Transcript Anomalous diffusion in generalised Ornstein

Clustering of Particles in
Turbulent Flows
Michael Wilkinson (Open University)
Senior collaborators:
Bernhard Mehlig, Stellan Ostlund (Gothenburg)
Students and postdoctoral workers:
V. Bezuglyy, K. Duncan, V. Uski (Open)
B. Anderson, K. Gustavsson, M. Lunggren, T. Weber (Chalmers)
Overview
Lecture 1:
Particles suspended in a turbulent fluid flow can cluster together. This
surprising observation has been most comprehensively explained
using models based upon diffusion processes.
Lecture 2:
Planet formation is thought to involve the aggregation of dust particles
in turbulent gas around a young star. Can the clustering of particles
facilitate this process?
The final lecture will discuss the problems of planet formation, and
consider whether aggregation of particles is relevant. Clustering
does not help to explain planet formation, but planet formation does
introduce new applications of diffusion processes.
Mixing
We expect that dust particles suspended in a randomly moving
fluid are mixed to a uniform density:
..but the opposite, unmixing, is also possible.
Unmixing
Clustering in Mixing Flows
Simulation of particles in a random twodimensional flow:
Reported experimental realisation:
Lycopodium powder in turbulent
channel flow.
This simulation from: M.Wilkinson and B.Mehlig,
Europhys. Lett. 71, 186, (2005).
J.R.Fessler, J.D.Kulick and J.K.Eaton,
Phys.Fluids, 6, 3742, (1994).
Aggregation versus clustering
For some choices of parameters the particles aggregate instead
of clustering. In the upper sequence particles have a lower mass.
Equations of motion
Particles move in a less dense fluid with velocity field
Particles do not affect the velocity field, or interact (until they make contact). The
equation of motion is assumed to be
Damping rate for a spherical particle of radius
:
Alternative form for very low density:
In our calculations
potential:
The components
is a random velocity field, obtained from a vector
have mean value zero, and correlation function
Dimensionless parameters
Parameters of the model:
From these we can form two independent dimensionless parameters:
Stokes number:
Kubo number:
For fully developed turbulence
.
Maxey’s centrifuge effect
Maxey suggested that suspended particles are centrifuged away from vortices:
M.R.Maxey, J. Fluid Mech., 174, 441, (1987).
If the Stokes number is too
large, the vortices are too shortlived. If the Stokes number is too
small, the particles are too
heavily damped to respond.
Clustering occurs when
Random walks
A good starting point to model randomly moving fluids is to analyse random
walks. A simple random walk is defined by:
Statistics of the random ‘kicks’:
Correlated random walks
The correlated random walk is the simplest model
showing a clustering effect: it exhibits path coalescence:
Equation of motion:
Statistics of the impulse field,
:
Lyapunov exponent for coreelated random
walk
The small separation between two nearby walks satisfies the linear equation:
The Lyapunov exponent is the rate of exponential increase of separation of nearby
trajectories: define
Find:
Lyapunov exponent and path coalescence
We found:
Expanding the logarithm, for weak kicks we obtain
The Lyapunov exponent becomes positive for large kicks:
The central limit theorem implies that the probability
distribution of the logarithm of the separations is
Gaussian distributed. If the Lyapunov exponent is
negative, paths almost surely coalesce:
Particles with inertia
Equation of motion
Can be written as two first-order equations:
Statistics of the noise:
We wish to determine the behaviour of the small separation
of two
trajectories. This approaches zero if the Lyapunov exponent is negative:
Linearised equations
Linearised equations for small separations of position and momentum:
A change of variables decouples one equation:
The Lyapunov exponent may be calculated by evaluating an average:
Diffusion approach
When the correlation time of the random force is sufficiently short, the equation of
motion is approximated by a Langevin equation:
Diffusion constant is obtained by the usual approach:
Generalised diffusion equation:
Exact solution in one dimension
Reduce to dimensionless variables:
Steady state diffusion equation then takes the form:
The steady-state solution has a constant probability flux:
Exact solution is determined by the integrating factor method
(Determine
by normalisation).
Lyapunov exponent and phase transition
The probability density is used to calculate the Lyapunov exponent:
There is a phase transition: particles cease to aggregate at
Transformation to a ‘quantum’ problem
In two or more dimensions there is no exact solution. In the limit
,
has a Gaussian solution,
. This suggests seeking a connection with the
quantum harmonic oscillator. Use Dirac notation for the Fokker-Planck equation:
Consider the transformation
Note that
,
.
is the Hamiltonian operator for a simple harmonic oscillator.
Raising and lowering operators
Let
be the
th eigenfunction of
Introduce annihilation and creation operators:
with the following properties:
In terms of these operators, the Fokker-Planck equation is
Perturbation theory
Expand solution as a power series:
Formal solution:
To produce concrete expressions, expand
Finally, use
to determine
in eigenfunctions of
Results of perturbation theory
Perturbation series has rapidly growing coefficients:
A finite expression may be obtained by Borel summation:
The Borel sum is replaced by one of its Pade
approximants.
Clustering in Mixing Flows
Simulation of particles in a random twodimensional flow:
Reported experimental realisation:
Lycopodium powder in turbulent
channel flow.
This simulation from: M.Wilkinson and B.Mehlig,
Europhys. Lett. 71, 186, (2005).
J.R.Fessler, J.D.Kulick and J.K.Eaton,
Phys.Fluids, 6, 3742, (1994).
Clustering and the Lyapunov dimension
The Lyapunov dimension or Kaplan-Yorke dimension is an estimate of the fractal
dimension of a clustered set which is generated by the action of a dynamical
system. The dimension
is estimated from the Lyapuov exponents,
.
Consider two-dimensional case. A small element of area is stretched
by the action of the flow. Schematically:
When estimating the fractal dimension, take:
Lyapunov dimension formula
in two-dimensions:
Quantifying clustering: the dimension deficit
We considered the fractal dimension of the set onto which the particles cluster.
We calculated the Lyapunov dimension (Kaplan-Yorke):
where the dimension deficit
the particle motion:
is expressed in terms of the Lyapunov exponents of
J.L.Kaplan and J.A.Yorke, Lecture Notes in Mathematics,
730, (1979).
More simply: clustering occurs when volume element
probability unity.
contracts with
Dimension deficit for turbulent flow
These are data from simulation of particles in a turbulent Navier-Stokes
flow. Data from:
J. Bec, Biferale, G.Boffetta, M.Cencini, S.Musachchio and F.Toschi, submitted to Phys. Fluid.,
nlin.CD/0606024, (2006).
0.4
0.2
0.0
-0.2
0.0
1.0
2.0
Calculating the Lyapunov exponent
Equations of motion:
Linearised equations for small separation of two particles:
Change of variable:
New equations of motion:
Extract Lyapunov exponent from an expectation value:
Theory for Lyapunov exponents
Lyapunov exponents may be obtained from elements of a
matrix
satisfying a stochastic differential equation:
random
Lyapunov exponents are expectation values of diagonal elements:
We consider the case where
varies rapidly: the probability density for
satisfies a Fokker-Planck equation:
Perturbation theory
Fokker-Planck equation is transformed to a perturbation problem:
perturbation parameter is a dimensionless measure of inertia of particles
Transformed operators are constructed from harmonic oscillator raising and
lowering operators:
Lowering and raising operators have simple action on eigenfunctions of
Perturbation series
We obtain exact series expansions for Lyapunov exponents:
K.P.Duncan, B.Mehlig, S.Ostlund and M.Wilkinson, Phys. Rev. Lett., 95, 240602, (2005).
It is very surprising that the coefficients are rational numbers. For one dimensional
case, series is
Same coefficients occur in studies of random graphs, hashing, e.g.
P. Flojolet et al, On the analysis of linear probing hashing, Algorithmica, 22, 490, (1998).
J. Spencer, Enumerating graphs and Brownian motion, Comm.Pure.Appl.Math.,1,291, (1997).
Comparison with Borel summation of series
We compared dimension deficit of particles in a Navier-Stokes flow with Borel
summation of our divergent series expansion. The strain-rate correlation function for
turbulent flow is not known, so we fitted the Kubo number to make the horizontal
scales agree.
Curves show good agreement for
0.258
The theoretical (lower) curve is
derived from a model which is
exact in the limit as
,
where the ‘centrifuge effect’
cannot operate.
Summary
The clustering of particles in a turbulent (random) flow can be explained by
considering diffusion processes, working at two different levels.
First, the turbulent flow is modelled as a random process, so that a particle
suspended in the flow undergoes a random walk.
Secondly, if the clustering effect is characterised by considering Lyapunov
exponents, the values of the Lyapunov exponents are determined by a diffusion
process.
Ours is the first quantitative theory for clustering effect: the fractal dimension is
obtained from series expansions of Lyapunov exponents, for short correlationtime flow.
We can explain 87% of the dimension deficit of particles embedded in NavierStokes turbulence from our analysis of a short-time correlated random flow:
the ‘centrifuge effect’ appears to be of minor importance.