Scaling of volume fraction and velocity fluctuations with domain size is investigated for high-mass-loading suspensions of finite-size inertial particles subject to gravity. Results from highly resolved Euler–Lagrange simulations are evaluated via an adaptive spatial filter with an averaging volume that varies with the local particle concentration. This filter enables the instantaneous particle velocity to be decomposed into a spatially correlated contribution used in defining the particle-phase turbulent kinetic energy (TKE), and a spatially uncorrelated contribution used in defining the granular temperature. The total granular energy is found to grow nearly linearly with the domain size, while the balance between the separate contributions remains approximately constant.

References

References
1.
Agrawal
,
K.
,
Loezos
,
P.
,
Syamlal
,
M.
, and
Sundaresan
,
S.
,
2001
, “
The Role of Meso-Scale Structures in Rapid Gas–Solid Flows
,”
J. Fluid Mech.
,
445
(
1
), pp.
151
185
.
2.
Vreman
,
B.
,
Geurts
,
B.
,
Deen
,
N.
,
Kuipers
,
J.
, and
Kuerten
,
J.
,
2009
, “
Two- and Four-Way Coupled Euler–Lagrangian Large-Eddy Simulation of Turbulent Particle-Laden Channel Flow
,”
Flow Turbul. Combust.
,
82
(
1
), pp.
47
71
.
3.
Capecelatro
,
J.
,
Pepiot
,
P.
, and
Desjardins
,
O.
,
2014
, “
Numerical Characterization and Modeling of Particle Clustering in Wall-Bounded Vertical Risers
,”
Chem. Eng. J.
,
245
, pp.
295
310
.
4.
Capecelatro
,
J.
,
Desjardins
,
O.
, and
Fox
,
R. O.
,
2014
, “
Numerical Study of Collisional Particle Dynamics in Cluster-Induced Turbulence
,”
J. Fluid Mech.
,
747
, pp.
R2 1
13
.
5.
Shaffer
,
F.
,
Gopalan
,
B.
,
Breault
,
R.
,
Cocco
,
R.
,
Karri
,
S.
,
Hays
,
R.
, and
Knowlton
,
T.
,
2013
, “
High Speed Imaging of Particle Flow Fields in CFB Risers
,”
Powder Technol.
,
242
, pp.
86
99
.
6.
Capecelatro
,
J.
,
Desjardins
,
O.
, and
Fox
,
R. O.
,
2015
, “
On Fluid–Particle Dynamics in Fully-Developed Cluster-Induced Turbulence
,”
J. Fluid Mech.
,
780
, pp.
578
635
.
7.
Guazzelli
,
E.
, and
Hinch
,
J.
,
2011
, “
Fluctuations and Instability in Sedimentation
,”
Annu. Rev. Fluid Mech.
,
43
(
1
), pp.
97
116
.
8.
Caflisch
,
R. E.
, and
Luke
,
J.
,
1985
, “
Variance in the Sedimentation Speed of a Suspension
,”
Phys. Fluids
,
28
(
3
), pp.
759
760
.
9.
Koch
,
D. L.
,
1994
, “
Hydrodynamic Diffusion in a Suspension of Sedimenting Point Particles With Periodic Boundary Conditions
,”
Phys. Fluids
,
6
(
9
), pp.
2894
2900
.
10.
Ladd
,
A.
,
1997
, “
Sedimentation of Homogeneous Suspensions of Non-Brownian Spheres
,”
Phys. Fluids
,
9
(
3
), pp.
491
499
.
11.
Cunha
,
F.
,
Sousa
,
A.
, and
Hinch
,
E.
,
2002
, “
Numerical Simulation of Velocity Fluctuations and Dispersion of Sedimentation Particles
,”
Chem. Eng. Commun.
,
189
(
8
), pp.
1105
1129
.
12.
Bergougnoux
,
L.
,
Ghicini
,
S.
,
Guazzelli
,
E.
, and
Hinch
,
J.
,
2003
, “
Spreading Fronts and Fluctuations in Sedimentation
,”
Phys. Fluids
,
15
(
7
), pp.
1875
1887
.
13.
Mucha
,
P.
,
Tee
,
S.
,
Weitz
,
D.
,
Shraiman
,
B.
, and
Brenner
,
M.
,
2004
, “
A Model for Velocity Fluctuations in Sedimentation
,”
J. Fluid Mech.
,
501
, pp.
71
104
.
14.
Nicolai
,
H.
,
Herzhaft
,
B.
,
Hinch
,
E.
,
Oger
,
L.
, and
Guazzelli
,
E.
,
1995
, “
Particle Velocity Fluctuations and Hydrodynamic Self-Diffusion of Sedimenting Non-Brownian Spheres
,”
Phys. Fluids
,
7
(
1
), pp.
12
23
.
15.
Nicolai
,
H.
, and
Guazzelli
,
E.
,
1995
, “
Effect of the Vessel Size on the Hydrodynamic Diffusion of Sedimenting Spheres
,”
Phys. Fluids
,
7
(
1
), pp.
3
5
.
16.
Nicolai
,
H.
,
Peysson
,
Y.
, and
Guazzelli
,
E.
,
1996
, “
Velocity Fluctuations of a Heavy Sphere Falling Through a Sedimenting Suspension
,”
Phys. Fluids
,
8
(
4
), pp.
855
862
.
17.
Segre
,
P. N.
,
Liu
,
F.
,
Umbanhowar
,
P.
, and
Weitz
,
D. A.
,
2001
, “
An Effective Gravitational Temperature for Sedimentation
,”
Nature
,
409
(
6820
), pp.
594
597
.
18.
Segre
,
P.
,
Herbolzheimer
,
E.
, and
Chaikin
,
P.
,
1997
, “
Long-Range Correlations in Sedimentation
,”
Phys. Rev. Lett.
,
79
(
13
), pp.
2574
2577
.
19.
Yin
,
X.
, and
Koch
,
D. L.
,
2007
, “
Hindered Settling Velocity and Microstructure in Suspensions of Solid Spheres With Moderate Reynolds Numbers
,”
Phys. Fluids
,
19
(
9
), p.
093302
.
20.
Koch
,
D. L.
,
1993
, “
Hydrodynamic Diffusion in Dilute Sedimenting Suspensions at Moderate Reynolds Numbers
,”
Phys. Fluids
,
5
(
5
), pp.
1141
1155
.
21.
Hinch
,
E.
,
1988
, “
Sedimentation of Small Particles
,”
Disorder and Mixing
,
Springer
, The Netherlands, pp.
153
162
.
22.
Yin
,
X.
, and
Koch
,
D. L.
,
2008
, “
Velocity Fluctuations and Hydrodynamic Diffusion in Finite-Reynolds-Number Sedimenting Suspensions
,”
Phys. Fluids
,
20
(
4
), p.
043305
.
23.
Maxey
,
M.
,
1987
, “
The Gravitational Settling of Aerosol Particles in Homogeneous Turbulence and Random Flow Fields
,”
J. Fluid Mech.
,
174
, pp.
441
465
.
24.
Dasgupta
,
S.
,
Jackson
,
R.
, and
Sundaresan
,
S.
,
1994
, “
Turbulent Gas–Particle Flow in Vertical Risers
,”
AIChE J.
,
40
(
2
), pp.
215
228
.
25.
Fox
,
R. O.
,
2014
, “
On Multiphase Turbulence Models for Collisional Fluid–Particle Flows
,”
J. Fluid Mech.
,
742
, pp.
368
424
.
26.
Cundall
,
P.
, and
Strack
,
O.
,
1979
, “
A Discrete Numerical Model for Granular Assemblies
,”
Geotechnique
,
29
(
1
), pp.
47
65
.
27.
Tenneti
,
S.
,
Garg
,
R.
, and
Subramaniam
,
S.
,
2011
, “
Drag Law for Monodisperse Gas–Solid Systems Using Particle-Resolved Direct Numerical Simulation of Flow Past Fixed Assemblies of Spheres
,”
Int. J. Multiphase Flow
,
37
(
9
), pp.
1072
1092
.
28.
Anderson
,
T.
, and
Jackson
,
R.
,
1967
, “
Fluid Mechanical Description of Fluidized Beds. Equations of Motion
,”
Ind. Eng. Chem. Fundam.
,
6
(
4
), pp.
527
539
.
29.
Capecelatro
,
J.
, and
Desjardins
,
O.
,
2013
, “
An Euler–Lagrange Strategy for Simulating Particle-Laden Flows
,”
J. Comput. Phys.
,
238
, pp.
1
31
.
30.
Einstein
,
A.
,
1906
, “
Eine neue Bestimmung der Moleküldimensionen
,”
Ann. Phys.
,
324
(
2
), pp.
289
306
.
31.
Thomas
,
D. G.
,
1965
, “
Transport Characteristics of Suspension: VIII. A Note on the Viscosity of Newtonian Suspensions of Uniform Spherical Particles
,”
J. Colloidal Sci.
,
20
(
3
), pp.
267
277
.
32.
Gibilaro
,
L.
,
Gallucci
,
K.
,
Di Felice
,
R.
, and
Pagliai
,
P.
,
2007
, “
On the Apparent Viscosity of a Fluidized Bed
,”
Chem. Eng. Sci.
,
62
(
1–2
), pp.
294
300
.
33.
Maxey
,
M.
,
Patel
,
B.
,
Chang
,
E.
, and
Wang
,
L.
,
1997
, “
Simulations of Dispersed Turbulent Multiphase Flow
,”
Fluid Dyn. Res.
,
20
(
1–6
), pp.
143
156
.
34.
Briley
,
W.
, and
McDonald
,
H.
,
1977
, “
Solution of the Multidimensional Compressible Navier–Stokes Equations by a Generalized Implicit Method
,”
J. Comput. Phys.
,
24
(
4
), pp.
372
397
.
35.
Eaton
,
J.
, and
Fessler
,
J.
,
1994
, “
Preferential Concentration of Particles by Turbulence
,”
Int. J. Multiphase Flow
,
20
(
Suppl. 1
), pp.
169
209
.
36.
Février
,
P.
,
Simonin
,
O.
, and
Squires
,
K.
,
2005
, “
Partitioning of Particle Velocities in Gas–Solid Turbulent Flows Into a Continuous Field and a Spatially Uncorrelated Random Distribution: Theoretical Formalism and Numerical Study
,”
J. Fluid Mech.
,
533
, pp.
1
46
.
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