Transport o of p particles i in fl fluid fl flows Transport o - - PowerPoint PPT Presentation

Transport o

• f p

particles i in fl fluid fl flows Transport o

• f p

particles i in fl fluid fl flows

Mas Massimo mo Mas Massimo mo Ce Cencini Ce Cencini

Istituto dei ei Sistemi emi Comp mples essi Istituto dei ei Sistemi emi Comp mples essi CNR Rome me Italy CNR Rome me Italy massimo. massimo.cencini@cnr cencini@cnr. .it it Conference/School on Anomalous Transport: from Billiards to Nanosystems Sperlonga Sept. 2010

Two k kinds o

• f p

particles Two k kinds o

• f p

particles

• same density of the fluid

same density of the fluid

• point-like

point-like

• same velocity of the underlying

same velocity of the underlying fluid velocity fluid velocity

Tracers= same as fluid elements Tracers= same as fluid elements

• density different from that of the fluid

density different from that of the fluid

• finite size

finite size

• friction (Stokes) and other forces should be included

friction (Stokes) and other forces should be included

• shape may be important (we assume spherical shape)

shape may be important (we assume spherical shape)

• velocity mismatch with that of the

velocity mismatch with that of the fluid fluid

Simplified dynamics under Simplified dynamics under some assumptions some assumptions Inertial particles= mass impurities of finite size Inertial particles= mass impurities of finite size

In Iner ertial par ial particles icles In Iner ertial par ial particles icles

Rain droplets Rain droplets

Sprays Sprays

Planetesimals Planetesimals

Marine Marine Snow Snow

Aerosols Aerosols: : sand sand, , pollution etc pollution etc

Finite-size & mass impurities in fluid flows Bubbles Bubbles

Go Goal al Go Goal al

Dynamical and statistical properties of particles evolving in turbulence Dynamical and statistical properties of particles evolving in turbulence focus on clustering observed in experiments focus on clustering observed in experiments Clustering important for Clustering important for

• particle interaction rates by enhancing contact probability

(collisions, chemical reactions, etc...)

• the fluctuations in the concentration of a pollutant (Bec’s talk)

Wood, Hwang & Eaton (2005)

Pheno Phenomeno menolo logy gy of f Turbulence Turbulence Pheno Phenomeno menolo logy gy of f Turbulence Turbulence

Basic proper erties es

• K41 energy cascade from large (!L) scale to the small dissipative

scales (!" = Kolmogorov length scale)

• inertial range "

"<< << r << L << << r << L Many characteristic time scales

• dissipative range r

r < r r < " " Fast Fast evo evolvi ving scal scale: charact e: characteri erist stic t c time - e --->

• ->

Fast Fast evo evolvi ving scal scale: charact e: characteri erist stic t c time - e --->

• ->

Si Simpli mplified fied parti particle cle dynami dynamics cs Si Simpli mplified fied parti particle cle dynami dynamics cs

Stokes time Fast fluid time scale

0#$ #$<1 <1 heavy heavy $ $=1 =1 neutral neutral 1< 1<$# $#3 3 light light

Stokes number Stokes number Density contrast Density contrast

Assumptions: Assumptions:

Small particles a<< Small particles a<<" " Small local Re a|u-V|/ Small local Re a|u-V|/% %<<1 <<1 No feedback on the fluid (passive particles) No feedback on the fluid (passive particles) No collisions (dilute suspensions) No collisions (dilute suspensions)

Ve Very heavy particle Ve Very heavy particle $ $=0 =0 =0 =0 (e (e.g .g. . water r dro rople lets in air r (e (e.g .g. . water r dro rople lets in air r $ $=1 =10 =1 =10-3

• 3
• 3
• 3)

)

Minimal interesting model Minimal interesting model

hea eavy hea eavy lig light ht

neutr neutral al

1 3

In Iner ertial al Par Particl cles as dyn es as dynam amical cal syst system ems In Iner ertial al Par Particl cles as dyn es as dynam amical cal syst system ems

Differentiable at Differentiable at small scales (r< small scales (r<" ") )

Particle in d-dimensional space Particle in d-dimensional space Well defined dissipative dynamical system in 2d-dimensional phase-space Well defined dissipative dynamical system in 2d-dimensional phase-space

Jacobian (stability matrix) Jacobian (stability matrix) Strain matrix Strain matrix

constant phase-space contraction rate, i.e. phase-space constant phase-space contraction rate, i.e. phase-space Volumes contract exponentially with rate -d/St Volumes contract exponentially with rate -d/St Motions evolve onto an attractor in phase space Motions evolve onto an attractor in phase space

Particles Particles in in turbulence turbulence Particles Particles in in turbulence turbulence

$ $=0 St =0 St& &1 1 $ $=0 St=0 =0 St=0 $ $=3 St =3 St& &1 1

0. 0.16- 6->4 0- 0->3 65 65 12 1283 0. 0.16- 6->3. 3.5 10 105 256 2563 0. 0.16- 6->3. 3.5 65 65 12 1283 0. 0.16- 6->3. 3.5 18 185 51 5123 0. 0.16- 6->70 70 400 400 2048 20483 0. 0.16- 6->4 0- 0->3 18 185 51 5123

St $ Re'

N3

DNS summary

Me Mechanisms Me Mechanisms at at at at wo work wo work

! Ejection/injection of heavy/light particles from/in vortices preferential concentration ! Dissipative dynamics in phase-space: volumes contraction & particles may arrive very close with very different velocities ! Finite response time to fluid fluctuations (filter of fast time scales)

$ $<1 hea eavy <1 hea eavy $ $>1 >1 light >1 >1 light

(Maxey 1987; (Maxey 1987; Balkovsky, Falkovich, Fouxon Balkovsky, Falkovich, Fouxon 2001) 2001)

Phase Phase space space dynamics dynamics Phase Phase space space dynamics dynamics

St<<1 St<<1 St>1 St>1 Enhanced relative velocity Enhanced encounters by clustering r=a1+a2 Collision Collision rate rate

Correlation Correlation with the flow with the flow

Strain rotation

Heavy particles like strain regions Heavy particles like strain regions Light particles like rotating regions Light particles like rotating regions

Bec et al (2006) Bec et al (2006)

(<0 (>0 P((>0) Preferential concentration

Lyapun Lyapunov

• v

Lyapun Lyapunov

• v Ex

Exponents Ex Exponents

Hea eavy Hea eavy St<<1 St<<1 ' '1

1(

(St

St) >

) > ' '1

1(

(St=0

St=0)

)

st stay l ay longer i er in st stay l ay longer i er in st strai rain-reg

• regions

st strai rain-reg

• regions

Li Ligh ght Li Ligh ght ' '1

1(

(St

St) <

) < ' '1

1(

(St=0

St=0)

)

st stayi aying aw away fro ay from st strai rain-reg regions st stayi aying aw away fro ay from st strai rain-reg regions

Another signature of the uneven distribution of particles Another signature of the uneven distribution of particles

Calzavarini, MC, Lohse & Toschi 2008

Two k kinds o

• f c

clustering Two k kinds o

• f c

clustering

Particle clustering is observed both in the di

dissipative di dissipative and in the in inertial in inertial range

Instantaneous p. distribution in a slice

• f width ! 2.5!. St! = 0.58 R" = 185

Bec, Biferale, MC, Lanotte, Musacchio & Toschi (2007)

Clu Clusteri tering ng at at Clu Clusteri tering ng at at r< r< r< r<" "

• Smooth flow -> fractal distribution
• Everything must be a function of St

St St St"

" &

& Re & & Re'

' only

correlation dimension

Here $=0 Heavy particles

Co Correlati rrelation n di dimens mension Co Correlati rrelation n di dimens mension

1

10 St

Re'=400 Re'=185

0.1

P((>0)

Clustering & Preferential concentration are correlated !Maximum of clustering for St"&1 !D2 almost independent of Re', ! Dq) D2 multifractality

Hea eavy particles es Hea eavy particles es ( ($ $=0 =0 =0 =0) )

Ka Kaplan- n-Yo York rke Ka Kaplan- n-Yo York rke di dimension di dimension

Re=75,185 Re=75,185

Light particles stronger clustering Light particles stronger clustering D D2

2&

&1 signature of vortex filaments 1 signature of vortex filaments

Light particles: neglecting collisions might be a problem!

Clu Clusteri tering ng Clu Clusteri tering ng at at at at in inertial scales in inertial scales

• Voids & structures

Voids & structures from from " " to L to L

• Distribution of

Distribution of particles over scales? particles over scales?

• What is the

What is the dependence on St dependence on St"

"?

? Or Or what is the proper what is the proper parameter? parameter?

Co Coars arse e grai rained ned Co Coars arse e grai rained ned de density de density

Algebraic tails signature

• f voids

Poisson ( (* *=0 =0) =0 =0)

*+ *+ *+ *+

Wh What is the relevant time Wh What is the relevant time s scale of s scale of in iner ertial r ial ran ange clust ge cluster erin ing in iner ertial r ial ran ange clust ge cluster erin ing

Effective compressibility Effective compressibility

We can estimate the phase-space contraction rate for We can estimate the phase-space contraction rate for A particle blob of size r when the Stokes time is A particle blob of size r when the Stokes time is * For St->0 we have that It relates to pressure

Non Nondim dimen ension sional con al contract raction ion Non Nondim dimen ension sional con al contract raction ion ra rate ra rate

Non-dimensional contraction rate Non-dimensional contraction rate

,=7.9 10-3 ,=2.1 10-3 ,=4.8 10-4

Inertial al par articles s & Stat atist stical al Physi ysics Inertial al par articles s & Stat atist stical al Physi ysics

Bec & Chétrite (2007) Developing statistical models for mass transport which retains phenomenological ingredients

Inertial al par articles s & Stat atist stical al Physi ysics Inertial al par articles s & Stat atist stical al Physi ysics

Brownian motion (Langevin)

Inertial (heavy) particles Brownian motion in a Disordered media with Nontrivial spatio-temporal Correlation (turbulence) Simplified model: retaining only spatial correlations mimicking turbulent ones (Kraichnan model for the velocity)

Inertial al par articles s & Stat atist stical al Physi ysics Inertial al par articles s & Stat atist stical al Physi ysics

e.g. separation between 2 particles R=X1-X2

For smooth velocities d=1 Equivalent to Anderson localization: time->space Localization length->Lyapunov exponent

Derevyanko, Falkovich, Turitsyn, Turitsyn JoT (2007) (1d) Horvai arXiv:nlin/0511023v1 [nlin.CD] (2d)

Fluid dynamics Dynamical Systems

Statistical Physics

In Iner ertial al par particl cles es In Iner ertial al par particl cles es

Chaos/Fractals Disordered systems Stochastic processes

Than Thanks ks Than Thanks ks

• J. Bec
• L. Biferale
• G. Boffetta
• E. Calzavarini
• A. Celani
• R. Hillerbrandt
• D. Lohse
• S. Musacchio
• F. Toschi
• K. Turitsyn

Readin Reading list g list Readin Reading list g list

Inertial Particles in Turbulence Inertial Particles in Turbulence

• E. Balkovsky, G. Falkovich, A. Fouxon, “Intermittent distribution of inertial

particles in turbulent flows”, PRL 86 2790-2793 (2001)

• G. Falkovich & A. Pumir, “Intermittent distribution of heavy particles in a

turbulent flow”, PoF 16, L47-L50 (2004)

• J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi

“Lyapunov exponents of heavy particles in turbulence” PoF 18 , 091702 (2006)

• J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi, “Heavy

particle concentration in turbulence at dissipative and inertial scales” PRL 98, 084502 (2007)

• E. Calzavarini, M. Cencini, D. Lohse and F. Toschi, “Quantifying turbulence

induced segregation of inertial particles”, PRL 101, 084504 (2008)

• J. Bec, L. Biferale, M. Cencini, A.S. Lanotte, F. Toschi, “Intermittency in the

velocity distribution of heavy particles in turbulence”, JFM 646, 527 (2010)

Readin Reading list g list Readin Reading list g list

Inertial particles in Stochastic flows Inertial particles in Stochastic flows

• M. Wilkinson & B. Mehlig, “Path coalescence transition and its application”,

PRE 68, 040101 (2003)

• J. Bec, “Fractal clustering of inertial particles in random flows”, PoF 15, L81

(2003)

• J. Bec, “Multifractal concentrations of inertial particles in smooth random

flows” JFM 528, 255 (2005)

• K. Duncan, B. Mehlig, S. Ostlund, M. Wilkinson “Clustering in mixing flows”, PRL

95, 240602 (2005)

• J. Bec, A. Celani, M. Cencini & S. Musacchio “Clustering and collisions of heavy

particles in random smooth flows” PoF 17, 073301, 2005

• G. Falkovich, S. Musacchio, L. Piterbarg & M. Vucelja “Inertial particles

driven by a telegraph noise”, PRE 76, 026313 (2007)

• G. Falkovich & M. Martins Afonso, “Fluid-particle separation in a random flow

described by the telegraph model” PRE 76 026312, 2007

Readin Reading list g list Readin Reading list g list

Inertial particles in Kraichnan model (uncorrelated flows) Inertial particles in Kraichnan model (uncorrelated flows)

• L.I. Piterbarg, “The top Lyapunov Exponent for stochastic flow modeling the

upper ocean turbylence” SIAM J App Math 62:777 (2002)

• S. Derevyanko, G.Falkovich , K.Turitsyn & S.Turitsyn, “Lagrangian and Eulerian

descriptions of inertial particles in random flows” JofTurb 8:1, 1-18 (2007)

• J. Bec, M. Cencini & R. Hillerbrand, “ Heavy particles in incompressible flows:

the large Stokes number asymptotics” Physica D 226, 11-22, 2007; “Clustering

• f heavy particles in random self-similar flow” PRE 75, 025301, 2007
• J. Bec, M. Cencini, R. Hillerbrand & K. Turitsyn “Stochastic suspensions of

heavy particles Physica D 237, 2037-2050, 2008

• M. Wilkinson, B. Mehlig & K. Gustavsson, “Correlation dimension of inertial

particles in random flows” EPL 89 50002 (2010)

• P. Olla, “Preferential concentration vs. clustering in inertial particle transport

by random velocity fields” PRE 81, 016305 (2010)