Concentration and segregation of particles and bubbles by turbulence : a numerical investigation

Transcription

1 Concentration and segregation of particles and bubbles by turbulence : a numerical investigation Enrico Calzavarini Physics of Fluids Group University of Twente The Netherlands with Massimo Cencini CNR-ISC Roma (IT) Federico Toschi IAC-CNR Roma (IT) Detlef Lohse University of Twente (NL) Slide 1

2 Phenomenology of particles in turbulence 1) Ejection/injection of heavy/light particles from/in vortices preferential concentrations Maxey (1987), Squires & Eaton (1991), Fessler Eaton (1994) 2) Finite response time to fluid fluctuations (smoothing of fast time scales) Maxey JFM87, Squires & Eaton PF91, Fessler Eaton IJMF94 Slice ~512x512x8 η from DNS heavy particles ρ p >> ρ f (red) and light particles ρ p << ρ f (blue) Stokes numbers St = τ p /τ η = 0.1,1,4.1 (left to right) Slide 2

3 Phenomenology of particles in turbulence 1) Ejection/injection of heavy/light particles from/in vortices preferential concentrations Maxey (1987), Squires & Eaton (1991), Fessler Eaton (1994) 2) Finite response time to fluid fluctuations (smoothing of fast time scales) Slice ~512x512x8 η from DNS heavy particles ρ >> ρ p f (red) and light particles ρ p << ρ f (blue) Stokes number St = 0.1 St = 1 St = 4.1 Slide 3

4 Objective study statistical properties and correlation with the carrier flow structures of inertial particle clusters for a wide range of density ratios and response times quantitative measure of the effects of particle inertia in turbulent flows. Slide 4

5 Numerical Simulations Incompressible flow field u Navier-Stokes - Homogeneous isotropic turbulence - Large scale forcing - Periodic cubic domain - Re λ 78 (L 3 = ) - Re λ 180 (L 3 = ) Slide 5

6 Particle s equation of motion Particles with a <<! Particle Reynolds: (Kolmogorov scale) Re = av! << 1 a Dilute suspensions (no collisions), no gravity a 2a (essentially: Maxey & Riley Phys. Fluids 1983, T.R.Auton et al. JFM 1988) Slide 6

7 Particle s equation of motion (II) Maxey & Riley Phys. Fluids (1983) T. R. Auton J. Fluid Mech. (1987, 1988) Slide 7

8 Numerical simulations: particles Total number of particles: N tot : Grouped in ~500 types on the β-st parameter-space: fluid tracer β=1 St=0 Particles tracked for 10s T eddy Database: particle s position, velocity, acceleration, fluid velocity and gradients at particle position Slide 8

9 Outline (i) Clusters dependence on β and St (small-scale features): - attractor (Kaplan-Yorke) dimension D KY - correlation dimension D 2 - Minkowski functionals. (ii) Clusters correlation to local flow properties - particle concentration conditioned to local flow topology. Slide 9

10 (i) Kaplan-Yorke dimension: D KY Particle equations of motion defines a dissipative dynamical system Attractor s dimension in the (x,v) space: Kaplan Yorke dimension D KY D KY 6 Lyapunov exponents computed by tracking Standard ortho-normalization Gram-Schmidt procedure adopted stretching rates As in Bec Phys. Fluids (2003), Bec JFM (2005), Bec et al. Phys. Fluids (2006) Slide 10

11 Kaplan-Yorke dimension Balance between contraction and expansion D KY Heavy min. at St 0.5, D KY 2.6 D KY Light min at St 1, D KY 1.4 D KY =3 ± 0.01 β St Slide 11

12 Projection of D KY Horizontal projection Vertical projection St <3 >3 <3 <2 D KY =3 ± 0.01 β Heavy min at St.5 St Light min at St 1 Close to fractal dimension of vortex filaments in turbulence (Moisy&Jimenez JFM 04) Slide 12

13 Correlation dimension D 2 P 2 (r) Probability to find a couple of particle whose distance is below r. At r << η P 2 (r) = A r D 2 r fractal dimension hierarchy: D 2 D 1 = D KY Same features as D KY More accessible in experiments Slide 13

14 Morphological analysis of point clouds - - Put balls particle i Let r increase with radius r around each - Measure total volume, surface, mean curvature and Euler characteristic of the emerging structure Minkowski functionals provide complete morphological characterization of point cloud! Slide 14

15 Visualization of A r particles with β=3 and St=1 radius = 0.5 η r radius = 3 η radius = 10 η Slide 15

16 Minkowski functionals V µ (r) in 3D µ V µ (r) V A/6 H/(3 π) χ geometric quantity V Volume A Surface H Mean curvature χ Euler characteristic In collaboration with M. Kerscher (Munich University, Dept. Mathematics) see also: Mecke, K.R., Buchert, T. and Wagner, H. (1994). Robust morphological measures for large scale structure in the universe. Astron. Astrophys., 288, Slide 16

17 Comparison of three extreme cases β = 0, heavy β = 1, tracer Poisson distribution β = 3, bubble bubble St= particles tracer heavy Slide 17

18 Bubbles β=3, St=0.6 and Re λ = particles Slide 18

19 Heavy particles β=0, St=0.6 and Re λ = particles Slide 19

20 Volume V 0 (r) space-filling space-filling less fast due to overlaps: remote clusters heavy passive (Poisson) bubble Slide 20

21 Surface V 1 (r) Mainly isolated balls Smaller maximal surface due to clustering heavy passive (Poisson) bubble space-filling: no surface left Slide 21

22 Mean curvature V 2 (r) Convex structure Less convex: sheet type clustering Concave: holes heavy passive (Poisson) bubble Never concave: filaments Slide 22

23 Euler characteristics V 3 (r) V 3 = χ = n. components - n. tunnels + n.cavities Isolated balls No tunnels: filaments Dramatic decrease: strong clustering tunnels get blocked tunnels form heavy passive (Poisson) bubble no cavities form: sheets Slide 23

24 (ii) Concentration conditioned to local flow geometry Eigenvalues of the strain matrix Discriminator: Hyperbolic non-hyperbolic (see Chong, Perry, Cantwell PF 1990) P hyper = < N(Δ<0) / N tot > Slide 24

25 Conditioned concentration: 1-P hyper Probability to be in non-hyperbolic regions segregation of species Slide 25

26 Conclusions Summary: 1) Small-scale clustering (dissipative range) quantified by: D KY, D 2 and Minkowski functionals. 2) Concentration conditioned to local flow geometry (-> segregation) quantified by: P hyper Ongoing and future work: - Quantifying clustering at larger scales (inertial-range). - Investigate spatial statistics of dilute bi-disperse solutions. - Trapping/ejection signature in temporal velocity statistics - Trapping/ejection signature in acceleration statistics. - Modeling needed. Slide 26

27 Laplacian of Pressure Squared vorticity (enstrophy) Linked to energy dissipiation: ε = νσ 2 Threshold on Δp used as criteria to identify vortexes Here we evaluate: (see Hunt,Wray,Moin 1988) Slide 27

28 Mean Pressure laplacian at particle position Δp Maximum around St 1 St β Slide 28

29 Other signatures of vortex ejection/trapping Temporal velocity statistics Slide 29

30 Lagrangian Structure Functions In the inertial range: Velocity statistics dimensional K41-style theory gives: Although, from Eulerian SF intermittency corrections expected Power-law scalings more evident by looking at: S (p) (τ) vs. S (2) (τ) on log-log scale (ESS) or d(logs (p) (τ) )/d(logs (2) (τ) ) vs τ Slide 30

31 The Lagrangian bottleneck in fluid tracers d(logs (4) )/d(logs (2) ) Plot as in Biferale et al. PRL (2004) Compatible with current experimental measurements: Mordant et al. PRL (2001) Xu et al. PRL (2006) τ/τ η Slide 31

32 Bottleneck for particles heavy heavy d(logs (4) )/d(logs (2) ) light St=0.1 heavy light St=0.2 heavy light light light St=0.6 St=1.0 τ/τ η τ/τ η Slide 32

33 Conclusions Summary: 1) Small-scale clustering (dissipative range): D KY and D 2 2) Segregation: concentration cond. to local flow topology and/or mean pressure laplacian 3) Trapping/ejection signature in temporal velocity statistics Future work: - Quantifying clustering at larger scales (inertial-range). - Investigate spatial statistics of dilute bi-disperse solutions. - Trapping/ejection signature in acceleration statistics. - Modeling needed. Slide 33

34 Other signatures of vortex ejection/trapping Acceleration statistics trapping by vortexes strong acceleration events ejection from vortexes reduced acceleration Slide 34

35 Acceleration root mean square same a rms as the fluid =1 Trapping/ejection signature also present in high-order moments and pdfs. Slide 35

36 Acceleration s Pdf Rescaled pdfs: Pdf(a) a rms St=0.6 No collapse at small St Slide 36

37 Acceleration s Pdf Rescaled pdfs: Pdf(a) a rms St=4 Partial collapse at large St Slide 37

38 Flatness of acceleration F particle F fluid if ß>1 F 3 non collapsing pdfs Slide 38

39 bubbles & heavy particles Slide 39