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Stochastic models for capturing dispersion in particle-laden flows

Published online by Cambridge University Press:  18 September 2020

Aaron M. Lattanzi Open the ORCID record for Aaron M. Lattanzi [Opens in a new window] ,
Vahid Tavanashad ,
Shankar Subramaniam Open the ORCID record for Shankar Subramaniam [Opens in a new window]  and
Jesse Capecelatro Open the ORCID record for Jesse Capecelatro [Opens in a new window]
Aaron M. Lattanzi*
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
Vahid Tavanashad
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA, USA
Shankar Subramaniam
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA, USA
Jesse Capecelatro
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
*
†Email address for correspondence: alattanz@umich.edu
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Abstract

This study provides a detailed account of stochastic approaches that may be utilized in Eulerian–Lagrangian simulations to account for neighbour-induced drag force fluctuations. The frameworks examined here correspond to Langevin equations for the particle position (PL), particle velocity (VL) and fluctuating drag force (FL). Rigorous derivations of the particle velocity variance (granular temperature) and dispersion resulting from each method are presented. The solutions derived herein provide a basis for comparison with particle-resolved direct numerical simulation. The FL method allows for the most complex behaviour, enabling control of both the granular temperature and dispersion. A Stokes number $St_F$ is defined for the fluctuating force that relates the integral time scale of the force to the Stokes response time. Formal convergence of the FL scheme to the VL scheme is shown for $St_F \gg 1$. In the opposite limit, $St_F \ll 1$, the fluctuating drag forces are highly inertial and the FL scheme departs significantly from the VL scheme.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A7
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Akiki, G., Jackson, T. & Balachandar, S. 2016 Force variation within arrays of monodisperse spherical particles. Phys. Rev. Fluids 1 (4), 044202.CrossRefGoogle Scholar
Akiki, G., Jackson, T. & Balachandar, S. 2017 Pairwise interaction extended point-particle model for a random array of monodisperse spheres. J. Fluid Mech. 813, 882–928.CrossRefGoogle Scholar
Andrews, A., Loezos, P. & Sundaresan, S. 2005 Coarse-grid simulation of gas-particle flows in vertical risers. Ind. Engng Chem. Res. 44 (16), 6022–6037.CrossRefGoogle Scholar
Balachandar, S., Liu, K. & Lakhote, M. 2019 Self-induced velocity correction for improved drag estimation in Euler–Lagrange point-particle simulations. J. Comput. Phys. 376, 160–185.CrossRefGoogle Scholar
Beetstra, R., van der Hoef, M. A. & Kuipers, J. A. M. 2007 Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres. AIChE J. 53 (2), 489–501.CrossRefGoogle Scholar
Capecelatro, J. & Desjardins, O. 2013 An Euler–Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 1–31.CrossRefGoogle Scholar
Chapman, S., Cowling, T. & Burnett, D. 1970 The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press.Google Scholar
Cundall, P. & Strack, O. 1979 A discrete numerical model for granular assemblies. Géotechnique 29 (1), 47–65.CrossRefGoogle Scholar
Elgobashi, S. 2006 An updated classification map of particle-laden turbulent flows. In IUTAM Symposium on Computational Approaches to Multiphase Flow (ed. Balachandar, S. & Prosperetti, A.), Fluid Mechanics and Its Applications, vol. 81, pp. 3–10. Springer.CrossRefGoogle Scholar
Esteghamatian, A., Bernard, M., Lance, M., Hammouti, A. & Wachs, A. 2017 Micro/meso simulation of a fluidized bed in a homogeneous bubbling regime. Intl J. Multiphase Flow 92, 93–111.CrossRefGoogle Scholar
Esteghamatian, A., Euzenat, F., Hammouti, A., Lance, M. & Wachs, A. 2018 A stochastic formulation for the drag force based on multiscale numerical simulation of fluidized beds. Intl J. Multiphase Flow 99, 363–382.CrossRefGoogle Scholar
Gardiner, C. 2009 Stochastic Methods: A Handbook for the Natural and Social Sciences, 4th edn.Springer.Google Scholar
Garzó, V. & Dufty, J. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59 (5), 5895–5911.CrossRefGoogle ScholarPubMed
Garzó, V., Tenneti, S., Subramaniam, S. & Hrenya, C. 2012 Enskog kinetic theory for monodisperse gas–solid flows. J. Fluid Mech. 712, 129–168.CrossRefGoogle Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic Press.Google Scholar
Haworth, D. & Pope, S. 1986 A generalized Langevin model for turbulent flows. Phys. Fluids 29 (2), 387–405.CrossRefGoogle Scholar
Hill, R., Koch, D. & Ladd, A. 2001 Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
van der Hoef, M. A., van Sint Annaland, M., Deen, N. G. & Kuipers, J. A. M. 2008 Numerical simulation of dense gas–solid fluidized beds: a multiscale modeling strategy. Annu. Rev. Fluid Mech. 40 (1), 47–70.CrossRefGoogle Scholar
Horwitz, J. & Mani, A. 2018 Correction scheme for point-particle models applied to a nonlinear drag law in simulations of particle-fluid interaction. Intl J. Multiphase Flow 101, 74–84.CrossRefGoogle Scholar
Huang, Z., Wang, H., Zhou, Q. & Li, T. 2017 Effects of granular temperature on inter-phase drag in gas–solid flows. Powder Technol. 321, 435–443.CrossRefGoogle Scholar
Iliopoulos, I., Mito, Y. & Hanratty, T. 2003 A stochastic model for solid particle dispersion in a nonhomogeneous turbulent field. Intl J. Multiphase Flow 29 (3), 375–394.CrossRefGoogle Scholar
Ireland, P. & Desjardins, O. 2017 Improving particle drag predictions in Euler–Lagrange simulations with two-way coupling. J. Comput. Phys. 338, 405–430.CrossRefGoogle Scholar
Kloeden, P. & Platen, E. 1992 Numerical Solution of Stochastic Differential Equations, corrected edn.Springer.CrossRefGoogle Scholar
Koch, D. 1990 Kinetic theory for a monodisperse gas–solid suspension. Phys. Fluids A 2 (10), 1711–1723.CrossRefGoogle Scholar
Koch, D. & Sangani, A. 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
Kriebitzsch, S., van der Hoef, M. A. & Kuipers, J. A. M. 2013 Fully resolved simulation of a gas-fluidized bed: a critical test of DEM models. Chem. Engng Sci. 91, 1–4.CrossRefGoogle Scholar
Lattanzi, A., Yin, X. & Hrenya, C. 2020 Heat and momentum transfer to a particle in a laminar boundary layer. J. Fluid Mech. 889, A6.CrossRefGoogle Scholar
Ma, D. & Ahmadi, G. 1988 A kinetic model for rapid granular flows of nearly elastic particles including interstitial fluid effects. Powder Technol. 56 (3), 191–207.CrossRefGoogle 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
Metzger, B., Rahli, O. & Yin, X. 2013 Heat transfer across sheared suspensions: role of the shear-induced diffusion. J. Fluid Mech. 724, 527–552.CrossRefGoogle Scholar
Na, Y., Papavassiliou, D. & Hanratty, T. 1999 Use of direct numerical simulation to study the effect of Prandtl number on temperature fields. Intl J. Heat Fluid Flow 20 (3), 187–195.CrossRefGoogle Scholar
Pai, M. & Subramaniam, S. 2009 A comprehensive probability density function formalism for multiphase flows. J. Fluid Mech. 628, 181–228.CrossRefGoogle Scholar
Pai, M. & Subramaniam, S. 2012 Two-way coupled stochastic model for dispersion of inertial particles in turbulence. J. Fluid Mech. 700, 29–62.CrossRefGoogle Scholar
Papavassiliou, D. & Hanratty, T. 1997 Transport of a passive scalar in a turbulent channel flow. Intl J. Heat Mass Transfer 40 (6), 1303–1311.CrossRefGoogle Scholar
Peng, C., Kong, B., Zhou, J., Sun, B., Passalacqua, A., Subramaniam, S. & Fox, R. 2019 Implementation of pseudo-turbulence closures in an Eulerian–Eulerian two-fluid model for non-isothermal gas–solid flow. Chem. Engng Sci. 207, 663–671.CrossRefGoogle Scholar
Pope, S. 1994 Lagrangian PDF methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1), 23–63.CrossRefGoogle Scholar
Pope, S. 2000 Turbulent Flows, 1st edn.Cambridge University Press.CrossRefGoogle Scholar
Pope, S. 2002 A stochastic Lagrangian model for acceleration in turbulent flows. Phys. Fluids 14 (7), 2360–2375.CrossRefGoogle Scholar
Pozorski, J. & Apte, S. 2009 Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Intl J. Multiphase Flow 35 (2), 118–128.CrossRefGoogle Scholar
Rao, A. & Capecelatro, J. 2019 Coarse-grained modeling of sheared granular beds. Intl J. Multiphase Flow 114, 258–267.CrossRefGoogle Scholar
Risken, H. & Frank, T. 1996 The Fokker–Planck Equation: Methods of Solution and Applications, 2nd edn.. Springer.CrossRefGoogle Scholar
Rubinstein, G., Derksen, J. & Sundaresan, S. 2016 Lattice Boltzmann simulations of low-Reynolds-number flow past fluidized spheres: effect of Stokes number on drag force. J. Fluid Mech. 788, 576–601.CrossRefGoogle Scholar
Sangani, A. & Mo, G. 1996 An O(N) algorithm for Stokes and Laplace interactions of particles. Phys. Fluids 8 (8), 1990–2010.CrossRefGoogle Scholar
Sangani, A., Mo, G., Tsao, H. & Koch, D. 1996 Simple shear flows of dense gas–solid suspensions at finite Stokes numbers. J. Fluid Mech. 313, 309–341.CrossRefGoogle Scholar
Sawford, B. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3 (6), 1577–1586.CrossRefGoogle Scholar
Shallcross, G., Fox, R. & Capecelatro, J. 2020 A volume-filtered description of compressible particle-laden flows. Intl J. Multiphase Flow 122, 103138.CrossRefGoogle Scholar
Tavanashad, V., Passalacqua, A., Fox, R. & Subramaniam, S. 2019 Effect of density ratio on velocity fluctuations in dispersed multiphase flow from simulations of finite-size particles. Acta Mech. 230 (2), 469–484.CrossRefGoogle Scholar
Taylor, G. 1922 Diffusion by continuous movements. Proc. Lond. Math. Soc. s2-20 (1), 196–212.CrossRefGoogle 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.CrossRefGoogle Scholar
Tenneti, S., Mehrabadi, M. & Subramaniam, S. 2016 Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions. J. Fluid Mech. 788, 695–729.CrossRefGoogle Scholar
Tsuji, Y., Kawaguchi, T. & Tanaka, T. 1993 Discrete particle simulation of two-dimensional fluidized bed. Powder Technol. 77 (1), 79–87.CrossRefGoogle Scholar
Vincenti, W. & Kruger, C. 1975 Introduction to Physical Gas Dynamics. Krieger Publishing Company.Google Scholar
Wylie, J., Koch, D. & Ladd, A. 2003 Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, 95–118.CrossRefGoogle Scholar
Yeung, P. & Pope, S. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar