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Particle transport in turbulence and the role of inertia

Particle transport in turbulence and the role of inertia. Singularities, fractals,and random uncorrelated motion. Michael Reeks School of Mechanical & Systems Engineering University of Newcastle-upon-Tyne, UK. Definition of particle inertia. Turbulent gas/solid flows.

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Particle transport in turbulence and the role of inertia

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

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

  3. Overview of scales in turbulent clouds Turbulence: Large scales: L0~ 100 m, t0~ 103 s, u0~ 1 m/s, Small scales: Lk~ 1 mm, tk~ 0.04 s, uk~ 0.025 m/s. Droplets: Radius: Inertia: Settling velocity: Formation: rd~ 10-7 m, St = td/tk~ 2 × 10-6, vT/uk~ 3 × 10-5 Microscales: rd~ 10-5 m, St = td/tk~ 0.02, vT/uk~ 0.3 Rain drops: rd~ 10-3 m, St = td/tk~ 200, vT/uk~ 3000 CONDENSATION COLLISIONS / COALESCENCE Collisions / coalescence process vastly enhanced if droplet size distribution at microscales is broad Workshop on Turbulence in Clouds

  4. 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 Workshop on Turbulence in Clouds

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

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

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

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

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

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

  11. Segregation in isotropic turbulence Workshop on Turbulence in Clouds

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

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

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

  15. Compression - fractional change in elemental volume of particles along a particle trajectory can be obtained directly from solving the eqns. of motion x(t),v(t),Jij(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 Workshop on Turbulence in Clouds

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

  17. Deformation Tensor J Workshop on Turbulence in Clouds

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

  19. Compressibility Workshop on Turbulence in Clouds

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

  21. Caustics - Wilkinson Workshop on Turbulence in Clouds

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

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

  25. r2 r1 Turbulent Agglomeration Saffman & Turner model Two colliding spheres volume v1, v2 test particle Collision sphere • 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 Workshop on Turbulence in Clouds

  26. Ghost = interpenetrating Finite particles = elastic particles DNS -5%, 25% agglomeration 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 rc Workshop on Turbulence in Clouds

  27. Probabalistic Methods Workshop on Turbulence in Clouds

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

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

  30. 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 Workshop on Turbulence in Clouds

  31. 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) Workshop on Turbulence in Clouds

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