inertial particles in self- similar random flows jérémie bec cnrs, observatoire de la côte...

Inertial particles in self-similar random

flowsJérémie Bec

CNRS, Observatoire de la Côte d’Azur, Nice

Massimo CenciniRafaela Hillerbrand

Rain initiation•Warm clouds1 raindrop = 109 dropletsGrowth by continued condensation way= too slow

• Collision/Coalescence:Polydisperse suspensions with a wide range of droplet sizes with different velocitiesLarger, faster droplets overtake smaller ones and collide Droplet growth by coalescence

Formation of the solar system

• Protoplanetary disk after the collapse of a nebula

(I) 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

• Problem = time

scales ?From Bracco et al. (Phys. Fluids 1999)

Very heavy particles• Impurities with size (Kolmogorov scale) and with mass density

viscous drag

• Passive suspensions: no feedback of the particles onto the fluid flow (e.g. very dilute suspensions)

• Stokes number: ratio between response time and typical timescale of the flow (turbulence:

)

with

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 and Large-scale Stokes number:No dependence for smooth velocity fields ( )

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

Phenomenology of the dynamics

The loops play a fundamental role:• Flux of probability from to , so that

• Events during which (and hence ) becomes very small

• Prevent from vanishing• Probable mechanism ensuring mixing of the dynamics

QuickTime™ et undécompresseur

sont requis pour visionner cette image.

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 inter-particle 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



Numericsdifferent colours = different

• Same qualitative picture reproduced for different values of

• Roughness weakens the maximum of clustering

Local Stokes number

Local correlation dimension

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 size” exponent

Numerics



Local Hölder exponent of

the particle velocity

Local Stokes number Free particles

Fluid tracers

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 systemExample: 1st Lyapunov exponent in the smooth case





Large Stokes - smooth flows

• Same argument applies to the large deviations of the stretching rate

Statistics of velocity differences

• PDFs of velocity differences also rescale at large Stokes numbers:



Power-law tails

Power law tails





Tails related to large loops

• Cumulative probability• Simplification of the dynamics: noise + loops

•

• 1st contribution: should be sufficientlysmall to initiate a large loopRadius estimated by

Prob to enter a sufficiently large loop

Fraction of time spent at



Prediction for the exponent• 2nd contribution:

Approximation of the dynamics by the deterministic driftFraction of time spent at is

• – confirmed by numerics

• Power law with sameexponent at large positive

and Smooth case

• Clustering weakens whenroughness increases

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 trajectoriesDifferent from tracers: breaking of Lipschitz continuity is “2nd order”

Add compressibility: what are the different regimes present? Are the regimes observed for tracers also present? Do they appear only in the singular limit ? Are there other regimes?

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•Particles do not sample uniformly the flow (their position correlates with fluid velocity statistics)

Open questions

inertial particles in self- similar random flows jérémie bec cnrs, observatoire de la côte...

Documents