i

i j i

D t ∂ x ∂ ∂ x x

j l l

5 Originally, Theodersen [THE 52] imagined a horseshoe-shaped
structure. See Chapter 4.