R. Benzi 1

Anomalous scaling in turbulence.

Roberto Benzi

Dip. Di Fisica, Univ. Roma Tor Vergata

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Starting point/1

The statistical properties of turbulence is multifractal and D(h) isequivalent to the “partition function” of the system, i.e. theknowledge of D(h) is enough to predict the whole set of scalingproperties of turbulent flows

The multifractal “theory” of turbulence is based on theassumption that intermittency (i.e. strong fluctuations of energydissipation) does not destroy the scaling properties of turbulence.

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Starting point/2Experimental and numerical simulations show anomalousscaling in homegenous and isotropic turbulent flows.

All anomalous exponents can be computed in terms of D(h).

D(h) does not depends on Re (ultraviolet stability). DNSshows the correct anomalous scaling by using an effectiveviscosity νeff at scale ld:

D(h) does not depends on the viscous cutoff.

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Non isotropic turbulence: the case of shear flows

For shear flows there exists a characteristic scale

For scale smaller than LS, the flow is dominated by isotropicturbulence (i.e. the same D(h) of homogenous and isotropic turb,)(Biferale, Procaccia, L’vov et. al. 1996 2002)

For scale larger than LS, the flow is no longer rotationallyinvariant: intermittency increases. (RB, Gualtieri, Casciola, Piva,Toschi, Succi 1998, 2003)

For scale larger than LS, intermittency does not depend on theviscous cutoff ld as far as ld < LS (Gualtieri, Jacob, Casciola, Piva 2005)

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All our knowledge is based on experimental data andnumerical simulations.

Can we prove that there exists a D(h)?

Can we prove that there exists anomalous scalingwhose properties are ultraviolet stable?

If this is the case, we can reach the conclusion that theturbulence problem is no longer a problem.

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A simpler problem: the case of a passive scalar

The velocity field u is prescribed. We want to compute thestatistical properties of φ

The simplest possible case: the Kraichnan model

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The scaling properties of φ, i.e. the computation of Cn, can beobtained by solving the linear equations:

Inertial operator forcing

Solution:

(Falkovich, Gawedzki, Vergassola 2002)

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Example: simplified model of the simplest passive scalar problem.

Anomalous scaling

------ analytical prediction

++++ numerical simulations

(RB, Biferale, Wirth 1996)

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Let us consider a passive scalar without forcing (decay)

One can show that the zero modes satisfy the equation:

Statistical Preserved Structure

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A sketch of the proof:

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Beyond the Kraichnan model: computing the SPS for the general caseof a passive scalar advected by a turbulent flow

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Numerical proof of existence of SPS for the passive scalar

φ(t=0) is choosen in the inertial range and the simulation isstopped after the large scale characteristic time TL when finite sizeeffect becomes dominant

(Arad, Biferale,Celani, Procaccia,Vergassola 2001)

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The existence of statistical preserved structures tells us that thestatistical property of turbulence (for the passive scalar) isdominated by the inertial operator and are stable with respect tothe viscous cutoff.

Can we say something for the Navier-Stokes equations?

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Let us consider the following problem

Some properties

The passive vector must have the same anomalous scaling of theNavier Stokes equations.

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Direct numerical simulations of a passive vector.

NS fields

Passive vector

(Angheluta,Benzi, Biferale,Procaccia, Toschi,2006)

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The limit for λ→0 is not singular

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Statistical preserved structures for the non linear problem.

The passive vector shows SPS. We can compute the NSsolution (numerically) and use the NS correlation function forcomputing the SPS in the passive case.

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Numerical verification of the SPS for thepassive vector with the NS solution (n=2,4)

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Some speculations

Let us consider a turbulent flow u and a set of passive vector wn:

…..

For each n we can define Dn(h) and we can define thefunctional map:

Then we can think of the Navier Stokes (?) scaling as the fixedpoint, if it exists, of the above equation.

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Let us use shell models

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From b=0.8 for the velocity field to b=0.4

Starting point (n=0) with b=0.8 (strong intermittency)

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Conclusions

The notion of SPS can be used for the non linear problem.

The velocity correlation functions of turbulent flows are eigenfunction (in the SPS sense) of the inertial operator.

The velocity correlation functions of turbulent flows areanomalous and independent on the viscous cutoff.

Close to the end?