Group-theoretical model of developed turbulence and renormalization of the Navier-Stokes equation V. L. Saveliev 1 and M. A. Gorokhovski 2 1. Institute.

Group-theoretical model of developed turbulence and renormalization of the Navier-Stokes equationV. L. Saveliev1 and M. A. Gorokhovski2

1. Institute of Ionosphere, Almaty 480020, Kazakhstan2. 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 ( )v r , we consider a field ( )sv r that is averaged on

the scale s , then the last one will “resemble” the original turbulent field ( )v r . 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 0s 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 ( ),tsv r , averaged over any scale s .

II. Renormalized averaging formula for <vu> Consider two functions ( )v r and ( )u r depending on radius vector r . The average of ( )v r is

( ) ( ) ( )v v d¢ ¢ ¢= Y -òr r r r r , (1)

When the weight function ( )Y r is Gaussian,

( )

( )

2

43

1

4e s

sps

-Y =

r

r , (2)

the average is defined by Gauss transform (filtering):

( ) ( ) ( ) ˆv v d G vs s¢ ¢ ¢= Y - =òr r r r r . (3)

The Gaussian distribution verifies the diffusion equation,

( ) ( )2s ss

¶Y = Ñ Y

¶r r , where ( )

22 ¶Ñ =

¶r. (4)

Thereby the Gauss transform operator (3) can be represented in the exponential form:

2

1 2 1 2ˆ ˆ ˆ ˆ,G e G G Gss s s s s

Ñ+= = . (5)

Operator for averaging the field product

Consider the averaged product of two fields:

( ) ( ) ( ) ( ) ( )2ˆvu G v u e v us

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,

1 2Ñ = Ñ +Ñ , (7)

with differentiation operators 1Ñ and 2Ñ acting on the first ( )v r and the second ( )u r

multipliers respectively. Setting Eq.(7) in Eq. (6) gives

( ) ( )( )

( ) ( ) ( )( ) ( )( )22 2 21 2 1 22e v u e v u e e v e u

ss s s sÑ +ÑÑ 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

1 22vu e v usÑ ×Ñ= . (9)

Simplest formula for averaging Instead of using the Taylor-series expansion with an infinite number of terms for the averaging

operator 1 22e sÑ ×Ñ , we use the Taylor series with a residual term,

( )

( )

( ) ( )

1 2

1 2

121 2 1 2

12

1 20

11 2 ... 2

1 !

12 ,!

n

nn

en

d q en

s

as

s s

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

( ) ( ) 11 nnq na a -= - ; ( )

1

0

1nd qa a =ò . (11)

For the case 1n = we have:

0

2vu v u d v us

s s s s s s ss ¢ ¢ ¢-¢= + Ñ ×Ñò . (12)

This averaging expression demonstrates that the integral part includes the contribution of all

scales less than s in an exact manner.

Example of averaging Consider the velocity field in the framework of the following model. Assume that in a region of

a typical size L , the velocity field can be represented as

( ) ( )rndi i ij j iv w a r v= + +r r , (13)

where iw and ija are constants and ( )rndiv r is an isotropic random field. This implies that the

considered region is rectilinearly moving as a whole, rotating and stretching. The liquid

particles inside this region have random isotropic velocities. For this field we have:

rndl m i l m iv vÑ Ñ = Ñ Ñ . (14)

Using the averaging formula (9), (10) with 2n = for the velocity field (13) yields

1 2 1 2

0

2

2 2 ,

i k i k l i l k

rnd rndl l i l l k

vv v v v v

d v v

s s s s s

s

s s s s

s

s s ¢ ¢ ¢-

= + Ñ Ñ +

¢ ¢+ Ñ Ñ Ñ Ñò (15)

where 1 2, 1,2,3l l = .

Deviatoric part of stress tensor

Accounting for isotropy of the random field, the integral term is proportional to the Kronecker

delta ikd ,

1 1

1 2 3 1 2 3

0

2

22 .

3

i k i k l i l k

rnd rndik l l l l l l

vv v v v v

d v v

s s s s s

s

s s s s

s

d s s ¢ ¢ ¢-

= + Ñ Ñ

¢ ¢+ Ñ Ñ Ñ Ñò (16)

Thus, for the model (13), the deviatoric part of the stress tensor can be expressed through the

averaged velocity field in closed form,

( )1 1

1 1 1 2 1 2

21 13 3

12 .

3

i k ik i k ik l l

l i l k ik l l l l

vv v v v v v

v v v v

s s s s ss

s s s s

d d

s d

- = -

+ Ñ Ñ - Ñ Ñ (17)

Important remark

Starting from Boussinesq, in papers devoted to deriving equations for averaged turbulent fields,

it is usual to use the averaging formula on the basis of the gradient-diffusion hypothesis,

( )2 21 13 3

i ki k ik ik tur i k k iv vvv v v vss ss s sss

d d n- - - = - Ñ +Ñv . (18)

However, if the velocity field ( )iv r in Eq.(18) is replaced by ( ) ( )i iv v= -r r% , we obtain

( )2 21 13 3i ki k ik ik tur i k k iv vvv v v vss ss s ss

sd d n- - - = + Ñ +Ñv% %%% % % %% . (19)

Inevitably, from Eq. (18) and (19), one of two fields ( ),iv tr or ( ),iv tr% has negative turbulent viscosity.

Thus, through averaging, it is impossible to derive the universal gradient-type formula of form (18). In

other words, one cannot obtain the turbulent viscosity in equations for turbulence by simple averaging of

the nonlinear term. The following sections show how the turbulent viscosity can be introduced in a natural

way.

III. Group-theoretical model of developed

turbulence Hereafter, we present a schematic description of the turbulent model, which is solely based on

the Euler equation for incompressible flow,

, 0 1pt

r¶= - Ñ × - Ñ Ñ × = =

¶v

vv v . (20)

We will consider continuous symmetry transformations of Eq. (20) where the time variable is

not transforming: scaling ,e etb tb® ®r r v v; rotation ,e et t´ ´® ®r r v vW W ; and

translation vt® +r r . These transformations constitute a group and induce the group of the

velocity field transformations. The generators of its one-parameter subgroups are the operators

that are the linear combination of scaling, translation, and rotation generators,

( ) ( ) ( )v vˆ , , 1q b b= - ×Ñ - ×Ñ - × ´ Ñ + ´r rW W W . (21)

Subgroup elements read as ˆˆ qg ett = , where t is a subgroup transformation parameter,

1 2 1 2ˆ ˆ ˆg g gt t t t+= . (22)

Symmetry parameters

The constants b , v , W have the following physical meaning: b is the rate (relative to

parameter t ) of the homogeneous scaling, v is the velocity of translation, and W is the angular

velocity of rotation. An action of these operators on velocity field ( )v r is as follows:

( )( ) ( )

( )( )( ) ( )

( ) ( )

1 ,

,

e e e

e e e

e

tb tb tb

t t t

- ×Ñ -

´ ×´ Ñ ´ - ´

- ×Ñ

=

=

= -

r

r

R

v r v r

v r v r

v r v r R

W - W W W . (23)

The symmetry of the Euler equation is then expressed by

( )( ) ( )ˆ ˆ ˆ ˆ ˆ, , 0q q q q qe e e e p p et t t t t¢Ñ × = Ñ × Ñ = Ñ Ñ × =vv v v v . (24)

Self-similar solutions

We propose to associate the phenomena of stationary turbulence with self-similar solutions of

the Euler equation (20) in relevance to symmetry subgroups with generators (21). The self-

similar solution implies its dependence on time through the parameter of the space symmetry

transformation only, i.e.,

( )( )

( )0 ˆ0, ,

t t qt e t

-=v r v r . (25)

The evolution in time of a such self-similar solution ( ),tv r is governed by the simple equation:

q̂t¶=

¶v

v (26)

and the spatial configuration, at each time moment, is defined by the equation:

ˆ , 0q p= - Ñ × - Ñ Ñ × =v vv v . (27)

Decomposition of turbulent field Requiring solution (25) to be dynamically stationary, the velocity field ( ),tv r must be

represented by a superposition of time harmonics i tew with real w. Then it follows from self-

similarity (26) that expansion coefficients ( )wv r are eigenfields of the generator q̂

corresponding to the pure imaginary eigenvalues,

( ) ( ), i tt eww

w

= åv r v r . (28)

( ) ( )ˆ ,q iw ww w= - ¥ < < ¥v r v r . (29)

Equation (28) indicates that only phases of eigenfields ( )wv r change during the time evolution

of turbulent velocity field ( ),tv r . The phases are stochastic and, in principle, unknown for

turbulent state. Therefore, only symmetry characteristics (rate of the homogeneous scaling, b ;

velocity of translation, v; angular velocity of rotation, W), and the power spectrum ( )*w wv v r

characterize the turbulent state in our model.

Invariance of turbulent state

The turbulent field with parameters v, ,b W is invariant under transformation ( )vˆ , ,qet bW

(accurate to the change of phases of ( )wv r , which are immaterial for the turbulent state),

( ) ( ) ( )v v vˆ , , , , , ,Avqet b b b=v vW W W . (30)

Indeed, applying the operator ( )vˆ , ,qet bW to the velocity field (28) leads only to the phase change

of eigenfields ( iewtw w®v v ) in the turbulent velocity field decomposition.

Hereafter we will skip the symbol Av, and writing "invariance of the turbulent velocity field,"

the sentence "accurate to the change of stochastic phases" will be omitted for the sake of

simplicity.

Renormalization-group invariance We introduce an averaged velocity field using the Gauss weight function with scale s ,

( )

( )

( )

( )2

24

3/ 2

14

e d es ssps

¢--Ñ ¢ ¢= = ò

r r

v v r r v r . (31)

The invariance (30) for turbulent fields provides the corresponding invariance for averaged

turbulent fields sv :

( )2 2 2 2 2ˆ 2 ˆ ˆq q qe e e e e e et bs s t s s t ss s s

- Ñ Ñ - Ñ Ñ Ñ= = = =v v v v v . (32)

This equation can be rewritten in an equivalent form:

0ˆ

0

1, lnqe st s s s

st

b s= =v v , (33)

The change of the scale of averaging from 0s to s is equivalent to the composition of scaling,

rotation, and translation transformations. Note also that the scaling coefficient 0/e st b s s=

depends only on initial 0s and final s scales and does not depend on turbulence parameters.

We call the property (33) a renormalization-group invariance of averaged turbulent fields.

IV.Renormalization of the averaged Navier-Stokes equation

Averaging over small scaleWe will study the case when characteristics of turbulence are changing in space and time

slowly. Consider now the Navier-Stokes equation

2 0, 0, 1pt

n r¶+Ñ × +Ñ - Ñ = Ñ × = =

¶v

vv v v . (34)

We average this equation with some small scale 0s . On the scales less than 0s , 0s s£ , we

assume that the velocity pulsations become purely isotropic (13). Making use of our averaging

formula in form (17), we have

( )0

0 0 0 0 0

0

202 ,

0.

l l pts

s s s s s

s

s n¶ ¢+Ñ × + Ñ Ñ +Ñ = Ѷ

Ñ × =

vv v v v v

v (35)

Inner threshold of turbulence We assume that 0s is the inner threshold of the turbulence such that, on scales larger than 0s ,

0s s> , we have Euler turbulence with self-similarity. The exact determination of 0s is an

open problem. Here we propose a simple method of estimating its value. The gradients of the

velocity field tend to destroy the continuous structure of flow due to nonlinear steeping while

the viscosity processes smooth over the flow. To derive the equation for 0s , we equate these

two factors on the basis of dimensional consideration. The antisymmetric part, which is

connected with the rotation as a whole, is excluded from gradient Ñv,

0 0

1/ 2

0ik ikc S Ss s

ns

é ù= ë û , ( )12ik i k k iS s s s= Ñ +Ñv v (36)

Here c is some unknown constant.

0 0

0 1/ 2ik ikc S Ss s

ns =

é ùë û . (37)

Equation for averaged turbulent fields

Excluding molecular viscosity n from the Navier-Stokes equation with the help of the equation

for 0s , we have:

( ) ( )00 0 0 0 0

20 02 l l turp

ts

s s s s ss n s¶ ¢+Ñ × + Ñ Ñ +Ñ = Ѷv

v v v v v , (38)

( )0 0

1/ 20 0tur ik ikc S Ss sn s s né ù= =ë û . (39)

Applying transformation q̂e st with 0

1lns

st

b s= , along with renormalization group

symmetry, we obtain the final equation for the averaged turbulent field for scales 0s s> ,

( ) ( )

( ) [ ]

2

1/ 2

2 ,

, 0.

l l tur

tur ik ik

pt

c S S

ss s s s s

ss s

s n s

n s s

¶ ¢+Ñ × + Ñ Ñ +Ñ = Ѷ

= Ñ × =

vv v v v v

v

(40)

Remarks about viscosity

1. The equation for averaged fields (40) are similar to the known equation of Leonard applied in

the LES approach ( the small difference is that the dissipative term with molecular viscosity is,

in principle, not present here).

2. The equation for averaged fields, due to absence of the molecular viscosity term, is invariant

under the scale transformation, which involves changing the averaging scale s :

( )

( ) ( )2ln 2 1 1

, ,e t ta s

ss a s a

a

¶×Ñ + -

¶ =r

v r v r (41)

3. It is important to stress that we have shown that the turbulent viscosity appeared not as a

result of averaging of the nonlinear term in the Navier-Stokes equation, but from the molecular

viscosity term with the help of renormalization-group transformation.

V. Conclusion 1. We have obtained the regularized averaging formula for averaging a two velocity field

product.

2. Assuming that on the small length scale 0s , the turbulent velocity field can be approximated

as the sum of a smooth velocity field and a random isotropic field, we averaged the Navier-

Stokes equation over this small scale by making use of our averaging formula.

3. We proposed to associate the phenomena of stationary turbulence with the special self-

similar solutions of the Euler equation – they represent the linear superposition of eigenfields of

the symmetry subgroup generators corresponding to the pure imaginary eigenvalues.

4. From this model, in particular, it follows that for stationary homogeneous turbulence, the

change of the scale of averaging from 0s to s is equivalent to the composition of scaling,

rotation, and translation transformations. We call this property a renormalization-group

invariance of averaged turbulent fields.

5. The renormalization-group invariance provides an opportunity to transform the averaged

Navier-Stokes equation over a small scale 0s to any scale s by simple scaling.

References

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