Smooth, rough, broken: From Lyapunov exponents and zero modes to caustics in the description of iner

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Smooth, rough, brokenFrom Lyapunov exponents
and zero modes to caustics in the description of
inertial particles.

• G. Falkovich
• Leiden, August 2006

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Smooth flow
1d
H is convex
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Multi-dimensional
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? singular (fractal) SRB Measure
entropy
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Coarse-grained density
An anomalous scaling corresponds to slower
divergence of particles to get more
weight. Statistical integrals of motion (zero
modes) of the backward-in-time evolution
compensate the increase in the distances by the
concentration decrease inside the volume. Bec,
Gawedzki, Horvai, Fouxon
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Inertial particles
u
v
Maxey
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Spatially smooth flow
One-dimensional model
Equivalent in 1d to Anderson localization
localization lengthLyapunov exponent
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Velocity gradient
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Fouxon, Stepanov, GF
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Lyapunov exponent
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Gawedzki, Turitsyn and GF.
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Statistics of inter-particle distance in 1d
high-order moments correspond effectively to
large Stokes
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Continuous flow
Piterbarg, Turitsyn, Derevyanko, Pumir, GF
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Derevyanko
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2d short-correlated
Baxendale and Harris, Chertkov, Kolokolov,
Vergassola, Piterbarg, Mehlig and Wilkinson
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Coarse-grained density
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-2
n
Falkovich, Lukaschuk, Denissenko
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3d
Short-correlated flow
Duncan, Mehlig, Ostlund, Wilkinson
Finite-correlated flow
Bec, Biferale, Boffetta, Cencini, Musacchio,
Toschi
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Clustering versus mixing in the inertial interval
Balkovsky, Fouxon, Stepanov, GF, Horvai, Bec
Cencini, Hillerbrand
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Fouxon, Horvai
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Fluid velocity roughness decreases clustering of
particles
Pdf of velocity difference has a power tail
Bec, Cencini, Hillerbrand
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Collision rate
Sundaram, Collins Balkovsky, Fouxon, GF
Fouxon, Stepanov, GF
Bezugly, Mehlig and Wilkinson
Pumir, GF
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Main open problems
1. To understand relations between the
Lagrangian and Eulerian descriptions. 2. To
sort out two contributions into different
quantities i) from a smooth dynamics and
multi-fractal spatial distribution, and ii) from
explosive dynamics and caustics. 3. Find how
collision rate and density statistics depend on
the dimensionless parameters (Reynolds, Stokes
and Froude numbers).