Mixing and segregation in two-phase flows

Gregory FalkovichWeizmann Institute of Science, Israel

Turbulent Mixing and Beyond, ICTP 2007

Mixing versus segregation in terms of an infinitesimal element.Lyapunov exponents.

entropy

→ singular (fractal) SRB Measure

Density in random compressible flows

Analogy: statistical distribution in phase

spaces (Sinai-Ruelle-Bowen measures)

Balkovsky, Fouxon, GF, Gawedzki, Bec, Horvai

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 mass decrease inside the volume.

Coarse-grained density

uv

Inertial particles

Spatially smooth flow

Stokes number

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

One-dimensional model

Super-symmetry broken

Lyapunov exponent

DNS, Bec et al

Fouxon, Stepanov, GF

Falkovich, Lukaschuk, Denissenko, Nature 2005

n-2

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