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)

OverviewLecture 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 FlowsSimulation of particles in a random two-dimensional flow:

Reported experimental realisation: Lycopodium powder in turbulent channel flow.

J.R.Fessler, J.D.Kulick and J.K.Eaton, Phys.Fluids, 6, 3742, (1994).

This simulation from: M.Wilkinson and B.Mehlig, Europhys. Lett. 71, 186, (2005).

Aggregation versus clusteringFor some choices of parameters the particles aggregate instead of clustering. In the upper sequence particles have a lower mass.

Equations of motionParticles 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 :

In our calculations is a random velocity field, obtained from a vector potential:

The components have mean value zero, and correlation function

Alternative form for very low density:

Dimensionless parametersParameters of the model:

From these we can form two independent dimensionless parameters:

Stokes number:

Kubo number:

For fully developed turbulence .

Maxey’s centrifuge effectMaxey 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 short-lived. If the Stokes number is too small, the particles are too heavily damped to respond. Clustering occurs when

Random walksA 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 coalescenceWe 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 inertiaEquation 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 equationsLinearised 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 approachWhen 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 dimensionReduce 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 transitionThe probability density is used to calculate the Lyapunov exponent:

There is a phase transition: particles cease to aggregate at

Transformation to a ‘quantum’ problem

has a Gaussian solution, . This suggests seeking a connection with the quantum harmonic oscillator. Use Dirac notation for the Fokker-Planck equation:

In two or more dimensions there is no exact solution. In the limit ,

Consider the transformation , .

Note that is the Hamiltonian operator for a simple harmonic oscillator.

Raising and lowering operatorsLet 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 theoryExpand solution as a power series:

Formal solution:

To produce concrete expressions, expand in eigenfunctions of

Finally, use to determine

Results of perturbation theoryPerturbation 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 FlowsSimulation of particles in a random two-dimensional flow:

Reported experimental realisation: Lycopodium powder in turbulent channel flow.

J.R.Fessler, J.D.Kulick and J.K.Eaton, Phys.Fluids, 6, 3742, (1994).

This simulation from: M.Wilkinson and B.Mehlig, Europhys. Lett. 71, 186, (2005).

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 deficitWe considered the fractal dimension of the set onto which the particles cluster. We calculated the Lyapunov dimension (Kaplan-Yorke):

where the dimension deficit is expressed in terms of the Lyapunov exponents of the particle motion:

More simply: clustering occurs when volume element contracts with probability unity.

J.L.Kaplan and J.A.Yorke, Lecture Notes in Mathematics, 730, (1979).

Dimension deficit for turbulent flowThese 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).

0.0

-0.2

0.2

0.4

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 random matrix satisfying a stochastic differential equation:

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 theoryFokker-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 seriesWe 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 seriesWe 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 correlation-time flow. We can explain 87% of the dimension deficit of particles embedded in Navier-Stokes turbulence from our analysis of a short-time correlated random flow: the ‘centrifuge effect’ appears to be of minor importance.