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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
From Kaneda & Yoshida (2004)
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From Kaneda & Yoshida (2004)
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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
From Ishida & Kaneda (2007)
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
From Ishida & Kaneda (2007)
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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.
From Ishida & Kaneda (2007)
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Results in our DNS
approaches to isotropic state. c(k)
Anisotropy increases.
From Ishida & Kaneda (2007)
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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
From Ishida & Kaneda (2007)
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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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The end