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Clustering and collisions of heavy particles in random smooth flows

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Abstract

Finite-size impurities suspended in incompressible flows distribute inhomogeneously, leading to a drastic enhancement of collisions. A description of the dynamics in the full position-velocity phase space is essential to understand the underlying mechanisms, especially for polydisperse suspensions. These issues are studied here for particles much heavier than the fluid by means of a Lagrangian approach. It is shown that inertia enhances collision rates through two effects: correlation among particle positions induced by the carrier flow and uncorrelation between velocities due to their finite size. A phenomenological model yields an estimate of collision rates for particle pairs with different sizes. This approach is supported by numerical simulations in random flows.

© 2005 American Institute of Physics
DOI: http://dx.doi.org/10.1063/1.1940367
Received 08 July 2004 Accepted 04 May 2005 Published online 24 June 2005
Acknowledgments:

The authors thank A. Vulpiani for motivating them to start this study. They warmly acknowledge F. Cecconi and B. Marani for their continuous support. A.C. acknowledges hospitality and support from the INFM Center SMC. This work was supported by the European Union Network “Fluid mechanical stirring and mixing” under Contract No. HPRN-CT-2002-00300.
Article outline:
I. INTRODUCTION
II. DYNAMICS AND STATISTICS OF DILUTE SUSPENSIONS
A. Lagrangian statistics
III. MODELFLOWS
IV. LOCAL DYNAMICS OF MONODISPERSE SUSPENSIONS
V. AN EXTENSION TO POLYDISPERSE SUSPENSIONS
VI. PHENOMENOLOGICAL MODEL FOR THE COLLISION KERNEL
VII. CONCLUDING REMARKS

Key Topics

Cluster analysis
29.0
Suspensions
26.0
Turbulent flows
21.0
Particle velocity
16.0
Attractors
15.0
Collision theories
15.0
Rheology and fluid dynamics
11.0
Lagrangian mechanics
8.0
Stokes flows
7.0
Fluid flows
6.0
0
29.0
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Clustering and collisions of heavy particles in random smooth flows
Phys. Fluids 17, 073301 (2005); http://dx.doi.org/10.1063/1.1940367
/content/figure/aip/journal/pof2/17/7/10.1063/1.1940367.f1
FIG. 1.

(Color online). (a) Scaling exponents μ of the density correlations and γ of the rates of approach as a function of the Stokes number for the time-correlated Gaussian random flow. Time averages are performed over (5×106)τf . (b) Same as in (a) for the shear flow with T=L=1 , U=23‾‾‾√ . Time averages are over (2×109)T . In both cases the exponents are measured as the mean logarithmic derivative of K over two to three decades in r .

aip/journal/pof2/17/7/1.1940367.online.f1_thmb.gif
aip/journal/pof2/17/7/1.1940367.online.f1.gif
Image of FIG. 1.

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FIG. 1.

(Color online). (a) Scaling exponents μ of the density correlations and γ of the rates of approach as a function of the Stokes number for the time-correlated Gaussian random flow. Time averages are performed over (5×106)τf . (b) Same as in (a) for the shear flow with T=L=1 , U=23‾‾‾√ . Time averages are over (2×109)T . In both cases the exponents are measured as the mean logarithmic derivative of K over two to three decades in r .
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FIG. 2.

Modulus of the particle velocities as a function of their positions for two different values of the Stokes number (a) S=10−3 and (b) S=1 . At small S the surface identified by the particles velocities is very close to that of the modulus of the Eulerian velocity field, meaning that at fixed spatial position the distribution of the particle velocity is sharply peaked on the fluid one. For larger Stokes numbers, the attractor folds in the velocity direction allowing particles to be very close with very different velocities.

aip/journal/pof2/17/7/1.1940367.online.f2_thmb.gif
aip/journal/pof2/17/7/1.1940367.online.f2.gif
Image of FIG. 2.

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FIG. 2.

Modulus of the particle velocities as a function of their positions for two different values of the Stokes number (a) S=10−3 and (b) S=1 . At small S the surface identified by the particles velocities is very close to that of the modulus of the Eulerian velocity field, meaning that at fixed spatial position the distribution of the particle velocity is sharply peaked on the fluid one. For larger Stokes numbers, the attractor folds in the velocity direction allowing particles to be very close with very different velocities.
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FIG. 3.

(Color online) Snapshot of the positions of N=4×105 particles associated to two different values of the Stokes number, S=0.4 (black) and S=0.5 (gray, red online) for the random flow (9) . The upper-left inset shows a zoom illustrating the effects induced by the difference in Stokes number (see text for details).

aip/journal/pof2/17/7/1.1940367.online.f3_thmb.gif
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FIG. 3.

(Color online) Snapshot of the positions of N=4×105 particles associated to two different values of the Stokes number, S=0.4 (black) and S=0.5 (gray, red online) for the random flow (9) . The upper-left inset shows a zoom illustrating the effects induced by the difference in Stokes number (see text for details).
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FIG. 4.

Rescaled cumulative probability P1,2(r) that two particles are at a distance smaller than r (lower curve) and cumulative approaching rate K1,2(r) (upper curve) as a function of r/r∗ . Results are shown for S=0.18 for different values of θ as in the label. Note the collapse of the different curves; the dashed lines indicate the scaling behavior below and above r∗ . It is worth also noticing that for r<r∗ both P and K scale as r2 while for r>r∗ they scale as rμ(S)+1 and rγ(S)+1 , respectively. As one can see for such small values of S γ≈μ+1 . Data refer to simulations in the random shear flow (10) with T=1 , L=1 , U=23‾‾‾√ . The curve for P1,2 has been shifted down for plotting purposes.

aip/journal/pof2/17/7/1.1940367.online.f4_thmb.gif
aip/journal/pof2/17/7/1.1940367.online.f4.gif
Image of FIG. 4.

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FIG. 4.

Rescaled cumulative probability P1,2(r) that two particles are at a distance smaller than r (lower curve) and cumulative approaching rate K1,2(r) (upper curve) as a function of r/r∗ . Results are shown for S=0.18 for different values of θ as in the label. Note the collapse of the different curves; the dashed lines indicate the scaling behavior below and above r∗ . It is worth also noticing that for r<r∗ both P and K scale as r2 while for r>r∗ they scale as rμ(S)+1 and rγ(S)+1 , respectively. As one can see for such small values of S γ≈μ+1 . Data refer to simulations in the random shear flow (10) with T=1 , L=1 , U=23‾‾‾√ . The curve for P1,2 has been shifted down for plotting purposes.
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FIG. 5.

Sketch of the different regions in the (a1,a2) plane corresponding to different contributions of κ1,2 to the effective collision rates.

aip/journal/pof2/17/7/1.1940367.online.f5_thmb.gif
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FIG. 5.

Sketch of the different regions in the (a1,a2) plane corresponding to different contributions of κ1,2 to the effective collision rates.
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FIG. 6.

(Color online). Typical functional shape (bold line) of the collision kernel, represented here as a function of a2 through a cut with a1=0.02 fixed. We chose here γ=1.8 and the numerical factors and constants to fit the order of magnitude obtained in our numerical experiments. The effective kernel takes one of the different functional forms (23), (25), and (26) (represented as dotted lines), depending on whether (a1,a2) is in the region A , B , or C .

aip/journal/pof2/17/7/1.1940367.online.f6_thmb.gif
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FIG. 6.

(Color online). Typical functional shape (bold line) of the collision kernel, represented here as a function of a2 through a cut with a1=0.02 fixed. We chose here γ=1.8 and the numerical factors and constants to fit the order of magnitude obtained in our numerical experiments. The effective kernel takes one of the different functional forms (23), (25), and (26) (represented as dotted lines), depending on whether (a1,a2) is in the region A , B , or C .
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FIG. 7.

Effective interparticle collision rate Q(a1,a2) obtained numerically by considering ghost collisions in the case of the time-correlated random flow. The Stokes number S and the particle radius a are related by S=2ρpa2/(9ρfL2) with the choice ρp/ρf=4.5×103 . To obtain these rates, interparticle statistics are computed for 100 different values of the Stokes number.

aip/journal/pof2/17/7/1.1940367.online.f7_thmb.gif
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FIG. 7.

Effective interparticle collision rate Q(a1,a2) obtained numerically by considering ghost collisions in the case of the time-correlated random flow. The Stokes number S and the particle radius a are related by S=2ρpa2/(9ρfL2) with the choice ρp/ρf=4.5×103 . To obtain these rates, interparticle statistics are computed for 100 different values of the Stokes number.
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FIG. 8.

(Color online). Collision rate Q(a1,a2) obtained with the same settings as for Fig. 7 and represented here for equal-size particles, i.e., a1=a2 (solid curve, black online) and as a function of a2 for two fixed values of a1 : a1=1.58×10−2 (dashed curve, blue online), (dotted curve, red online) a1=3.54×10−2 .

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FIG. 8.

(Color online). Collision rate Q(a1,a2) obtained with the same settings as for Fig. 7 and represented here for equal-size particles, i.e., a1=a2 (solid curve, black online) and as a function of a2 for two fixed values of a1 : a1=1.58×10−2 (dashed curve, blue online), (dotted curve, red online) a1=3.54×10−2 .
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FIG. 9.

(Color online). Solid curve (black online): collision rate Q(a,a) for equal-size particles, normalized by that obtained when neglecting particle inertia, i.e., the Saffman and Turner result equation (20) . Dashed curve (blue online): collision rate obtained when neglecting correlations between the velocity difference and the density (i.e., assuming that the first is just proportional to the separation between the particles), normalized in the same way. These curves were obtained numerically with the same setting as for Fig. 7 ; the Saffman–Turner kernel was evaluated by following tracer particles advected by the same flow.

aip/journal/pof2/17/7/1.1940367.online.f9_thmb.gif
aip/journal/pof2/17/7/1.1940367.online.f9.gif
Image of FIG. 9.

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FIG. 9.

(Color online). Solid curve (black online): collision rate Q(a,a) for equal-size particles, normalized by that obtained when neglecting particle inertia, i.e., the Saffman and Turner result equation (20) . Dashed curve (blue online): collision rate obtained when neglecting correlations between the velocity difference and the density (i.e., assuming that the first is just proportional to the separation between the particles), normalized in the same way. These curves were obtained numerically with the same setting as for Fig. 7 ; the Saffman–Turner kernel was evaluated by following tracer particles advected by the same flow.
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Clustering and collisions of heavy particles in random smooth flows
Phys. Fluids 17, 073301 (2005); http://dx.doi.org/10.1063/1.1940367
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Scitation: Clustering and collisions of heavy particles in random smooth flows
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