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Multiphase flows: Rich physics, challenging theory, and big simulations
Phys. Rev. Fluids 5, 110520 – Published 24 November, 2020
DOI: https://doi.org/10.1103/PhysRevFluids.5.110520
Abstract
Understanding multiphase flows is vital to addressing some of our most pressing human needs: clean air, clean water, and the sustainable production of food and energy. This article focuses on a subset of multiphase flows called particle-laden suspensions involving nondeforming particles in a carrier fluid. The hydrodynamic interactions in these flows result in rich multiscale physics, such as clustering and pseudo-turbulence, with important practical implications. Theoretical formulations to represent, explain, and predict these phenomena encounter peculiar challenges that multiphase flows pose for classical statistical mechanics. A critical analysis of existing approaches leads to the identification of key desirable characteristics that a formulation must possess in order to be successful at representing these physical phenomena. The need to build accurate closure models for unclosed terms that arise in statistical theories has motivated the development of particle-resolved direct numerical simulations (PR-DNS) for model-free simulation at the microscale. A critical perspective on outstanding questions and potential limitations of PR-DNS for model development is provided. Selected highlights of recent progress using PR-DNS to discover new multiphase flow physics and develop models are reviewed. Alternative theoretical formulations and extensions to current formulations are outlined as promising future research directions. The article concludes with a summary perspective on the importance of integrating theoretical, modeling, computational, and experimental efforts at different scales.
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2020 Invited Papers
Physical Review Fluids publishes a collection of papers associated with the invited talks presented at the 72nd Annual Meeting of the APS Division of Fluid Dynamics.
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References (154)
- K. Chu, B. Wang, A. Yu, and A. Vince, CFD-DEM modelling of multiphase flow in dense medium cyclones, Powder Technol. 193, 235 (2009).
- K. Chu, B. Wang, A. Yu, A. Vince, G. Barnett, and P. Barnett, CFD-DEM study of the effect of particle density distribution on the multiphase flow and performance of dense medium cyclone, Miner. Eng. 22, 893 (2009), special issue: Computational Modelling.
- S. McGovern, G. Harish, C. Pai, W. Mansfield, J. Taylor, S. Pau, and R. Besser, Multiphase flow regimes for hydrogenation in a catalyst-trap microreactor, Chem. Eng. J. 135, S229 (2008).
- J. Latham, Control of global warming? Nature (London) 347, 339 (1990).
- J. Latham, K. Bower, T. Choularton, H. Coe, P. Connolly, G. Cooper, T. Craft, J. Foster, A. Gadian, L. Galbraith et al., Marine cloud brightening, Philos. Trans. R. Soc. A 370, 4217 (2012).
- G. Cooper, J. Foster, L. Galbraith, S. Jain, A. Neukermans, and B. Ormond, Preliminary results for salt aerosol production intended for marine cloud brightening, using effervescent spray atomization, Philos. Trans. R. Soc. A 372, 20140055 (2014).
- Center for Multiphase Flow Research and Education (CoMFRE), Iowa State University, https://comfre.iastate.edu/
- J. P. Polin, H. D. Carr, L. E. Whitmer, R. G. Smith, and R. C. Brown, Conventional and autothermal pyrolysis of corn stover: Overcoming the processing challenges of high-ash agricultural residues, J. Anal. Appl. Pyrolysis 143, 104679 (2019).
- J. P. Polin, C. A. Peterson, L. E. Whitmer, R. G. Smith, and R. C. Brown, Process intensification of biomass fast pyrolysis through autothermal operation of a fluidized bed reactor, Appl. Energy 249, 276 (2019).
- J. P. Polin, Process intensification of biomass fast pyrolysis via autothermal operation, Ph.D. thesis, Iowa State University (2019).
- T. Knowlton, S. Karri, and A. Issangya, Scale-up of fluidized-bed hydrodynamics, Powder Technol. 150, 72 (2005).
- R. W. Breault and C. P. Guenther, Mass transfer in the core-annular and fast fluidization flow regimes of a CFB, Powder Technol. 190, 385 (2009).
- B. Sun, S. Tenneti, and S. Subramaniam, Modeling average gas–solid heat transfer using particle-resolved direct numerical simulation, Int. J. Heat Mass Transf. 86, 898 (2015).
- F. Shaffer, B. Gopalan, R. W. Breault, R. Cocco, S. R. Karri, R. Hays, and T. Knowlton, High speed imaging of particle flow fields in CFB risers, Powder Technol. 242, 86 (2013).
- J. McMillan, F. Shaffer, B. Gopalan, J. W. Chew, C. Hrenya, R. Hays, S. R. Karri, and R. Cocco, Particle cluster dynamics during fluidization, Chem. Eng. Sci. 100, 39 (2013).
- F. Shaffer and B. Gopalan, The science and beauty of fluidization, arXiv:1311.1058 [physics.flu-dyn].
- H. A. Jakobsen, Chemical reactor modeling, Multiphase Reactive Flows (Springer International Publishing, Switzerland, 2008).
- H. Anglart and O. Nylund, CFD application to prediction of void distribution in two-phase bubbly flows in rod bundles, Nucl. Eng. Des. 163, 81 (1996).
- A. Serizawa, K. Huda, Y. Yamada, and I. Kataoka, Experiment and numerical simulation of bubbly two-phase flow across horizontal and inclined rod bundles, Nucl. Eng. Des. 175, 131 (1997).
- N. Kantarci, F. Borak, and K. O. Ulgen, Bubble column reactors, Process Biochem. 40, 2263 (2005).
- A. Haghtalab, M. Nabipoor, and S. Farzad, Kinetic modeling of the Fischer–Tropsch synthesis in a slurry phase bubble column reactor using Langmuir–Freundlich isotherm, Fuel Process. Technol. 104, 73 (2012).
- D. Storm and R. Köpsel, Modelling of catalytic coal hydrogenation in a bubble column reactor: 2. Model calculations of bubble column cascades, Fuel 71, 681 (1992).
- M. Ferrer, R. David, and J. Villermaux, Homogeneous oxidation of n-butane in a self-stirred reactor, Chem. Eng. Process. 19, 119 (1985).
- H. Debellefontaine and J. N. Foussard, Wet air oxidation for the treatment of industrial wastes. Chemical aspects, reactor design and industrial applications in Europe, Waste Manage. 20, 15 (2000).
- R. Ranjbar, R. Inoue, H. Shiraishi, T. Katsuda, and S. Katoh, High efficiency production of astaxanthin by autotrophic cultivation of Haematococcus pluvialis in a bubble column photobioreactor, Biochem. Eng. J. 39, 575 (2008).
- M. Kosseva, V. Beschkov, and R. Popov, Biotransformation of d-sorbitol to l-sorbose by immobilized cells Gluconobacter suboxydans in a bubble column, J. Biotechnol. 19, 301 (1991).
- E. K. Nauha and V. Alopaeus, Modeling method for combining fluid dynamics and algal growth in a bubble column photobioreactor, Chem. Eng. J. 229, 559 (2013).
- C. Posten, Design principles of photo-bioreactors for cultivation of microalgae, Eng. Life Sci. 9, 165 (2009).
- J. L. Muñoz-Cobo, S. Chiva, M. A. A. E. A. Essa, and S. Mendes, Simulation of bubbly flow in vertical pipes by coupling Lagrangian and Eulerian models with 3D random walks models: Validation with experimental data using multi-sensor conductivity probes and laser Doppler anemometry, Nucl. Eng. Des. 242, 285 (2012).
- G. Tryggvason, S. Dabiri, B. Aboulhasanzadeh, and J. Lu, Multiscale considerations in direct numerical simulations of multiphase flows, Phys. Fluids 25, 031302 (2013).
- B. Gvozdić, E. Alméras, V. Mathai, X. Zhu, D. P. van Gils, R. Verzicco, S. G. Huisman, C. Sun, and D. Lohse, Experimental investigation of heat transport in homogeneous bubbly flow, J. Fluid Mech. 845, 226 (2018).
- Z. Wang, V. Mathai, and C. Sun, Self-sustained biphasic catalytic particle turbulence, Nat. Commun. 10, 3333 (2019).
- Y. Ling, J. Wagner, S. Beresh, S. Kearney, and S. Balachandar, Interaction of a planar shock wave with a dense particle curtain: Modeling and experiments, Phys. Fluids 24, 113301 (2012).
- E. P. DeMauro, J. L. Wagner, S. J. Beresh, and P. A. Farias, Unsteady drag following shock wave impingement on a dense particle curtain measured using pulse-burst PIV, Phys. Rev. Fluids 2, 064301 (2017).
- Y. Mehta, C. Neal, T. L. Jackson, S. Balachandar, and S. Thakur, Shock interaction with three-dimensional face centered cubic array of particles, Phys. Rev. Fluids 1, 054202 (2016).
- S. Subramaniam, Multiphase flows: Rich physics, challenging theory, and big simulations, 72nd Annual Meeting of the American Physical Society, Division of Fluid Dynamics, Seattle, WA (2019).
- S. Balachandar and J. K. Eaton, Turbulent dispersed multiphase flow, Annu. Rev. Fluid Mech. 42, 111 (2010).
- V. Tavanashad, A. Passalacqua, R. O. Fox, and S. Subramaniam, Effect of density ratio on velocity fluctuations in dispersed multiphase flow from simulations of finite–size particles, Acta Mech. 230, 469 (2019).
- V. Tavanashad and S. Subramaniam, Fully resolved simulation of dense suspensions of freely evolving buoyant particles using an improved immersed boundary method, Int. J. Multiphase Flow 132, 103396 (2020).
- M. Mehrabadi, J. Horwitz, S. Subramaniam, and A. Mani, A direct comparison of particle-resolved and point-particle methods in decaying turbulence, J. Fluid Mech. 850, 336 (2018).
- R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles (Academic Press, New York, 1978).
- M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids 26, 883 (1983).
- D. L. Marchisio and R. O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems (Cambridge University Press, 2013).
- S. Apte, K. Mahesh, and T. Lundgren, Accounting for finite-size effects in simulations of disperse particle-laden flows, Int. J. Multiphase Flow 34, 260 (2008).
- S. Subramaniam, Lagrangian–Eulerian methods for multiphase flows, Prog. Energy Combust. Sci. 39, 215 (2013).
- S. Elghobashi, On predicting particle–laden turbulent flows, Appl. Sci. Res. 52, 309 (1994).
- D. McQuarrie, Statistical Mechanics (University Science Books, Sausalito, CA, 2000).
- R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 3rd ed. (Springer-Verlag, New York, 2003).
- A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics II (MIT Press, 1975).
- S. B. Pope, Turbulent Flows (Cambridge University Press, Port Chester, NY, 2000).
- H. Brenner, Suspension rheology, in Progress in Heat and Mass Transfer (Elsevier, 1972), pp. 89–129.
- E. Hinch, An averaged-equation approach to particle interactions in a fluid suspension, J. Fluid Mech. 83, 695 (1977).
- D. L. Koch, Kinetic theory for a monodisperse gas–solid suspension, Phys. Fluids A 2, 1711 (1990).
- S. Tenneti and S. Subramaniam, Particle-resolved direct numerical simulation for gas-solid flow model development, Annu. Rev. Fluid Mech. 46, 199 (2014).
- L. Jofre, Z. R. del Rosario, and G. Iaccarino, Data-driven dimensional analysis of heat transfer in irradiated particle-laden turbulent flow, Int. J. Multiphase Flow 125, 103198 (2020).
- W. Moore, S. Balachandar, and G. Akiki, A hybrid point-particle force model that combines physical and data-driven approaches, J. Comput. Phys. 385, 187 (2019).
- J. L. Lumley, Whither Turbulence? Turbulence at the Crossroads, Lecture Notes in Physics Vol. 357 (Springer-Verlag, New York, 1990).
- R. J. Adler, The Geometry of Random Fields (SIAM, Chichester, 1981).
- S. Panchev, Random Functions and Turbulence (Pergamon Press, New York, 1971).
- D. A. Drew, Mathematical modeling of two–phase flow, Annu. Rev. Fluid Mech. 15, 261 (1983).
- M. G. Pai and S. Subramaniam, A comprehensive probability density function formalism for multiphase flows, J. Fluid Mech. 628, 181 (2009).
- T. B. Anderson and R. Jackson, A fluid mechanical description of fluidized beds, Ind. Eng. Chem. Fundam. 6, 527 (1967).
- M. Ishii, Thermo-fluid Dynamic Theory of Two-Phase Flow (Eyrolles, Paris, 1975).
- D. D. Joseph, T. S. Lundgren, R. Jackson, and D. A. Saville, Ensemble averaged and mixture theory equations for incompressible fluid–particle suspensions, Int. J. Multiphase Flow 16, 35 (1990).
- J. Smagorinsky, General circulation experiments with the primitive equations: I. The basic experiment, Mon. Weather Rev. 91, 99 (1963).
- D. Lilly, The representation of small–scale turbulence in numerical simulation experiments, in Proc. IBM Scientific Computing Symp. on Environmental Sciences, edited by H. H. Goldstein (IBM, Yorktown Heights, NY, 1967), pp. 195–210.
- P. Moin and J. Kim, Numerical investigation of turbulent channel flow, J. Fluid Mech. 118, 341 (1982).
- K. Goc, S. Bose, and P. Moin, Wall-modeled large eddy simulation of an aircraft in landing configuration, in AIAA Aviation 2020 Forum (American Institute of Aeronautics and Astronautics, Reston, VA, 2020), p. 3002.
- J. Capecelatro and O. Desjardins, An Euler–Lagrange strategy for simulating particle-laden flows, J. Comput. Phys. 238, 1 (2013).
- S. Subramaniam, Statistical representation of a spray as a point process, Phys. Fluids 12, 2413 (2000).
- R. O. Fox, Large-eddy-simulation tools for multiphase flows, Annu. Rev. Fluid Mech. 44, 47 (2012).
- S. Tenneti, M. Mehrabadi, and S. Subramaniam, Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions, J. Fluid Mech. 788, 695 (2016).
- J. K. Eaton and J. R. Fessler, Preferential concentration of particles by turbulence, Int. J. Multiphase Flow 20, 169 (1994).
- V. Garzó, S. Tenneti, S. Subramaniam, and C. M. Hrenya, Enskog kinetic theory for monodisperse gas–solid flows, J. Fluid Mech. 712, 129 (2012).
- B. Kong and R. O. Fox, A solution algorithm for fluid–particle flows across all flow regimes, J. Comput. Phys. 344, 575 (2017).
- J. C. Heylmun, B. Kong, A. Passalacqua, and R. O. Fox, A quadrature-based moment method for polydisperse bubbly flows, Comput. Phys. Commun. 244, 187 (2019).
- J. Capecelatro, O. Desjardins, and R. O. Fox, Numerical study of collisional particle dynamics in cluster-induced turbulence, J. Fluid Mech. 747, R2 (2014).
- N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1973).
- D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields, Wiley Series in Probability and Mathematical Statistics (John Wiley and Sons, New York, 1995).
- D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and its Applications, 2nd ed., Wiley Series in Probability and Mathematical Statistics (John Wiley and Sons, New York, 1995).
- S. Sundaram and L. R. Collins, A numerical study of the modulation of isotropic turbulence by suspended particles, J. Fluid Mech. 379, 105 (1999).
- S. Markutsya, R. O. Fox, and S. Subramaniam, Coarse-graining approach to infer mesoscale interaction potentials from atomistic interactions for aggregating systems, Ind. Eng. Chem. Res. 51, 16116 (2012).
- S. L. Rani, R. Dhariwal, and D. L. Koch, A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence, J. Fluid Mech. 756, 870 (2014).
- A. Cartellier and N. Rivière, Bubble-induced agitation and microstructure in uniform bubbly flows at small to moderate particle Reynolds numbers, Phys. Fluids 13, 2165 (2001).
- A. A. Amsden, P. J. O'Rourke, and T. D. Butler, KIVA–II: A computer program for chemically reactive flows with sprays, Tech. Rep. LA–11560–MS, Los Alamos National Laboratory (1989).
- R. Garg, C. Narayanan, D. Lakehal, and S. Subramaniam, Accurate numerical estimation of interphase momentum transfer in Lagrangian–Eulerian simulations of dispersed two-phase flows, Int. J. Multiphase Flow 33, 1337 (2007).
- R. Garg, C. Narayanan, and S. Subramaniam, A numerically convergent Lagrangian–Eulerian simulation method for dispersed two-phase flows, Int. J. Multiphase Flow 35, 376 (2009).
- S. Subramaniam, Statistical modeling of sprays using the droplet distribution function, Phys. Fluids 13, 624 (2001).
- D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids (Springer, 1998).
- I. Kataoka and A. Serizawa, Basic equations of turbulence in gas–liquid two–phase flow, Int. J. Multiphase Flow 15, 843 (1989).
- R. Jackson, Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid, Chem. Eng. Sci. 52, 2457 (1997).
- C. Hrenya and J. Sinclair, Effects of particle-phase turbulence in gas-solid flows, AIChE J. 43, 853 (1997).
- O. Simonin, Two–fluid model approach for turbulent reactive two–phase flows, Summer school on numerical modeling and prediction of dispersed two-phase flows, IMVU, Meserburg, Germany (1995).
- G. Ahmadi and D. Ma, A thermodynamical formulation for dispersed multiphase turbulent flows—1: Basic theory, Int. J. Multiphase Flow 16, 323 (1990).
- Y. Xu and S. Subramaniam, A multiscale model for dilute turbulent gas-particle flows based on the equilibration of energy concept, Phys. Fluids 18, 033301 (2006).
- R. O. Fox, On multiphase turbulence models for collisional fluid-particle flows, J. Fluid Mech. 742, 368 (2014).
- S. Sundaram and L. R. Collins, Spectrum of density fluctuations in a particle–fluid system—I. Monodisperse spheres, Int. J. Multiphase Flow 20, 1021 (1994).
- S. Torquato and G. Stell, Microstructure of two-phase random media. I. The -point probability functions, J. Chem. Phys. 77, 2071 (1982).
- S. Sundaram and L. R. Collins, Spectrum of density fluctuations in a particle–fluid system—II. Polydisperse spheres, Int. J. Multiphase Flow 20, 1039 (1994).
- B. Lu and S. Torquato, Local volume fraction fluctuations in heterogeneous media, J. Chem. Phys. 93, 3452 (1990).
- E. Murphy, Analysis and modeling of structure formation in granular and fluid-solid flows, Ph.D. thesis, Iowa State University (2017).
- M. Syamlal, W. Rogers, and T. J. O'Brien, MFIX documentation: Theory guide, Tech. Rep. DOE/METC-95/1013, NTIS/DE95000031, National Energy Technology Laboratory, Department of Energy (1993); see also http://www.mfix.org
- V. Kumaran, Stability of a sheared particle suspension, Phys. Fluids 15, 3625 (2003).
- M. Mehrabadi, S. Tenneti, R. Garg, and S. Subramaniam, Pseudo-turbulent gas-phase velocity fluctuations in homogeneous gas-solid flow: Fixed particle assemblies and freely evolving suspensions, J. Fluid Mech. 770, 210 (2015).
- B. Sun, S. Tenneti, S. Subramaniam, and D. L. Koch, Pseudo-turbulent heat flux and average gas–phase conduction during gas–solid heat transfer: Flow past random fixed particle assemblies, J. Fluid Mech. 798, 299 (2016).
- G. Akiki, T. L. Jackson, and S. Balachandar, Force variation within arrays of monodisperse spherical particles, Phys. Rev. Fluids 1, 044202 (2016).
- H. H. Hu, N. A. Patankar, and M. Y. Zhu, Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian–Eulerian technique, J. Comput. Phys. 169, 427 (2001).
- T. Nomura and T. J. R. Hughes, An arbitrary Lagrangian–Eulerian finite element method for interaction of fluid and a rigid body, Comput. Meth. Appl. Mech. Eng. 95, 115 (1992).
- P. Bagchi and S. Balachandar, Effect of turbulence on the drag and lift of a particle, Phys. Fluids 15, 3496 (2003).
- P. Bagchi and S. Balachandar, Response of the wake of an isolated particle to an isotropic turbulent flow, J. Fluid Mech. 518, 95 (2004).
- T. M. Burton and J. K. Eaton, Fully resolved simulations of particle-turbulence interaction, J. Fluid Mech. 545, 67 (2005).
- N. Patankar, P. Singh, D. D. Joseph, R. Glowinski, and T. W. Pan, A new formulation of the distributed Lagrange multipliers/fictitious domain method for particulate flow, Int. J. Multiphase Flow 26, 1509 (2000).
- R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph, and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow, J. Comput. Phys. 169, 363 (2001).
- A. J. C. Ladd and R. Verberg, Lattice–Boltzmann simulations of particle-fluid suspensions, J. Stat. Phys. 104, 1191 (2001).
- N. Sharma and N. Patankar, A fast computation technique for the direct numerical simulation of rigid particulate flows, J. Comput. Phys. 205, 439 (2005).
- S. Apte, M. Martin, and N. Patankar, A numerical method for fully resolved simulation (FRS) of rigid particle–flow interactions in complex flows, J. Comput. Phys. 228, 2712 (2009).
- C. S. Peskin, The fluid dynamics of heart valves: experimental, theoretical, and computational methods, Annu. Rev. Fluid Mech. 14, 235 (1981).
- J. Mohd-Yusof, Interaction of massive particles with turbulence, Ph.D. thesis, Cornell University (1996).
- K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys. 225, 2118 (2007).
- H. Oguz and A. Prosperetti, PHYSALIS: A new () method for the numerical simulation of disperse systems. Part I: Potential flow of spheres, J. Comput. Phys. 167, 196 (2001).
- M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209, 448 (2005).
- R. Garg, Modeling and simulation of two-phase flows, Ph.D. thesis, Iowa State University (2009).
- D. Kim and H. Choi, Immersed boundary method for flow around an arbitrarily moving body, J. Comput. Phys. 21, 662 (2006).
- F. Lucci, A. Ferrante, and S. Elgobashi, Modulation of isotropic turbulence by particles of Taylor length-scale size, J. Fluid Mech. 650, 5 (2010).
- R. Scardovelli and S. Zaleski, Direct numerical simulation of free–surface and interfacial flow, Annu. Rev. Fluid Mech. 31, 567 (1999).
- J. J. Wylie, D. L. Koch, and A. J. Ladd, Rheology of suspensions with high particle inertia and moderate fluid inertia, J. Fluid Mech. 480, 95 (2003).
- S. Tenneti, R. Garg, C. M. Hrenya, R. O. Fox, and S. Subramaniam, Direct numerical simulation of gas–solid suspensions at moderate Reynolds number: Quantifying the coupling between hydrodynamic forces and particle velocity fluctuations, Powder Technol. 203, 57 (2010).
- S. Kriebitzsch, M. van der Hoef, and J. Kuipers, Fully resolved simulation of a gas-fluidized bed: A critical test of DEM models, Chem. Eng. Sci. 91, 1 (2013).
- G. Zhou, Q. Xiong, L. Wang, X. Wang, X. Ren, and W. Ge, Structure-dependent drag in gas–solid flows studied with direct numerical simulation, Chem. Eng. Sci. 116, 9 (2014).
- K. Luo, J. Tan, Z. Wang, and J. Fan, Particle-resolved direct numerical simulation of gas–solid dynamics in experimental fluidized beds, AIChE J. 62, 1917 (2016).
- Y. Tang, E. A. J. F. Peters, and J. A. M. Kuipers, Direct numerical simulations of dynamic gas–solid suspensions, AIChE J. 62, 1958 (2016).
- G. J. Rubinstein, J. J. Derksen, and S. Sundaresan, Lattice Boltzmann simulations of low-Reynolds-number flow past fluidized spheres: Effect of Stokes number on drag force, J. Fluid Mech. 788, 576 (2016).
- G. J. Rubinstein, A. Ozel, X. Yin, J. J. Derksen, and S. Sundaresan, Lattice Boltzmann simulations of low-Reynolds-number flows past fluidized spheres: Effect of inhomogeneities on the drag force, J. Fluid Mech. 833, 599 (2017).
- A. A. Zaidi, Study of particle inertia effects on drag force of finite sized particles in settling process, Chem. Eng. Res. Des. 132, 714 (2018).
- R. Garg, S. Tenneti, J. Mohd-Yusof, and S. Subramaniam, 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, edited by S. Pannala, M. Syamlal, and T. J. O'Brien (IGI Global, 2011), pp. 245–276.
- S. Tenneti, R. Garg, and S. Subramaniam, Drag law for monodisperse gas–solid systems using particle–resolved direct numerical simulation of flow past fixed assemblies of spheres, Int. J. Multiphase Flow 37, 1072 (2011).
- A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow (Cambridge University Press, 2007).
- Y. Xu and S. Subramaniam, Effect of particle clusters on carrier flow turbulence: A direct numerical simulation study, Flow Turbul. Combust. 85, 735 (2010).
- B. Sun, Modeling heat and mass transfer in reacting gas-solid flow using particle-resolved direct numerical simulation, Ph.D. thesis, Iowa State University (2016).
- M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1989).
- K. Zhou and S. Balachandar, Investigation of direct forcing immersed boundary method, 71st Annual Meeting of the American Physical Society, Division of Fluid Dynamics, Atlanta, GA (2018).
- H. Weller, C. Greenshields, and B. Santos, OpenFOAM (2018), http://www.openfoam.org
- Y. Igci, A. T. Andrews IV, S. Sundaresan, S. Pannala, and T. O'Brien, Filtered two-fluid models for fluidized gas-particle suspensions, AIChE J. 54, 1431 (2008).
- S. S. Ozarkar, X. Yan, S. Wang, C. C. Milioli, F. E. Milioli, and S. Sundaresan, Validation of filtered two-fluid models for gas–particle flows against experimental data from bubbling fluidized bed, Powder Technol. 284, 159 (2015).
- D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys. 187, 137 (2003).
- C. Soize and R. G. Ghanem, Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields, Comput. Methods Appl. Mech. Eng. 198, 1926 (2009).
- A. C. Öztireli and M. Gross, Analysis and synthesis of point distributions based on pair correlation, ACM Trans. Graphics (TOG) 31, 1 (2012).
- P. Colucci, F. Jaberi, P. Givi, and S. Pope, Filtered density function for large eddy simulation of turbulent reacting flows, Phys. Fluids 10, 499 (1998).
- R. O. Fox, Computational Models for Turbulent Reacting Flows (Cambridge University Press, 2003).
- Y. Klimontovich, Statistical Theory of Open Systems: Volume 1: A Unified Approach to Kinetic Description of Processes in Active Systems, Fundamental Theories of Physics (Springer Netherlands, Amsterdam, 2012).
- T. P. C. Van Noije, M. H. Ernst, R. Brito, and J. A. G. Orza, Mesoscopic Theory of Granular Fluids, Phys. Rev. Lett. 79, 411 (1997).
- D. R. Nicholson, Introduction to Plasma Theory (Krieger Publishing, Malabar, FL, 1992).
- Y. Klimontovich, Statistical Physics (Taylor & Francis, 1986).
- J. Quintanilla and S. Torquato, Local volume fraction fluctuations in random media, J. Chem. Phys. 106, 2741 (1997).