Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T05:02:15.507Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows

Published online by Cambridge University Press:  11 January 2019

A. Innocenti Open the ORCID record for A. Innocenti [Opens in a new window] ,
R. O. Fox Open the ORCID record for R. O. Fox [Opens in a new window] ,
M. V. Salvetti  and
S. Chibbaro
A. Innocenti
Affiliation:
Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond D’Alembert, F-75005 Paris, France Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, Via G. Caruso 8, 56122 Pisa, Italy
R. O. Fox
Affiliation:
Department of Chemical and Biological Engineering, 618 Bissell Road, Iowa State University, Ames, IA 50011-1098, USA
M. V. Salvetti
Affiliation:
Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, Via G. Caruso 8, 56122 Pisa, Italy
S. Chibbaro*
Affiliation:
Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond D’Alembert, F-75005 Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence (CIT). In the present work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian probability-density-function formalism. This framework includes the Eulerian approach, and hence can be useful also for the development of two-fluid models. A rigorous formalism and a general model have been put forward focusing, in particular, on the two ingredients that are key in moderately dense flows, namely, two-way coupling in the carrier phase, and the decomposition of the particle-phase velocity into its spatially correlated and uncorrelated components. Specifically, this last contribution allows us to identify in the stochastic model the contributions due to the correlated fluctuating energy and to the granular temperature of the particle phase, which determine the time scale for particle–particle collisions. The model is then validated and assessed against direct-numerical-simulation data for homogeneous configurations of increasing difficulty: (i) homogeneous isotropic turbulence, (ii) decaying and shear turbulence and (iii) CIT.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 449 - 489
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmed, A. M. & Elghobashi, S. 2000 On the mechanisms of modifying the structure of turbulent homogeneous shear flows by dispersed particles. Phys. Fluids 12 (11), 2906–2930.Google Scholar
Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111–133.Google Scholar
Balkovsky, E., Falkovich, G. & Fouxon, A. 2001 Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86 (13), 2790–2794.Google Scholar
Bosse, T., Kleiser, L. & Meiburg, E. 2006 Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling. Phys. Fluids 18 (2), 027102.Google Scholar
Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A. 1998 Hydrodynamics for granular flow at low density. Phys. Rev. E 58 (4), 4638.Google Scholar
Brilliantov, N. V. & Pöschel, T. 2010 Kinetic Theory of Granular Gases. Oxford University Press.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2014 Numerical study of collisional particle dynamics in cluster-induced turbulence. J. Fluid Mech. 747, R2.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2015 On fluid–particle dynamics in fully developed cluster-induced turbulence. J. Fluid Mech. 780, 578–635.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2016a Strongly coupled fluid–particle flows in vertical channels. Part I. Reynolds-averaged two-phase turbulence statistics. Phys. Fluids 28 (3), 033306.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2016b Strongly coupled fluid–particle flows in vertical channels. Part II. Turbulence modeling. Phys. Fluids 28 (3), 033307.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R. O. 2018 On the transition between turbulence regimes in particle-laden channel flows. J. Fluid Mech. 845, 499–519.Google Scholar
Chibbaro, S. & Minier, J.-P. 2011 A note on the consistency of hybrid Eulerian/Lagrangian approach to multiphase flows. Intl J. Multiphase Flow 37 (3), 293–297.Google Scholar
Chouippe, A. & Uhlmann, M. 2015 Forcing homogeneous turbulence in DNS of particulate flow with interface resolution and gravity. Phys. Fluids 27 (12), 123301.Google Scholar
Crowe, C. T., Schwarzkopf, J. D., Sommerfeld, M. & Tsuji, Y. 2011 Multiphase Flows With Droplets and Particles. CRC Press.Google Scholar
Dasgupta, S., Jackson, R. & Sundaresan, S. 1994 Turbulent gas-particle flow in vertical risers. AIChE J. 40 (2), 215–228.Google Scholar
Eaton, J. K. & Fessler, J. R. 1994 Preferential concentration of particles by turbulence. Intl J. Multiphase Flow 20, 169–209.Google Scholar
Elghobashi, S. 1994 On predicting particle-laden turbulent flows. Appl. Sci. Res. 52 (4), 309–329.Google Scholar
Elghobashi, S. & Truesdell, G. C. 1992 Direct simulation of particle dispersion in a decaying isotropic turbulence. J. Fluid Mech. 242, 655–700.Google Scholar
Elghobashi, S. E. & Abou-Arab, T. W. 1983 A two-equation turbulence model for two-phase flows. Phys. Fluids 26 (4), 931–938.Google Scholar
Ferrante, A. & Elghobashi, S. 2003 On the physical mechanisms of two-way coupling in particle-laden isotropic turbulence. Phys. Fluids 15 (2), 315–329.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
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 1–24.Google Scholar
Fox, R. O. 2003 Computational Models for Turbulent Reacting Flows. Cambridge University Press.Google Scholar
Fox, R. O. 2014 On multiphase turbulence models for collisional fluid–particle flows. J. Fluid Mech. 742, 368–424.Google Scholar
Gardiner, C. W. 1990 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer.Google Scholar
Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 1 (2), 143–160.Google Scholar
Glasser, B. J., Sundaresan, S. & Kevrekidis, I. G. 1998 From bubbles to clusters in fluidized beds. Phys. Rev. Lett. 81 (9), 1849.Google Scholar
Gualtieri, P., Battista, F. & Casciola, C. M. 2017 Turbulence modulation in heavy-load suspensions of tiny particles. Phys. Rev. Fluids 2 (3), 034304.Google Scholar
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics. Cambridge University Press.Google Scholar
Hinze, J. O. 1975 Turbulence, 2nd edn. McGraw Hill.Google Scholar
Innocenti, A., Marchioli, C. & Chibbaro, S. 2016 Lagrangian filtered density function for LES-based stochastic modelling of turbulent particle-laden flows. Phys. Fluids 28 (11), 115106.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.Google Scholar
Jenny, P., Roekaerts, D. & Beishuizen, N. 2012 Modeling of turbulent dilute spray combustion. Prog. Energy Combust. Sci. 38 (6), 846–887.Google Scholar
Kloeden, P. E. & Platen, E. 1992 Numerical Solution of Stochastic Differential Equations. Springer.Google Scholar
Launder, B. E., Reece, G. Jr & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68 (03), 537–566.Google Scholar
Marconi, U. M. B., Puglisi, A., Rondoni, L. & Vulpiani, A. 2008 Fluctuation–dissipation: response theory in statistical physics. Phys. Rep. 461 (4–6), 111–195.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., Horwitz, J. A. K., Subramaniam, S. & Mani, A. 2018 A direct comparison of particle-resolved and point-particle methods in decaying turbulence. J. Fluid Mech. 850, 336–369.Google Scholar
Minier, J.-P., Chibbaro, S. & Pope, S. B. 2014 Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows. Phys. Fluids 26 (11), 113303.Google Scholar
Minier, J.-P. & Peirano, E. 2001 The pdf approach to turbulent polydispersed two-phase flows. Phys. Rep. 352, 1–214.Google Scholar
Minier, J-P., Peirano, E. & Chibbaro, S. 2004 PDF model based on Langevin equation for polydispersed two-phase flows applied to a bluff body-body gas–solid flow. Phys. Fluids 16 (7), 2419.Google Scholar
Muradoglu, M., Pope, S. B. & Caughey, D. A. 2001 The hybrid method for the PDF equations of turbulent reactive flows: consistency conditions and correction algorithms. J. Comput. Phys. 172 (2), 841–878.Google Scholar
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313–358.Google Scholar
Peirano, E., Chibbaro, S., Pozorski, J. & Minier, J.-P. 2006 Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Prog. En. Comb. Sci. 32 (3), 315–371.Google Scholar
Peirano, E. & Minier, J.-P. 2002 Probabilistic formalism and hierarchy of models for polydispersed turbulent two-phase flows. Phys. Rev. E 65, 046301.Google Scholar
Picano, F., Breugem, W.-P. & Brandt, L. 2015 Modulation of isotropic turbulence by particles of Taylor length-scale size. J. Fluid Mech. 650, 5–55.Google Scholar
Picano, F., Breugem, W.-P., Mitra, D. & Brandt, L. 2013 Shear thickening in non-Brownian suspensions: an excluded volume effect. Phys. Rev. Lett. 111 (9), 098302.Google Scholar
Pope, S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.Google Scholar
Pope, S. B. 1994 On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Phys. Fluids 6 (2), 973–985.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Post, S. L. & Abraham, J. 2002 Modeling the outcome of drop–drop collisions in diesel sprays. Intl J. Multiphase Flow 28 (6), 997–1019.Google Scholar
Pozorski, J. & Minier, J.-P. 1998 On the Lagrangian turbulent dispersion models based on the Langevin equation. Intl J. Multiphase Flow 24, 913–945.Google Scholar
Puglisi, A. 2014 Transport and Fluctuations in Granular Fluids: From Boltzmann Equation to Hydrodynamics, Diffusion and Motor Effects. Springer.Google Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.Google Scholar
Sundaram, S. & Collins, L. R. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.Google Scholar
Tanaka, M. 2017 Effect of gravity on the development of homogeneous shear turbulence laden with finite-size particles. J. Turb. 18 (12), 1144–1179.Google Scholar
Tchen, C. M.1947 Mean value and correlation functions connected with the motion of small particles suspended in a turbulent fluid. PhD Thesis, Delft.Google Scholar
Ten Cate, A., Derksen, J. J., Portela, L. M. & Van Den Akker, H. E. A. 2004 Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence. J. Fluid Mech. 519, 233–271.Google Scholar
Viollet, P. L. & Simonin, O. 1994 Modelling dispersed two-phase flows: closure, validation and software development. Appl. Mech. Rev. 47 (6), S80–S84.Google Scholar
Wang, L. P. & Stock, D. E. 1993 Dispersion of heavy particles by turbulent motion. J. Atmos. Sci. 50 (13), 1897–1913.Google Scholar