Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 368

can be engendered because of the spanwise symmetry
(in direction z ). Indeed, the value of wx

∂∂ wx

is:

∂∂

∞∞ ∞

∂ w

1

∫∫∫

[5.107]

(, ,,)

x y z t =−

(

x − x

)

F dx dy dz

' ' '

∂ x

4 π

−∞ −∞ z '0 =

where, in light of equation [5.106]

∂
∂

ω

y +

x '

F = ⎡

3/2

( ) (

2

)

2

( )

2

⎤

xx −+− ++ ' yy ' zz '

⎣ ⎦

[5.108]

∂
∂

ω

y −

x '

− ⎡

3/ 2

( ) ( ) ( )

⎤

2 2 2

xx −+−+− ' yy ' zz '

⎣ ⎦

Thus, if spatially and temporally, ,

∂ω ∂ x ' = ∂ω ∂ x '

y + y −

then will be null in the zone in question. The creation

∂∂ wx

of ∂∂ wx depends directly on the

difference . We can show

this rigorously by averaging equation [5.107] at a given
distance y across a surface S which is axisymmetrical in the
plane x

Δ ( , , ,) x yzt = ∂ω ∂ x ' − ∂ω ∂ x '

y + y −

− z (Figure 5.45). The mean value of across ∂∂ wx

S is then:

∂ w 1

∫∫

=− GdxdydzdS Δ ' ' '

∂ x 4 π S

SR

where

⎡ 1 ⎤

⎢ ⎥

3/2

⎡

( ) (

2

)

2

( )

2

⎤

xx −+− ++ ' yy ' zz '

⎢ ⎥

⎣ ⎦

⎢ ⎥

Gxx =− ⎢ ( ')

1 ⎥

−

⎢ ⎥ 3/2

⎡

( ) ( ) ( )

⎤

2 2 2

⎢ xx −+− +− ' yy ' zz ' ⎥

⎣ ⎦

⎣ ⎦