Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 246

We assume that the fluid particle whose Lagrangian
coordinate is y 2 t 0 , t , passing through the plane x 2 at t 0 ,

maintains its associated quantity q , and engenders a
fluctuation

()

dQ

( ) ( )

[4.42]

qt t , =− y t t dx ,

0 2 0

2

as Fi gu re 4.30 shows. Readers should note that the mean
field Q is supposed to be linear in x 2 .

Figure 4.30. Fluctuation caused by a particle in the plane x 2

As, by definition, the component v 2 is

d
dt

() = ()

v 2 t 0 y 2 t 0 , t

equation [4.41] becomes

⎛ ⎞

T

2
2

dQ 1 d dQ 1 dy t ()

() ( ) ( )

∫

[4.43]

vq t =− y t t , y t t dt , =− ⎜ ⎟

2 2 0 2 0 0

⎜ ⎟

dx T dt dx 2 dt

⎝ ⎠

2 0 2