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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