Download Heavy particle clustering in turbulent flows

Heavy particle clustering in
turbulent flows
Alessandra Lanotte
with:
Jeremie Bec
Luca Biferale
Guido Boffetta
Antonio Celani
Massimo Cencini
Stefano Musacchio
Federico Toschi
back to the real world
Space d=3
flow is fully developed turbulent Re>>1
Ku = u Tu/Lu ≈ 1
St = 1/ (γTu) not large not small
Outline
• Introduction
Physical systems
Observations
Model
• Core of the talk
Small scales clustering
Inertial scales clustering
Motivation
a part from our interest
Strong particle concentration fluctuations
have an impact on climate in different ways
Desert dusts are particularly active
ice-forming agents.
They can affect clouds formation.
Reflective power of the atmosphere
due to aerosols scattering and absorption
is crucial for climatological models
Heavy particles in wind tunnel turbulence
particles can be accurately tracked !
Cornell experiment
Reλ ≈ 250
High-speed camera: 2D frames
Sampling time 1/100 τη
water droplets in air
ρf/ ρp ~ 0.001
drops have different sizes:
clearly polydisperse flow
Something we know about inertia
1. Ejection of heavy particles from vortices
--> experience smaller acceleration
2. Particle have finite response time to fluid fluctuations
--> smoothing or filtering of fast time scales
3. Very strong concentration fluctuation
--> particle distribute on clusters
Understand physical mechanisms
and relevant parameters for statistical description…
Choose the simplest model
 much smaller than the flow dissipative scale a << "
 much heavier than the fluid
 particle Reynolds number low
Re a = a | V - u | " << 1
!
 very dilute suspension : no role of collisions
 no back reaction on the
! flow, no gravity
(M. Maxey & J. Riley,
Phys Fluids 1983)
X
Stokes time
-->
Stokes number
only Stokes drag
(ex. water in air β=0.001)
3d Direct Numerical Simulations
+ statistical homogeneity & isotropy
Reλ= 65, 105, 185
Viscous scales well resolved
+
Tracers & Particles
Accurate time sampling
Energy spectrum
Spectral flux
k-5/3
How long do we wait for the stationary mass
distribution?
St=0.9
St=1.6
St=0.48
St=3.3
St=0.27
St=0.16
St=0
Coarse-grained mass in the j-th cell of side l=2Δx
Clustering: only a small scale feature?
Pressure gradient
modulus
B/W coding of low/high
intermittency
Particles at Stokes
St=0.16 (b);
St= 0.8 (c); and
St=3.3 (d)
Slice of width ≈ 2.5η.
Particle clusters & voids are observed both
in the dissipative and in inertial range of turbulent scales
Observables at small scales r < η
Tools of dissipative dynamical system theory
Space density of particles pairs (useful for collisions, pair dynamics)
Probability to find 2 particles at a distance smaller than r
r
D2 is the correlation dimension (Grassberger 1983 ; Hentschel Procaccia, 1983)
Another common observable is the radial distribution function g(r)
It is O(1) for tracers, it diverges as r--> 0 for inertial particles
(or in compressible flows).
Probability and D2
At r < η the flow is differentiable, no-scale dependency
we expect that particles have fractal distribution (with power law tails)
Also, at these scales, the is a unique relevant time scale is τη
Does variation of Kolmogorov scale η matter?
Shape of correlation dimension D2 in 3d turbulence
Optimal Stokes number for maximal clusterization
No Reynolds dependence (as in Collins & Keswani 2004)
Particles positions correlate with low values of acceleration
( prefer stay in hyperbolic regions)
Maximum of clustering seems to be connected to
preferential concentration, confirming classical scenario….
What happens at larger scales η < r < L?
Can particles of Stokes τ feel effects of time scales tr>> τ ?
How do particles distribute out of vortical regions?
What are the proper parameters to describe voids & clusters?
Observations for 2d flows
G. Boffetta, F. De Lillo, and A. Gamba , Phys. Fluids (2004)
----> void characterization
Chen, Goto, Vassilicos, JFM 2006
“clustering of zero-acceleration points determines the
clustering of inertial particles”
large-scale sweeping plays a role, but it is not clear….
Inertial range observable: bad
Probability Distribution Function
of the coarse-grained mass mr
does not work well to measure the porpability to have cells with
few particles (statistics of empty region)
Inertial range observable
Probability Distribution Function
of the coarse-grained particle density,
r
Given N particles, we compute number density ρ of particles within
a cell of scale r,
weighting each cell with the mass it contains:
Quasi-Lagrangian measure
a natural measure to reduce finite N effects, at ρ<<1 due to voids
QL mass density distribution
r=L/16
τ
Tracers behave according to uniform Poisson distribution
Particle show deviations, already there for very small τ
such deviations become stronger with τ
Algebraic tails at low density ρ <<1
we have
(tracers limit, uniform)
we have
St
(non zero prob. to have empty areas)
At large Stokes, we should also tend to uniformity
At difference from small scales, for St=3.3 we are still very far from uniformity
These empty regions can play a relevant role in many physical issues
How do we understand this PDFs?
Particles should not distribute self-similarly
i.e. Deviations from a uniform distribution are not scale-invariant
(Balkovsky, Falkovich & Fouxon 2001)
No simple rescaling of the mass distributions
However for the mass PDF,these two limits are
equivalent:
• fixed τ and r
• fixed r and τ
∞ (large observation scale)
0 (small inertia)
Both limits give a uniform particle distribution….
So there could be a parameter, rescaling
the mass distribution , which relates
Stokes times τ and observation scales r
At scale r, the eddy-turn-over time scale is τr=ε-1/3r2/3,
in analogy with dissipative scales, we could define:
Is this time scale relevant
particle clustering in
the inertial range?
Unfortunately not so simple!
This simple analogy works in synthetic flows:
e.g. Kraichnan flows
• no time correlation
• no spatial structures
• no large scale-sweeping
(Bec, Cencini & Hillerbrand 2006; Rafaela talk!)
But it does not work in real turbulence where
these features are present…
X
A different observation
[Maxey (1987)]
--> effective compressible field felt by the particle:
Effective compressibility
good for r<<η for Stη<<1
[Balkovsky, Falkovich & Fouxon (2001)]
particle
flow
Suppose we can apply it also at Str -> 0
(large r and not too large τ)
This is the contraction- rate of a particle volume of
of size r3 and Stokes time τ
K41 argum.
or
Although this might be a finite Re effect
for Re=65,105,185, 280
Tsuji and Ishihara, PRE 2003
Collapse of the
coarse-grained mass PDF for different values of Γ
Γ =4.8e-4
(St=0.16, 0.27, 0.37, 0.48)
Γ =2.1e−3
(St=0.58, 0.69, 0.80, 0.91, 1.0)
Γ=7.9e−3
(St=1.60, 2.03, 2.67, 3.31)
Non-dimensional
contraction rate
Uniformity is recovered going to the large scales
But very slowly
Conclusions
We gave a description of particle clustering
for moderate St ≤ 3.3 and moderate Reλ ≤200 numbers
1. clustering at small scales r < η
 The only relevant number for particle dynamics is
Stη=τ/τη
 Particles concentrate onto a multi-fractal set, whose
dimension depends on the Stokes number only
(or just very weakly depends on Reynolds)
 Optimal finite Stokes number for clusterization: Stη ~ 0.6
(unpredictable..)
This global picture is the same as in smooth random flow
(see Bec 2005; Bec, Celani, Cencini,Musacchio 2005)
…continue
2. clustering at inertial range scales η < r < L
 concentration fluctuations are relevant also for the
inertial range scales
 uniformity of mass distribution is recovered at large scale
but very slowly
 if the contraction rate Γ, and not Str, is the proper number to
rescale mass statistics ----> sweeping is important
(Bec, Biferale, Cencini, AL, Musacchio, Toschi PRL submitted 2006)
Perspectives
A better understanding of the statistics
of fluid acceleration (rather than vorticity) seems
crucial to understand clustering
Larger Re studies are necessary to confirm the picture
or eventually see what happens with K41 scaling for
pressure.
Rapid rain initiation ?
(warm) cloud large scale L=100m;
dissipative scale η = 1mm; Re=107
Enhanced collision rate is due to:
i) clustering ?
If yes, at which scales ?
ii) caustics ?
Rain Drops formation
in warm clouds
1.
2.
3.
CCN activation
Condensation St=5.e-4
Coalescence St=0.05 -0.2
where there are voids, there is lot of vapor at disposal
so spectrum size of particles can be broad
--> then also clustering might count
END