1. Workshop on Turbulence in Clouds Particle transport in turbulence and the role of inertia Michael Reeks School of Mechanical & Systems Engineering University of Newcastle-upon-Tyne, UK Singularities, fractals,and random uncorrelated motion

2. Workshop on Turbulence in Clouds Definition of particle inertia Turbulent gas/solid flows Dilute mixture/ one way coupling Scaling Parameters in Shear Flows

3. Workshop on Turbulence in Clouds Overview of scales in turbulent clouds Turbulence: Large scales:L 0 ~ 100 m,  0 ~ 10 3 s,u 0 ~ 1 m/s, Small scales:L k ~ 1 mm,  k ~ 0.04 s,u k ~ 0.025 m/s. Droplets:Radius:Inertia:Settling velocity: Formation:r d ~ 10 -7 m,St =  d /  k ~ 2 × 10 -6,v T /u k ~ 3 × 10 -5 Microscales:r d ~ 10 -5 m,St =  d /  k ~ 0.02,v T /u k ~ 0.3 Rain drops:r d ~ 10 -3 m,St =  d /  k ~ 200,v T /u k ~ 3000 COLLISIONS / COALESCENCE CONDENSATION Collisions / coalescence process vastly enhanced if droplet size distribution at microscales is broad

4. Workshop on Turbulence in Clouds Purpose and Objectives Overview / Historical Development Relevance to Cloud Physics –Segregation / demixing /collisions/ agglomeration Analogies and similarities to related processes –Deposition in a turbulent boundary layer Role of KS and DNS Awareness and appreciation

5. Workshop on Turbulence in Clouds Outline Turbulent diffusion –Homogeneous turbulence Particle diffusion coefficients Crossing trajectories –Simple shear –Inhomogeneous turbulence Turbulent boundary layer Segregation –Characteristics –Agglomeration

6. Workshop on Turbulence in Clouds Particle dispersion in homogeneous stationary turbulence Fundamental result

7. Workshop on Turbulence in Clouds Diffusion coefficient versus inertia K. Squires PhD thesis

8. Workshop on Turbulence in Clouds Diffusion coefficient versus drift Crossing trajectories Yudine (1959) Csanady (1970?) Wells & Stock (1983) Wang-Stock (1988)

9. Workshop on Turbulence in Clouds Segregation Quantifying segregation –Historical development –Compressibility –Singularities –Random uncorrelated motion –Radial distribution function Agglomeration –Simulation –Probabalistic methods

10. Workshop on Turbulence in Clouds particle motion in vortex and straining flow Stokes number St ~1

11. Workshop on Turbulence in Clouds Segregation in isotropic turbulence

12. Workshop on Turbulence in Clouds Segregation simple random flow field

13. Workshop on Turbulence in Clouds Settling in homogeneous turbulence, Maxey 1988, Maxey & Wang 1992, Davila & Hunt Maxey & Wang v g >v g (0) Davila & Hunt: settling around free vortices v g >,<v g (0 ) g

14. Workshop on Turbulence in Clouds Compressibility of a particle flow Falkovich, Elperin,Wilkinson, Reeks zero for particles which follow an incompressible flow non zero for particles with inertia measures the change in particle concentration Divergence of the particle velocity field along a particle trajectory particle streamlines Compressibility (rate of compression of elemental particle volume along particle trajectory)

15. Workshop on Turbulence in Clouds can be obtained directly from solving the eqns. of motion x(t),v(t),J ij (t),J(t))  Avoids calculating the compressibility via the particle velocity field  Can determine the statistics of ln J(t) easily.  The process is strongly non-Gaussian Compression - fractional change in elemental volume of particles along a particle trajectory

16. Workshop on Turbulence in Clouds Particle trajectories in a periodic array of vortices

17. Workshop on Turbulence in Clouds Deformation Tensor J

18. Workshop on Turbulence in Clouds Singularities in a particle concentration

19. Workshop on Turbulence in Clouds Compressibility

20. Workshop on Turbulence in Clouds Intermittency – Balkovsky, Falkovich (2001), Ijzermans et al (2008) Moments of the spatially averaged number density, St=.5

21. Workshop on Turbulence in Clouds Caustics - Wilkinson

22. Random uncorrelated motion Quasi Brownian Motion - Simonin et al Decorrelated velocities - Collins Crossing trajectories - Wilkinson RUM - Ijzermans et al. Free flight to the wall - Friedlander (1958) Sling shot effect - Falkovich Falkovich and Pumir (2006)

23. Workshop on Turbulence in Clouds Radial distribution function g(r) r g(r)

24. Workshop on Turbulence in Clouds Compressibility in DNS isotropic turbulence Piccioto and Soldati (2005)

25. Workshop on Turbulence in Clouds Turbulent Agglomeration Two colliding spheres volume v 1, v 2 r1r1 r2r2 test particle Saffman & Turner model Agglomeration in DNS turbulence L-P Wang et al. critically examined S&T model Frozen field versus time evolving flow field Absorbing versus reflection Brunk et al. – used linear shear model to asess influence of persistence of strain rate, boundary conditions, rotation Collision sphere

26. Workshop on Turbulence in Clouds Agglomeration of inertial particles Sundarim & Collins(1997), Reade & Collins (2000): measurement of rdfs and impact velocities as a function of Stokes number St Net relative velocity between colliding spheres along their line of centres RDF at r c Ghost = interpenetrating Finite particles = elastic particles DNS -5%, 25% agglomeration

27. Workshop on Turbulence in Clouds Probabalistic Methods

28. Workshop on Turbulence in Clouds Kinetic Equation and its Moment equations Zaichik, Reeks,Swailes, Minier) momentum w = relative velocity between identical particle pairs, distance r apart Δu(r) = relative velocity between 2 fluid pts, distance r apart Structure functions Net turbulent Force (diffusive) mass convection β = St -1 Probability density(Pdf) mass

29. Workshop on Turbulence in Clouds Kinetic Equation predictions Zaichik and Alipchenkov, Phys Fluids 2003

30. Workshop on Turbulence in Clouds Dispersion and Drift in compressible flows ( Elperin & Kleorin, Reeks, Koch & Collins, Reeks ) w(r,t) the relative velocity between particle pairs a distance r apart at time t Particles transported by their own velocity field w(r,t) Conservation of mass (continuity) Random variable Only works for St<<1

31. Workshop on Turbulence in Clouds Summary Conclusions Overview –Transport, segregation, agglomeration dependence on Stokes number –Use of particle compressibility d/dt(lnJ) –Singularities, caustics, fractals, random uncorrelated motion –Measurement) and modeling of agglomeration (RDF and de-correlated velocities PDF (kinetic) approach, diffusion / drift in a random compressible flow field –New PDF approaches – statistics of acceleration points( sweep/stick mechanisms)(Coleman & Vassilicos)