Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 122

and the viscous terms are then negligible for similar reasons.
Thus, the problem is reduced to solving the linearized
equations without viscosity, which determine the temporal
evolution of a homogeneous turbulence subjected to
manipulatable initial conditions, like the original anisotropy.
The RDT has been widely studied in the area of turbulence,
and yields information which is very useful for modeling.
Obviously, we will not go into the technical details of the
RDT in this topic; interested readers can consult [TOW 76],
and the references given therein, along with [HUN 90] for an
overview of the subject. Here, we limit ourselves to a
presentation of the results found by Maxey [MAX 82], which
are instructive for the understanding of certain aspects of
wall turbulence.

The linearized equations without viscosity, in connection
with equation [2.86], are

∂ u i

∂

x 2 ∂

u i 1

ρ

∂ p

() () δ i 1 =−

+ β t

x 1 +

u 2 β t

t ∂ ∂ x i

[2.91]

∂ u i

x i =

0

∂

It is useful to work in the Fourier space. Let us use the
notation

G G

G G G

( ) ( )

( )

∫

akt , = uxt , exp − ik xdx •

3

i i

for the Fourier coefficients of the fluctuating velocity field.
The solution of the linear system [2.91] in the Fourier space
is written, in matrix form, as

G G G

G G

( ) ( ) ( )

⎡

()

⎤ ⎡ ⎤

akt , = A mkt , , α t a mkt , ,0

⎣ ⎦ ⎣ ⎦

i ij j

( ) ( ) ( )

mm m , , = kk k , , + k α t

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