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Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions

Published online by Cambridge University Press:  12 January 2016

Sudheer Tenneti ,
Mohammad Mehrabadi  and
Shankar Subramaniam
Sudheer Tenneti
Affiliation:
Department of Mechanical Engineering, Center for Multiphase Flow Research and Education, Iowa State University, Ames, IA 50011, USA
Mohammad Mehrabadi
Affiliation:
Department of Mechanical Engineering, Center for Multiphase Flow Research and Education, Iowa State University, Ames, IA 50011, USA
Shankar Subramaniam*
Affiliation:
Department of Mechanical Engineering, Center for Multiphase Flow Research and Education, Iowa State University, Ames, IA 50011, USA
*
†Email address for correspondence: [email protected]
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Abstract

The acceleration of an inertial particle in a gas–solid flow arises from the particle’s interaction with the gas and from interparticle interactions such as collisions. Analytical treatments to derive a particle acceleration model are difficult outside the Stokes flow regime, but for moderate Reynolds numbers (based on the mean slip velocity between gas and particles) particle-resolved direct numerical simulation (PR-DNS) is a viable tool for model development. In this study, PR-DNS of freely-evolving gas–solid suspensions are performed using the particle-resolved uncontaminated-fluid reconcilable immersed-boundary method (PUReIBM) that has been extensively validated in previous studies. Analysis of the particle velocity variance (granular temperature) equation in statistically homogeneous gas–solid flow shows that a straightforward extension of a class of mean particle acceleration models (drag laws) to their corresponding instantaneous versions, by replacing the mean particle velocity with the instantaneous particle velocity, predicts a granular temperature that decays to zero, which is at variance with the steady particle granular temperature that is obtained from PR-DNS. Fluctuations in particle velocity and particle acceleration (and their correlation) are important because the particle acceleration–velocity covariance governs the evolution of the particle velocity variance (characterized by the particle granular temperature), which plays an important role in the prediction of the core annular structure in riser flows. The acceleration–velocity covariance arising from hydrodynamic forces can be decomposed into source and dissipation terms that appear in the granular temperature evolution equation, and these have already been quantified in the Stokes flow regime using a combination of kinetic theory closure and multipole expansion simulations. From PR-DNS data we show that the fluctuations in the particle acceleration that are aligned with fluctuations in the particle velocity give rise to a source term in the granular temperature evolution equation. This approach is used to quantify the hydrodynamic source and dissipation terms of granular temperature from PR-DNS results for freely-evolving gas–solid suspensions that are performed over a wide range of solid volume fraction ($0.1\leqslant {\it\phi}\leqslant 0.4$), Reynolds number based on the slip velocity between the solid and the fluid phase ($10\leqslant \mathit{Re}_{m}\leqslant 100$) and solid-to-fluid density ratio ($100\leqslant {\it\rho}_{p}/{\it\rho}_{f}\leqslant 2000$). The straightforward extension of drag law models does not give rise to any source in the granular temperature due to hydrodynamic effects. This motivates the development of better Lagrangian particle acceleration models that can be used in Lagrangian–Eulerian formulations of gas–solid flow. It is found that a Langevin equation for the increment in the particle velocity reproduces PR-DNS results for the stationary particle velocity autocorrelation in freely-evolving suspensions. Based on the data obtained from the simulations, the functional dependence of the Langevin model coefficients on solid volume fraction, Reynolds number and solid-to-fluid density ratio is obtained. This new Lagrangian particle acceleration model reproduces the correct steady granular temperature and can also be adapted to gas–solid flow computations using Eulerian moment equations.

JFM classification

Mathematical Foundations: Computational methods Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 695 - 729
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Copyright
© 2016 Cambridge University Press 

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Footnotes

‡

Present address: CD-adapco, Lebanon, NH 03766, USA.

References

Beetstra, R., van der Hoef, M. A. & Kuipers, J. A. M. 2007 Drag force of intermediate Reynolds number flows past mono- and bidisperse arrays of spheres. AIChE J. 53, 489–501.CrossRefGoogle Scholar
Calzavarini, E., Volk, R., Bourgoin, M., Lévêque, E., Pinton, J.-F. & Toschi, F. 2009 Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxén forces. J. Fluid Mech. 630, 179–189.CrossRefGoogle Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2014 Numerical study of collisional particle dynamics in cluster-induced turbulence. J. Fluid Mech. 747.Google Scholar
Cocco, R., Shaffer, F., Hays, R., Reddy Karri, S. B. & Knowlton, T. 2010 Particle clusters in and above fluidized beds. Powder Technol. 203 (1), 3–11.Google Scholar
Cundall, P. A. & Strack, O. D. L. 1979 A discrete numerical model for granular assemblies. Gèotechnique 29, 47–65.Google Scholar
Février, P., Simonin, O. & Squires, K. D. 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, 1–46.Google Scholar
Garg, R., Galvin, J., Li, T. & Pannala, S.2010 Documentation of open-source MFIX–DEM software for gas–solids flows. Tech. Rep. National Energy Technology Laboratory, Department of Energy.Google Scholar
Garg, R., Tenneti, S., Mohd-Yusof, J. & Subramaniam, S. 2011 Direct numerical simulation of gas–solids flow based on the immersed boundary method. In Computational Gas–Solids Flows and Reacting Systems: Theory, Methods and Practice (ed. Pannala, S., Syamlal, M. & O’Brien, T. J.), pp. 245–276. IGI Global.CrossRefGoogle Scholar
Garzo, V., Dufty, J. W. & Hrenya, C. M. 2007 Enskog theory for polydisperse granular mixtures. I. Navier–Stokes order transport. Phys. Rev. E 76 (3), 031303.CrossRefGoogle ScholarPubMed
Garzó, V., Tenneti, S., Subramaniam, S. & Hrenya, C. M. 2012 Enskog kinetic theory for monodisperse gas–solid flows. J. Fluid Mech. 712, 129–168.Google Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization. Academic.Google Scholar
Goswami, P. & Kumaran, V. 2010a Particle dynamics in a turbulent particle gas suspension at high Stokes number. Part 1. Velocity and acceleration distributions. J. Fluid Mech. 646, 59–90.CrossRefGoogle Scholar
Goswami, P. & Kumaran, V. 2010b Particle dynamics in a turbulent particle gas suspension at high Stokes number. Part 2. The fluctuating-force model. J. Fluid Mech. 646, 91–125.CrossRefGoogle Scholar
Goswami, P. & Kumaran, V. 2010c Particle dynamics in the channel flow of a turbulent particle gas suspension at high Stokes number. Part 2. Comparison of fluctuating force simulations and experiments. J. Fluid Mech. 687, 41–71.CrossRefGoogle Scholar
Goswami, P. & Kumaran, V. 2011 Particle dynamics in the channel flow of a turbulent particle gas suspension at high Stokes number. Part 1. DNS and fluctuating force model. J. Fluid Mech. 687, 1–40.Google Scholar
Haworth, D. C. & Pope, S. B. 1986 A generalized Langevin model for turbulent flows. Phys. Fluids 29, 387–405.Google Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001a The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–241.Google Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001b Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.Google Scholar
van 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 sphere: results for the permeability and drag force. J. Fluid Mech. 528, 233–254.Google Scholar
Holloway, W., Yin, X. & Sundaresan, S. 2010 Fluid-particle drag in inertial polydisperse gas–solid suspensions. AIChE J. 56, 1995–2004.CrossRefGoogle Scholar
Homann, H. & Bec, J. 2010 Finite-size effects in the dynamics of neutrally buoyant particles in turbulent flow. J. Fluid Mech. 651, 81–91.Google Scholar
Hrenya, C. M. & Sinclair, J. L. 1997 Effects of particle-phase turbulence in gas–solid flows. AIChE J. 43 (4), 853–869.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Kloeden, P. E. & Platen, E. 1992 Numerical Solution of Stochastic Differential Equations. Springer.Google Scholar
Koch, D. L. 1990 Kinetic theory for a monodisperse gas–solid suspension. Phys. Fluids A 2 (10), 1711–1723.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.Google Scholar
Liboff, R. L. 2003 Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 3rd edn. Springer.Google Scholar
Marchisio, D. L. & Fox, R. O. 2013 Computational Models for Polydisperse Particulate and Multiphase Systems. Cambridge University Press.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.Google Scholar
Mehrabadi, M., Tenneti, S., Garg, R. & Subramaniam, S. 2015 Pseudo-turbulent gas-phase velocity fluctuations in homogeneous gas–solid flow: fixed particle assemblies and freely evolving suspensions. J. Fluid Mech. 770, 210–246.CrossRefGoogle Scholar
Pai, M. G. & Subramaniam, S. 2009 A comprehensive probability density function formalism for multiphase flows. J. Fluid Mech. 628, 181–228.Google Scholar
Pope, S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidization: part 1. Trans. Inst. Chem. Engrs 32, 35–53.Google Scholar
Sawford, B. L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3 (6), 1577–1586.Google Scholar
Shen, L., Zheng, M., Xiao, J. & Xiao, R. 2008 A mechanistic investigation of a calcium-based oxygen carrier for chemical looping combustion. Combust. Flame 154 (3), 489–506.CrossRefGoogle Scholar
Subramaniam, S. 2000 Statistical representation of a spray as a point process. Phys. Fluids 12 (10), 2413–2431.Google Scholar
Subramaniam, S. 2001 Statistical modeling of sprays using the droplet distribution function. Phys. Fluids 13 (3), 624–642.Google Scholar
Subramaniam, S. 2013 Lagrangian-Eulerian methods for multiphase flows. Prog. Energy Combust. Sci. 39, 215–245.Google Scholar
Subramaniam, S., Mehrabadi, M., Horwitz, J. & Mani, A. 2014 Developing improved Lagrangian point particle models of gas–solid flow from particle-resolved direct numerical simulation. In Studying Turbulence Using Numerical Simulation Databases-XV, Proceedings of the CTR 2014 Summer Program, pp. 5–14. Center for Turbulence Research, Stanford University, CA.Google Scholar
Tenneti, S., Garg, R., Hrenya, C. M., Fox, R. O. & Subramaniam, S. 2010 Direct numerical simulation of gas–solid suspensions at moderate Reynolds number: quantifying the coupling between hydrodynamic forces and particle velocity fluctuations. Powder Technol. 203 (1), 57–69.Google Scholar
Tenneti, S., Garg, R. & Subramaniam, S. 2011 Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. Intl J. Multiphase Flow 37 (9), 1072–1092.Google Scholar
Tenneti, S. & Subramaniam, S. 2014 Particle-resolved direct numerical simulation for gas–solid flow model development. Annu. Rev. Fluid Mech. 46, 199–230.Google Scholar
Tenneti, S., Sun, B., Garg, R. & Subramaniam, S. 2013 Role of fluid heating in dense gas–solid flow as revealed by particle-resolved direct numerical simulation. Intl J. Heat Mass Transfer 58, 471–479.Google Scholar
Williams, F. A. 1958 Spray combustion and atomization. Phys. Fluids 1 (6), 541–545.Google Scholar
Xu, Y. & Subramaniam, S. 2007 Consistent modeling of interphase turbulent kinetic energy transfer in particle-laden turbulent flows. Phys. Fluids 19.Google Scholar
Yin, X. & Sundaresan, S. 2009a Drag law for bidisperse gas–solid suspensions containing equally sized spheres. Ind. Engng Chem. Res. 48 (1), 227–241.Google Scholar
Yin, X. & Sundaresan, S. 2009b Fluid–particle drag in low-Reynolds-number polydisperse gas–solid suspensions. AIChE J. 55 (6), 1352–1368.Google Scholar
Yusof, J. Mohd.1996 Interaction of massive particles with turbulence. PhD thesis, Cornell University.Google Scholar
Zhang, X. S., Yang, G. X., Jiang, H., Liu, W. J. & Ding, H. S. 2013 Mass production of chemicals from biomass-derived oil by directly atmospheric distillation coupled with co-pyrolysis. Sci. Rep. 3 (3).Google ScholarPubMed
Zick, A. A. & Homsy, G. M. 1982 Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13–26.Google Scholar