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Lagrangian Phenomenology of Turbulence

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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Lagrangian Phenomenology of Turbulence

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  1. 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

  2. integral (pumping) scale cascade dissipation scale Field formulation (Eulerian) Particles(“QM”) (Lagrangian) Passive scalar turbulence

  3. 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

  4. velocity gradient tensor coarse-grained over the blob tensor of inertia of the blob “Particle’s” objects

  5. Kolmogorov 4/5 law • Richardson law • Intermittency • rare events • more (structures) Fundamentals ofNS turbulence

  6. Less known facts Isotropic, local (Draconian appr.) Restricted Euler equation Viellefosse ‘84 Leorat ‘75 Cantwell ‘92,’93

  7. 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 **

  8. 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

  9. 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

  10. Enstrophy density DNS Model

  11. Enstrophy production DNS Model

  12. DNS Model Energy flux

  13. Statistical Geometry of the Flow

  14. Scaling of Energy Transfer

  15. Anomalous Scaling

  16. 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.

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