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Particle dynamics in a turbulent particle–gas suspension at high Stokes number. Part 1. Velocity and acceleration distributions

Published online by Cambridge University Press:  08 March 2010

PARTHA S. GOSWAMI  and
V. KUMARAN
PARTHA S. GOSWAMI
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
V. KUMARAN*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
*
†Email address for correspondence: [email protected]
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Abstract

The effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas–solid suspension is analysed in the low-Reynolds-number and high Stokes number limits, where the particle relaxation time is long compared with the correlation time for the fluid velocity fluctuations, and the drag force on the particles due to the fluid can be expressed by the modified Stokes law. The direct numerical simulation procedure is used for solving the Navier–Stokes equations for the fluid, the particles are modelled as hard spheres which undergo elastic collisions and a one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. The particle mean and root-mean-square (RMS) fluctuating velocities, as well as the probability distribution function for the particle velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette (where the velocity profile is linear and the RMS velocities are nearly constant), are examined. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the spanwise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the particle velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. It is also found that the acceleration distribution on the particles is in very good agreement with the distribution that is calculated from the velocity fluctuations in the fluid, using the Stokes drag law, indicating that there is very little correlation between the fluid velocity fluctuations and the particle velocity fluctuations in the presence of one-way coupling. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 59 - 90
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Armenio, V. & Fiorotto, V. 2001 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 13, 2437.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2003 Shear versus vortex-induced lift on a rigid sphere at moderate Re. J. Fluid Mech. 473, 379–388.Google Scholar
Balachandar, S. & Maxey, M. R. 1989 Methods for evaluating fluid velocities in spectral simulations of turbulence. J. Comput. Phys. 83, 96–125.CrossRefGoogle Scholar
Bech, K. H., Tilmark, N., Alfredsson, P. H. & Anderson, H. I. 1995 An investigation of turbulent plane Couette flow at low Reynolds number. J. Fluid Mech. 286, 291–325.CrossRefGoogle Scholar
Berlemont, A., Desjonqueres, P. & Gouesbet, G. 1990 Particle Lagrangian simulation in turbulent flows. Intl J. Multiphase Flow 16, 19–34.CrossRefGoogle Scholar
Burton, T. M. & Eaton, J. K. 2005 Fully resolved simulations of particle–turbulence interaction. J. Fluid Mech. 545, 67–111.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. & Zang, T. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carlier, J. Ph., Khaliji, M. & Osterle, B. 2005 An improved model for anisotropic dispersion of small particles in turbulent shear flows. Aerosol Sci. Tech. 39, 196–205.CrossRefGoogle Scholar
Choi, J., Yeo, K. & Lee, C. 2004 Lagrangian statistics in turbulent channel flow. Phys. Fluids 16, 777–793.CrossRefGoogle Scholar
Crowe, C. T., Troutt, T. R. & Chung, J. N. 1996 Numerical models for two-phase turbulent flows. Annu. Rev. Fluid Mech. 28, 11–43.CrossRefGoogle Scholar
Elghobashi, S. E. 1994 On predicting particle laden turbulent flows. Appl. Sci. Res. 52, 309.CrossRefGoogle Scholar
Elghobashi, S. E. & Abou-Arab, T. W. 1983 A two-equation turbulence model for two-phase flows. Phys. Fluids 26, 931–938.Google Scholar
Elghobashi, S. & Truesdell, G. C. 1992 Direct simulation of particle dispersion in a decaying isotropic turbulence. J. Fluid Mech. 242, 655.CrossRefGoogle Scholar
Elghobashi, S. & Truesdell, G. C. 1993 On the two-way interaction between homogeneous turbulence and the dispersed solid particles 1: turbulence modification. Phys. Fluids A 5, 1790.CrossRefGoogle Scholar
Fessler, J. R., Kulick, J. D. & Eaton, J. K. 1994 Preferential concentration of heavy particles in a turbulent channel flow. Phys. Fluids 6, 3742–3749.CrossRefGoogle Scholar
Fevrier, P., Simonin, O. & Squires, K. D. 2005 Partitioning of the particle velocities in gas–solid turbulent flows in to a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. J. Fluid Mech. 533, 1–46.CrossRefGoogle Scholar
Gibson, J. F. 2007 Channel flow: a spectral Navier–Stokes simulator in C++. Tech rep. Georgia Institute of Technology. http://www.channelflow.orgGoogle Scholar
Gore, R. A. & Crowe, C. T. 1989 Effect of particle size on modulating turbulent intensity. Intl J. Multiphase Flow 15, 279.CrossRefGoogle Scholar
Goswami, P. S. & Kumaran, V. 2010 Particle dynamics in a turbulent particle-gas suspension at high Stokes number. Part 2. The fluctuating force model. J. Fluid Mech. 646, 91–125.Google Scholar
Hetsroni, G. 1989 Particle–turbulence interaction. Intl J. Multiphase Flow 15, 735.CrossRefGoogle Scholar
Hoockney, R. W. & Eastwood, J. W. 1988 Computer Simulation Using Particles. Institute of Physics.CrossRefGoogle Scholar
Hwang, W. & Eaton, J. K. 2006 Homogeneous and isotropic turbulence modulation by small heavy (St 50) particles. J. Fluid Mech. 564, 361–393.CrossRefGoogle Scholar
Hyland, K. E., McKee, S. & Reeks, M. W. 1999 Exact analytic solution to turbulent particle flow equation. Phys. Fluids 11, 1249–1261.CrossRefGoogle Scholar
Kallio, G. A. & Reeks, M. W. 1989 A numerical simulation of particle deposition in turbulent boundary layers. Intl J. Multiphase Flow 15, 433.CrossRefGoogle Scholar
Khalitov, D. A. & Longmire, E. K. 2003 Effect of particle size on the velocity correlations in turbulent channel flow. Fourth ASME/JSME Joint Fluid Engineering Conference, Honolulu. FEDSM2003-45730.Google Scholar
Kim, I., Elghobashi, S. & Sirignano, W. A. 1998 On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers. J. Fluid Mech. 367, 221–253.CrossRefGoogle Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-d numerical spectral simulations of plane channel flows. Proc. Third GAMM Conf. on Numerical Methods in Fluid Mechanics (ed. Hirschel, E. H.), Vieweg, 165–173.CrossRefGoogle Scholar
Kominnaho, J., Lundbladh, A. & Johansson, A. V. 1996 Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285.CrossRefGoogle Scholar
Kontomaris, K., Hanratty, T. J. & McLaughlin, J. B. 1992 An algorithm for tracking fluid particles in a spectral simulation of turbulent channel flow. J. Comput. Phys. 103, 231–242.CrossRefGoogle Scholar
Kuerten, J. G. M. 2006 Subgrid modelling in particle-laden channel flow. Phys. Fluids 18, 025108.CrossRefGoogle Scholar
Kulick, J. D., Fessler, J. R. & Eaton, J. K. 1994 Particle response and turbulence modification in a fully developed channel flow. J. Fluid Mech. 277, 109–134.CrossRefGoogle Scholar
Kumaran, V. 2003 Stability of a sheared particle suspension. Phys. Fluids 15, 3625–3637.CrossRefGoogle Scholar
Kumaran, V. & Koch, D. L. 1993 a Properties of a bidisperse particle-gas suspension. Part 1. Collision time small compared to viscous relaxation time. J. Fluid Mech. 247, 623–642.CrossRefGoogle Scholar
Kumaran, V. & Koch, D. L. 1993 b Properties of a bidisperse particle-gas suspension. Part 2. Viscous relaxation time small compared to collision relaxation time. J. Fluid Mech. 247, 643–660.CrossRefGoogle Scholar
Li, Y. & McLaughlin, J. B. 2001 Numerical simulation of particle-laden turbulent channel flow. Phys. Fluids 13, 2957.CrossRefGoogle Scholar
Louge, M. Y., Mastorakos, E. & Jenkins, J. T. 1991 The role of particle collisions in pneumatic transport. J. Fluid Mech. 231, 345–359.CrossRefGoogle Scholar
Luu, Q. Q., Fontaine, J. R. & Aubertin, G. 1993 Numerical study of the solid particle motion in grid-generated turbulence. Intl J. Heat Mass Transf. 36, 79–87.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26, 883.CrossRefGoogle Scholar
McLaughlin, J. B. 1989 Aerosol particle deposition in numerically simulated channel flow. Phys. Fluids A 1, 1211.CrossRefGoogle Scholar
Rauson, D. W. I. & Eaton, J. K. 1994 Direct numerical simulation of turbulent channel flow with immersed particles. Numer. Methods Multiphase Flows FED 185, 47–57.Google Scholar
Reeks, M. W. 1991 On a kinetic equation for the transport of particles in turbulent flows. Phys. Fluids A 3, 446–456.CrossRefGoogle Scholar
Reeks, M. W. 1992 On the continuum equations for dispersed particles in non-uniform flows. Phys. Fluids 4, 1290–1303.CrossRefGoogle Scholar
Reeks, M. W. 1993 On the constitutive relations for dispersed particles in non-uniform flows. Phys. Fluids 5, 750–761.CrossRefGoogle Scholar
Reeks, M. W. 2005 On model equation for particle dispersion in inhomgeneous turbulence. Intl J. Multiphase Flow 31, 93–114.CrossRefGoogle Scholar
Riley, J. J. & Patterson, G. S. 1974 Diffusion experiments with numerically integrated isotropic turbulence. Phys. Fluids 17, 292.CrossRefGoogle Scholar
Rizik, M. A. & Elghobashi, S. E. 1989 A two-equation turbulence model for dispersed dilute confined two-phase flows. Intl J. Multiphase Flow 15, 119–133.CrossRefGoogle Scholar
Rouson, D. W. I. & Eaton, J. K. 2001 On the preferential concentration of solid particles in turbulent channel flow. J. Fluid Mech. 428, 149.CrossRefGoogle Scholar
Shin, M. & Lee, J. W. 2002 Nonequilibrium Reynolds stress for the dispersed phase of solid particles in turbulent flows. Phys. Fluids 14, 2898–2916.CrossRefGoogle Scholar
Sommerfeld, M. 1995 The importance of inter-particle collision in horizontal gas–solid channel flows. Gas-Particle Flows FED 228, 335–345.Google Scholar
Squires, K. D. & Eaton, J. K. 1990 Particle response and turbulence modification in isotropic turbulence. Phys. Fluids A 2, 1191.CrossRefGoogle Scholar
Squires, K. D. & Eaton, J. K. 1991 a Lagrangian and Eulerian statistics obtained from direct numerical simulations of homogeneous turbulence. Phys. Fluids A 3, 130–143.CrossRefGoogle Scholar
Squires, K. D. & Eaton, J. K. 1991 b Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1994 a Spectrum of density fluctuation in a particle fluid system 1. Monodisperse spheres. Intl J. Multiphase Flow 20, 1021–1037.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1996 Numerical consideration in simulating a turbulent suspension of finite-volume particles. Phys. Fluids 13, 2437.Google Scholar
Sundaram, S. & Collins, L. R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulation. J. Fluid Mech. 335, 75.CrossRefGoogle Scholar
Swailes, D. C. & Reeks, M. W. 1994 Particle deposition from a turbulent flow. I. A steady-state model for high inertia particles. Phys. Fluids 6, 3392–3403.CrossRefGoogle Scholar
Tanaka, T. & Eaton, J. K. 2008 Classification of turbulence modification by dispersed spheres using a novel dimensionless number. Phys. Rev. Lett. 101, 114502.CrossRefGoogle ScholarPubMed
Tsao, H.-K. & Koch, D. L. 1995 Shear flows of a dilute gas–solid suspension. J. Fluid Mech. 296, 211–245.CrossRefGoogle Scholar
Wang, Q. & Squires, K. D. 1996 Large eddy simulation of particle-laden channel flow. Phys. Fluids 8, 1207–1223.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
Yeung, P. K. & Pope, S. B. 1988 An algorithm for tracking fluid particles in numerical simulation of homogeneous turbulence. J. Comput. Phys. 79, 373–416.CrossRefGoogle Scholar
Yuu, S., Yasukouchi, N. I., Hirosawa, Y. & Jotaki, T. 1978 Particle turbulent diffusion in a duct laden jet. AIChE J. 24, 509–519.Google Scholar
Zaichik, L. I., Alipchenkov, V. M. & Avetissian, A. R. 2006 Modelling turbulent collision rates of inertial particles. Intl J. Heat Fluid Flow 27, 937–944.CrossRefGoogle Scholar