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

Prof. Sauro Succi

Turbulence

Turbulence modeling

Effects of small (unresolved) scaleson large (resolved) ones

Energy Cascade

Turbulent energy spectrum: broad and gapless!

Turbulence

• Kolmogorov length

Turbulence

• Kolmogorov length

Faucet, Re=10^4, DOF=10^9,Work=10^12 Car , Re=10^6, DOF=10^14,Geo , Re=10^9, DOF=10^20Astro , Re=10^10,DOF=10^22,Work=10^30

Why is Reynolds so large?

Transition to turbuence

Small and Large Eddies

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

Dissipative: No Hamiltonian, no standard statistical ensembles

Non-gaussian fluctuations, intermittency,bursts, rare events

Turbulence Cost

Memory CPU

Modeling vs Simulation

Eddy size

Direct Numerical Simulation (DNS)

All significantly excited scales of motion are computed - WORK = O(R3)

Reynolds Averaged Navier-Stokes (RANS)

All scales of motion are described by semi-empirical models

Large Eddy Simulation (LES)

D (grid size) All eddies larger than grid size are computed

Very Large Eddy Simulation (VLES)

Dissipative eddies Inertial range eddies Anisotropic eddiesOnly statistically anisotropic eddies outside the Kolmogorov range are computed

Theory/Model ComputeApproaches:

All CR’s

All-sim’s

Least-computing Multiscale

Principle of Least-Computing!

Complex Fluids

Beyond NSE

Strong gradients: molecular details

Small volumes, large S/V: molecular

Internal structure: complex rheology

Non-Newtonian Fluids

Internal structure: complex rheology

Local, Non-linearNon-localTensor.....

Hydrophobicity: slip flow

Constitutive Relation

Constitutive: sigma=A+B*S^n

Newton: A=0,n=1Yield-Stress A>0n>1 shear-thickening (paints)n<1 shear-thinning (blood,ketch-up…)

Boundary Conditions

Periodic: (Free-flows)

Non-slip: zero velocity (Solid walls)

Prescribed pressure/density,Zero velocity: (Open flows)

Moving Boundaries (Pistons, bioflows..)

End of Lecture 3

Multiscale allies: Universality & Forgiveness

Large Kn allow large Dx and dt

Weak departure from local equilibrium (herd effect)

From Boltzmann to Navier-Stokes: weak non-equilibrium

T

n=n(r,t) u=u(r,t)T=T(r,t)

Order params:

The evershifting battle: stream and collide

Macro field

Macroscopic persistence: the coherence length

a>1/2a=1/2a=3/4

Below l_c microphysics takes over

weak-> strong

Coupling strength

(Turbulence)

(Compressibility)

How big is g? Turbulence

Reynolds ~ Length/molecular mean free path!

Bernouilli

Clebsch representation