Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 227

not illogical to suppose that, apart from intermittent periods
of regeneration 9 , the local flow induced by the QSVs should
be independent of x . If this is the case, the instantaneous
streamwise vorticity

( )

transport equation ω z y , z ; t

D ω ∂ w ∂ w ∂ w 2

[4.9]

z

= ω + ω + ω + ν ∇ ω

x y z z

Dt ∂ x ∂ y ∂ z

is reduced to

∂ω ∂ω ∂ω ∂∂ u w ∂ w

[4.10]

z

+ v

z

+ w

z

= + ω + ν ∇

2

ω

z z

∂ t ∂ y ∂ z ∂ z ∂ y ∂ z

uyz G in direction x reduces
the components of the vorticity field to

( )

because the independence of ,;

z , ω x = ∂ w ∂ y −∂ v ∂

z and

y . Using the continuity equation,

ω y = ∂ u ∂

ω =−∂ u ∂

z

which is reduced to y , and noting that

∂ w ∂ z =−∂ v ∂

∂ω

∂ ∂ w ∂ ∂ v

( ) ( )

w

z

= w ωω − = w ωω +

z z z z

∂ z ∂ z ∂ z ∂ z ∂ y

⎛ ⎞ ⎛ ⎞

∂∂ u w ∂ ∂ w ∂ ∂ w

= u − u

⎜ ⎟ ⎜ ⎟

∂∂ zy ∂ z ∂ y ∂ ∂ z y

⎝ ⎠ ⎝ ⎠

we can show that equation [4.10] assumes the form:

∂ω ∂ω ∂ v ∂ ⎛ ⎞ ∂ w ∂ ⎛ ⎞ ∂ w

z

+ v

z

= − 2 ω − u + u

⎜ ⎟ ⎜ ⎟

z

∂ t ∂ y ∂ y ∂ z ∂ y ∂ z ∂ y

⎝ ⎠ ⎝ ⎠

[4.11]

∂

( )

− w ωνω + ∇

2

z z

∂ z

We are going to find a spatial mean for the above equation
in the spanwise direction. The spanwise mean of the

9 The mechanisms governing the regeneration of the coherent structures
will be discussed in Chapter 5.