A kinetic theory for the transport of small particles in turbulent flows

1. A kinetic theory for the transport of small particles in turbulent flows Michael W Reeks School of Mechanical & Systems Engineering, Newcastle University

2. Environmental /industrial processes Mixing & combustion pollutant dispersion fouling / deposition clean up radioactive releases slurry /pneumatic conveying aerosol formation

3. Modelling Particle Flows Particle Tracking (Lagrangian) track particles through a random flow by solving particle equation of motion Two-Fluid Model (Eulerian) Continuum equations for continuous (carrier flow) and dispersed phase (particles) constitutive relations /closure approximations boundary conditions

4. Objectives application of formal closure methods in dilute flows to derive continuum equations /constitutive relations for the particle phase compare with traditional heuristic approach criteria for their validity incorporate the influence of turbulent structures on particle motion into continuum equations One particle dispersion Two particle (pair) dispersion Drift in inhomogeneous turbulence

5. particle motion in plain vortex and straining flow

7. Caustics – Mehlig & Wilkinson

8. Settling in homogeneous turbulence Maxey 1988, Maxey & Wang 1992

9. Two-Fluid Model Reynolds stresses Inter-phase momentum transfer Mass X accel • Begin with particle equation of motion e.g. for gas-solid flows • Separate particle velocities & aerodynamic forces into mean & fluctuating components • Average over all realisations of the flow mean and fluctuating carrier flow velocities

10. Kinetic/ PDF Approach • analogous to the kinetic theory of gases • uses an equation analogous to the Maxwell-Boltzmann Equation to derive the two-fluid equations for a dispersed flow • mass-momentum and energy equations (c.f. RANS for continuous phase) • constitutive relations • boundary conditions

11. Two PDF approaches • Apply closure to transport equations for • <W(,x,t)> particle phase space density • <P(,up,x,t)> up is the fluid velocity seen by the particle at x (Simonin approach) 11 Modelling of Particle Flows

12. Closure approximations forReeks, Zaichek, Swailes, Minier

13. Momentum Equation - PDF Approach Turbulent stress Body force Mass x acceleration Equation of state 13 Modelling of Particle Flows

14. Momentum equation as a diffusion equation

15. Particle Reynolds stress transport eqns - Reynolds stresses depend on shearing of both phases - Requires closure for Reynolds stress flux 15

16. Chapman Enskog Approximation 16

17. Application- PDF solutions • Transport and deposition in turbulent boundary layers Deposition velocity Velocity distribution at a wall for particles settling under gravity in a turbulent flow with particla absorption at the wall particle relaxation time

18. particle streamlines along a particle trajectory • measures the change in particle concentration • zero for particles which follow an incompressible flow • non zero for particles with inertia Divergence of the particle velocity field

19. Application to kinetic approach 19

20. g(r) r r2 r1 Two colliding spheres radii r1, r2 Pair dispersion and segregation Collision sphere

21. Probability density(Pdf) mass Net turbulent Force convection β = St-1 , St=Stokes number mass momentum Kinetic Equation for P(w,r,t)and moment equations w = relative velocity between identical particle pairs, distance r apart Δu(r) = relative velocity between 2 fluid pts, distance r apart Structure functions

22. Kinetic Equation predictions Zaichik and Alipchenkov, Phys Fluids 2003

23. Summary / Conclusions • Particle transport and segregation in a turbulent flow • Kinetic / pdf approach (single particle transport) • Treatment of the dispersed particle phase as a fluid • Continuum equations • Constitutive relations • Boundary conditions (perfectly / partially absorbing) • Kinetic approach for particle pair transport • radial distribution function • Role of compressibility in the formulation of a kinetic equation • Net relative drift velocity between particle pairs • enhancement local concentration of neighbouring particles

24. Moments of particle number density St=0.05 St=0.5 • Particle number density is spatially strongly intermittent • Sudden peaks indicate singularities in particle velocity field

25. Influence of turbulent structures Consider the instantaneous concentration (x,t)derived from an initial concentration (x,0) and a particle velocity field vp(x,t). The conservation of mass equation is x,t Xp,s

26. Net particle flux Drift velocity(Maxey) Diffusion tensor D (Taylor’s theory) Dispersion in a random compressible flow

27. Segregation of inertial particles in turbulent flows ‘Fractals, singularities, intermittency, and random uncorrelated motion’ M. W. Reeks, R. IJzermans, E. Meneguz, Y.Ammar Newcastle University, UK M. Picciotto, A. Soldati University of Udine, It

28. De-mixing of particles • Particles suspended in a turbulent flow do not mix but segregate • depends upon the particles inertial response to: • structure and persistence of the turbulence • Important in mixing and particle collision processes • growth of PM10 and cloud droplets in the atmosphere • the onset of rain. • Presentation is about quantifying segregation • analysing statistics and morphology of the segregation • using a Full Lagrangian Method (FLM) • use of compressibility to reveal • fractal nature • intermittency • random uncorrelated motion

29. particle motion in a vortex and straining flow Stokes number St = τp/τf~1

30. Segregation in isotropic turbulence

31. Segregation - dependence on Stokes number St=τp/τf

32. Segregation in counter-rotating vortices Flow pattern translated randomly in space with finite life- time

33. Caustics – Mehlig & Wilkinson

34. 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

35. Compression - fractional change in elemental volume of particles along a particle trajectory Deformation of elemental volume can be obtained directly from solving the particle eqns. of motion - xp(t),vp(t),Jij(t),J(t)) - Fully Lagrangian Method • Avoids calculating the compressibility via the particle velocity field • Can determine the statistics of ln J(t) easily. • The process is strongly non-Gaussian – highly intermittent Measurement of the compressibility

36. Particle trajectories in a periodic array of vortices

37. Deformation Tensor J

38. Singularities in particle concentration

39. Compressibility KS random Fourier modes: distribution of scales, turbulence energy spectrum Simple 2-D flow field of counter rotating vortices

40. Moments of particle number density • Along particle trajectory: particle number density n related to J by: • Particle averaged value of is related to spatially averaged value: • Any space-averaged momentis readily determined, if J is known for all particles in the sub-domain (equivalent to counting particles) Trivial limits:

41. Moments of particle number density St=0.05 St=0.5 • Particle number density is spatially strongly intermittent • Sudden peaks indicate singularities in particle velocity field

42. If St is sufficiently small: Comparison with analytical estimate • For first time, numerical support for theory of Balkovsky et al (2001, PRL): “is convex function of”.

43. 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)

44. g(r) r Radial distribution function (RDF) g(r)

45. DNS: details of the code • NSE for an incompressible viscous turbulent flow: • In a DNS of HIT, the solution domain is in a cube of size L, and: • Statistically stationary HIT • Pseudo-spectral code • Grid 128x128x128 • Re =65 • Forcing is applied at the lowest wavenumbers • 100.000 inertial particles are random distributed at t=0 in a box of L=2 • Interpolation of the velocity fluid @ the particle position with a 6th order Lagrangian polynomial • Trajectories and equations calculated by RK4 method • Initial conditions so that volume is initially a cube

46. Averaged value of compressibility vs time WHAT CAUSES THE POSITIVE VALUES??? • Qualitatively the same trend with respect to KS • We expect a different threshold value Elena Meneguz 46

47. Moments of particle concentration intermittency due to the presence of singularities in the pvf

48. r2 r1 Turbulent Agglomeration Two colliding spheres radii r1, r2 Saffman & Turner model test particle Collision sphere • Agglomeration in DNS turbulence • L-P Wang et al. - examined S&T model • Frozen field versus time evolving flow field • Absorbing versus reflection • Brunk et al. - used linear shear model to assess influence of persistence of strain rate, boundary conditions, rotation

49. 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

50. r2 r1 Inertial collisions (RUM) rc=r1+r2 particle Stokes number St Ratio of the RMS of the relative velocity of colliding particles over the corresponding RMS of the relative fluid velocity; collision radius rc/ηk =0.1