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Inertial particles in selfsimilar random flows
Jérémie Bec
CNRS, Observatoire de la Côte d’Azur, Nice
Massimo Cencini
Rafaela Hillerbrand
Rain initiation
• Warm clouds
1 raindrop = 109 droplets
Growth by continued condensation way
= too slow
• Collision/Coalescence:
Polydisperse suspensions with a wide
range of droplet sizes with different
velocities
Larger, faster droplets overtake
smaller ones and collide  Droplet
growth by coalescence
Formation of the solar system
• Protoplanetary disk after the collapse of a
nebula
Migration of dust toward the equatorial plane of
the star
(II) Accretion 109 planetesimals
from 100m to few km
(III) Merger and growth
 planetary embryos  planets
(I)
From Bracco et al. (Phys. Fluids
• Problem =
time
scales ?
Very heavy particles
• Impurities with size
(Kolmogorov scale)
and with mass density
viscous
drag
with
• Passive suspensions: no feedback of the
particles onto the fluid flow (e.g. very dilute
suspensions)
• Stokes number: ratio between response time
Clustering of inertial particles
• Different mechanisms involved in clustering:
 Delay on the flow dynamics (smoothing)
 Ejection from eddies by centrifugal forces
Dissipative dynamics due to Stokes drag
• Idea: find models to disentangle these effects
in order to understand their signature on the
spatial distribution and dynamical properties
of particles.
Fluid flow = Kraichnan
• Gaussian carrier flow with no time correlation
Incompressible, homogeneous, isotropic
= Hölder exponent of the flow
• -correlation in time  no structure, no
sweeping
• Relevant when
(Fouxon-Horvai)
Reduced dynamics
• Two-point motion can be written as a system of
SDE with additive noise
(smooth case: Piterbarg
2D, Wilkinson-Mehlig 3D)
+ Time
2D:
+ Boundary conditions on
Large-scale Stokes number:
and
Phenomenology of the dynamics
•
stable fixed line
• Close to this line, noise dominates 
and
behave as two independent Ornstein–
Uhlenbeck processes
• Far away, the quadratic terms dominate
and trajectories perform loops from
to
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Phenomenology of the dynamics
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The loops play a fundamental role:
• Flux of probability from
to
that
• Events during which (and hence
becomes very small
, so
)
Smooth case
• Single dimensionless parameter: Stokes
number
• Exponential separation of the particles
Rough case
• For
, the dynamics can be
rescaled and depends only on a local Stokes
number
[Falkovich et al.]
• If we drop the boundary condition, the only
lengthscale is the initial value of
. The interparticle separation is given by
Correlation dimension
• Behaviour of
when
• Fractal mass distribution:
• Smooth case:
both when
and when
• Rough case: scale-dependent Stokes number
when
and thus
Information on clustering is given by the local
correlation dimension:
expected to depend only upon
and
Numerics
Local correlation
dimension
different
colours =
different
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Local Stokes
number
• Same qualitative picture reproduced for
different values of
Velocity differences
• Typical velocity difference between particles
separated by Important for applications
(approaching rate + multiphasic models)
small-scale behaviour:
Hölder exponent for the “particle velocity
field”
• Smooth case: function of the Stokes number
• Rough case:
(infinite inertia at small
scales)
Relevant information contained in the “finite
Numerics
Local Hölder exponent of the
particle velocity
Fluid
tracers
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Local Stokes
number
Free
particles
Large Stokes number behaviour
• Relevant asymptotics for smooth flows
+ gives the small-scale behaviour in the rough
case
• Idea: [Horvai]
with
fixed
Any statistical quantity should depend only on
in this limit but depends also only on
for
the original system
Example: 1st Lyapunov exponent in the smooth
Large Stokes - smooth flows
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• Same argument
applies to the large
deviations of the
stretching rate
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Statistics of velocity differences
• PDFs of velocity differences
also rescale at large Stokes numbers:
Powerlaw
tails
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Power law tails
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Tails related to large loops
• Cumulative probability
• Simplification of the dynamics: noise + loops
•
Prob to enter a
sufficiently
large loop
Fraction of time
spent at
• 1st contribution:
should be sufficiently
small to initiate a large loop
Radius estimated by
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Prediction for the exponent
• 2nd contribution:
Approximation of the dynamics by the deterministic
drift
Fraction of time spent at
is
– confirmed by
•
numerics
• Power law with same
exponent at large positive
and
Smooth case
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• Clustering weakens when
Open questions
More on Kraichnan flows:
Move to mass dynamics instead of two-point
motion. Does this model catch the formation of
voids in the particle distribution?
Understanding of the dynamical flow
singularity at
and
 Questions related to the uniqueness of
trajectories
Different from tracers: breaking of Lipschitz continuity is
“2nd order”
Add compressibility: what are the different
Open questions
Toward realistic flows:
Does large-Stokes rescaling apply in turbulent
flows?
 Important for planet formation (density ratio
)
Measure of relative velocity PDFs in real
flows: are the algebraic tails also present?
Effect of time correlation?
Problems =
• Rescaling with the turnover time is wrong