Journal of Modern Physics, 2013, 4, 56-63
http://dx.doi.org/10.4236/jmp.2013.41010 Published Online January 2013 (http://www.scirp.org/journal/jmp)
On One Possibility of Closuring the Chain of Equations for
Statistical Moments in Turbulence Theory
Edward V. Teodorovich
The Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia
Received September 25, 2012; revised October 28, 2012; accepted November 7, 2012
ABSTRACT
The paper concerns the problem on statistical description of the turbulent velocity pulsations by using the method of
characteristic functional. The equations for velocity covariance and Green’s function, which describes an average ve-
locity response to external force action, have been obtained. For the nonlinear term in the equation for velocity covari-
ance, it has been obtained an exact representation in the form of two terms, which can be treated as describing a mo-
mentum transport due to turbulent viscosity and action of effective random forces (within the framework of traditional
phenomenological description, the turbulent viscosity is only accounted for). Using a low perturbation theory approxi-
mation for high statistical moments, a scheme of closuring the chain of equations for statistical moments is proposed.
As the result, we come to a closed set of equations for velocity covariance and Green’s function, the solution to which
corresponds to summing up a certain infinite subsequence of total perturbation series.
Keywords: Random Velocity Field; Statistical Description; Characteristic Functional; Chain of Equations for Moments;
Problem of Closuring
1. Introduction
The problem on statistical description of turbulence, even
at the simplest assumptions of stationary state and ho-
mogeneity of the velocity field, so far remaines to be
unsolved in spite of its scientific and practical impor-
tance. The point at hand is a search for statistical solution
of the Navier-Stokes equation with the availability of
external random force (the Langevene force), which si-
mulates the generation of turbulent energy due to the
development of instability of large-scale flows at high
Reynolds numbers.
Statistical description of turbulent flows means a
knowledge of probability to find a given realization of
the velocity field in the elementary volume
of the space of velocity field realizations
that are solutions to hydrodynamic equations (the Na-
vier-Stokes equations) [1,2]. The relevant probabilistic
measure may be written as

,tur
d


,dtur

,tdt

ur

,Pt  ur

,,1dt 

ur

,t


ur





,,dtP  

ur ur (1.1)
where  is the functional probability density
being subject to the normalization condition

Pt 

ur (1.2)
where the integration is carried out over all possible ve-
locity field realizations. An experimental determination
of the probability density of the velocity field realization
is a challenging (in fact impractable) task.
A knowledge of the probability density is equivalent to
one of velocity field statistical moments (of all orders).
An experimental determination of statistical moments (of
finite order) seems to be more actual for approximate
describing the field of turbulent velocity pulsations. How-
ever, the attemps to obtain equations for statistical moments
appear to be unsuccessful since, due to the nonlinearity of
Navier-Stokes equations, the equation for statistical moment
of given order includes statistical moments of more high
orders, and thus an infinite chain of coupled equations arises.
It is the chain by Keller and Friedman [3], which is similar
to the well-known in statistical physics chain by Bogoly-
ubov-Born-Green-Ivon-Kirkwood [4].
Thus, the problem of cutting off this chain (closure
problem) arises. There exist a large body approaches to
solving this problem, among which the most simple is
neglecting the terms containing moments of order higher
then some chosen. Another possibility reduces to attempt
to express high-order moments in terms of low-order
ones, for example, using the congecture of quasinormal-
ity or log-normality, however, this often leads to non-
physical results of a type of negative probability [5].
Another widely used possibility of solving the closure
problem reduces to a certain phenomenological hypothe-
sis on connection between high-order and low-order
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E. V. TEODOROVICH 57
moments of the type the hypotheses of turbulent viscos-
ity or of mixing length. As the result, it occurs to be pos-
sible to achieve a satisfactory agreement with experi-
mental data in a limited domain of wavenumbers and
frequencies.
The method of “direct interaction approximation” pro-
posed by R. Kraichnan [6] (see also [7]) can be treated as
a further development of the moment method. Within the
framework of this method, in addition to statististical
moments, Kraichnan introduces into consideration a
function of averaged response of velocity field to exter-
nal force action (Green’s function), which is defined as a
solution of the problem on diffusion transfer in a random
velocity field at given statistics of the velocity field.
It should be also mentioned the method of constructing
a statistical solution to the Navier-Stokes equations using
the perturbation theory when the nonlinear term in Na-
vier-Stokes equation is considered as perturbation, and,
while culculating statistical moments, a solution for ve-
locity is presented in the form of series in the parameter
of nonlinear interaction with subsequent multiplication of
series obtained and term-by-term averaging the resulting
expression (“simple perturbation theory”). However, in
fact the nonlinear term is not small, the dimensionless
parameter of nonlinear interaction is defined by Reynolds
number, which is very large in the case of fully devel-
oped turbulence. Due to this fact, the convergence of
perturbation series is slow, i.e. a knowledge of several
first terms in perturbation series appears to be not suffi-
cient for estimating the behaviour of turbulent character-
istics being a subject of interest. The problem turns out to
be similar to that of strong interactions in quantum field
theory, where the methods of improving perturbation
theory with the help of partial summing up an infinite
subsequence of total perturbation series by solving Dy-
sons’ equations were developed (“improved perturbation
theory”). As applied to description of turbulence, a rele-
vant approach has been proposed by H. Wyld [8]. In his
analysis of pertubation series, he used Feynmann’s dia-
gram techniques from quantum field theory, where to
every term of perturbation series there are assigned some
graphic symbols with prescribed pictorial physical
meaning as processes of quanta production, annihilation
or propagation. Note that the effective Reynolds number
defined as a ratio of energy input to given mode from
large-scale modes to the value of viscous absorbtion
caused by interaction with smale-scale modes appears to
be approximately unit under the conditions of stationary
developed turbulence. This provides more rapid conver-
gence of perturbation series. Also note that, within the
framework of “improved perturbation theory”, the per-
tubation method is used for finding some quantities con-
nected with a solution (for example, the self-energy op-
erator in quantum field theory). A substitution of this
quantity into the equation and sequent solving that cor-
responds to summing up a certain infinite subsequence of
total perturbation series.
It should be also pointed out an alternative possibility
to improve the perturbation theory related to application
of the renormalization-group (RG) method (see reviews
[9,10]), this approach corresponds to summing up a cer-
tain (but yet another) infinite subsequence of total series.
One more peculiarity of developed turbulence, which
enables one to reveal the key features of the energy spec-
trum of turbulent velocity pulsations in some wave-
number range, is a conjecture that the turbulent spectrum
is formatted due to nonlinear interactions between modes
of close scales, whereas interactions between modes with
significantly different scales are performed through the
cascade sequence of interactions between modes of all
intermediate scales (Richardson’s cascade), in other
words, one has to tell about “a locality of intermode in-
teractions in the wave-number space” [11]. This is ex-
plained by the fact that the interaction between modes of
significantly different scales reduces to a simple me-
chanical translation of small-scale modes by large-scale
ones without energy redistribution [12]. This brings up
the problem of separating strong interactions between
modes with essentially different scales, which reduce to
translation, from weak interactions between modes of
close scales that execute the energy transfer along the
wave-number spectrum [13].
The concept of locality of intermode couplings lies at
the basis of describing turbulence within the framework
of renormalization-group method [14], and, in particular,
the method of

-expansion, which makes an integral
part of the RG-technique, should be treated as a way of
separating off local interactions from nonlocal interations,
which do not take part in formatting the spectrum [14].
The hypothesis of locality enables one to predict the
shape of energy spectrum in a certain wave-number
range (Kolmogorov’s spectrum), which turns out to be in
good agreement with experimental data.
The present investigation aims to obtain a closed set of
equations for two functions of direct interest, namely, the
covariance of velocity field and the function of averaged
responce to external force action (Green’s function).
Similarly to the formulation of turbulence theory in terms
of statistical moments, in the approach proposed, a chain
of equations for quantities through which statisticl mo-
ments of velocity field can be expressed arises. A closure
of this chain is performed by using the perturbation the-
ory at a certain stage. This enables one to obtain a set of
equations for two scalar functions. A solution to these
equations corresponds a sum of some infinite subse-
quence of the total pertubation-theory series. The method
of space-time characteristical functional and the equation
in functional derivatives for that are used in probabilistic

E. V. TEODOROVICH 58
description of turbulent velocity field.
2. Mathematical Statement of the Problem
One way of describing random processes and fields is the
method of characteristic functional. In turbulence theory
the space-time characteristic functional, which has been
first introduced R.Lewis and R.Kraichnan [15], is a func-
tional Fourier-transform of the probability density of a
random velocity field [1, Ch.3]
 

 
,exp, ,
,exp, ,d
Wt itu
Pti tut




 

 



rr
urrr r


dd
dd ,
tt
t t

 

rr
ur

,t

 r
(2.1)
here are the components of arbutrary vector
function of space and time coordinates,


,Pt 

ur

,tur

,, 1t  r

is
the functional probability density of realizing velocity
field and integration is carried out over all ve-
locity field realizations allowed by the set of equations
for this field.
To made the record more simple, we will further use
digital notations for a set of coordinates of space-time
point and subscripts of vector or tensor component
11 1, according to which


,1utur
111

, and
integration over space-time coordinates and summing
over component numbers with repeated digital notations
will be implied (the Eistein rule), i.e.








11
11 11
tv t

rrr
  
1
1 1
11 dduv tu

 (2.2)
When the vector or tensor subscripts are written ex-
plicitly, the digital notation will only relate to space-time
coordi- nates.
According to these notation

1exW  
 p11iu
 

10
(2.3)
From the definition of characteristic functional it fol-
lows that statistic moments of velocity field are ex-
pressed as functional derivatives of the functional W
with respect at 

.
  
   
0
0
12
W

2
11
12
uW
i
uu ii





 


 

lnW


(2.4)
and so on. Semi-invariants (cumulants or irreducible
means) of the velocity field are defined through func-
tional derivating the characteristc functional logarithm
.
After excluding the pressure by means of the equation
of continuity, the Navier-Stokes equation can be written
in the form
  
01, 22
112,3231 1
2
Lu
VuufF



 

011
0
1, 212
t
L 

(2.5)
here

 

(2.6)


1

f
is the component of external regular force,

1F

1
—
the component of external random force specified sta-
tistically,

is the Laplace-operator construted of
derivatives with respect to components of the vector ,
1
r









123
12 313 2
11223 3
23
12,3,, ; ,
1, 41, 4
42 43
VVttt
PP
 
 




 


 
rr r
 
 

14 1414
11 1
14 14
1, 4P
 


rr rr


14 1
1
14 0P  
(2.7)
here
(2.8)
is the operator of transversal projection satisfied the con-
dition
rr (2.9)



1
rr


11
14 14


 rr rr
14
is the reverse Laplace operator defined by
the relation
(2.10)
i.e. Green’s function for Laplace operator.
If there exists external regular force
f
, the character-
istic functional will be a functional of two functional
arguments
f
.

and
While calculating the characteristic functional the av-
eraging is carried out over external random force
F
.
Without loss of generality one may assume that this force
is solenoidal and statistically described by centered nor-
mal distribution with a covariance of the form

 


12 12
00
1, 21,332DPD
 
(2.11)




Under these assumptions and using a representation of
characteristic functional in the form of two-fold func-
tional integral one can obtain for that an equation in
functional derivatives [16,17]










 

0
0
,
1, 22
,
112,3
232
,
1,21, .
2
Wf
Li
Wf
Vii
Wf
iDf Wf
f




 
 



(2.12)
This equation is an analogous to Hopf’s equation for
spatial characteristic functional [1].

E. V. TEODOROVICH 59


In addition we present the equation for lnW

  
 
 

  
ln
2 3
1
3
WW
i

 
0
(0)
ln 1
1, 212,3
22
ln
1, 22
1lnln
12,3
22
LV
ii
W
iD f
WW
Vf
ii




 



(2.13)
Now let us turn from variables

and
f
to new
functional variables ˆ

and ˆ
f
definded by relations
 


  
ˆ1ln
ˆ1ln
, 1,
, 1
Wfi
f
Wfif





 (2.14)
When 0

, we get tends to

ˆ1



1u, which
is a mean value of velocity component 1

at the
space- time point in the external force field
.

11
,tr

ˆˆ
ˆˆ
,f iiff  

,ftr
222
Going to new variables can be performed with the help
of functional Legendre transformation through introduc-
ing new characteristic functional

,l nf W 


 (2.15)
In this case
   
1
ˆ1
1,
ˆ1
f
i


 


if
 

0
(2.16)
The limit

 corresponds to the exremality con-
dition of the functional : ˆ0 i  

ˆ
, whereas the
extremality condition of functional with respect to
f
corresponds to absense of external regular force.
The functional ˆ
ˆ,
f



 obeys the following equa-
tion in functional derivatives


  


 
0
0
ˆ 1, 22
1ˆˆ
12,3 2
ˆ
1, 22
L
V
iD f





ˆ3
3
22
ˆ1
i
if








0
(2.17)
In the limit

, Equation (2.17) transforms into
the equation for mean velocity field with accounting for
Reynolds tension.
For further analysis the mixed derivatives should be
considered, namely,
 


  

  
22
ˆˆ ˆ
231
12
ii

 



 
2
ˆ3
2 1
ˆ3
ˆ
2ˆ
31
f
fi

 

 



(2.18)
 

  
22
2
ˆ3
ˆˆ
2ˆ
21 31
ˆ30
ˆˆ
231
ff
f
ff



 

 




(2.19)
Here



 
12 12 12
12 tt

  rr (2.20)
If the extremality conditions for functional

are
satisfied, the expression that includes functional deriva-
tives of functional ˆ

only with respect to fields

vanish as well as functional derivatives of functional
only with respect to fields lnW
f
. This exact property
can be easily proved within the framework of diagramm
technique [14].
In particular
 
22
ln
0, 0
ˆˆ
121 2
W
ff

 
 (2.21)


In this case one will have


   

  
ˆ1ln1, 2,
212
ˆ1lnln 2,1
212
WC
iii
fWWG
fi
 


 
 

(2.22)





 
1, 21212Cuuuu Here is the co-
variance function of velocity field and


2, 1G
 
is
Green’s function that describes averaged response of
velocity field at the space-time point 2 to unit force ac-
tion localizated at point 1.
From Equations (2.18) and (2.19) we obtain

2
11, 2
ˆˆ
12
iG
f




 
(2.23)

 
2
11
1, 2
ˆˆ
21
1,32, 43, 4
D
ff
GG C





(2.24)

Here

11, 2G is the reverse Green’s function de-
fined by the relation




1
1, 33,212GG 
 (2.25)
The function D


1, 2 can be treated as a variance of
effective random forces describing averaged effect on
given mode from other modes due to nonlinear inter-
mode interactions.
Relation (2.24) can be rewritten in the form of Wyld’s
equation obtained previously by summing up the perturb-
bation-theory series [8]



 
1, 21,32, 43, 4CGGD (2.26)

E. V. TEODOROVICH 60
To find and let us act on this equation by
operators
1
GD

ˆ4  and

ˆ4if
 .
In the limit 0

 and we will have 0f
 


 

 
2
0
11, 4
ˆˆ
14
1,412, 3ˆ
24
GL
if
V



 


 
2
1, 41, 4
1ln
23
W


(2.27)
 






 
2
(1)
1, 41, 4
ˆˆ
41
1
1, 412,3ˆ4
DD
ff
DV
if





01
2
1, 4
ln
223
D
W 
 



11, 2D
(2.28)
here and are corrections to the
terms of molecular viscosity and covariance of external
random forces induced by an influence of turbulent mix-
ing.

1, 2
To obtain the equation for velocity field variance,
which describes the energy spectrum of turbulent pulsa-
tions, let us act to Equation (2.13) by operator


4i 
0 In the limit

0f, we get

  
, 32, 3, 4C

2,3, 4C

 
0
0
1
1, 22, 412
2
1, 22, 40
LC V
DG


(2.29)
here is the three-point statistical moment of
third order
  
2ln
423
W 
 
2,3,4Ci

  (2.30)
To calculate this function, one should apply Equations
(2.18) and (2.19) and the identity


 

 
ˆ
22
ˆ
1
2f



ˆ
ˆ 11
2
f  

 
 (2.31)
As the result, the nonlinear term in (2.29) appears to
be presented in the following form


112,32,3,4
2
1,
VC 





1
1, 22, 4
22,4
C
GD




12
1, 22, 4
TC

(2.32)



The obtained representation of nonlinear term in the
equation for velocity field variance is exact one, since,
when finding Equation (2.32), no approximation has
been made.
Equation (2.32) needs for some comments. Within the
framework of phenomenological approach to closure
problem, this term is treated as a turbulent viscosity and,
with assumption of the gradient hypothesis type, it is
commonly written as



1, 2 

1, 212  
(2.33)
where the integral kernel T is the turbulent vis-
cosity which shape can be found from the experimental
data by the trial-and-error method. As the simplest ex-
ample it may be the choice TT , where
T

is some empirical constant. Another possibility of
choice the representation for turbulent viscosity is






1, 2tt    rr
 1
D

12 12TT known as “Marcovian
approximation”. The result (2.32) means that the inter-
pretation of nonlinear term as turbulent viscosity is un-
safficient, and, when applying the phenomenological
closure scheme, one has to account for influence of ef-
fective random forces being subject to normal distribu-
tion with the covariance .


1
3. Equations for Functions and
D
and the Closuring Procedure
The set of Equations (2.27) and (2.28) is not close since
it includes two unknown functions  and . To
find these functions, one needs to calculate the functional
derivatives of Wyld’s Equation (2.26) with respect to

1
D
ˆ

and ˆ
f
. The functional derivatives of Green’s function
emerged can be found from the formula obtained
after differentiating relation (2.18) with respect to
G
ˆ

and ˆ


f
. If the extremality condition 0   is satis-
fied, one gets


   



3
1, 21,45, 2
ˆ
ˆ3ˆˆ
453
1,443, 55,2
GGG
if
GG

 


 
(3.1)
In a similar manner


    
  
  
3
1, 21, 45, 2
ˆˆˆ
ˆ 3345
1, 443,55,2
1, 434,55, 2
GGG
ifif if
GG
CG




 

ˆ
(3.2)
Differentiating Wyld’s Equation (2.26) with respect to
ˆ
f
leads to following relations

and


 


 
1, 221,4 43,5 5,2
ˆ3
1,42, 54,53
CGG
GG


 

(3.3)


 


 
1, 2=21,43,4 55,2
ˆ3
1, 42,53, 4,5
CGC
if
GG




(3.4)
where we introduced three new functions

E. V. TEODOROVICH 61



 
11, 2
ˆ3
3
G 



3
12 ,3 ˆˆˆ
12if

 



 

 

3
1, 23ˆˆ
1, 2
ˆ3
ifi





1
ˆ
12 3
1, 3
ˆ2
f
DG
if






  


31, 2
ˆ
3 3
D
if


1, 2, 3ˆˆˆ
12if ifif



 (3.5)
In the diagramm technique of quantum field theory
these functions are referred to as vertices.
The resulting expressions for desired functions

and take the form

1
D



 


65,6 2



1
1, 21 3, 43, 5
2
24,65 6,24,
VG
CG


(3.6)
and





 
11 3, 4
,62,5 6
2, 5,6

 

ˆ

11,22
2 3,54
3, 54, 6
DV
GC
GG



 (3.7)
The set of equations obtained is exact one. However,
this set is not closed because it contains the vertices of
three types. In turn, using functional differentiating
Equations (2.27) and (2.28) with respect to

and ˆ
f
,
one can obtain equations which will contain 4-th deriva-
tives of characteristic functional . If to continue this
procedure, it will arise an infinite chain of equations for
vertices of various orders being similar to the chain for
statistic moments by Keller and Freidman. To cut off this
chain in the way that differs from commonly used clo-
sure schemes, we apply the low-order perturbation theory
approximation for vertices by putting


1 2,31 2,3,1,23V 

1,2,3 0


1
D
1
G
ˆ0
(3.8)
Such approximation for and after substitute-
ing those into the equations for and C and solv-
ing the set obtained will correspond to summing up a
certain infinite subsequence of total perturbation-theory
series for the quantities desired.
In the case



(averaged velocity vanishes when
), we come to the closed set of equations for func-
tions G and C
0f
  
 


0
1
4,656,2CV








1, 21,21,2
1, 21213, 43,5
GL
VG

 (3.9)
and
  
01
1
1, 21,32,43, 43,4
1
1, 213,43, 54, 625, 6
2
CGGDD
DVCCV






0u

(3.10)
The set of tensor Equations (3.9)-(3.10) can be essen-
tiely simplified by using the condition of velocity field
solenoidality,

, from which it follows that func-
tions


12 1,2G


12 1,2C

and have the form


1212
1, 21,33, 2,GPG
 



1212
1, 21,33,2CPC
 
 (3.11)
With account for the unimodality property of the trans-
versal projection operator




13 3212
1, 33, 21,2PPP
 
 (3.12)
and from Wyld’s Equation (2.26) it also follows


 
12 12
12 12
11
1, 21, 33, 2,
1, 2=1, 33, 2
GPG
DPD
 
 



(3.13)
Thus, the problem is formulated in terms of two scalar
functions that depend on modulus of vector coordinates.
4. Equations in the Space of Fourier
Transforms
After going to the space of Fourier-transforms, for the
operator of transversal projection, vertex, Green’s func-
tion, and velocity covariance one gets
2
kk
Pk


 
 k

 
,,;,Vikk

 
 
 kq kq



2
,,GPGk
 

 kk



2
,,CPCk
 

 kk (4.1)
(while performing the Fourier transformation for vertex
(2.5) the operators of transveral projection


P

k were
changed by the Kronecker delta



due to the fact that
in all formulas one only meets the convolution


 
,,;,PV
 



kkqkq
and there exists the identity




PP PP

  

kk kk (4.2)
—the property of unimodality.
Due to the isotropy of the system considered the tensor



has the form





1
22
,, ,qqqq
 

  q (4.3)
and, since the operator



is convoluted with the op-

E. V. TEODOROVICH 62
erator of transversal projection, the term proportional to
gives no contribution and can be omitted.

1

In the Fourier space Formula (3.9) for


 takes the
form

 


2
,,
,,

2
dd
2π
2πdA
Gq C
 






kk q
kq
d
 
V
   
kqq



q
(4.4)
where is the space dimension and
 
,()AqVPP
  
kkq (4.5)
The relevant calculation gives
 


2
22
2
,kq
2
2
A
qq
q




 

kq
kq kq

kq (4.6)
The correction to the variance of effective random
force is calculated symilarly. According to Equation
(3.10) the Fourier-transform of this quantity has the form

  

,,
2π
DBq






kk
 
V
 

q k


1
2
2
dd
2π
,,
d
Cq C





q
kq
(4.7)
where
 

,
2,
BVPP
A





kqk qk
kq (4.8)
Thus, in combination with the equations




12
0
,Gki2 2 ,,k k



 
 
22
,,Gk

22
,,CkGk Dk

   
G C2 k
one comes to a closed set of equations for two scalar
functions and of two scalar variables and

.
5. Conclusions
One of the main exactly obtained results is a statement
that a traditional treatment of nonlinear term in the equa-
tion for velocity covariance as turbulent viscosity is not
safficient in modeling this term within the framework of
phenomenological approach, and it should be accounted
for additional term that describes the contribution from
effective random forces. The proposed expression for
this term may occur to be useful in finding the energy
spectrum of turbulent pulsations


2
1d ,
22πCk

d
Ek



The equation of energy balance in the wave-number
space can be obtained after Fourier transformation of the
equation for velocity covariance and integrating over

.
In this case the nonlinear term in the equation obtained
may be written in the form of divergence (in the space of
wave numbers) of a certain vector treated as the energy
flux along the wave-number spectrum. From the dimen-
sionality arguments and the assumption on locality of
intermode interactions in the wave-number space it fol-
lows that the spectral energy flux defines the spectrum
shape, however, unlike the Kolmogorov’s theory, this
quantity is not a constant due to dissipation effects.
The closed set of equations for two scalar functions
and G of two scalar variables obtained by applying
the perturbation theory approximation for high-order
statistical moments proves to be more simple in numeri-
cal sumulation the turbulence problem. When solving
these equations, the renormalization-group method may
appear to be useful since this method is based on the
property of locality of intermode interactions in the
wave-number space (see, [18] and, for example, [19]).
C
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