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L4: The Navier-Stokes equations III: Turbulence and Non- Newtonian

L4: The Navier-Stokes equations III: Turbulence and Non- Newtonian. Prof. Sauro Succi. Turbulence. Turbulence modeling. Effects of small (unresolved) scales o n large (resolved) ones. Energy Cascade. Turbulent energy spectrum: broad and gapless!. Turbulence. Kolmogorov length.

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L4: The Navier-Stokes equations III: Turbulence and Non- Newtonian

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  1. L4: The Navier-Stokesequations III:Turbulence and Non-Newtonian Prof. Sauro Succi

  2. Turbulence

  3. Turbulence modeling Effects of small (unresolved) scales on large (resolved) ones

  4. Energy Cascade

  5. Turbulent energy spectrum: broad and gapless!

  6. Turbulence • Kolmogorovlength

  7. Turbulence • Kolmogorovlength • Faucet, Re=10^4, DOF=10^9,Work=10^12 • Car , Re=10^6, DOF=10^14, • Geo , Re=10^9, DOF=10^20 • Astro , Re=10^10,DOF=10^22,Work=10^30

  8. Whyis Reynolds so large?

  9. Transition to turbuence

  10. Small and Large Eddies

  11. Turbulence: NO scale separation Small eddies are swept away by large eddies (Advection) Large eddies experience random collisions from small ones(Diffusion) Brownian motion? NO! Advection/Diffusion is scale-dependent Non-gaussian fluctuations, intermittency,bursts, rare events Dissipative: No Hamiltonian, no standard statistical ensembles

  12. TurbulenceCost Memory CPU

  13. Modeling vs Simulation All-sim’s Eddy size Theory/Model Compute Approaches: Direct Numerical Simulation (DNS) All significantly excited scales of motion are computed - WORK = O(R3) Large Eddy Simulation (LES) D(grid size) All eddies larger than grid size are computed Least-computing Multiscale Very Large Eddy Simulation (VLES) Dissipative eddies Inertial range eddies Anisotropic eddies Only statistically anisotropic eddies outside the Kolmogorov range are computed Reynolds Averaged Navier-Stokes (RANS) All CR’s All scales of motion are described by semi-empirical models Principle of Least-Computing!

  14. ComplexFluids

  15. Beyond NSE Strong gradients: molecular details Small volumes, large S/V: molecular Internal structure: complex rheology

  16. Non-NewtonianFluids Internal structure: complexrheology Local, Non-linear Non-local Tensor .....

  17. Hydrophobicity: slip flow

  18. Constitutive Relation Constitutive: sigma=A+B*S^n Newton: A=0,n=1 Yield-Stress A>0 n>1 shear-thickening (paints) n<1 shear-thinning (blood,ketch-up…)

  19. BoundaryConditions Periodic: (Free-flows) Non-slip: zero velocity (Solid walls) Prescribed pressure/density, Zero velocity: (Open flows) MovingBoundaries (Pistons, bioflows..)

  20. End of Lecture 3

  21. Multiscale allies: Universality & Forgiveness Large Kn allow large Dxand dt

  22. From Boltzmann to Navier-Stokes: weak non-equilibrium Weak departure from local equilibrium (herd effect) T n=n(r,t) u=u(r,t) T=T(r,t) Order params:

  23. The evershifting battle: stream and collide Macro field

  24. Macroscopic persistence: the coherence length Below l_c microphysics takes over Coupling strength a>1/2 a=1/2 a=3/4 weak-> strong

  25. How big is g? Turbulence (Turbulence) (Compressibility) Reynolds ~ Length/molecular mean free path!

  26. Bernouilli

  27. Clebschrepresentation

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