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Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds

Published online by Cambridge University Press:  31 August 2007

J. J. DERKSEN  and
S. SUNDARESAN
J. J. DERKSEN
Affiliation:
Multi-Scale Physics Department, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, The Netherlandsj.j.derksen@tudelft.nl
S. SUNDARESAN
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
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Abstract

We present results of direct numerical simulations of travelling waves in dense assemblies of monodisperse spherical particles fluidized by a liquid. The cases we study have been derived from the experimental work of others. In these simulations, the flow of interstitial fluid is solved by the lattice-Boltzmann method (LBM) and the particles move under the influence of gravity, hydrodynamic forces stemming from the LBM, subgrid-scale lubrication forces and hard-sphere collisions. We first show that the propagating inhomogeneous structures seen in the simulations are in agreement with those observed experimentally. We then use the detailed information contained in the simulation results to assess aspects of two-fluid model closures, namely, fluid–particle drag, and the various contributions to the effective stresses. We show that the rates of compaction and dilation of the particle phase in the travelling waves are comparable to the rate at which the microstructure relaxes, and that there is a pronounced effect of the rate of compaction on the average collisional normal stress. Although this effect can be expressed as an effective bulk viscosity term, this approach would require the use of a path-dependent bulk viscosity. We also find that the effective fluid–particle drag coefficient can be described well with the often-used closure motivated by the experiments of Richardson & Zaki (Trans. Inst. Chem. Engng vol. 32, 1954, p. 35). In this respect, the effect of the system size for determining the drag requires specific care. The shear viscosity of the particle phase manifests small, but clearly noticeable dependence on the rate of compaction/dilation of the particle phase. Our observations point to the need for higher-order closures that recognize the slow evolution of the microstructure in these flows and account for the effects of non-equilibrium microstructure on the stresses.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 587 , 25 September 2007 , pp. 303 - 336
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S. 2001 The role of meso-scale structures in rapid gas–solid flows. J. Fluid Mech. 445, 151–185.Google Scholar
Aidun, C. K., Lu, Y. & Ding, E. J. 1998 Direct analysis of particle suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech. 373, 287–311.CrossRefGoogle Scholar
Anderson, K., Sundaresan, S. & Jackson, R. 1995 Instabilities and the formation of bubbles in fluidized beds. J. Fluid Mech. 303, 327–366.CrossRefGoogle Scholar
Anderson, T. B. & Jackson, R. 1969 A fluid mechanical description of fluidized beds – comparison of theory and experiment. Indust. Engng Chem. Fund. 8, 137–144.CrossRefGoogle Scholar
Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.CrossRefGoogle Scholar
Batchelor, G. K. 1988 A new theory of the instability of a uniform fluidized bed. J. Fluid Mech. 193, 75–110.CrossRefGoogle Scholar
Brady, J. F., Khair, A. S. & Swaroop, M. 2006 On the bulk viscosity of suspensions. J. Fluid Mech. 554, 109–123.CrossRefGoogle Scholar
Chen, S. & Doolen, G. D. 1998 Lattice–Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Cichocki, B.. & Jones, R. B. 1998 Image representation of a spherical particle near a hard wall. Physica A 258, 273–302.CrossRefGoogle Scholar
Derksen, J. & Vanden Akker, H. E. A. den Akker, H. E. A. 1999 Large-eddy simulations on the flow driven by a Rushton turbine. AIChE J. 45, 209–221.CrossRefGoogle Scholar
Didwania, A. K. & Homsy, G. M. 1981 Flow regime and flow transitions in liquid-fluidized beds. Intl J. Multiphase Flo. 7, 563–580.CrossRefGoogle Scholar
Duru, P. & Guazelli, E. 2002 Experimental investigations on the secondary instability of liquid-fluidized beds and the formation of bubbles. J. Fluid Mech. 470, 359–382.CrossRefGoogle Scholar
Duru, P., Nicolas, M., Hinch, J. & Guazelli, E. 2002 Constitutive laws in liquid-fluidized beds. J. Fluid Mech. 452, 371–404.CrossRefGoogle Scholar
Eggels, J. G. M. & Somers, J. A. 1995 Numerical simulation of free convective flow using the lattice-Boltzmann scheme. Intl J. Heat Fluid Flo. 16, 357–364.CrossRefGoogle Scholar
El-Kaissy, M. M. & Homsy, G. M. 1976 Instability waves and the origin of bubbles fluidized beds. Part 1: Experiments. Intl J. Multiphase Flo. 2, 379–395.CrossRefGoogle Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization. Academic Press, CA.Google Scholar
Glasser, B. J., Kevrekidis, I. G. & Sundaresan, S. 1996 One- and two-dimensional travelling wave solutions in gas-fluidized beds. J. Fluid Mech. 306, 183–221.CrossRefGoogle Scholar
Glasser, B. J., Kevrekidis, I. G. & Sundaresan, S. 1997 Fully developed travelling wave solutions and bubble formation in fluidized beds. J. Fluid Mech. 334, 157–188.CrossRefGoogle Scholar
Goldstein, D., Handler, R. & Sirovich, L. 1993 Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105, 354–366.CrossRefGoogle Scholar
Gondret, P., Lance, M. & Petit, L. 2002 Bouncing motion of spherical particles in fluids. Phys. Fluid. 14, 643–652.CrossRefGoogle Scholar
Griffith, B. E. & Peskin, C. S. 2005 On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208, 75–105.CrossRefGoogle Scholar
Ham, J. M., Thomas, S., Guazzelli, E., Homsy, G. M. & Anselmet, M. C. 1990 An experimental study of the stability of liquid-fluidized beds. Intl J. Multiphase Flo. 16,171–185.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
vander Hoef, M. A. der Hoef, M. A., Beetstra, R. & Kuipers, J. A. M. 2005 Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254.Google Scholar
Jackson, R. 2000 Dynamics of Fluidized particles. Cambridge University Press.Google Scholar
Joseph, G. G., Zenit, R., Hunt, M. L. & Rosenwinkel, A. M. 2001 Particle–wall collisions in a viscous fluid. J. Fluid Mech. 433, 329–346.CrossRefGoogle Scholar
Kandhai, D., Derksen, J. J. & Vanden Akker, H. E. A. den Akker, H. E. A. 2003 Interphase drag coefficients in gas–solid flows. AIChE J. 49, 1060–1065.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.Google Scholar
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for a homogeneous monodisperse gas fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229–263.CrossRefGoogle Scholar
Ladd, A. J. C. 1994a Numerical simulations of particle suspensions via a discretized Boltzmann equation. Part 1. Theoretical Foundation. J. Fluid Mech. 271, 285–309.CrossRefGoogle Scholar
Ladd, A. J. C. 1994b Numerical simulations of particle suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311–339.CrossRefGoogle Scholar
Ladd, A. J. C. 1997 Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluid. 9, 491–499.CrossRefGoogle Scholar
Li, J. & Kuipers, J. A. M. 2003 Gas–particle interactions in dense gas-fluidized beds. Chem. Engng Sci. 58, 711–718.CrossRefGoogle Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561–605.CrossRefGoogle Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow – inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluid. 26, 883–889.CrossRefGoogle Scholar
Nguyen, N.-Q. & Ladd, A. J. C. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. 1rm E. 66, 046708.CrossRefGoogle ScholarPubMed
Pan, T. W., Joseph, D. D., Bai, R., Glowinski, R. & Sarin, V. 2002 Fluidization of 1204 spheres: simulation and experiment. J. Fluid Mech. 451, 169–191.CrossRefGoogle Scholar
Poletto, M., Bai, R. & Joseph, D. D. 1995 Propagation of voidage waves in a two-dimensional liquid-fluidized bed. Intl J. Multiphase Flo. 21, 223–239.CrossRefGoogle Scholar
Potapov, A. V., Hunt, M. L. & Campbell, C. S. 2001 Liquid–solid flows using smoothed particle hydrodynamics and the discrete element method. Powder Technol. 116, 204–213.CrossRefGoogle Scholar
Qian, Y. H., d'Humieres, D. & Lallemand, P. 1992 Lattice BGK for the Navier–Stokes equations. Europhys. Lett. 17, 479–484.CrossRefGoogle Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidization. Part 1. Trans. Inst. Chem. Engn. 32, 35–53.Google Scholar
Schiller, L. & Naumann, A. 1933 Uber die grundlagenden Berechnungen bei der Schwerkraftaufbereitung. Ver. Deut. Ing. Z. 77, 318–320.Google Scholar
Singh, P. & Joseph, D. D. 1995 Dynamics of fluidized suspensions of spheres of finite size. Intl J. Multiphase Flo. 21, 1–26.CrossRefGoogle Scholar
Singh, P., Hesla, T. I. & Joseph, D. D. 2003 Distributed Lagrange multiplier method for particulate flows with collisions. Intl J. Multiphase Flo. 29, 495–509.CrossRefGoogle Scholar
Somers, J. A. 1993 Direct simulation of fluid flow with cellular automata and the lattice-Boltzmann equation. Appl. Sci. Res. 51, 127–133.CrossRefGoogle Scholar
Sundaresan, S. 2003 Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, 63–88.CrossRefGoogle Scholar
TenCate, A. Cate, A. & Sundaresan, S. 2006a Analysis of flow in inhomogeneous particle beds using the spatially averaged two-fluid equations. Intl J. Multiphase Flo. 36, 106–131.Google Scholar
TenCate, A. Cate, A. & Sundaresan, S. 2006b Analysis of unsteady forces in ordered arrays. J. Fluid Mech. 552, 257–287.Google Scholar
Ten Cate, A., Nieuwstad, C. H., Derksen, J. J. & Vanden Akker, H. E. A. den Akker, H. E. A. 2002 PIV experiments and lattice-Boltzmann simulations on a single sphere settling under gravity. Phys. Fluid. 14, 4012–4025.CrossRefGoogle Scholar
Ten Cate, A., Derksen, J. J., Portela, L. M. & Vanden Akker, H. E. A. den Akker, H. E. A. 2004 Fully resolved simulations of colliding spheres in forced isotropic turbulence. J. Fluid Mech. 519, 233–271.CrossRefGoogle Scholar
Torquato, S., Truskett, T. M. & Debenedetti, P. G. 2000 Is random close packing of spheres well defined? Phys. Rev. Lett. 84, 2064–2067.CrossRefGoogle ScholarPubMed
Verberg, R. & Koch, D. L. 2006 Rheology of particle suspensions with low to moderate fluid inertia at finite particle inertia. Phys. Fluid. 18, 083303-1-16.CrossRefGoogle Scholar
Wang, J. W. & Ge, W. 2005 Collisional particle phase pressure in particle–fluid flows at high particle inertia. Phys. Fluid. 17,128103-1-3.CrossRefGoogle Scholar
Wylie, J. J., Koch, D. L. & Ladd, A. J. C. 2003 Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, 95–118.CrossRefGoogle Scholar
Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, T. & Tsuji, Y. 2001 Large-eddy simulation of turbulent gas–particle flow in a vertical channel: effect of considering inter-particle collisions. J. Fluid Mech. 442, 303–334.CrossRefGoogle Scholar
Yang, F.-L. & Hunt, M. L. 2006 Dynamics of particle–particle collisions in a viscous liquid. Phys. Fluid. 18, 121506-1-11.CrossRefGoogle Scholar
Zenit, R. & Hunt, M. L. 2000 Solid fraction fluctuations in solid-liquid flows. Intl J. Multiphase Flo. 26, 763–781.CrossRefGoogle Scholar
Zenit, R., Hunt, M. & Brennen, C. E. 1997 Collisional particle pressure measurements in solid–liquid flows. J. Fluid Mech. 353, 261–283.CrossRefGoogle Scholar