1. Measuring segregation of inertial particles in turbulent flows by a Full Lagrangian approach E. Meneguz Ph.D. project: Rain in a box of turbulence Supervisor: M. W. Reeks PostDoc: R. H. A. IJzermans School of Mechanical and Systems Engineering UNIVERSITY OF NEWCASTLE 4 th IMS Workshop on Clouds and Turbulence Institute for Mathematical Sciences Imperial College London 23-25 March 2009

2. Outline Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 2 Introduction; Equations of motion of inertial particles and other equations; Compressibility of the Particle Velocity Field: - MEPVF (Eulerian) - FLA (Lagrangian) Method - flow field description - numerical simulations Results: - Compressibility of the particle vel. field in both approaches - preliminary results in DNS of HIT - moments of the particle number density (comparison with theor. predictions) Conclusions and future developments

3. State of the art Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 3 Importance of de-mixing of particles in turbulent flows: many environmental-industrial and statistical interest Preferential concentration (Crowe et al, 1993 and Maxey (1987); Sundaram and Collins (1997) and Wang et al. (1998); Recently studied from different viewpoints (Chen et al, 2006, Balkowsky et al, 2001, Sommerer & Ott (1993) and Wilkinson et al, 2005;2007) Random Uncorrelated Motion - “sling effect” (Falkovich et al. 2002) or “crossing trajectories effect” (Wilkinson et al. 2005) Fevrier et al. MEF as sum of two contributions: MEPVF and RUM; based on box-counting (EULERIAN); Osiptsov’s method (Reeks 2004, Healy and Young 2005) LAGRANGIAN

4. Equations of motion (Lagrangian frame) Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 4 Particles/droplets: spherical, rigid, identical and heavy Dilute system Effect of gravity and Brownian motion not included particle position particle velocity fluid velocity at the position of the particle normalized particle relaxation number

5. Compressibility of the PVF Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 5 Flow field incompressible: Preferential concentration: PVF compressible non-zero gradients in the particle number density Continuity equation: For sufficiently small Stokes number:

6. MEPVF and compressibility of PVF Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 6 According to Fevrier et al. 2005: MEPVF = PVF + RUM PVF = i-th cell To be obtained from finite differences

7. Lagrangian quantification of the compressibility Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 7 2D EXAMPLE: Evol. eqts. Continuity equation and averaging over all particle trajectories:

8. Model of synthetic turbulent flow 2D carrier flow field (Babiano et al. 2000): Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 8 To study effect of RUM: Threshold value: St=0.25

9. Numerical methods ( Lagrangian vs Eulerian ) Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 9 10,000 inertial particles, uniformly distributed at t=0; Periodic boundary conditions for particles; Trajectories and equations for calculated by RK4 method; Initial conditions: (Volume is initially a cube)‏. 1,000,000 inertial particles, uniformly distributed at t=0; Trajectories calculated by RK4 method; Periodic boundary conditions for particles; Divergence of PVF calculated using 2 nd order finite difference scheme; Numerical resolution varied between 10 2 and 60 2 cells.

10. Compressibility in time Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 10 St=0.05St=0.2 St=0.5 St=2

11. Influence of numerical resolution Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 11 Lagrangian method corresponds to limiting case for infinitely fine grid in Eulerian box-counting method Resolution: 10 2 cells ---- 20 2 cells ---- 60 2 cells ---- d /dt ---- St=0.2

12. Compressibility in DNS of HIT Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 12 Good agreement between Lagrangian and Eulerian method Singularities seem to be detected better by Lagrangian method St=1 Picciotto et al. 2005

13. Moments of particle number density Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 13 Along particle trajectory: particle number density n related to J by : Particle averaged value of is related to spatially averaged value: Trivial limits: (equivalent to counting particles) Any space-averaged moment is readily determined, if J is known for all particles in the sub-domain

14. Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 14 St=0.2St=0.5 Particle number density is spatially strongly intermittent Sudden peaks indicate singularities in particle velocity field Moments of particle number density

15. Comparison with analytical estimate Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 15 If St is sufficiently small: In agreement with Balkovsky et al (2001, PRL) Trivial limits:

16. Conclusions Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 16 The Lagrangian and Eulerian method show good agreement in the compressibility of the PVF for a wide range of St; Singularities in the PVF can be detected by Lagrangian method but not by Eulerian method, due to finite grid size in the latter; Lagrangian method allows for determination of any space-averaged moment of the particle number density, in contrast with Eulerian which would have too limited spatial resolutions; The determination of moments of the particle number density have shown very high spatial intermittency due to RUM. For first time, numerical support for theory of Balkovsky et al (2001, PRL): “  is convex function of  ”.

17. Further developments Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 17 Open question: high intermittency in particle number density in DNS? 3D DNS of stationary HIT for different St numbers pdf methods for two particle dispersion at higher Re numbers

18. Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 18 Thank you for your attention