Multifractal acceleration statistics in turbulence Benjamin Devenish Met Office, University of Rome...

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

57.400

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