Small-Scale Anisotropy in High Reynolds Number Turbulence

-- Universality of the 2nd Kind --

Y. KanedaNagoya University

August 23rd, 2007, Summer School at Cargèse,Small-Scale Turbulence : Theory, Phenomenology and Applications

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

Analogy with Statistical 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

Equilibrium state

at thermal equilibrium state

disturbance

response

Universality in Response to disturbances, near equilibrium state

1905, Einstein, D=μkT,the first example of FD-elation Perrin’s experiment.

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

1931, Onsager’s reciprocal theorem:

J = C X, C=TC

generalized flux, generalized force

J = CX

Generalized Flux vs. Generalized Force

e.g.Density Flux vs. Density gradient ； J = C grad ρ

Heat Flux vs. Temperature gradient ； J = C grad T

Electric Current vs. External electric field ; J = σ E = σ grad φ , (I=E/R, Ohm’s law)

---

Momentum Flux vs. Strain rate ;

coupling only between tensors of the same order.

(Newton’s law)

Thermal Equilibrium system

Linear response

Equilibrium state

Disturbance

Xgrad φgrad Tgrad c

Xgrad φgrad Tgrad c

F=CX (Hooke’s law)J = C’ grad φ= σE (Ohm’s law)J = C’’ grad T (Fourier’s law)J = C’’’ grad c (Fick’s law)

F=CX (Hooke’s law)J = C’ grad φ= σE (Ohm’s law)J = C’’ grad T (Fourier’s law)J = C’’’ grad c (Fick’s law)

History:Universality in Response

to disturbances, near equilibrium state

1905, Einstein, D=μkT,the first example of FD-elation Perrin’s experiment.

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

1931, Onsager’s reciprocal theorem:J = C X, C=TC

generalized flux, generalized force

1950-60, Nakano, KuboLinear Response Theory

Equilibrium State of Turbulence

Equilibrium state

?

disturbance

response

Turbulence ?

DNS

Series1Series1 Series2Series2

Energy Spectrum a la Kolmogorov (K41)

From Kaneda &Ishihara (2006)

Equilibrium state

a la Kolmogorov K41

disturbance

response

Disturbance

1. Mean Shear2. Buoyancy by Stratification3. Magneto-Hydrodynamic Force

I. Mean Shear

See Ishihara et al. (2002)

Local co-ordinateNS-equation

Effect of mean shear

Turbulent Shear Flow

Local strain rate of mean flow

Effect of Mean FlowHomogenous Mean Shear Flow:

• in the inertial subrange of homogeneous turbulent shear flow;

i ij jU S x=

( ) ( ) ( )ij i jQ u u= −k k k = ?

Let δ(k) =τk/T, (where τk ＝１/(ε 1/3 k 2/3), T=1/S)

Assume (1) δ <<1, for large enough k(2) expand in powers of δ

k=1/l

Two kind of measures characterizing the universal equilibrium state

0( ) ( ) ( )ij ij ijQ Q C Sαβ αβ= +k k k<ui(k)uj(-k)> =

1) Equilibrium state itself

2) Response to disturbanceSimilar to stress vs. rate of strain relation:

τ ij = Cijαβ Sαβ(justified by the Linear response theory for non-equilibrium system)

Not only

but also

(0) 2 /3 11/3( ) ( )4

Kij ij

CQ k Pεπ

−=k k

ˆ ˆ ˆ( ) , /ij ij i j i iP kk k k kδ= − =k

Anisotropic part

ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ij i j i j ijC a k P P P P b k P k kαβ α β β α α β⎡ ⎤= + +⎣ ⎦k k k k k k1/3 13/3 1/3 13/3( ) , ( )a k A k b k B kε ε− −= =

Only 2 (universal) parameters, A and B

Is this correct? What are the values of A and B?

0( ) ( ) ( )ij ij ijQ Q C Sαβ αβ= +k k k<ui(k)uj(-k)> =

at small scale of turbulent shear flowKolmogorov’s isotropic equilibrium spectrum

response

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

Method of numerical experiment

max 1k η =

Initial condition: isotropic turbulence

max 241k =

DNS of homogeneous turbulence with the simple shear flow

2

00

Sx⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

U

Resolution: 5123

ˆ ˆ( ) ( ), ( ) ( )abij ij ij a b ij

p k p kE k Q E k p p Q

= =

= =∑ ∑p pObserve

especially，12 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 =

1212 12

4 1 4( ) (7 ) , ( ), ( ) ( ) ,15 2 15iiE k A B E k E k A Bπ πξ ξ= − = − + L 1/3 7 /3k Sξ ε −=

Anisotropic Energy spectrum of homogeneous turbulent shear flow

ˆ ˆ( ) ( ) ( )abij a b i j

p kE k p p u u

=

= −∑ p p

( ) ( ) ( )ij i jp k

E k u u=

= −∑ p p

124( ) (7 )15

E k A Bπ ξ= −

121 4( ) ( )2 15iiE k A Bπ ξ= − +

12 ( )E k

121 ( )2 iiE k−

1 ( )2 iiE k

2

00

Sx⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

UMean Flow

3312

4( ) (23 ) ,105

E k A Bπ ξ= − L

11 2212 12

4( ) ( ) (13 3 )105

E k E k A Bπ ξ= = −

1/3 7 /3S kξ ε −=

Theoretical predictions:

where

Ishihara, et al. (2002)

Consistency11 22 3312 12 12( ), ( ), ( )E k E k E k− − − 12 12 12

11 22 33( ), ( ), ( )E k E k E k− − −

3312

4( ) (23 )105

E k A Bπ ξ= −

11 2212 12

4( ) ( ) (13 3 )105

E k E k A Bπ ξ= = − 12 1211 22

16( ) ( ) ( 2 )105

E k E k A Bπ ξ= = − +

1233

8( ) ( 3 )105

E k A Bπ ξ= +

Theoretical predictions:

1.02.0

SSt

==

1.02.0

SSt

==

Ishihara et al.(2002)

See also Yoshida et al. (2003)

II. Stratification

See Kaneda & Yoshida (2004)

Boussinesq approximation

Buoyancy by Stratification

Kolmogorov scaling:

Boussinesq Turbulence Scaling

From Kaneda & Yoshida (2004)

Angular dependence

From Kaneda & Yoshida (2004)

From Kaneda & Yoshida (2004)

III. MHD

See Ishida & Kaneda (2007)

MHD approximation

quasi-static approximation

＋O(b2)

Magnetic force

Local co-ordinateNS-equation

Effect of mean shear

Turbulent Shear Flow

Local strain rate of mean flow

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”

response of isotropic turbulence

3 (universal) parameters in the inertial subrange

anisotropic parts

• To detect , define

< f (k) >k= k

Theoretical prediction (1)• Angular dependence

Theoretical prediction (2)• Scale dependence

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

Comparison between theory and DNS

• Angular dependence– Theory

– DNS

– for the DNS• Every ３°• 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)

Comparison between theory and the DNS

• Angular dependence– Theory

– DNS

From Ishida & Kaneda (2007)

Comparison between theory and DNS

• Scale dependence

Theory

DNS-5/3

-7/3

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

From Ishida & Kaneda (2007)

anisotropy remains at small scale.

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

Results in our DNS

approaches to isotropic state. （c(k)→１）

Anisotropy increases.

From Ishida & Kaneda (2007)

Results in our DNS

approaches to isotropic state. （c(k)→１）

Anisotropy increases.

From Ishida & Kaneda (2007)

• Anisotropy : c(k)– DNS

– Our theory

Comparison between theory and the DNS

T=TM=16 magnetic field OFF T=TF=32 magnetic field ON

Comparison with the DNS and the theory

≒0.1～0.2

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

From Ishida & Kaneda (2007)

Summary -1For turbulence,We don’t know the pdf nor Hamiltonian in contrast to thermal equilibrium state

But we may assume 1) 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.

Summary -2

Equilibrium state

?

Disturbance; X

Response: J

Mean shearStratificationMHD

Jij= Δ Qij= Cijkm Xkm

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.

The end