Smooth, rough, broken:

From Lyapunov exponents and zero modes to

caustics in the description of inertial particles.

G. Falkovich

Leiden, August 2006

Smooth flow

1d

dze zNSzX )(

0)0( H

H is convex

Multi-dimensional

entropy

→ singular (fractal) SRB Measure )exp( tn mm

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

Coarse-grained density

u

v

Maxey

Inertial particles

Spatially smooth flow

One-dimensional model

Equivalent in 1d to Anderson localization: localization length=Lyapunov exponent

Velocity gradient

Fouxon, Stepanov, GF

Lyapunov exponent

Gawedzki, Turitsyn and GF.

dnndtdnnPtttn 20 )()/(1)(

Statistics of inter-particle distance in 1d

high-order moments correspond effectively to large Stokes

Rdt

dR

Piterbarg, Turitsyn, Derevyanko, Pumir, GF

||/1 RnContinuous flow

?|| mm Rn

Derevyanko

2d short-correlated

Baxendale and Harris, Chertkov, Kolokolov ,Vergassola, Piterbarg, Mehlig and Wilkinson

Coarse-grained density:

Falkovich, Lukaschuk, Denissenko

n-2

3d

Bec, Biferale, Boffetta, Cencini, Musacchio, Toschi

Finite-correlated flow

Short-correlated flow

Duncan, Mehlig, Ostlund, Wilkinson

Clustering versus mixing in the inertial interval:

Balkovsky, Fouxon, Stepanov, GF, Horvai, Bec Cencini, Hillerbrand

2/)1( ruUvdiv r

Fouxon, Horvai

Fluid velocity roughness decreases clustering of particles

Pdf of velocity difference has a power tail

Bec, Cencini, Hillerbrand

Collision rate

Fouxon, Stepanov, GF

Bezugly, Mehlig and Wilkinson

Pumir, GF

Sundaram, Collins; Balkovsky, Fouxon, GF

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).

Main open problems