General Points - Transport and Coherent Structures in Wall Turbulence - page 43

spanwise directions (respectively, and ), is

xi ( ) = zi ( ) =

nothing but the spanwise component

()

. We

Ω= −

z y

dUdy

have

( )

Ω x y Vx Uy , = ∂∂ ∂∂ −

z

in a canonical turbulent boundary layer on a n infinite flat
plate. Given that the order of magnitude of Vx ∂∂ is lesser

than that of the shear Uy ∂∂ , the average spanwise

component is approximately

( )

in these

Ω x y Uy , ≈ − ∂∂

z

conditions.

The components of the fluctuating vorticity field are:

∂∂ w v

( )

ω xyzt ,,, =−

x

∂∂
∂∂

y z
u w

( )

[1.56]

ω xyzt ,,, =−

y

∂∂
∂∂

z x
v u

( )

ω xyzt ,,, =−

z

∂∂ x y

respectively, in the streamwise, wall-normal and spanwise
directions. The vector form of the transport equation [1.55] is
useful, particularly if we want to express that equation in
the general curvilinear coordinates,

G

K GG

G

∂ω G G G

( )

2

+•∇= •∇ +∇ u ωω u νω [1.57]

∂ t

where the details of the decomposition into an average and
fluctuating value are deliberately omitted for the sake of
conciseness. In order to save the readers from needlessly
wasting time, we now briefly recap the vectorial operators
appearing in the above expression in a curvilinear
coordinates system

G GG and the transformations

( )

eee ,,

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