Transport o of t tracers & & p particles Transport o of - - PowerPoint PPT Presentation
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
Similarly to absolute dispersion one can work with
But But we e can an und under erstand tand muc uch re reaso soning on !Ru u which depends on the fl flow
2
4
6
spatially & temporally regular or weakly disordered regular or weakly disordered
few spatio-temporal scales are active
strongly disordered both spatially & temporally temporally
many active spatio/temporal scales
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
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
Forcing Nonlinear Transfer "# "# Dissipation "# "# L.
Richar ards dson
L.
Richar ards dson
universal statistics Velocity regularized by viscosity (differentiable velocities)
energy
Kolmo mogorov Kolmo mogorov 19
1941 19 1941
smooth at small scales
large scales system dependent
few active temporal & spatial scales scales
smooth at small scales r<<$ $
non-smooth & universal at intermediate scales intermediate scales $ $<<r<<L <<r<<L
r>>L system dependent but almost uncorrelated velocities almost uncorrelated velocities
spatiotemporally disordered
many spatiotemporal scales excited excited
Universal Universal behavior behavior
Uncorrelated Velocities and Possibly effects
System dependent possibly effects of boundaries
1) short time behavior when particles start very close R(t=0)<<$ 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
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!!!
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
time time
position
velocity Example from yesterday
Lagrangian Chaos is a generic phenomenon present
in 2d flows if time-dependent and in 3d also for steady flows
Chaos: infinitesimally small separations are exponentially amplified in time Strain matrix Tangent space / linearized dynamics
Law of large numbers ergodicity Finite time Lyapunov exponent Lyapunov exponent
Evolution matrix (time ordered exponential) We need to generalize the d=1 treatment to matricess (Oseledec theorem (1968))
Positive & symmetric Positive & symmetric Finite time Lyapunov exponents
Lyapunov exponents Lyapunov exponents % %1
1 =>
=> growth rate of infinitesimal segments
growth rate of infinitesimal segments
% %1
1+
+% %2
2 =>
=> growth rate of infinitesimal surfaces
growth rate of infinitesimal surfaces
% %1
1+
+% %2
2+
+% %3
3 =>
=> growth rate of infinitesimal volumes
growth rate of infinitesimal volumes
: : : : : : % %1
1+
+% %2
2+
+% %3
3+
+… …+ +% %d
d =>
=> growth rate of infinitesimal phase-space volumes
growth rate of infinitesimal phase-space volumes
suggesting
However % catches only the typical growth rate while fluctuations may be important Generalized Lyapunov exponents
Again the cellular flow
Two asymptotics
BUT in general there are several possible effects
correlations as in turbulence
Growth rate & fluctuates: spurious anomalous superdiffusion Intermediate regime due to superposition of exponential and linear behavior
Physics is in the scale not in the time we need to disentangle the separation growth at different scales
Di Dispersions ns of tr tracers In a dis isk wit ith 4 poin int vort rtic ices Anomalous dispersion? Bounded domain no time asyptotics and thus difficulty of Interpretation: is the
physically interesting?
Idea: instead of averaging of separation at fixed times performing averages of first exit time to reach a given separation Fixed time Fixed separation R rR rR Tr(R) (R)
R(t) R(t) R(t+ R(t+'t)
Aurell et al (1996)
Finite Size Lyapunov Exponent
Fixed time Fixed separation
Relative separation
FSLE
Scale dependent diff coeff
Artale et al Phys. Fluids 1997
With fixed separation analysis No more spurious regime
Artale et al Phys. Fluids 1997
Simply assuming exponential relaxation to uniform distribution with time scale (
Artale et al Phys. Fluids 1997
Boffetta, MC, Espa & Querzoli EPL 1999 & Phys. Fluids 2000 Experiment (PIV) & data analysis
Uncorrelated velocities
Chaotic dispersion Richardson Dispersion Standard Dispersion Richardson 1926
<R <R2(t (t)> 1/< 1/<%(R (R))> )> t3 R2/3 Usual Fixed time analysis Fixed scale analysis Biferale et al Phys. Fluids 2005 DNS 10243
Finite extension of inertial range & Memory of initial separation
FSLE only depends
effect of initial separation Decreases the effect of Crossovers revealing the Interesting regime
5
but short times & large but short times & large separations separations
Science 2006
Pros:
separation
Cons:
Pros:
inertial range Cons:
Deviations are obseved->intermittency->broken selfsimilarity
Boffetta & Sokolov PRL 88, 094501 (2002)
Falkovich, Gawedzky & Vergassola Rev. Mod. Phys. 73, 913 (2001) complete review on particle transport with link with field transport
Artale, Boffetta, Celani, Cencini & Vulpiani Phys. Fluids 9, 3162 (1997) Boffetta, Celani, Cencini, Lacorata & Vulpiani, Chaos 10, 50 (2000) Jullien, Paret & Tabeling PRL 82, 2872 (1999) Boffetta & Celani Physica A 280, 1 (2000) Boffetta & Sokolov PRL 88, 094501 (2002) Biferale, Boffetta, Celani, Devenish, Lanotte & Toschi,
M Bourgoin, NT Ouellette, H Xu, J Berg, E Bodenschatz, Science 311, 835 (2006)