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

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

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 ?

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?

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

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

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=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

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 ++ . . .

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!

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 . . .

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

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

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

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

✟✟✯✁✁✁✁✁✕

✂✂✂✂✂✂✂✂✍

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

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

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

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

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

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