Document

Group-theoretical model of developed turbulence
and renormalization of the Navier-Stokes equation
V. L. Saveliev1and M. A. Gorokhovski2
1. Institute of Ionosphere, Almaty 480020, Kazakhstan
2. Laboratoire de Mécanique des Fluides et AcoustiqueEcole Centrale de Lyon, France
PHYSICAL REVIEV E 72, 016302 (2005)

Introduction
In relation to turbulence, we can roughly express Kadanoff’s idea of “block picture” for the Ising spin
field as follows: If, instead of turbulent field
( )
vr
, we consider a field
( )
s
vr
that is averaged on
the scale
s
, then the last one will “resemble” the original turbulent field
( )
vr
. The exact sense of
“resemble” must be defined by the group of transformations for both fields and equations for these
fields.
In this paper, solely on the basis of the Navier-Stokes equation, we propose a simple model of
stationary developed turbulence. This model allows us:
1. To explicitly derive the group of renormalization transformations.
2. We average the Navier–Stokes equation over some small length scale
0
s
with the help of our
averaging formula.
3. Then we transform this averaged equation by a renormalization group transformation to the
equation for the velocity field
( )
,t
s
vr
, averaged over any scale
s
.

II. Renormalized averaging formula for <vu>
Consider two functions
( )
vr
and
( )
ur
depending on radius vector
r
. The average of
( )
vr
is
( )
( ) ( )
v v d
¢ ¢ ¢
= Y -
ò
r r r r r
, (1)
When the weight function
( )
Yr
is Gaussian,
( )
( )
2
4
3
1
4es
sps
-
Y= r
r
, (2)
the average is defined by Gauss transform (filtering):
( )
( ) ( )
ˆ
v v d G v
ss
¢ ¢ ¢
= Y - =
ò
r r r r r
. (3)
The Gaussian distribution verifies the diffusion equation,
( ) ( )
2
ss
s
¶Y = Ñ Y
¶rr
, where
( )
2
2¶
Ñ= ¶r
. (4)
Thereby the Gauss transform operator (3) can be represented in the exponential form:
2
1 2 1 2
ˆ ˆ ˆ ˆ
,G e G G G
s
s s s s s
Ñ+
==
. (5)

Operator for averaging the field product
Consider the averaged product of two fields:
( ) ( ) ( ) ( ) ( )
2
ˆ
vu G v u e v u
s
sÑ
==r r r r r
. (6)
The differentiation of the product of two multipliers can be done by the Leibnitz rule. To this
end, we decompose the differentiation operator
Ñ
into two parts,
12
Ñ = Ñ + Ñ
, (7)
with differentiation operators
1
Ñ
and
2
Ñ
acting on the first
( )
vr
and the second
( )
ur
multipliers respectively. Setting Eq.(7) in Eq. (6) gives
( ) ( )
( )
( ) ( ) ( )
( )
( )
( )
2
2 2 2
12 12
2
e v u e v u e e v e u
s
s s s s
Ñ + Ñ
Ñ Ñ ×Ñ Ñ Ñ
==r r r r r r
. (8)
It then follows that the average of the product of two fields can be written as
12
2
vu e v u
sÑ ×Ñ
=
. (9)

Simplest formula for averaging
Instead of using the Taylor-series expansion with an infinite number of terms for the averaging
operator
12
2
esÑ ×Ñ
, we use the Taylor series with a residual term,
( )
( )
( )
( )
12
12
1
21 2 1 2
12
120
1
1 2 ... 2
1!
12,
!
n
nn
en
d q e
n
s
as
ss
s a a
-
Ñ ×Ñ
Ñ ×Ñ
= + Ñ ×Ñ + + Ñ ×Ñ +
-
+ Ñ ×Ñ ò
(10)
where
n
is an arbitrary natural number (
1,2,3,....n=
), and
( )
( )
1
1n
n
qnaa
-
=-
;
( )
1
0
1
n
dqaa=
ò
. (11)
For the case
1n=
we have:
0
2vu v u d v u
s
s s s s s ss
s¢¢
¢
-
¢
= + Ñ ×Ñ
ò
. (12)
This averaging expression demonstrates that the integral part includes the contribution of all
scales less than
s
in an exact manner.