Transport o of t tracers & & p particles Transport o of t tracers & & p particles in in fluid fluid flow flows (II) s (II) in fluid in fluid flow flows (II) s (II) Mas Massimo mo Ce Ce Cencini Mas Massimo 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
Absolut Absolute vs Relat e vs Relative dispersion ive dispersion Absolut Absolute vs Relat e vs Relative dispersion ive dispersion Absolute dispersion Relative dispersion Similarly to absolute dispersion one can work with But But we e can an und under erstand tand muc uch re reaso soning on ! R u u which depends on the fl flow
Lamin Lam inar vs Turbulen ar vs Turbulent flow flows Lam Lamin inar vs Turbulen ar vs Turbulent flow flows +BC +BC Re=10 -2 Re=10 -2 Laminar motion Laminar motion • spatially & temporally • spatially & temporally Re=20 Re=20 regular or weakly disordered regular or weakly disordered • few spatio-temporal scales are active few spatio-temporal scales are active • Re=10 2 Re=10 2 Turbolent motion Re=10 4 Turbolent motion Re=10 4 • • strongly disordered both spatially & strongly disordered both spatially & temporally temporally Re=10 6 Re=10 6 • • many active spatio/temporal scales many active spatio/temporal scales
Examples of Examples f Laminar Laminar flo flows Examples Examples of f Laminar Laminar flo flows Typically there is a single characteristic scale L below which the flow is differentiable and above which there is a system depedent behavior with, possibly, the effect of boundaries
Turbulent fl flows Turbulent fl flows Re >>1 at scales small enough the statistical properties expected to be independent of the forcing mechanism --universal statistics-- What is the behavior of At changing the scale R?? <= Velocity in a point
Turbulence p phenomenology Turbulence p phenomenology energy Forcing Nonlinear universal statistics L. F L. F. R Richar ards dson on L. F L. F. R Richar ards dson on Transfer "# "# Dissipation "# "# Velocity regularized by viscosity (differentiable velocities)
Turbulence p phenomenology Turbulence p phenomenology “Exact result” Kolmo mogorov 19 Kolmo mogorov 19 1941 1941 experiments
Lamin Lam inar vs Turbulen ar vs Turbulent flow flows Lamin Lam inar vs Turbulen ar vs Turbulent flow flows Uncorrelated System dependent Velocities and possibly Possibly effects effects of of boundaries boundaries Universal Universal behavior behavior L "$ r L r r L r "$ L $ $ • smooth at small scales smooth at small scales • • smooth at small scales r<< • smooth at small scales r<< $ $ • large scales system dependent large scales system dependent • • non-smooth & universal at non-smooth & universal at • • few active temporal & spatial few active temporal & spatial • intermediate scales intermediate scales $ $ <<r<<L <<r<<L scales scales • r>>L system dependent but r>>L system dependent but • almost uncorrelated velocities almost uncorrelated velocities • • spatiotemporally disordered spatiotemporally disordered • many spatiotemporal scales many spatiotemporal scales • excited excited
Relat Relative dispersion ive dispersion Relat Relative dispersion ive dispersion Asymptotic regimes 1) short time behavior when particles start very close R(t=0)<< $ i.e. at scales where the velocity field (laminar or turbulent) is smooth/differentiable (chaos at play) 2) Very large times (large separations R>>L) in the absence of boundaries expected diffusive behavior if uncorrelated velocities Non-asymptotic regimes 1) Presence of boundaries preventing or the possibility to reach an asymptotic diffusive behavior 2) Presence of non-trivial correlations in the velocity as in the inertial range of turbulence 3) Spurious effects!!!
Relat ative di disp spersi sion at at sm smal all se separ arat ations Relat ative di disp spersi sion at at sm smal all se separ arat ations We saw already that even regular velocity fields can generate irregular particle trajectories so that both the evolution of their position and of the lagrangian velocity is very irregular Example from yesterday velocity position time time Lagrangian Chaos is a generic phenomenon present in 2d flows if time-dependent and in 3d also for steady flows What happen to two very close trajectories in the presence of Lagrangian Chaos?
Sensi Sen sitive depen ve dependen dence on ce on i initial al con condi dition ons Sensi Sen sitive depen ve dependen dence on ce on i initial al con condi dition ons Chaos: infinitesimally small separations are exponentially amplified in time Strain matrix Tangent space / linearized dynamics d= d=1 Lyapunov exponent Finite time Law of large ergodicity Lyapunov exponent numbers d>1 d>1 Evolution matrix (time ordered exponential) We need to generalize the d=1 treatment to matricess (Oseledec theorem (1968)) �
Sensi Sen sitive depen ve dependen dence on ce on i initial al con condi dition ons Sensi Sen sitive depen ve dependen dence on ce on i initial al con condi dition ons Positive & symmetric Positive & symmetric Finite time Lyapunov exponents Oseledec--> if ergodic Oseledec--> if ergodic Lyapunov exponents Lyapunov exponents Physical meaning of LEs => growth rate of infinitesimal segments 1 => growth rate of infinitesimal segments % 1 % 1 + + % 2 => => growth rate of infinitesimal surfaces growth rate of infinitesimal surfaces % 1 % 2 % + % + % => growth rate of infinitesimal volumes 1 + 2 + 3 => growth rate of infinitesimal volumes % 1 % 2 % 3 % : : : : : : 1 + + % 2 + + % 3 + +… …+ + % d => => growth rate of infinitesimal phase-space volumes growth rate of infinitesimal phase-space volumes % 1 % 2 % 3 % d %
Relat ative di disp spersi sion at at sm smal all se separ arat ations Relat ative di disp spersi sion at at sm smal all se separ arat ations suggesting However % catches only the typical growth rate while fluctuations may be important Generalized Lyapunov exponents
Asym symptotic behav avior: an an ide deal al case ase Asym symptotic behav avior: an an ide deal al case ase Two asymptotics Again the cellular flow L BUT in general there are several possible effects • impossibility to reach asymptotic regimes due to boundaries • non-trivial non-asymptotic regimes as, e.g., due to velocity correlations as in turbulence • Spurious effects due to fluctuactions of the growth rate
Spurious r Spur ous regi egimes due t es due to fl o fluct uctuat uation ons Spurious r Spur ous regi egimes due t es due to fl o fluct uctuat uation ons Simple hand made example Growth rate & fluctuates: spurious anomalous superdiffusion Intermediate regime due to superposition of exponential 4Dt and linear behavior t 1.75 Physics is in the scale not in the time we need to disentangle the separation growth at different scales
A less h A less han and m d made exam ade example ple A less h A less han and m d made exam ade example ple Dispersions Di ns of tr tracers In a dis isk wit ith 4 poin int vort rtic ices Anomalous dispersion? t 1.8 Bounded domain no time asyptotics and thus difficulty of Interpretation: is the observed anomalous physically interesting?
Scale dependent Scale dependent descri descripti ption Scale Scale dependent dependent descri descripti ption Idea: instead of averaging of separation at fixed times performing averages of first exit time to reach a given separation Fixed separation Fixed time r>1 rR rR R(t+ R(t+ ' t) R R(t) R(t) T r (R) (R) Finite Size Lyapunov Exponent Aurell et al (1996)
Scale dependent Scale dependent descri descripti ption Scale Scale dependent dependent descri descripti ption Fixed time Fixed separation Scale dependent FSLE Relative separation diff coeff advantage of % (R) & D(R) no more spurios contaminations between regimes which pertain to different scales and thus different physics Artale et al Phys. Fluids 1997
Re-exam xamining previous s exam xamples Re-exam xamining previous s exam xamples In good cases the two methods are equivalent
Re-exam xamining previous s exam xamples Re-exam xamining previous s exam xamples 4Dt t 1.75 < & >= % DR -2 With fixed separation analysis No more spurious regime Artale et al Phys. Fluids 1997
Re-exam xamining previous s exam xamples Re-exam xamining previous s exam xamples t 1.8 Anomalous scaling is here a spurious effect % Simply assuming exponential relaxation to uniform distribution with time scale ( Artale et al Phys. Fluids 1997
Easy asy to compute in exp xperiments Easy asy to compute in exp xperiments Experiment (PIV) & data analysis Boffetta, MC, Espa & Querzoli EPL 1999 & Phys. Fluids 2000
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