L4: The Navier-Stokes equations III: Turbulence and Non- Newtonian
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Transcript of L4: The Navier-Stokes equations III: Turbulence and Non- Newtonian
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L4: The Navier-Stokes equations III:Turbulence and Non-Newtonian
Prof. Sauro Succi
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Turbulence
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Turbulence modeling
Effects of small (unresolved) scaleson large (resolved) ones
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Energy Cascade
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Turbulent energy spectrum: broad and gapless!
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Turbulence
• Kolmogorov length
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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
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Why is Reynolds so large?
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Transition to turbuence
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Small and Large Eddies
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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
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Turbulence Cost
Memory CPU
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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!
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Complex Fluids
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Beyond NSE
Strong gradients: molecular details
Small volumes, large S/V: molecular
Internal structure: complex rheology
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Non-Newtonian Fluids
Internal structure: complex rheology
Local, Non-linearNon-localTensor.....
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Hydrophobicity: slip flow
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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…)
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Boundary Conditions
Periodic: (Free-flows)
Non-slip: zero velocity (Solid walls)
Prescribed pressure/density,Zero velocity: (Open flows)
Moving Boundaries (Pistons, bioflows..)
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End of Lecture 3
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Multiscale allies: Universality & Forgiveness
Large Kn allow large Dx and dt
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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:
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The evershifting battle: stream and collide
Macro field
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Macroscopic persistence: the coherence length
a>1/2a=1/2a=3/4
Below l_c microphysics takes over
weak-> strong
Coupling strength
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(Turbulence)
(Compressibility)
How big is g? Turbulence
Reynolds ~ Length/molecular mean free path!
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Bernouilli
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Clebsch representation