Transport o of p particles i in fl fluid fl flows Transport o of p particles i in fl fluid fl flows Mas Massimo mo Ce Cencini Ce Massimo Mas mo 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.cencini@cnr cencini@cnr. .it it massimo. Conference/School on Anomalous Transport: from Billiards to Nanosystems Sperlonga Sept. 2010
Two k kinds o of p particles Two k kinds o of p particles Tracers= same as fluid elements Tracers= same as fluid elements • 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 Inertial particles= mass impurities of finite size Inertial particles= mass impurities of finite size • 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
Iner In ertial par ial particles icles In Iner ertial par ial particles icles Rain droplets Aerosols: : sand sand, , pollution etc pollution etc Rain droplets Aerosols Sprays Sprays Marine Snow Snow Marine Bubbles Bubbles Planetesimals Planetesimals Finite-size & mass impurities in fluid flows
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 Wood, Hwang & Eaton (2005) 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)
Phenomeno Pheno 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 Fast evo Fast evolvi evolvi ving scal ving scal scale: charact scale: charact e: characteri e: characteri erist erist stic t stic t c time - c time - e ---> e ---> --> -->
Si Simpli mplified fied parti particle cle dynami dynamics cs Si Simpli mplified fied parti particle cle dynami dynamics cs Assumptions: Assumptions: Stokes number Stokes number hea eavy light lig ht hea eavy Small particles a<< " Small particles a<< " neutral al Small local Re a|u-V|/ % <<1 Small local Re a|u-V|/ % <<1 neutr No feedback on the fluid (passive particles) No feedback on the fluid (passive particles) No collisions (dilute suspensions) No collisions (dilute suspensions) Stokes time Fast fluid 0 1 3 time scale Density contrast Density contrast 0 #$ 0 #$ <1 <1 heavy heavy =1 neutral neutral $ =1 $ light 1< $# 1< $# 3 3 light Minimal interesting model Minimal interesting model Very heavy particle $ Ve Very heavy particle Ve $ =0 =0 =0 =0 (e (e.g .g. . water r dro rople lets in air r $ =1 =10 -3 ) (e (e.g .g. . water r dro rople lets in air r $ =1 =10 -3 -3 ) -3
Iner In ertial al Par Particl cles as dyn es as dynam amical cal syst system ems Iner In ertial al Par Particl cles as dyn es as dynam amical cal syst system ems Particle in d-dimensional space Particle in d-dimensional space Differentiable at Differentiable at small scales (r< " ) small scales (r< " ) 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 =0 St=0 & 1 1 & 1 1 =0 St & =3 St & $ =0 St $ =3 St $ $ $ DNS summary N 3 Re ' St $ 512 3 51 18 185 0- 0->3 0. 0.16- 6->4 128 3 12 65 65 0->3 0- 0.16- 0. 6->4 2048 3 2048 400 400 0 0. 0.16- 6->70 70 512 3 51 18 185 0 0.16- 0. 6->3. 3.5 256 3 256 105 10 0 0. 0.16- 6->3. 3.5 128 3 12 65 65 0 0. 0.16- 6->3. 3.5
Me Mechanisms at at at wo wo work Me Mechanisms at work Ejection/injection of heavy/light particles ! from/in vortices preferential concentration <1 hea eavy $ <1 hea eavy $ >1 light >1 $ >1 >1 light $ 2001) (Maxey 1987; Balkovsky, Falkovich, Fouxon (Maxey 1987; Balkovsky, Falkovich, Fouxon 2001) 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)
Phase Phase space space dynamics dynamics Phase Phase space space dynamics dynamics St<<1 St>1 St<<1 St>1 Collision Collision r=a 1 +a 2 rate rate Enhanced relative velocity Enhanced encounters by clustering
Correlation with the flow with the flow Correlation Preferential concentration Strain rotation Heavy particles like strain regions Heavy particles like strain regions Light particles like rotating regions Light particles like rotating regions ( <0 ( >0 P( ( >0) Bec et al (2006) Bec et al (2006)
Lyapun Lyapunov ov Ex Ex Exponents Lyapun Lyapunov ov Exponents Another signature of the uneven distribution of particles Another signature of the uneven distribution of particles Hea eavy Hea eavy St<<1 St<<1 ( St ) > ' ( St=0 ) 1 ( St ) > 1 ( St=0 ) ' 1 ' 1 ' st stay l ay longer i er in stay l st ay longer i er in Ligh Li ght Li Ligh ght strai st rain-reg -regions strai st rain-reg -regions ( St ) < ' ( St=0 ) 1 ( St ) < 1 ( St=0 ) ' 1 ' 1 ' st stayi aying aw away fro ay from st strai rain-reg regions stayi st aying aw away fro ay from st strai rain-reg regions Calzavarini, MC, Lohse & Toschi 2008
Two k kinds o of c clustering Two k kinds o of c clustering Particle clustering is observed both di dissipative and in the in in inertial range in the di dissipative inertial Instantaneous p. distribution in a slice of width ! 2.5 ! . St ! = 0.58 R " = 185 Bec, Biferale, MC, Lanotte, Musacchio & Toschi (2007)
Clusteri Clu tering ng at at r< r< r< " Clu Clusteri tering ng at at r< " • Smooth flow -> fractal distribution • Everything must be a function of St St St " & & Re ' ' only St & & Re " correlation dimension Here $ =0 Heavy particles
Co Correlati rrelation n di dimens mension Correlati Co rrelation n di dimens mension ! Maximum of clustering for St " & 1 ! D 2 almost independent of Re ' , Re ' =400 ! D q ) D 2 multifractality Re ' =185 1 10 0.1 St P( ( >0) Clustering & Preferential Hea eavy particles es ( ( $ =0 =0) ) Hea eavy particles es $ =0 =0 concentration are correlated
Kaplan- Ka n-Yo York rke di di dimension Kaplan- Ka n-Yo York rke dimension Light particles stronger clustering Light particles stronger clustering D 2 1 signature of vortex filaments D & 1 signature of vortex filaments 2 & Re=75,185 Re=75,185 Light particles: neglecting collisions might be a problem!
Clu Clusteri tering ng at at in at inertial scales in Clu Clusteri tering ng at inertial scales •Voids & structures Voids & structures • from " to L from " to L •Distribution of Distribution of • particles over scales? particles over scales? •What is the What is the • dependence on St " ? Or Or dependence on St " ? what is the proper what is the proper parameter? parameter?
Co Coars arse e grai rained ned de density de Coars Co arse e grai rained ned density Poisson ( * =0) =0 ( * =0 =0) *+ *+ *+ *+ Algebraic tails signature of voids
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