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Lagrangian Phenomenology of Turbulence. Misha Chertkov (CNLS, Los Alamos) Alain Pumir (CNRS, Nice) Boris Shraiman (Lucent). Phys. Fluids 11, 2394 (1999) PRL 85, 5324 (2000) + Euro.Phys.Lett. in press. integral (pumping) scale. cascade. dissipation scale.

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slide1

Lagrangian Phenomenology of Turbulence

Misha Chertkov (CNLS, Los Alamos)

Alain Pumir (CNRS, Nice)

Boris Shraiman (Lucent)

Phys. Fluids 11, 2394 (1999)

PRL 85, 5324 (2000)

+ Euro.Phys.Lett. in press

slide2

integral

(pumping)

scale

cascade

dissipation

scale

Field formulation

(Eulerian)

Particles(“QM”)

(Lagrangian)

Passive scalar turbulence

slide3

cascade

viscous

(Kolmogorov)

scale

integral

(pumping)

scale

No exact reduction

Field formulation

(Eulerian)

Fluid blob(“QM”)

(Lagrangian)

Approximate ???

  • The idea :
  • To construct PhenomenologicalLagrangian Model ofNS
  • To verify the model against Direct Numerical Simulations of NS
  • proper variables
  • satisfy known facts

Navier-Stokes Turbulence

slide4

velocity gradient tensor

coarse-grained over the blob

tensor of inertia of the blob

“Particle’s” objects

slide5

Kolmogorov 4/5 law

  • Richardson law
  • Intermittency
  • rare events
  • more (structures)

Fundamentals ofNS turbulence

slide6

Less known facts

Isotropic, local

(Draconian appr.)

Restricted Euler equation

Viellefosse ‘84

Leorat ‘75

Cantwell ‘92,’93

slide7

Still

  • Finite time singularity (unbounded energy)
  • No structures (geometry)
  • No statistics

Restricted Euler. Partial validation.

  • DNS on statistics of vorticity/strain alignment is compatible
  • with RE**Ashurst et all ‘87
  • DNS for PDF in Q-R variables respect the RE assymetry
  • ** Cantwell ‘92,’93; Borue & Orszag ‘98
  • DNS for Lagrangian average flow resembles
  • the Q-R Viellefosse phase portrait **
slide8

To count for concomitant evolution of and !!

How to fix deterministic blob dynamics?

  • Energy is bounded
  • No finite time sing.

* Exact solution of Euler in the domain bounded by perfectly elliptic

isosurface of pressure

slide9

self-advection

small scalepressure and

velocity fluctuations

coherent

stretching

Stochastic minimal model

+ assumption:velocity statistics

is close to Gaussian

at the integral scale

Where is statistics ?

Verify against DNS

slide12

DNS

Model

Energy flux

slide16

Conclusions

  • Lagrangian approach borrowed from the PS-studies is co-intuitive
  • with Kolmogorovcascade ideas and is applicable to NS
  • Vieillefosse (R,Q) plane densities of the 2nd and 3rd invariants are
  • illuminating in detecting the role of vorticity and strain intermittency
  • The tetrad model is surprisingly reach in physics offering an insight
  • both into geometry and dynamics, statistics and scaling of variety of
  • the internal range fields
  • It is suggestive to study tetrad statistics (four-points Lagrangian
  • measurements) experimentally
  • Large Eddy Simulations may be build on the basis of the model
  • Gap between the phenomenological modeling (particles) and
  • the original microscopic problem (fields) is well contoured.
  • All the theoretical reserves (1/d, instanton, operator product
  • expansion) should be called to bridge it.