Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 249

0 0 0

dU 1 ∂ P

() ()

[4.50]

∫ ∫ ∫

2

ds =− sds + ∇ ν U sds

dt ρ∂ x

− t − t − t

where U and P are, respectively, the local instantaneous
velocity and pressure (which include their mean and
fluctuating values). All the quantities are now a function of
the curvilinear coordinate s . Equation [4.50] becomes

0 0

1 ∂ P

() ( ) () () () ()

[4.51]

⎦ ∫

∫

2

u 0 =−+ u t ⎡ U −− t U 0 ⎤ − sds + ∇ ν U sds

⎣

ρ∂ x

− t − t

The notation ( s ) indicates the quantities evaluated along
the trajectory of the particle at time t . By multiplying the
above equation by v 0

()

and finding the overall mean, we find

0

1 ∂ P

()() ()() () () () () ()

⎡ ⎤

∫

uv 00 =− ut v 0 + v Ut 0 −− U 0 − v 0 s s

⎣ ⎦

ρ ∂ x

− t

[4.52]

0

() ()

∫

2

+ ν v 0 ∇ Us ds

− t

Th e velocity U of the fluid particle is decomposed into

is evaluated at
the point traversed by the fluid particle at time

u . The mean Eulerian velocity U

()

U = U +

− t

t . It is −

expressed in a Taylor series by

0

dU

() () () ()

[4.53]

∫

Ut −= U 0 − vs 0 s

dy

− t

of the first order. Equat io n [4 .53] is a direct consequence of
equation [4.42], with Q

U . After substituting equation
[4.53] back into equation [4.51] and multiplied the relation
thus obtained by v , we find

=