Multifractal acceleration statistics in turbulence

Benjamin Devenish

Met Office, University of Rome

L. Biferale, G. Boffetta, A. Celani, A.Lanotte, F. Toschi

Intermittency

K41

• Kolmogorov (1941)• Landau’s objection

- fluctuating energy dissipation

• Kolmogorov’s refined hypothesis (1962)

• Random beta model• Multifractal model

Eulerian velocity structure function

Multifractal formalism (1)

Frisch (1995)

Multifractal formalism (2)

• Eulerian reference frame

• Lagrangian reference frame

- time scale of eddy of scale r

- velocity difference at scale r

vur

Fractal dimension

Superposition of contributions from eddies of all sizes

Contributions from eddies smaller than scale r are uncorrelated

Fluctuatingur r /

h

r L

ruu

00 0Lr )(hD

),( maxmin hhhUniversal

Acceleration in multifractal framework

• Acceleration

• Fluctuating Kolmogorov scale

Acceleration pdf (1)

• Pdf of acceleration

• Probabililty of observing h in

• Pdf of large scale velocity 0v

0000 )()|(),|( dhdvvvhvha PPP

Acceleration pdf (2)

• Multifractal

• K41 (h=1/3, D(h) = 3)

dhRaRaaP hzhhyhDh

)(3/)1(2)(3/))(5( ~2

1exp~)~(

Ra 2aaa /~ 14.1

No additional free parameters D(h) derived from She-Leveque model

Direct numerical simulation

• Homogeneous isotropic turbulence• cubic lattice• Taylor-scale Reynolds number • Two million Lagrangian particles• Sampling rate 07.0

31024

280R

Acceleration pdf (3)

K41prediction

Multifractal prediction

Conditional acceleration variance

Acceleration variance conditional on velocity

Lagrangian stochastic models

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Conditional acceleration variance

Multifractal prediction

B.L. Sawford et al., Phys. Fluids 15, 3478 (2003).

Lagrangian structure functions

)( p

L

L

T

Multifractal prediction for the Lagrangian structure functions

whereSame D(h) as in Eulerian model

LT

Lagrangian structure functions

P=8

P=6

P=4

Plotted using Extended Self Similarity

Bottleneck at

2.72

2.16

1.71

Multifractal Lagrangianexponents

Conclusions

• Acceleration exhibits fluctuations up to • Multifractal formalism captures this behaviour

with no additional free parameters• Conditional acceleration variance• Velocity structure functions only for • Read more in Physical Review Letters vol. 93,

no.6 p.064502-1

a80

10t

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