1. 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)

2. 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.

3. 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.

4. Unmixing

5. Clustering in Mixing Flows Simulation 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).

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

7. 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 : 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:

8. Dimensionless parameters Parameters of the model: From these we can form two independent dimensionless parameters: Stokes number: Kubo number: For fully developed turbulence.

9. 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 short- lived. If the Stokes number is too small, the particles are too heavily damped to respond. Clustering occurs when

10. 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’:

11. 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, :

12. 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:

13. 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:

14. 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:

15. 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:

16. 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:

17. 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).

18. 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

19. 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.

20. 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

21. Perturbation theory Expand solution as a power series: Formal solution: To produce concrete expressions, expand in eigenfunctions of Finally, use to determine

22. 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.

23. Clustering in Mixing Flows Simulation 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).

24. 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:

25. 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 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).

26. 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). 0.0 -0.2 0.2 0.4 0.0 1.02.0

27. 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:

28. 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:

29. 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

30. 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).

31. 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.

32. 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.