y. kaneda- small-scale anisotropy in high reynolds number turbulence: universality of the 2nd kind

8/3/2019 Y. Kaneda- Small-Scale Anisotropy in High Reynolds Number Turbulence: Universality of the 2nd Kind

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Small-Scale Anisotropyin High Reynolds Number Turbulence

-- Universality of the 2nd Kind --

Y. Kaneda

Nagoya University

August 23rd, 2007, Summer School at Cargse,

Small-Scale Turbulence :

Theory, Phenomenology and Applications


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Turbulence

= a system of huge degree of freedom

A paradigm of

Studies of systems of huge degree of freedom

Thermodynamics and Statistical Mechanics

for thermal equilibrium state


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Analogy withStatistical Mechanics for Near Equilibrium System

I. Two kinds of universalitycharacterizing the macroscopic state of the equilibriumsystemnot only

a) Equilibrium state itself, like Boyle-Charles lawbut also

b) Response to disturbance

II . Thermal equilibrium state Universal equilibrium state at small scaleinfluence of external force, mean flow, etc. at small scale

may be regarded as disturbance.How dose the equilibrium state respond to the disturbance ?

Universality of the 2nd Kind


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Equilibrium state

at thermal equilibrium state

disturbance

response


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Universality in Response

to disturbances, near equilibrium state

1905, Einstein, D=kT,the first example of FD-elation Perrins experiment.

1928, Nyquists theorem on thermal noise:P(f)=4kT Re(Z(f))

1931, Onsagers reciprocal theorem:

J = C X, C=TC

generalized flux, generalized force


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J = CX

Generalized Flux vs. Generalized Force

e.g.

Density Flux vs. Density gradient J = C grad

Heat Flux vs. Temperature gradient J = C grad T

Electric Current vs. External electric field ; J = E = grad ,(I=E/R, Ohms law)

---

Momentum Flux vs. Strain rate ;

coupling only between tensors of the same order.

(Newtons law)


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Thermal Equilibrium system

Linear response

Equilibrium state

Disturbance

Xgrad grad T

grad c

Xgrad grad T

grad c

F=CX (Hookes law)J = C grad = E (Ohms law)J = C grad T (Fouriers law)J = C grad c (Ficks law)

F=CX (Hookes law)J = C grad = E (Ohms law)

J = C grad T (Fouriers law)J = C grad c (Ficks law)


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History:

Universality in Responseto disturbances, near equilibrium state

1905, Einstein, D=kT,the first example of FD-elation Perrins experiment.

1928, Nyquists theorem on thermal noise:P(f)=4kT Re(Z(f))

1931, Onsagers reciprocal theorem:J = C X, C=TC

generalized flux, generalized force

1950-60, Nakano, KuboLinear Response Theory


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Equilibrium State

of Turbulence


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Equilibrium state

?

disturbance

response

Turbulence ?


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DNS

Series1Series1 Series2Series2

Energy Spectrum a la Kolmogorov (K41)

From Kaneda &Ishihara (2006)


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Equilibrium state

a la KolmogorovK41

disturbance

response


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Disturbance

1. Mean Shear

2. Buoyancy by Stratification

3. Magneto-Hydrodynamic Force


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I. Mean Shear

See Ishihara et al. (2002)


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Local co-ordinateNS-equation

Effect of mean shear

Turbulent Shear Flow

Local strain rate of mean flow


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Effect of Mean FlowHomogenous Mean Shear Flow:

in the inertial subrange of homogeneous turbulent shearflow;

i ij jU S x=

( ) ( ) ( )ij i jQ u u= k k k = ?

Let (k) =k /T, (where k/( 1/3 k2/3), T=1/S)

Assume (1)


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Two kind of measurescharacterizing the universal equilibrium state

0( ) ( ) ( )ij ij ijQ Q C S = +k k k =

1) Equilibrium state itself

2) Response to disturbance

Similar to stress vs. rate of strain relation:

ij = Cij S(justified by the Linear response theory for non-equilibrium system)

Not only

but also

(0) 2 /3 11/3

( ) ( )4

K

ij ij

C

Q k P

=k k ( ) , / ij ij i j i iP kk k k k = =k


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Anisotropic part

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ij i j i j ijC a k P P P P b k P k k = + + k k k k k k

1/3 13/3 1/3 13/3( ) , ( )a k A k b k B k = =

Only 2 (universal) parameters, A and B

Is this correct?

What are the values of A and B?

0

( ) ( ) ( )ij ij ijQ Q C S = +k k k =

at small scale of turbulent shear flow

Kolmogorovs isotropic equilibrium spectrum

response

cf. Lumley(1967)Cambon & Rubinstein (2006)


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Method of numerical experiment

max 1k =

Initial condition:isotropic turbulence

max 241k =

DNS of homogeneous turbulence with the simple shear flow

2

0

0

Sx =

U

Resolution: 5123

( ) ( ), ( ) ( )abij ij ij a b ij p k p k

E k Q E k p p Q= =

= = p pObserve

especially12 12 12 12 11 22 33

12 11 22 33 12 12 12( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )iiE k E k E k E k E k E k E k E k

Estimate A and B, and check consistency.

0.5,1.0S =

12

12 12

4 1 4( ) (7 ) , ( ), ( ) ( ) ,

15 2 15iiE k A B E k E k A B

= = + L 1/3 7 /3k S =


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Anisotropic Energy spectrum ofhomogeneous turbulent shear flow

( ) ( ) ( )abij a b i jp k

E k p p u u=

= p p

( ) ( ) ( )ij i jp k

E k u u=

= p p

12 4( ) (7 )15

E k A B =

121 4( ) ( )2 15

ii E k A B

= +

12 ( )E k

121 ( )2

iiE k

1( )

2iiE k

2

0

0

Sx

=

UMean Flow

33

12

4( ) (23 ) ,

105

E k A B

= L

11 22

12 12

4( ) ( ) (13 3 )

105 E k E k A B

= =

1/3 7 /3S k =

Theoretical predictions:

where

Ishihara, et al. (2002)


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Consistency11 22 33

12 12 12( ), ( ), ( )E k E k E k 12 12 12

11 22 33( ), ( ), ( )E k E k E k

33

12

4( ) (23 )

105 E k A B

=

11 22

12 12

4( ) ( ) (13 3 )

105E k E k A B

= = 12 1211 22

16( ) ( ) ( 2 )

105 E k E k A B

= = +

12

33

8( ) ( 3 )

105 E k A B

= +

Theoretical predictions:

1.0

2.0

S

St

=

=

1.0

2.0

S

St

=

=

Ishihara et al.(2002)

See also Yoshida et al. (2003)


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II. Stratification

See Kaneda & Yoshida (2004)


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Boussinesq approximation

Buoyancy by Stratification


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Kolmogorov scaling:


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Boussinesq TurbulenceScaling

From Kaneda & Yoshida (2004)


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Angular dependence



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III. MHD

See Ishida & Kaneda (2007)


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MHD approximation

quasi-static approximation

O(b2)

Magnetic force


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Local co-ordinateNS-equation

Effect of mean shear

Turbulent Shear Flow

Local strain rate of mean flow


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Response; Anisotropic

Isotropic turbulence( Equilibrium state )

Disturbance X

External forcedue to the magnetic field

External forcedue to the magnetic field

C XC X

Approximately linear in X


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response ofisotropic turbulence

3 (universal) parametersin the inertial subrange


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anisotropic parts

To detect , define

k= k


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Theoretical prediction (1)

Angular dependence


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Theoretical prediction (2) Scale dependence

Assumptions

are functions of only and in the inertial subrange)(kqi

C i b h


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Comparison between theory

and DNS Angular dependence

Theory

DNS

for the DNS

Every Shells

[k0,k1)=[8,16)

[k0,k1)=[16,32)

[k0,k1)=[32,64) [k0,k1)=[64,128)

[k0,k1)=[128,241)

3

From Ishida & Kaneda (2007)


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Comparison between

theory and the DNS Angular dependence

Theory

DNS


C i b t


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Comparison between

theory and DNS Scale dependence

Theory

DNS-5/3

-7/3

k-7/3/k-5/3=k-2/3



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anisotropy remains at small scale.

From Vorobev et al.( Phys. Fluids 17, 125105 (2005))


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Results in our DNS

approaches to isotropic state. c(k)

Anisotropy increases.



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Results in our DNS

approaches to isotropic state. c(k)

Anisotropy increases.



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Anisotropy : c(k)

DNS

Our theory

Comparison between


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Comparison between

theory and the DNS

T=TM=16 magnetic field OFF T=T

F=32 magnetic field ON

Comparison with the DNS and the theory

0.10.2

k-7/3/k-5/3=k-2/3



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Summary -1For turbulence,We dont know the pdf nor Hamiltonian in contrast to thermal equilibrium state

But we may assume1) the existence of equilibrium state at small scale

(Universal) Local equilibrium state a la K41,disturbance can be treated as perturbation to theinherent equilibrium state determined by the NS-dynamics

and see

2) the response small disturbancedisturbances tested:

Mean Shear, Stratification, MHD

(others, e.g., scalar field under mean scalar gradient.

But3) The anisotropy may remain large even at small scales,

if Re is not high enough, so that the inertial subrange is not wide.


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Summary -2

Equilibrium state

?

Disturbance; X

Response: J

Mean shearStratification

MHD

Jij= Qij= Cijkm Xkm


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References

Ishida T. and Kaneda Y., (2007)

Small-scale anisotropy in MHD turbulence under a strong uniform magnetic field,

Physics of Fluids, Vol.19, No. 7, 2007/07, Art. No. 075104-1_10

Kaneda Y. and Ishihara T., (2006)

High-resolution direct numerical simulation of turbulence,Journal of Turbulence, 7 (20): 1-17.

Kaneda Y. and Yoshida K., (2004)

Small-scale anisotropy in stably stratified turbulence,

New Journal of Physics, Vol.6, Page/Article No.34-1_16.

Yoshida K., Ishihara T. and Kaneda Y., (2003)

Anisotropic spectrum of homogeneous turbulent shear flow in a Lagrangian renormalized

approximation, Physics of Fluids, Vol.15, No. 8, pp.2385-2397.

Ishihara T., Yoshida K. and Kaneda Y., (2002)Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow,

Physical Review Letters, Vol. 88, No. 15, Art. No. 154501-1_4.


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y. kaneda- small-scale anisotropy in high reynolds number turbulence: universality of the 2nd kind

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