General Points - Transport and Coherent Structures in Wall Turbulence - page 6

where U , V and W are, respectively, the instantaneous
components of the velocity vector in directions x y and z ,
and P is the (instantaneous) pressure. The final term
includes all the viscosity terms. This equation is valid at any
time t (with the appropriate initial and boundary
conditions). The instantaneous components of the velocity
vector

G vary in time and space, while the flow, at
any given time, is highly unsteady and three-dimensional
(3D). If we look at the behavior of u ρ over long periods, then
we need to average equation [1.10] over time. To do so, we
decompose each physical value

( )

UUxt = ,

i i

Qx ρ into a temporal ( )

,

Q G and a fluctuating value () (

G

)

average value , where

qxt ,

. Thus we have, for example,

q = 0

( ) (

)

UU U u U u UU uu =+ += + , and the correlation

i j i i j j i j i j

between the fluctuations uu is generally non-null.

Equation [1.10], subjected to this treatment and
appropriately arranged, is written as

i j

∂ U ∂ U ∂ U 1 ∂ P ∂ uu ∂ uv ∂ uw

[1.11]

UV W U
x

+ + = − + ν ∇

2

− − −

∂ ∂ y ∂ z ρ ∂ x ∂ x ∂ y ∂ z

and generally:

∂ uu

∂ U 1 ∂ P

i j

2

[1.12]

U

i

=− + ∇ ν U −

i

j

∂ x ρ ∂ x ∂ x

j i j

The continuity equation, for its part, is of the same form
fo r the average field and the fluctuating field; in other words,
0

, with the latter identity being
valid instantaneously. What we need to take away from
these equations (and by comparison with a laminar flow) is
the existence of the t erm s of inter-correlation or cross-
correlation of the type

∂∂= Ux and ∂∂= ux 0

i i i i

uu . These terms, called Reynolds

i j