General Points - Transport and Coherent Structures in Wall Turbulence - page 49

simulations) the fluctuating components in the viscous
sublayer, also at distances very near to the walls (

y + ≤

2 ).

Let us now consider the instantaneous and local
momentum balance equations directly at the wall. In a
canonical flow in the absence of any action exerted at the
walls (such as distributed blowing/suction for example),
these equations can be written as

D Uu 1 ∂ ∂

2

⎡ ⎤

( ) ( ) ( )

+ = 0 = − Pp + + ν Uu +

⎢ ⎥

i i i i

Dt ρ∂ x ∂ ∂ x x

0 0

⎣ ⎦

0 i l l

Noting that the diffusion terms can be expressed as:

2

∂ ∂

( ) ( )

ν Uu

xx

+=− νε Ω+ ω

i i ijk k k

∂∂ ∂ x

l l j

we obtain:

∂ ∂

( ) ( )

Pp + − με Ω+ ω

ijk k k

∂ x ∂ x

0 0

i j

which finally gives us

∂ ⎛ ⎞ ∂ω

() p =−

μ

z

⎜ ⎝ ⎠
⎛ ⎞

∂

x

0

∂

y

0

[1.69]

∂ ∂ω

()

p =−

μ

x

⎜ ⎝ ⎠

∂

z

0

∂

y

0

This relation clearly indicates that the fluxes of
fluctuating spanwise and streamwise components of vorticity
are directly linked to the instantaneous and local pressure
gradients at the wall, respectively, in the directions x and z .
Equation [1.69] is accurate, and it is at the heart of certain
strategies for controlling wall turbulence. It is easy to see