Frontier Science Pavia, 2003

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Frontier SciencePavia, 2003
Luca Biferale Dept. of Physics, University of
Tor Vergata, Rome. INFN-INFM biferale_at_roma2.infn.i
t
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NAVIER-STOKES eqs.
boundary conditions
Shear Flow Pseudo Spectral Code
3d Convection Lattice Boltzmann Sim. APE
Turbulent jet (exp)
Mathematicians Existence/Uniqueness of weak
solutions. Physicists understanding statistical
properties and universality. Engineers
controlling small-scales parametrization.
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Physical Complexity
Reynolds number (Non-Linear)/(Linear terms)

• Fully Developed Turbulence
• Strongly out-of-equilibrium non-perturbative
  system

Many-body problem

• Power laws

Energy spectrum

• Small-scales PDFs strongly non-Gaussian

acceleration
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Numbers.
Astrophys. flow
atmosph. flow
laboratory flow
state-of-the-art DNS
Isotropic, homogeneous Fully Periodic
Flows Pseudo-Spectral Methods. Resolution
4096x4096x4096 (Earth Sim.)
Reynolds 106, Storage of 1 velocity
configuration (float) 1 Tbyte RAM requirements
for time marching 10 Tbyte
Moral easy to saturate any computing
power (present and/or future)
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NS eqs.
Multi-scale correlation functions
infinite hierarchy
Integro-differential operator of d(n-1) variables
Anomalous scaling
Non-gaussian small-scales PDFs
Universality robustness against changing of
forcing and/or b.c.
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Mean-field scaling
Local slope of 6th order structure function in
the isotropic sector, at changing Reynolds and
large scale set-up.
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Turbulent Transport
passive scalar field
Temperature fluctuations
Kraichnan Model
Stochastic velocity field Gaussian and
white-in-time
correlation function
Elliptic differential operator of d(n-1) variables
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If stationary

• Universality zero-mode is a property of the LHS
  of (1)
• Anomalous behaviour zero-modes do not feel any
  dimensioanl constraints

NS eqs
Integro-differential operator of d(n-1) variables
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conclusions
How to prove universality of anomalous
scaling for velocity correlation
functions? Nobel Prize Problem Calculate
anomalous exponents Temporal
properties? Statistical properties of particle
velocites?
Chernobyl cloud