Transcript Слайд 1 - International Centre for Theoretical Physics

Group-theoretical model of developed turbulence
and renormalization of the Navier-Stokes equation
V. L. Saveliev1 and 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 v ( r ) , we consider a field v
s
(
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 s 0 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 v
s
( r , t ),
averaged over any scale s .
II. Renormalized averaging formula for <vu>
Consider two functions v ( r
and u ( r
)
v
When the weight function Y ( r
)
(
r
)
)
=
depending on radius vector r . The average of v ( r
- r ¢)d r ¢,
ò v ( r ¢) Y ( r
)
is
(1)
is Gaussian,
Y s (r
)
1
=
(
-
e
3
r
2
4s
(2)
,
4p s )
the average is defined by Gauss transform (filtering):
v
(
r
)
=
ò v ( r ¢) Y s ( r
- r ¢)d r ¢ = Gˆ s v
(3)
.
The Gaussian distribution verifies the diffusion equation,
¶
¶s
Y s (r
)
2
= Ñ Y s (r ),
where
Ñ
2
=
¶
2
(¶ r ) .
(4)
Thereby the Gauss transform operator (3) can be represented in the exponential form:
2
Gˆ s = e s Ñ ,
Gˆ s 1Gˆ s 2 = Gˆ s 1 + s 2 .
(5)
Operator for averaging the field product
Consider the averaged product of two fields:
vu
(
r
)
= Gˆ s v ( r )u ( r
2
)
= e s Ñ v ( r )u ( 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,
Ñ = Ñ1 + Ñ2,
(7)
with differentiation operators Ñ 1 and Ñ 2 acting on the first v ( r
)
and the second u ( r
)
multipliers respectively. Setting Eq.(7) in Eq. (6) gives
2
e s Ñ v ( r )u ( r
)
= e
s ( Ñ 1 + Ñ 2 )2
2
v ( r )u ( r
)
= e 2 s Ñ 1 ×Ñ 2 (e s Ñ v ( r
)
)(e s Ñ
2
u (r
)
).
(8)
It then follows that the average of the product of two fields can be written as
vu
= e 2 s Ñ 1 ×Ñ 2 v
u .
(9)
Simplest formula for averaging
Instead of using the Taylor-series expansion with an infinite number of terms for the averaging
operator e 2 s Ñ 1 ×Ñ 2 , we use the Taylor series with a residual term,
e 2 s Ñ 1 ×Ñ 2 = 1 + 2 s Ñ 1 ×Ñ 2 + ... +
1
n- 1
( n - 1 )!
( 2 s Ñ 1 ×Ñ 2 )
(10)
1
+
1
n
n!
ò d a qn
( 2 s Ñ 1 ×Ñ 2 )
+
(
a
)
e 2 a s Ñ 1 ×Ñ 2 ,
0
where n is an arbitrary natural number ( n = 1, 2, 3, .... ), and
1
qn ( a
)
= n ( 1 - a )n - 1 ;
ò d a qn
(
a
)
= 1.
(11)
0
For the case n = 1 we have:
s
vu
s
=
v
s
u
s
+ 2ò d s ¢ Ñ v
s¢
×Ñ u
s ¢ s - s ¢.
(12)
0
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
vi (r
)
= w i + a ij r j + v ir n d ( r
where w i and a ij are constants and v ir n d ( r
)
)
(13)
,
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:
Ñ l Ñ m v i = Ñ l Ñ m v ir n d .
(14)
Using the averaging formula (9), (10) with n = 2 for the velocity field (13) yields
vivk
s
=
vi
s
vk
s
+ 2s Ñ l vi
s
Ñ l vk
s
+
s
+ 2 ò d s ¢2 s ¢
0
where l1 , l2 = 1, 2, 3 .
Ñ l1 Ñ l2 v ir n d
s¢
Ñ l1 Ñ l2 v kr n d
(15)
s¢ s- s¢
,
Deviatoric part of stress tensor
Accounting for isotropy of the random field, the integral term is proportional to the Kronecker
delta dik ,
vivk
=
s
vi
s
vk
s
+ 2 s Ñ l1 v i
s
Ñ l1 v k
s
s
+
2
3
dik
ò d s ¢2 s ¢
Ñ l1 Ñ l2 v lr3n d
rn d
Ñ
Ñ
v
l
l
l
1
2
3
s¢
(16)
s¢ s- s¢
.
0
Thus, for the model (13), the deviatoric part of the stress tensor can be expressed through the
averaged velocity field in closed form,
vivk
+ 2s
s
(Ñ
-
l1
1
3
dik v 2
vi
s
s
Ñ l1 v k
=
vi
s
-
s
1
3
vk
s
-
1
3
dik Ñ l1 v l2
dik v l1
s
v l1
s
(17)
s
Ñ l1 v l 2
s
).
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,
vivk
s
-
1
3
dik v 2
s
-
vi
However, if the velocity field v i ( r
v%%
ivk
s
-
1
3
dik v%2
s
-
v%i
vk
s
s
)
s
-
1
3
dik v
= - ntur ( Ñ i vk
2
s
in Eq.(18) is replaced by v%i ( r
v%k
s
-
1
3
s
+ Ñ k vi
s
) . (18)
s
)
= - v i ( r ) , we obtain
= + n t u r ( Ñ i v%k
dik v% 2s
s
+ Ñ k v%i
s
) . (19)
s
Inevitably, from Eq. (18) and (19), one of two fields v i ( r , t ) or v%i ( r , t ) 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,
¶v
¶t
= - Ñ ×v v - Ñ p ,
Ñ ×v = 0
r = 1.
(20)
We will consider continuous symmetry transformations of Eq. (20) where the time variable is
not transforming: scaling r ® e t b r , v ® e t b v ; rotation r ® e t W´ r , v ® e t W´ v ; and
translation r ® r + t v . 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,
qˆ ( b , W, v ) = b ( 1 - r ×Ñ ) - v ×Ñ - W ×r ´ Ñ + ( W ´ ).
(21)
Subgroup elements read as gˆ t = e t qˆ , where t is a subgroup transformation parameter,
gˆ t 1 gˆ t 2 = gˆ t 1 + t 2 .
(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
etb
( 1 - r ×Ñ )
v (r
)
et
( ( W´ )- W×r ´ Ñ )
e-
R ×Ñ
= e t b v (e -
v (r
v (r
)
)
tb
)
is as follows:
r ),
= e t W´ v (e -
t W´
(23)
r ), .
= v (r - R )
The symmetry of the Euler equation is then expressed by
e t qˆ Ñ ×v v = Ñ ×(e t qˆ v
)(e t qˆ v ),
e t qˆ Ñ p = Ñ p ¢,
Ñ ×(e t qˆ v
)=
0.
(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 selfsimilar solution implies its dependence on time through the parameter of the space symmetry
transformation only, i.e.,
v (r , t ) = e
( t - t 0 )qˆ
v ( r , t 0 ).
(25)
The evolution in time of a such self-similar solution v ( r , t ) is governed by the simple equation:
¶v
¶t
= qˆ v
(26)
and the spatial configuration, at each time moment, is defined by the equation:
qˆ v = - Ñ ×v v - Ñ p ,
Ñ ×v = 0 .
(27)
Decomposition of turbulent field
Requiring solution (25) to be dynamically stationary, the velocity field v ( r , t ) must be
represented by a superposition of time harmonics e i wt with real w . Then it follows from selfsimilarity (26) that expansion coefficients v w ( r
)
are eigenfields of the generator qˆ
corresponding to the pure imaginary eigenvalues,
v (r , t ) =
å
e i wt v w ( r ) .
(28)
w
qˆ v w ( r
)
= i wv w ( r ),
- ¥
< w < ¥ .
Equation (28) indicates that only phases of eigenfields v w ( r
)
(29)
change during the time evolution
of turbulent velocity field v ( r , t ). 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 v w v w* ( r
characterize the turbulent state in our model.
)
Invariance of turbulent state
The turbulent field with parameters b , W, v is invariant under transformation e t qˆ ( b , W, v )
(accurate to the change of phases of v w ( r ) , which are immaterial for the turbulent state),
e t qˆ
( b , W, v )
Av
v ( b , W, v ) = v ( b , W, v ) .
(30)
Indeed, applying the operator e t qˆ ( b , W, v ) to the velocity field (28) leads only to the phase change
of eigenfields ( v w ® e i wt v w ) in the turbulent velocity field decomposition.
Hereafter we will skip the symbol A v , 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
1
2
v
= e s Ñ v (r
s
)
=
ò
3/ 2
(4p s )
d r ¢e
-
( r - r ¢)
v ( r ¢) .
4s
(31)
The invariance (30) for turbulent fields provides the corresponding invariance for averaged
turbulent fields v
e
s
:
t ( qˆ - 2 b s Ñ 2
)
2
v
s
= e s Ñ e t qˆ e -
sÑ2
2
v
s
2
= e s Ñ e t qˆ v = e s Ñ v =
v
s
(32)
.
This equation can be rewritten in an equivalent form:
e t s qˆ v
s
0
=
v
s
,
t
s
=
1
b
ln
s
s
(33)
,
0
The change of the scale of averaging from s 0 to s is equivalent to the composition of scaling,
rotation, and translation transformations. Note also that the scaling coefficient e t s b =
s / s
0
depends only on initial s 0 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 scale
We will study the case when characteristics of turbulence are changing in space and time
slowly. Consider now the Navier-Stokes equation
¶v
¶t
+ Ñ ×v v + Ñ p - n Ñ 2v = 0,
Ñ ×v = 0,
r = 1.
(34)
We average this equation with some small scale s 0 . On the scales less than s 0 , s £ s 0 , we
assume that the velocity pulsations become purely isotropic (13). Making use of our averaging
formula in form (17), we have
¶ v
s0
¶t
Ñ × v
+ Ñ ×( v
s0
= 0.
s
0
v
s
0
+ 2s 0Ñ l v
s
0
Ñl v
s
0
) + Ñ p ¢ = nÑ 2 v
s
0
,
(35)
Inner threshold of turbulence
We assume that s 0 is the inner threshold of the turbulence such that, on scales larger than s 0 ,
s > s 0 , we have Euler turbulence with self-similarity. The exact determination of s
0
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 s 0 , 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 Ñ v ,
n
s
0
= c éë S ik
s
0
1/ 2
S ik
s
0
ù
û
,
S ik
s
1
=
2
(Ñ i v k
s
+ Ñ k vi
s
)
(36)
Here c is some unknown constant.
s
0
=
n
c éë S ik
s
0
S ik
1/ 2
s
0
ù
û
.
(37)
Equation for averaged turbulent fields
Excluding molecular viscosity n from the Navier-Stokes equation with the help of the equation
for s 0 , we have:
¶ v
s0
¶t
+ Ñ ×( v
v
s0
s0
+ 2s 0Ñ l v
s0
n t u r ( s 0 ) = c s 0 éë S ik
Applying transformation e t s qˆ with t s =
Ñl v
s0
1
S ik
s
ln
b
s0
s
) + Ñ p ¢ = ntur ( s
1/ 2
s0
ù
û
0
)Ñ
2
v
s0
,
(38)
(39)
= n.
, along with renormalization group
0
symmetry, we obtain the final equation for the averaged turbulent field for scales s > s 0 ,
¶ v
¶t
s
+ Ñ ×( v
s
v
s
+ 2s Ñ l v
s
Ñl v
s
) + Ñ p ¢ = ntur
(
s
)
Ñ2 v
s
,
(40)
ntur ( s
)
= c s [ S ik
s
S ik
1/ 2
s
]
,
Ñ × v
s
= 0.
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 :
(
ln a r ×Ñ + 2 s
e
¶
¶s
- 1
)
v
s
(r , t ) =
1
a
v
a 2s
(a r , t )
(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 s 0 , the turbulent velocity field can be approximated
as the sum of a smooth velocity field and a random isotropic field, we averaged the NavierStokes 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 selfsimilar 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 s 0 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 s 0 to any scale s by simple scaling.
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