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Slide of the Seminar
Measuring anisotropy and universality in turbulence using SO(3) decomposition
Prof. Kartik P. Iyer, Fabio Bonaccorso, Luca Biferale, Federico Toschi
ERC Advanced Grant (N. 339032) “NewTURB” (P.I. Prof. Luca Biferale)
Università degli Studi di Roma Tor Vergata C.F. n. 80213750583 – Partita IVA n. 02133971008 - Via della Ricerca Scientifica, 1 – 00133 ROMA
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Measuring anisotropy and universality in
turbulence using SO(3) decomposition
Kartik P. Iyer1, Fabio Bonaccorso1, Luca Biferale1,Federico Toschi2
1Department of Physics, University of Rome, Tor Vergata2Department of Applied Physics, University of Eindhoven
July 18, 2016
Supported by ERC Grant No 339032Supercomputing resources at CINECA
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source: Wikipedia
! Incompressible NSE invariant under Rotation + Translation
At large scales:
forcing, B.C
break rotation invariance
⎧
⎨
⎩
∂tv + v · ∂v = −∂p + ν∂2v + f
∂ · v = 0+ boundary conditions
! Is breaking of rotational symmetry passed down-scale ?
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source: Wikipedia
! Incompressible NSE invariant under Rotation + Translation
At large scales:
forcing, B.C
break rotation invariance
⎧
⎨
⎩
∂tv + v · ∂v = −∂p + ν∂2v + f
∂ · v = 0+ boundary conditions
! Is breaking of rotational symmetry passed down-scale ?
Universal signatures in small-scale fluctuations?
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Longitudinal structure function: S (n)(r) ≡!
"#
v(x+ r)− v(x)$
· r̂%n&
(0-rank tensor)
S (n)(r) =∞∑
j=0
m=j∑
m=−j
S(n)jm (r)Yjm(r̂) Arad et. al. PRL’98
Y00=1/√4π Y20=
!
516π (3cos2θ−1) Y22=
!
1516π (sin2 θ cos 2φ)
Y40 Y42 Y44
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rotational invariant operators✟✟✟✟✙❅❅❘
∂tS(2)(r) + Γ(3)S (3)(r)− 2ν∇2S (2)(r) = f (2)(r)
❄
r ≪ LfUniverality at small scales
∂tS (2)(r) + Γ(3)S (3)(r)− 2ν∇2S (2)(r) ∼ 0
❄
+ SO(3)
S(2)j (r) =
m=+j∑
m=−j
Yjm(r̂)
∫
S(2)jm (r)Yjm(r̂)d r̂
Weak Anisotropy
∂tS(2)j (r) + Γ(3)j S
(3)j (r)− 2ν∇2S
(2)j (r) ∼ 0 j = 0, 1, 2, . . .
FOLIATION
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10−4 10−2 10010−20
10−15
10−10
10−5
100
105
kη
E(k)
❆❆
❆❆❑
large scales:all sectors coupled byforcing/B.C
k−5/3(()(()❍❍❥❍❍❥
j=0
j=2
j=4
j=6
(1) Univerality: leading isotropic sector
(2) Foliation of j-sectors for k ≫ kF
(3) different physics in different sectors
(4) return-to-isotropy
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10−4 10−2 10010−20
10−15
10−10
10−5
100
105
kη
E(k)
❆❆
❆❆❑
large scales:all sectors coupled byforcing/B.C
k−5/3(()(()❍❍❥❍❍❥
j=0
j=2
j=4
j=6
(1) Univerality: leading isotropic sector
(2) Foliation of j-sectors for k ≫ kF
(3) different physics in different sectors
(4) return-to-isotropy
10−4 10−2 10010−20
10−15
10−10
10−5
100
105
kη
E(k)
❆❆
❆❆❑
large scales:all sectors coupled byforcing/B.C
k−5/3
j=6
j=0j=2
j=4
(1) NO Univerality:sub-leading isotropic sector
(2) NO Foliation:j-sectors coupled at k ≫ kF
(3) Isotropy not recovered at small-scales
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S (n)(r) =∞∑
j=0
m=j∑
m=−j
S(n)jm (r)Yjm(r̂)
Working Hypothesis
S(n)jm (r) = A
(n)jm
( r
L
)ςnj
! Projection on sector-j has universal scaling exponent ςnj ininertial range depending on that sector only
! Power law behavior only in each separated sector
! Prefactors depend on large scale physics
S (n)(r) ∼ A0( r
L
)ςn0 + A1( r
L
)ςn1 + A2( r
L
)ςn2 ++ . . .
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S (n)(r) =∞∑
j=0
m=j∑
m=−j
S(n)jm (r)Yjm(r̂)
S (n)(r) ∼ A0( r
L
)ςn0 + A1( r
L
)ςn1 + A2( r
L
)ςn2 ++ . . .
S (n)(L) ∼ A0 + A1 + A2 + . . .
Prefactors cannot be universal!
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S (n)(r) =∞∑
j=0
m=j∑
m=−j
S(n)jm (r)Yjm(r̂)
S (n)(r) ∼ A0( r
L
)ςn0 + A1( r
L
)ςn1 + A2( r
L
)ςn2 ++ . . .
S (n)(L) ∼ A0 + A1 + A2 + . . .
Prefactors cannot be universal!
Open Questions
! Are scaling exponents ςnj in j-sector m-independent ?
! Are scaling exponents ςnj universal ?
! Return-to-Isotropy ?
ςn0 ≤ ςn2 ≤ ςn4 ≤ . . .
! No rigorous inferences from NSE at least thus far . . .
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Random Kolmogorov Flow (Biferale, Toschi PRL’01)
! RKF is stationary, homogeneous on average and anisotropic
t/TE=1 t/TE=3
10243,Rλ = 280
! Anisotropic forcing:
fi (k1,2) = δi ,2f1,2(t)eiθ1,2(t) , k1 = (1, 0, 0), k2 = (2, 0, 0)
! fi (t) random time-varying amplitudes
! θi (t) random time-varying phases
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RKF: test-bed for anisotropic turbulence
! Reynolds Stress Tensor bij ≡ ⟨uiuj⟩/⟨ukuk⟩ − δij/3
! I2 ≡ b2ii/6 I3 ≡ (bijbjkbki )1/3/6
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
6 8 10 12 14 16 18t/TE
I2(t)
I3(t)
20483
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Lumley Triangle
0
0.05
0.1
0.15
0.2
0.25
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25I3
I2
20483
I2 = ( 127 + 2I 33 )
1/2
Isotropic✏✏✏✮
I3 > 0
I3 < 0
! Different large scale configurations (I2-I3) possible in RKF
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Isotropic sector vs undecomposed structure functionS (n)(r) = A0
( r
L
)ςn0 + A1( r
L
)ςn1 + . . .
Isotropy in Inertial Range (η ≪ r ≪ L) S (3)(r) = −4
5⟨ϵ⟩r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
100 101 102 103
20483,Rλ = 450
iso sector: A0
(
rL
)ςn0
X−dir✛
Y−dir✛
Z−dir✛
−S (3)(r)/r⟨ϵ⟩
r/η
IR
Avg(X/Y/Z dirs)((()
before SO(3)
decomposition
✟✟✯✁✁✁✁✁✕
✂✂✂✂✂✂✂✂✍
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Local slopes: Isotropic sector vs Cartesian dirsς(n)0 (r) = d [log(Sn
0 (r))]/
d [log r ]
0 0.5
1 1.5
2 2.5
3 3.5
4
100 101 102 103
0.5 0.6 0.7 0.8 0.9
1 1.1 1.2 1.3 1.4
102
r/η
20483,Rλ = 450
ς404/3
2/3
✛
ς4X✛
ς4Y✛ς4Z✟✟✯
IR
! Power law scaling in isotropic sectors! Intermittency exponent - contaminated by anisotropy
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Universality of scaling exponent in isotropic sector
S (n)(r) = A0( r
L
)ς(n)0 + A1
( r
L
)ς(1)n + . . . =⇒ Is ς(n)0 universal?
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3
I3 > 0I3 < 0
20483,Rλ = 450
log(r)
ς(4)0 /ς(2)0
ς(6)0 /ς(2)0
ς(8)0 /ς(2)0
IR
! Different I2-I3 confs in Lumley triangle have similar InertialRange scaling at least up to order 8
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Ansatz for projection coeffs: Snjm(r) ∼ An
jmrςnj
10-810-710-610-510-410-310-210-1100
100 101 102 103
20483,Rλ = 450
|S300|
|S320|
|S344|
❄
|S340|❅
❅❅■
|S366|
r/η
IR
ς300 = 0.97 (1.00)
ς320 = 1.67 (1.67)
ς340 = 1.76 (2.33)
ς344 = 1.76 (2.33)
ς366 = 2.49 (3.00)
Dimensional scaling exponent (Biferale et. al. PRE’02): ςnj = (j + n)/3❅❅❘
! Scaling exponents m-independent: ςnjm = ςnj
! ςn0 < ςn20 < ςn40 < ςn66 ⇒ (r/L)ςn0 > (r/L)ς
njm for r/L ≪ 1
! Isotropic sector dominant. Anisotropic part: sub-leading
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Scaling in anisotropic sectors
10-910-810-710-610-510-410-310-210-1100101
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
20483,Rλ = 450
S440(r)
S441(r)
S442(r)
slope = 1
❄
! Scaling of Snjm(r) within same j-sector is m-independent
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Univerality of scaling exponents in anisotropic sectors
10-810-710-610-510-410-310-210-1100101102
10-810-710-610-510-410-310-210-1 100 101 102
20483,Rλ = 450
|S444(r)|
∣
∣
I3>0
|S444(r)|
∣
∣
I3<0
! j-sectors of different large-scale configurations scale similarlyin inertial range
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Conclusions
! SO(3) decomposition- useful tool for decomposing differentsectors, for a systematic investigation of anisotropic effects
! Isotropic sector: power law behavior
! Scaling in anisotropic sectors (j ,m) is intermittent andm-independent
! Scaling exponents in given j-sector might be universal