1 / 71

Elena Meneguz * M. W. Reeks

Elena Meneguz * M. W. Reeks. *School of Mechanical & Systems Engineering, Newcastle University, U.K. Statistical properties of particle segregation in homogeneous isotropic turbulence Singularities , intermittency and random uncorrelated motion. DUST STORM/TORNADOS. VOLCANIC ERUPTIONS.

korbin
Download Presentation

Elena Meneguz * M. W. Reeks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Elena Meneguz* M. W. Reeks *School of Mechanical & Systems Engineering, Newcastle University, U.K. Statistical properties of particle segregation in homogeneous isotropic turbulence Singularities , intermittency and random uncorrelated motion DUST STORM/TORNADOS VOLCANIC ERUPTIONS RAIN CLOUDS

  2. Unmixing by turbulent flows particles vorticity Stokes no St=0.1 St=1.0 St=10.0 Wang & Maxey JFM 1993 Caustics (Wilkinson & Mehlig 2007) ‘Particle segregation ’ Open Stats Phys 6-3-12

  3. Random uncorrelated motion (RUM) MEPVF+RUM Random uncorrelated motion (RUM) Mesoscopic Eulerian particle velocity field Février et. al JFM, 2005

  4. particle streamlines Compressibility of a particle flow 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 ‘Particlesegregation’ Mathematics, Strathclyde, 14/2/12

  5. Compression - fractional change in elemental volume of particles along a particle trajectory Deformation of elemental volume Measurement of the compressibility can be obtained directly from solving the particle eqns. of motion - xp(t),vp(t),Jij(t),J(t)) - Fully Lagrangian Method (FLM) Osiptsov(200) • Avoids calculating the compressibility via the particle velocity field • Can determine the statistics of ln J(t) easily. Evolution equations: ‘Particlesegregation’ Mathematics, Strathclyde, 14/2/12

  6. Particle trajectories in a periodic array of vortices ‘Particle segregation ’ Open Stats Phys 6-3-12

  7. Deformation Tensor J ‘Particle segregation ’ Open Stats Phys 6-3-12

  8. Particle average compressibility Compressibility of the pvf: Simple 2-D flow field of counter rotating vortices ‘Particlesegregation ’ Maths, Strathclyde, 13/2/12

  9. Model of synthetic turbulent flow • 3D Carrier flow field: • Fully described by 200 random Fourier modes • (Spelt & Biesheuvel 1997) • Incompressible, periodic in space • Smoothly varying in time and space • Not a solution to Navier-Stokes equations • Relatively small separation of scales • Valid for low Reynolds number turbulence Energy spectrum from Kraichnan (1970) ‘Particle segregation ’ Open Stats Phys 6-3-12

  10. Particle average compressibility Compressibility of the pvf: KS divergence compression For a given flow field, there is a threshold St below which the segregation goes on indefinitely with time, and above which the dilation prevails over segregation. ‘Particle segregation ’ Open Stats Phys 6-3-12

  11. Moments of particle number density

  12. Moments of particle number density St=0.5 St=0.1 DNS • Particle number density is spatially strongly intermittent • The segregation goes on with time! • The peaks reveal the presence of singularities! ‘Particle segregation ’ Open Stats Phys 6-3-12

  13. Singularities in the ptcl concentration field Singularities correspond to |J|=0 events Frequency of singularities Distribution of singularities St=0.5 St=1 max at St=5 : The distribution of singularities follows a Poisson curve The maximum frequency of singularity events occurs for ‘Particle segregation ’ Open Stats Phys 6-3-12

  14. Statistics of the compression C=ln|J| St=0.5 St=0.5 • The PDF of the compression looks Gaussian but..it is not! • Singularities correspond to ln|J|-> -inf..what is the cause for the deviation from Gaussianity visible on the left tail of the curve? ‘Particle segregation ’ Open Stats Phys 6-3-12

  15. Random uncorrelated motion (RUM) MEPVF+RUM Random uncorrelated motion (RUM) Mesoscopic Eulerian particle velocity field Février et. al JFM, 2005

  16. Decomposition of the compression (C) • We want to separate the mesoscopic and RUM component of the compression C=ln|J| • The domain is subdivided in 80x80x80 cells • mesoscopic contribution is evaluated as: • RUM contribution as: • where: j-th cell Compression experienced by the i-th particle in the j-th cell Average of compression for the j-th cell Average of compression calculated all over the domain

  17. The effect of RUM on C=ln|J| • It is the RUM component of the compression which causes the deviation from Gaussianity! =>Singularities and RUM are intrinsically related ‘Particle segregation ’ Open Stats Phys 6-3-12

  18. Singularities- ”sling shot”- RUM Trajectories Velocity (RUM) |J| (singularities) ‘Particle segregation ’ Open Stats Phys 6-3-12

  19. Conclusions The study is relevant to particle dispersion and de-mixing processes, first of all droplet coalescence and the onset of rain: • Segregation of particles: • the accumulation of particles in preferred • zones and shapes • Spatially random contribution (RUM) The RUM component of the velocity of nearby particle pairs The clustering of inertial particles can be quantified for any given St 2. The deviation from Gaussianity of the clustering process has shown to be due to the presence of this component 2. It is not a stationary process (as so far assumed) but goes on until particles collide with one another • Singularities instantaneous events which correspond to very large concentration The FLM is a very powerful technique; Extension to in-homogeneous case is possible Easily detected in contrast to traditional box-counting methods (due to spatial resolution limits) 2. For the first time, their distribution and has been studied and found to be Poisson ‘Particle segregation ’ Open Stats Phys 6-3-12

  20. THANKS FOR YOUR ATTENTION Any questions? Elena Meneguz , 13th European Turbulence Conference, 12-15th September 2011 20

  21. Particle trajectories in a periodic array of vortices ‘Particle segregation ’ Open Stats Phys 6-3-12

  22. Deformation Tensor J ‘Particle segregation ’ Open Stats Phys 6-3-12

  23. Singularities in particle concentration ‘Particle segregation ’ Open Stats Phys 6-3-12

  24. Model of synthetic turbulent flow • 3D Carrier flow field: • Fully described by 200 random Fourier modes • (Spelt & Biesheuvel 1997) • Incompressible, periodic in space • Smoothly varying in time and space • Not a solution to Navier-Stokes equations • Relatively small separation of scales • Valid for low Reynolds number turbulence Energy spectrum from Kraichnan (1970) ‘Particle segregation ’ Open Stats Phys 6-3-12

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

  26. 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:

  27. 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 ‘Particle segregation ’ Open Stats Phys 6-3-12

  28. 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) ‘Particle segregation ’ Open Stats Phys 6-3-12

  29. g(r) r Radial distribution function (RDF) g(r) ‘Particle segregation ’ Open Stats Phys 6-3-12

  30. Legitimate questions … What happens if we take into consideration a more complex flow field model e.g. a model with a broader range of scales and which takes into account the sweeping of the small scales by the large ones? • e.g. Do we find the same threshold value? ? Direct Numerical Simulations will give us the answer!!!

  31. 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

  32. 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 32

  33. Moments of particle number density (KS/RUM) St=0.5 St=0.05 St=0.5 DNS • Particle number density is spatially strongly intermittent • The segregation goes on with time! • The peaks reveal the presence of singularities! ‘Particle segregation ’ Open Stats Phys 6-3-12

  34. PDF of SINGULARITIES: Poisson distr.! St=0.5 St=1 St=5 Elena Meneguz34

  35. i-th cell Compressibility in the MEPVF • MEPVF = PVF + RUM (According to Février et al. 2005) • PVF = Spatially uncorrelated component (for large inertia) “sling effect” (Falkovich et al. 2002) and “crossing of trajectories” (Wilkinson & Mehlig 2005) Smoothly varying ‘Particle segregation ’ Open Stats Phys 6-3-12 Elena Meneguz35

  36. Random Uncorrelated Motion (RUM) It is the manifestion of the decorrelation of two nearby particles. Février et. al (2005) Elena Meneguz36

  37. SINGULARITIES and RUM For other Stokes numbers… Here is where singularities take place! The distribution of log|J| approaches a Gaussian for (St=0.5) ‘Particle segregation ’ Open Stats Phys 6-3-12

  38. SINGULARITIES St=1 Deviation from the Gaussian trend!!! What is the cause for this behaviour which manifests itself in the limit of log|J|->-inf, in other words |J|=0 (singularities)? ‘Particle segregation ’ Open Stats Phys 6-3-12 Elena Meneguz38

  39. Decomposition of the compression (C) • We want to separate the mesoscopic and RUM component of the compression C=ln|J| • The domain is subdivided in 80x80x80 cells • Mesoscopic contribution is evaluated as: • RUM contribution as: • where: j-th cell Compression experienced by the i-th particle in the j-th cell Average of compression for the j-th cell Average of compression calculated all over the domain

  40. Singularities and RUM(C) St=1 It’s the RUM Component that is causing the deviation! Singularities are related to RUM, but how? C=Log|J|

  41. The answer is… CAUSTICS! Wilkinson & Mehlig 2005 • The particle velocity field is multivalued as a consequence of folding; • The frequency of singularities is ultimately the frequency of activation of the caustics events • This is related to RUM (crossing of trajectory) as this takes place between caustics (1D example) • This has an impact on the rate of collision of particles: preferential concentration is not the only effect (dependency on the St number) 41

  42. Caustics, singularities and RUM

  43. Conclusions… INITIAL PROBLEM: To investigate and quantify the clustering of inertial particles in turbulent flow from a theoretical and numerical point of view. We have exploited a FLM • We have calculated quantities such as particle averaged compressibility and moments of the particle number density • We have compared results in different models of turbulent flows – from simple to more complex ones • We have investigated some detailed features such as the presence of singularities – not detected with box counting methods • We were trying to conclude our investigation by looking at a way to establish a link between the singularities and the occurrence of Random Uncorrelated Motion. • Caustics are the answer!

  44. Thank you for you attention

  45. Kinetic stress flux Work done by total stresses Particle kinetic stress transport equation kinetic stresses stresses from turbulent forces Particle-fluid velocity covariances

  46. Chapman Enskog Approximation Swailes, Sergeev at al ‘Particle segregation ’ Open Stats Phys 6-3-12

  47. Predictions versus experimental measurements Simonin et al. Work done by shear stresses ‘Particle segregation ’ Open Stats Phys 6-3-12

  48. Near-wall behaviour • influence of particle wall interactions • scattering & absorption/deposition/ bounce /resuspension particle resuspension • p, the resuspension rate constant • ω, the typical frequency of vibration, • Q height of adhesive potential well, • <PE>average potential energy of particle in the well. Q boundary conditions Particle escape from potential well ‘Particle segregation ’ Open Stats Phys 6-3-12

  49. Particle wall impacts with absorption Swailes & Reeks (1997) turbulence Vg=5 particle P(v) Critical impact velocity vc=5, settling velocity vg=5 (normalized on particle rms velocity for perfect reflection) Velocity v ‘Particle segregation ’ Open Stats Phys 6-3-12

  50. Deposition in turbulent boundary layer deposition velocity τ+ van Dijk JCP (2010) τ+ -1 van Dijk JCP (2010) -2 • rapid decay of turbulence near the wall -3 • particle not in local • equilibrium with flow 4 5 • break down of gradient • transport Zaichik & Alipchenkov, (2009) • comparison of PDF results • with experimental results 0 y+ 1 2 -2 -1 y+

More Related