Transport of tracers & particles in fluid flows Massimo Cencini - PowerPoint PPT Presentation

Transport of tracers & particles in fluid flows Massimo Cencini Istituto dei Sistemi Complessi CNR, Rome massimo.cencini@cnr.it Conference/School on Anomalous Transport: from Billiards to Nanosystems Sperlonga Sept. 2010

transport in fluids flows Biology & environment Pollution

transport in fluids flows

transport in fluids flows Enhanced Mixing design efficient mixers in microfluids

Two points of view Aim: understanding properties of Lagrange trajectories X (t) given u ( x ,t) Aim: understanding properties of fields θ ( x ,t) given u ( x ,t) Euler

The two descriptions are connected (x,t) time z y(0;x,t| η ) Studying particle trajectories is thus relevant also to understand the transport of fields we will focus on particle motion

Two kind of particles Tracers ! same density of the fluid ! point-like ! move with the same velocity of the fluid ! essentially they move like fluid elements (Inertial) Particles e size e size ! density different from the fluid ! finite size uded uded ! inertia & other forces are acting shape) shape) velocity different from the fluid one We shall only consider passive particles: i.e. the velocity field is not modified by their presence

Outline (I) Single particle motion (absolute dispersion) conditions for standard & anomalous diffusion, examples in simple laminar flows (II) Two particle motion (relative dispersion) focus on relative dispersion in laminar & turbulent flows, relative dispersion at changing the scale & characterization of non-asymptotic regimes (III) Clustering of inertial particles in turbulence characterization of clustering & preferential concentration for particles which do not follow fluid motion (I) & (II) focus on tracers

Single particle dynamics thermal noise prescribed fluid velocity Lagrangian velocity We are interested in the long time behavior of and how it depends on the properties of * Typically we expect standard diffusive behaviors * D E effective diffusion coefficient, D E [u]>>D 0 *Which properties must be present to have non-standard behaviors? *effective macroscopic description of transport?

Green-Kubo-Taylor relation Lagrangian velocity correlation function ||| Everything is written in the Lagrangian velocity correlation function

Green-Kubo-Taylor formula conditions for standard & anomalous diffusione To understand absolute dispersion we just need to know the velocity autocorrelation function Standard diffusion anomalous diffusion superdiffusion subdiffusion

Standard diffusion essentially CLT holds displacement velocity correlation Diffusive motion Ballistic motion

Standard Diffusion macroscopic time T M if diffusive behavior Effective macroscopic description at large t & Δ X D E >>D 0 will depend non trivially on u and D 0 Various techniques to derive D E in periodic or random velocity fields based on perturbative expansions - Multiscale methods - Idea : slow (X M ,T M ) & fast (x,t) variables (1) It comes an effective equation for Bensoussan, Lions & Papanicolaou, Asymptotic Analysis for Periodic Structures (1978) Biferale, Crisanti, Vergassola & Vulpiani PoF 7, 2725 (1995) Majda & Kramer Phys. Rep. 314, 237 (1999)

Non-Standard diffusion anomalous superdiffusion anomalous subdiffusion long negative tails long positive tails 1 1 10 0 t - η 10 -1 0.8 t -1 0.8 10 -1 10 -2 10 -3 0.6 0.6 t -1 10 -2 C(t)/C(0) t - η C(t)/C(0) 10 -4 0.4 0.4 10 -5 10 -3 10 0 10 1 10 2 10 3 10 -2 10 -1 10 0 10 1 10 2 10 3 0.2 0.2 0 0 -0.2 -0.2 0 5 10 15 20 0 100 200 300 400 500 t t if D 0 =0 impossible in incompressible flows

Physical origin of long correlations? Long spatial correlations of the velocity field diffusive superdiffusive time independent flows:Avellaneda & Majda, Commun. Math. Phys. 138, 339 (1991) time dependent flows: Avellaneda &Vergassola, Phys. Rev. E 52, 3249 (1995) The velocity field has finite correlation length but particle dynamics generate very long Lagrangian velocity correlations We will see these two mechanisms with some example

Random shear flows (strong spatial correlations) V=const U(y) random & gaussian Power spectrum spatial correlation function V=0 D 0 >0 G. Matheron & G. de Marsily, Wat. Resour. Res. 16, 901 (1980) V ! 0 D 0 =0 F .W . Elliott, D.J. Horntrop & A. Majda, Chaos 7, 39 (1997) Absolute dispersion in the x-direction?

Random shear: D 0 =0 V ! 0 to simplify temporal correlation spatial correlation At large times ϒ behavior <( Δ Y(t)) 2 > t 0 trapping ϒ ≥ 1 t 1- ϒ subdiffusion 0< ϒ <1 Elliott, Horntrop & Majda, Chaos 7, 39 (1997) t diffusion ϒ =0 t 1- ϒ superdiffusion ϒ <0

Random shear: D 0 =0 V ! 0 ϒ =-0.5 10 2 ϒ =0 10 0 10 -2 ϒ =0.5 S(k) 10 -4 ϒ =1.5 10 -6 Analytically solvable 10 -8 10 -4 10 -2 10 0 10 2 k 1 tails 0.8 t -(1+ ϒ ) 0.6 C(t)/C(0) ϒ =1 trapping 0.4 0.2 in incompressible flows trapping & subdiffusion do not happen when D 0 ! 0 0 -0.2 0 5 10 15 20 Elliott, Horntrop & Majda, Chaos 7, 39 (1997) t

Random shear: D 0 ! 0 V=0 temporal correlation spatial correlation ϒ >1 standard -1< ϒ <1 anomalous Matheron & de Marsily, Wat. Resour. Res. 16, 901 (1980)

Time dependent Cellular flows (Lagrangian persistency) convection-> 2d model (Solomon & Gollub PRA 38 , 6280 (1988)) vorticity u has a single mode no spatial persistency

Lagrangian chaos steady flow/ particles cannot escape the vortex 3.5 3.5 Lagrangian motion is 3 3 regular 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 3.5 time periodic flow: Lagrangian chaos induces motion along x even if D 0 =0 Lagrangian velocity is irregular even if eulerian velocity is regular 20 3.5 3 15 2.5 10 2 5 1.5 0 1 -5 0.5 -10 0 0 100 200 300 400 500 600 700 800 900 1000 -10 -5 0 5 10 15 20

Resonances ( synchronization ) between particle circulation time (T c ) & cell oscillation can cause persistence of the motion in the same direction ( ballistic channel ) for long time causing long tail in the velocity correlation responsible for anomalous diffusion when D 0 =0 Long tails due to non-trivial Lagrangian motion For D 0 ! 0 synchronization is imperfect and asymptotically diffusion is standard but D E depends as a power law on D 0 Castiglione et al J.Phys. A 31, 7197 (1998); Castiglione et al. Physica D 134, 75 (1999) Solomon et al. Physica D 157, 40 (2001)

“Strong” anomalous diffusion diffusion superdiffusion signature of persistent ballistic motion What about higher moments? when “strong” for pure diffusion anomalous diffusion or superdiffusion the core rescale with < Δ 2 (t)> the tails not rescale with < Δ 2 (t)> Castiglione et al. Physica D 134, 75 (1999) Andersen et al Europ. Phys. J. B 18, 447 (2000)