Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 294

( )

its velocity [LAM 32]. The u = dx dt v , = dy dt w , = dz dt

Lagrangian equations are

2

∂ u ∂ U ∂ U ∂ u

= U

∞

+ W

∞

+

∞ ∞

∂ t ∂ x ∂ z ∂ y

*2

[5.21]

2

∂ w ∂ W ∂ W ∂ w

= U

∞

+ W

∞

+

∞ ∞

*2

∂ t ∂ x ∂ z ∂ y

( )

These relations contain x yz , whereas the independent

,,

( ) ( ) ( )

variables are abc is

made through the procedure indicated in [LAM 32], [VAN
80], [VAN 90], [PER 91a] and [PER 91b]. To take an
example, the gradient

abc . The switch from x yz at ,, ,, ,,

*

is thus expressed by [ATI 04]:

∂∂ y

∂ ∂ ∂ x z ∂ ∂ x z ∂ ∂ ∂ x z ∂ ∂ x z ∂

⎛ ⎞ ⎛ ⎞

= − + −

⎜ ⎟ ⎜ ⎟

*

∂ y ∂ ∂ c b ∂ ∂ b c ∂ a ∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂
∂∂ ∂∂ ∂

a c c a b

⎝ ⎠ ⎝ ⎠

[5.22]

x z x z

⎛ ⎞

+ −

⎜ ⎟

ba ab c

⎝ ⎠

Using equation [5.21] combined with and ∂∂= x tu

, associated with the appropriate boundary
conditions, it is possible to determine

∂∂= ztw

( ) ( )

uabct , w abct ,,; ,,;

( ) ( )

and the positions

z abct . The continuity
equation, for its part, implies that the Jacobian of the
transformation is constant and equal to 1 [LAM 32]:

x abct and ,,; ,,;

∂∂∂
∂∂∂

x x x
abc

( )

∂ xyz ,,

∂∂∂ y y y

[5.23]

= = 1

( )

∂ abc ,, ∂∂∂

∂∂∂
∂∂∂

a b c
z z z
abc