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Modeling and Direct Simulation of Velocity Fluctuations and Particle-Velocity Correlations in Sedimentation

[+] Author and Article Information
F. R. Cunha, G. C. Abade, A. J. Sousa

Department of Mechanical Engineering, University of Brası́lia, Campus Universitário, 70910-900 Brası́lia-DF, Brazil

E. J. Hinch

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

J. Fluids Eng 124(4), 957-968 (Dec 04, 2002) (12 pages) doi:10.1115/1.1502665 History: Received April 21, 2000; Received April 30, 2002; Online December 04, 2002
Copyright © 2002 by ASME
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References

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Figures

Grahic Jump Location
Representation of a typical lattice used in the simulations. The particles are randomly distributed in a periodic cell with ϕ=0.03. (a) Side view; (b) three-dimensional perspective view.
Grahic Jump Location
Time evolution of the dimensionless gap between two unequal sedimenting spheres. The figure is for an aspect ratio of λls=1.75 with upstream impact parameter λl. In the inset are represented three steps of the time evolution, being (b) the step of minimum interparticle gap.
Grahic Jump Location
Dimensionless settling velocity as a function of ϕ1/3 for a simple cubic arrangement of particles. The numerical results for point-particle approximation (○) and including the finite size of the particle (•) are shown in comparison with the low ϕ asymptotic solution of and Sangani-Acrivos 40 (solid curve).
Grahic Jump Location
Dimensionless settling velocity as a function of the solid volume fraction. Simulations results (•) are shown in comparison with the low ϕ asymptotic result of Batchelor 3 (solid curve), the Brady-Durlofsky 41 result (dashed curve) and the Richardson-Zaki correlation 2 (dashed-dotted curve).
Grahic Jump Location
The settling velocity, nondimensionalized by U0, as a function of the total solid volume fraction for a bimodal size suspension. Simulation results for small (□) and large (•) species are shown in comparison with the low ϕ asymptotic result of Batchelor-Wen 43 (solid curve) and the Davis-Gecol correlation 42 (dashed curve). The simulations were performed over 100 random and equally probable configurations. The system is comprised of 1000 particles in a cubic periodic cell. The results are for ϕsl=ϕ/2 and λls=2.
Grahic Jump Location
Dimensionless horizontal density fluctuation obtained over 100 random and independent configurations as a function of the number of particles.
Grahic Jump Location
Dimensionless velocity fluctuation for a monodisperse suspension as a function of the system parameter ϕl/a. The simulations were performed over 100 random and equally probable configurations. The system is comprised of 300 particles in the unit cell with periodic sides and impenetrable boundaries perpendicular to gravity. The dashed lines are the linear fit: (a) 〈U2〉/U0=0.79ϕl/a; (b) 〈U2〉/U0=0.20ϕl/a.
Grahic Jump Location
Dimensionless vertical velocity fluctuation for a bidisperse suspension as a function of the system parameter ϕl/a. The simulations were performed over 100 random configurations. The system is comprised of 300 particles in the unit cell with periodic sides and impenetrable box. The results are for ϕsl=ϕ/2 and λls=2. The dashed lines are the linear fit: (a) 〈U2〉/U0=1.400ϕl/a; (b) 〈U2〉/U0=1.345ϕl/a.
Grahic Jump Location
Typical dynamic simulation of particle configuration at different times during sedimentation: (a) monodisperse sedimentation for a/l=0.05,h/l=3,N=286;ϕ=0.05; (b) bimodal sedimentation for ϕ=0.05(N=185),ϕsl=0.025 and aspect ratio λls=1.5;h/l=3.
Grahic Jump Location
Time evolution of the dimensionless horizontal density number fluctuations at different conditions of the simulated system with the aspect ratio h/l=3. (□): a/l=0.05;ϕ=0.03(N=172), (•): a/l=0.06;ϕ=0.02(N=66).
Grahic Jump Location
Normalized velocity fluctuation auto-correlation functions parallel, C (□) and perpendicular, C (▵) to the gravity direction. (a) Computer simulations for h/l=3,a/l=0.05,N=114⇒ϕ=0.02; (b) Computer simulations for h/l=3,a/l=0.05,N=172⇒ϕ=0.03. The error bars represent experimental data 25 with ϕ=0.05,h/l=4,h/d=10 and d/a≈100. The dashed lines indicate the uncertainly range of the present computer simulations.
Grahic Jump Location
Dimensionless hydrodynamic self-diffusivities for h/l=3,a/l=0.05, and ϕ=3%. The dashed lines are the error bars.
Grahic Jump Location
Vertical dimensionless hydrodynamic self-diffusivity as a function of the scaling ϕ1/2(l/a)3/2. The dot line is the linear fit D=0.19aU0ϕ1/2(l/a)3/2.
Grahic Jump Location
Time developing of three-dimensional velocity-fluctuation fields across the numerical box (20×20×60) during the sedimentation process of monodisperse particles at ϕ=0.05. The dimensionless time corresponds to multiples of Stokes time a/Uo. Large-scale motions (i.e., convective currents) dominate the sedimentation process with large swirl depending on the numerical box.

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