Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-05-02T11:19:06.984Z Has data issue: false hasContentIssue false
  • English
  • Français

A multiphase model for compressible granular–gaseous flows: formulation and initial tests

Published online by Cambridge University Press:  18 January 2016

Ryan W. Houim  and
Elaine S. Oran
Ryan W. Houim*
Affiliation:
Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, USA
Elaine S. Oran
Affiliation:
Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A model for predicting the behaviour of a compressible flow laden with shocks interacting with granular material has been developed and tested. The model consists of two sets of coupled Euler equations, one for the gas phase and the other for the granular phase. Drag, convective, heat transfer and non-conservative terms couple the two sets of governing equations. Intergranular stress acting on the grains is modelled using granular kinetic theory in dilute regimes where particle collisions are dominant and frictional–collisional pressure in dense regions where layers of granular material slide over one another. The two-phase granular–gaseous model, as a result, is valid from dilute to densely packed granular regimes. The solution of these nonlinearly coupled Euler equations is challenging due to the presence of the non-conservative nozzling and work terms. A numerical technique, based on Godunov’s method, was designed for solving these equations. This method takes advantage of particle incompressibility to simplify the nozzling terms. It also uses the observation that a Riemann problem is valid in the region where gas can flow between particles and can be used to provide a physically accurate approximation of the non-conservative terms. The model and solution method are verified by comparisons to test problems involving granular shocks and two-phase shock-tube problems, and they are validated against experimental measurements of shock and dense particle-curtain interactions and transmitted oblique granular shocks.

JFM classification

Granular media: Granular media Multiphase and Particle-laden Flows: Multiphase flow Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 166 - 220
Check for updates
Copyright
© 2016 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.)

Article purchase

Temporarily unavailable

References

Abgrall, R. & Karni, S. 2010 A comment on the computation of non-conservative products. J. Comput. Phys. 229 (8), 2759–2763.Google Scholar
Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S. 2001 The role of meso-scale structures in rapid gas–solid flows. J. Fluid Mech. 445 (1), 151–185.CrossRefGoogle Scholar
Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.Google Scholar
ASC FLASH Center 2012 Flash User’s Guide. University of Chicago.Google Scholar
Baer, M. R. & Nunziato, J. W. 1986 A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Intl J. Multiphase Flow 12 (6), 861–889.Google Scholar
Balsara, D. S. & Shu, C.-W. 2000 Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160 (2), 405–452.CrossRefGoogle Scholar
Benkiewicz, K. & Hayashi, A. K. 2002 Aluminum dust ignition behind reflected shock wave: two-dimensional simulations. Fluid Dyn. Res. 30 (5), 269–292.CrossRefGoogle Scholar
Billet, G. & Abgrall, R. 2003 An adaptive shock-capturing algorithm for solving unsteady reactive flows. Comput. Fluids 32 (10), 1473–1495.Google Scholar
Brilliantov, N. V. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.Google Scholar
Capecelatro, J. & Desjardins, O. 2013 An Euler–Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 1–31.CrossRefGoogle Scholar
Chang, C.-H. & Liou, M.-S. 2007 A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM $^{+}$ -up scheme. J. Comput. Phys. 225 (1), 840–873.CrossRefGoogle Scholar
Chuanjie, Z., Baiquan, L., Bingyou, J., Qian, L. & Yidu, H. 2012 Simulation of dust lifting process induced by gas explosion disaster in underground coal mine. Disaster Adv. 5 (4), 1407–1413.Google Scholar
Colella, P. & Woodward, P. R. 1984 The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54 (1), 174–201.Google Scholar
Collins, J. P., Ferguson, R. E., Chien, K., Kuhl, A. L., Krispin, J. & Glaz, H. M. 1994 Simulation of shock-induced dusty gas flows using various models. In 25th AIAA Fluid Dynamics Conference, Colorado Springs, CO.Google Scholar
Crochet, M. W. & Gonthier, K. A. 2013 Numerical investigation of a modified family of centered schemes applied to multiphase equations with nonconservative sources. J. Comput. Phys. 255, 266–292.Google Scholar
Dacombe, P., Pourkashanian, M., Williams, A. & Yap, L. 1999 Combustion-induced fragmentation behavior of isolated coal particles. Fuel 78 (15), 1847–1857.Google Scholar
Drew, D. A. & Lahey, R. T. Jr. 1987 The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Intl J. Multiphase Flow 13 (1), 113–121.Google Scholar
Edwards, J. C. & Ford, K. M.1988 Model of coal dust explosion suppression by rock dust entrainment. Tech. Rep. RI 9206, US Department of the Interior, Bureau of Mines.Google Scholar
Einfeldt, B., Munz, C. D., Roe, P. L. & Sjögreen, B. 1991 On Godunov-type methods near low densities. J. Comput. Phys. 92 (2), 273–295.Google Scholar
Fan, B. C., Chen, Z. H., Jiang, X. H. & Li, H. Z. 2007 Interaction of a shock wave with a loose dusty bulk layer. Shock Waves 16 (3), 179–187.Google Scholar
Fedorov, A. V. & Fedorchenko, I. A. 2010 Numerical simulation of shock wave propagation in a mixture of a gas and solid particles. Combust. Explos. Shock Waves 46 (5), 578–588.Google Scholar
Fedorov, A. V., Kharlamova, Y. V. & Khmel’, T. A. 2007 Reflection of a shock wave in a dusty cloud. Combust. Explos. Shock Waves 43 (1), 104–113.Google Scholar
Gerber, S., Behrendt, F. & Oevermann, M. 2010 An Eulerian modeling approach of wood gasification in a bubbling fluidized bed reactor using char as bed material. Fuel 89 (10), 2903–2917.Google Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization. Academic.Google Scholar
Goos, E., Burcat, A. & Rusnic, B.2010 Ideal gas thermochemical database with updates from active thermochemical tables. http://garfield.chem.elte.hu/Burcat/burcat.html.CrossRefGoogle Scholar
Grinstein, F. F., Margolin, L. G. & Rider, W. J. 2007 Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Gunn, D. J. 1978 Transfer of heat of mass to particles in fixed and fluidized beds. Intl J. Heat Mass Transfer 21, 467–476.Google Scholar
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomena. J. Fluid Mech. 134, 401–430.Google Scholar
Harten, A., Lax, P. D. & van Leer, B. 1983 On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1), 35–61.Google Scholar
Helland, E., Occelli, R. & Tadrist, L. 2000 Numerical study of cluster formation in a gas–particle circulating fluidized bed. Powder Technol. 110 (3), 210–221.Google Scholar
Horio, M. & Kuroki, H. 1994 Three-dimensional flow visualization of dilutely dispersed solids in bubbling and circulating fluidized beds. Chem. Engng Sci. 49 (15), 2413–2421.Google Scholar
Houim, R. W. & Kuo, K. K. 2011 A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios. J. Comput. Phys. 230 (23), 8527–8553.Google Scholar
Houim, R. W. & Oran, E. S. 2015a Numerical simulation of dilute and dense layered coal-dust explosions. Proc. Combust. Inst. 35 (2), 2083–2090.Google Scholar
Houim, R. W. & Oran, E. S. 2015b Structure and flame speed of dilute and dense layered coal-dust explosions. J. Loss Prev. Process. Ind. 36, 214–222.Google Scholar
Huang, K., Wu, H., Yu, H. & Yan, D. 2011 Cures for numerical shock instability in HLLC solver. Intl J. Numer. Meth. Fluids 65, 1026–1038.CrossRefGoogle Scholar
Igci, Y., Andrews, A. T., Sundaresan, S., Pannala, S. & O’Brien, T. 2008 Filtered two-fluid models for fluidized gas–particle suspensions. AIChE J. 54 (6), 1431–1448.Google Scholar
Ishii, M. & Hibiki, T. 2006 Thermo-Fluid Dynamics of Two-Phase Flow. Springer.Google Scholar
Jenike, A. W. 1987 A theory of flow of particulate solids in converging and diverging channels based on a conical yield function. Powder Technol. 50 (3), 229–236.Google Scholar
Johnson, P. C. & Jackson, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93.Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727–730.Google Scholar
Kamenetsky, V., Goldshtein, A., Shapiro, M. & Degani, D. 2000 Evolution of a shock wave in a granular gas. Phys. Fluids 12, 3036.Google Scholar
Karni, S. & Hernández-Dueñas, G. 2010 A hybrid algorithm for the Baer–Nunziato model using the Riemann invariants. J. Sci. Comput. 45 (1–3), 382–403.Google Scholar
Khmel’, T. A. & Fedorov, A. V. 2014a Description of dynamic processes in two-phase colliding media with the use of molecular-kinetic approaches. Combust. Explos. Shock Waves 50 (2), 196–207.Google Scholar
Khmel’, T. A. & Fedorov, A. V. 2014b Modeling of propagation of shock and detonation waves in dusty media with allowance for particle collisions. Combust. Explos. Shock Waves 50 (5), 547–555.CrossRefGoogle Scholar
Khmel’, T. & Fedorov, A. 2015 Numerical simulation of dust dispersion using molecular-kinetic model for description of particle-to-particle collisions. J. Loss Prev. Process. Ind. 36, 223–229.CrossRefGoogle Scholar
Kim, K. H. & Kim, C. 2005 Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II: multi-dimensional limiting process. J. Comput. Phys. 208 (2), 570–615.Google Scholar
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for a homogenous monodisperse gas fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229–263.Google Scholar
Koo, J. H. & Kuo, K. K.1977 Transient combustion in granular propellant beds. Part I. Theoretical modeling and numerical solution of transient combustion processes in mobile granular propellant beds. Tech. Rep. DAAG 29-74-G-0116. US Army Research Office.Google Scholar
Kuhl, A. L., Bell, J. B. & Beckner, V. E. 2010 Heterogeneous continuum model of aluminum particle combustion in explosions. Combust. Explos. Shock Waves 46 (4), 433–448.Google Scholar
LeVeque, R. J. 2004 The dynamics of pressureless dust clouds and delta waves. J. Hyperbolic Diff. Equ. 1 (2), 315–327.Google Scholar
Lhuillier, D., Chang, C.-H. & Theofanous, T. G. 2013 On the quest for a hyperbolic effective-field model of disperse flows. J. Fluid Mech. 731, 184–194.CrossRefGoogle Scholar
Ling, Y., Haselbacher, A. & Balachandar, S. 2011a Importance of unsteady contributions to force and heating for particles in compressible flows. Part 1: Modeling and analysis for shock-particle interaction. Intl J. Multiphase Flow 37 (9), 1026–1044.CrossRefGoogle Scholar
Ling, Y., Haselbacher, A. & Balachandar, S. 2011b Importance of unsteady contributions to force and heating for particles in compressible flows. Part 2: Application to particle dispersal by blast waves. Intl J. Multiphase Flow 37 (9), 1013–1025.Google Scholar
Ling, Y., Wagner, J. L., Beresh, S. J., Kearney, S. P. & Balachandar, S. 2012 Interaction of a planar shock wave with a dense particle curtain: modeling and experiments. Phys. Fluids 24 (11), 113301.Google Scholar
Liou, M. S., Chang, C. H., Nguyen, L. & Theofanous, T. G. 2008 How to solve compressible multifluid equations: a simple, robust, and accurate method. AIAA J. 46 (9), 2345–2356.Google Scholar
Liou, M.-S. 1996 A sequel to AUSM: $\text{AUSM}^{+}$ . J. Comput. Phys. 129 (2), 364–382.Google Scholar
Liou, M. S. 2006 A sequel to AUSM, Part II: AUSM $^{+}$ -up for all speeds. J. Comput. Phys. 214 (1), 137–170.Google Scholar
Liu, Q., Hu, Y., Bai, C. & Chen, M. 2013 Methane/coal dust/air explosions and their suppression by solid particle suppressing agents in a large-scale experimental tube. J. Loss Prev. Process. Ind. 26 (2), 310–316.CrossRefGoogle Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.Google Scholar
MacNeice, P., Olson, K. M., Mobarry, C., de Fainchtein, R. & Packer, C. 2000 PARAMESH: a parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun. 126 (3), 330–354.Google Scholar
Markatos, N. C. 1986 Modelling of two-phase transient flow and combustion of granular propellants. Intl J. Multiphase Flow 12 (6), 913–933.Google Scholar
Markatos, N. C. & Kirkcaldy, D. 1983 Analysis and computation of three-dimensional, transient flow and combustion through granulated propellants. Intl J. Heat Mass Transfer 26 (7), 1037–1053.Google Scholar
Martín, M. P., Taylor, E. M., Wu, M. & Weirs, V. G. 2006 A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220 (1), 270–289.Google Scholar
Miura, H. & Glass, I. I. 1982 On a dusty-gas shock tube. Proc. R. Soc. Lond. A 382 (1783), 373–388.Google Scholar
Neri, A., Ongaro, T. E., Macedonio, G. & Gidaspow, D. 2003 Multiparticle simulation of collapsing volcanic columns and pyroclastic flow. J. Geophys. Res. 108 (B4), 2202.Google Scholar
Nigmatulin, R. I. 1990 Dynamics of Multiphase Media, vol. 1. Taylor & Francis.Google Scholar
Nusca, M. J., Horst, A. W. & Newill, J. F.2004 Multidimensional, two-phase simulations of notional telescoped ammunition propelling charge. Tech. Rep. ARL-TR-3306. US Army Research Laboratory.Google Scholar
Nussbaum, J., Helluy, P., Hérard, J.-M. & Carriére, A. 2006 Numerical simulations of gas–particle flows with combustion. Flow Turbul. Combust. 76, 403–417.CrossRefGoogle Scholar
Oran, E. S. & Gamezo, V. N. 2007 Origins of the deflagration-to-detonation transition in gas-phase combustion. Combust. Flame 148 (1–2), 4–47.Google Scholar
Pandolfi, M. & D’Ambrosio, D. 2001 Numerical instabilities in upwind methods: analysis and cures for the ‘carbuncle’ phenomenon. J. Comput. Phys. 166 (2), 271–301.CrossRefGoogle Scholar
Parmar, M., Haselbacher, A. & Balachandar, S. 2010 Improved drag correlation for spheres and application to shock-tube experiments. AIAA J. 48 (6), 1273–1276.Google Scholar
Pelanti, M. & LeVeque, R. J. 2006 High-resolution finite volume methods for dusty gas jets and plumes. SIAM J. Sci. Comput. 24, 1335–1360.Google Scholar
Poludnenko, A. Y. & Oran, E. S. 2010 The interaction of high-speed turbulence with flames: global properties and internal flame structure. Combust. Flame 157 (5), 995–1011.CrossRefGoogle Scholar
Poludnenko, A. Y. & Oran, E. S. 2011 The interaction of high-speed turbulence with flames: turbulent flame speed. Combust. Flame 158 (2), 301–326.Google Scholar
Porterie, B. & Loraud, J. C. 1994 An investigation of interior ballistics ignition phase. Shock Waves 4, 81–93.Google Scholar
Rogue, X., Rodriguez, G., Haas, J. F. & Saurel, R. 1998 Experimental and numerical investigation of the shock-induced fluidization of a particles bed. Shock Waves 8 (1), 29–45.Google Scholar
Saito, T., Marumoto, M. & Takayama, K. 2003 Numerical investigations of shock waves in gas–particle mixtures. Shock Waves 13 (4), 299–322.CrossRefGoogle Scholar
Sapko, M. J., Weiss, E. S., Cashdollar, K. L. & Zlochower, I. A. 2000 Experimental mine and laboratory dust explosion research at NIOSH. J. Loss Prev. Process. Ind. 13 (3–5), 229–242.Google Scholar
Saurel, R. & Abgrall, R. 1999 A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (2), 425–467.Google Scholar
Schneiderbauer, S., Aigner, A. & Pirker, S. 2012 A comprehensive frictional–kinetic model for gas–particle flows: analysis of fluidized and moving bed regimes. Chem. Engng Sci. 80 (1), 279–292.Google Scholar
Schwendeman, D. W., Wahle, C. W. & Kapila, A. K. 2006 The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212 (2), 490–526.Google Scholar
Serna, S. & Marquina, A. 2005 Capturing shock waves in inelastic granular gases. J. Comput. Phys. 209 (2), 787–795.Google Scholar
Spiteri, R. J. & Ruuth, S. J. 2003 A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40 (2), 469–491.Google Scholar
Srivastava, A. & Sundaresan, S. 2003 Analysis of a frictional–kinetic model for gas–particle flow. Powder Technol. 129 (1), 72–85.Google Scholar
Syamlal, M., Rogers, W. & O’Brien, T. J.1993 MFIX Documentation, Vol. 1, Theory Guide. Tech. Rep. DOE/METC-9411004, NTIS/DE9400087. National Technical Information Service.Google Scholar
Talbot, L., Cheng, R. K., Schefer, R. W. & Willis, D. R. 1980 Thermophoresis of particles in a heated boundary layer. J. Fluid Mech. 101 (4), 737–758.Google Scholar
Taylor, E. M., Wu, M. & Martín, M. P. 2007 Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. J. Comput. Phys. 223 (1), 384–397.CrossRefGoogle Scholar
Thornber, B., Mosedale, A. & Drikakis, D. 2007 On the implicit large eddy simulations of homogeneous decaying turbulence. J. Comput. Phys. 226 (2), 1202–1929.Google Scholar
Thornber, B., Mosedale, A., Drikakis, D., Youngs, D. & Williams, R. J. R. 2008 An improved reconstruction method for compressible flows with low Mach number features. J. Comput. Phys. 227 (10), 4873–4894.Google Scholar
Toro, E. F. 1989 Riemann-problem-based techniques for computing reactive two-phased flows. In Numerical Combustion (ed. A., Dervieux & B., Larrouturou), Lecture Notes in Physics, vol. 351, pp. 472–481. Springer.Google Scholar
Toro, E. F. 1999 Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn. Springer.Google Scholar
Toro, E. F., Spruce, M. & Speares, W. 1994 Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4 (1), 25–34.CrossRefGoogle Scholar
Van der Weele, K. 2008 Granular gas dynamics: how Maxwell’s demon rules in a non-equilibrium system. Contemp. Phys. 49 (3), 157–178.CrossRefGoogle Scholar
Van Wachem, B. G. M., Schouten, J. C., van den Bleek, C. M., Krishna, R. & Sinclair, J. L. 2001 Comparative analysis of CFD models of dense gas–solid systems. AIChE J. 47 (5), 1035–1051.Google Scholar
Wagner, J. L., Beresh, S. J., Kearney, S. P., Trott, W. M., Castaneda, J. N., Pruett, B. O. & Baer, M. R. 2012 A multiphase shock tube for shock wave interactions with dense particle fields. Exp. Fluids 52 (6), 1507–1517.Google Scholar
Wayne, P. J., Vorobieff, P., Smyth, H., Bernard, T., Corbin, C., Maloney, A., Conroy, J., White, R., Anderson, M., Kumar, S. et al. 2013 Shock-driven particle transport off smooth and rough surfaces. Trans. ASME J. Fluids Engng 135 (6), 061302.Google Scholar
Wilson, L. 1980 Relationships between pressure, volatile content and ejecta velocity in three types of volcanic explosion. J. Volcanol. Geotherm. Res. 8 (2), 297–313.Google Scholar
Zèmerli, C.2013 Continuum mechanical modeling of dry granular systems: from dilute flow to solid-like behavior. PhD thesis, Technical University of Kaiserslautern.Google Scholar
Zhang, D. Z. 2005 Evolution of enduring contacts and stress relaxation in a dense granular medium. Phys. Rev. E 71, 041303.Google Scholar
Zhao, Z. & Fernando, H. J. S. 2007 Numerical simulation of scour around pipelines using an Euler–Euler coupled two-phase model. Environ. Fluid Mech. 7 (2), 121–142.Google Scholar
Zheng, Y.-P., Feng, C.-G., Jing, G.-X., Qian, X.-M., Li, X.-J., Liu, Z.-Y. & Huang, P. 2009 A statistical analysis of coal mine accidents caused by coal dust explosions in China. J. Loss Prev. Process. Ind. 22 (4), 528–532.Google Scholar
Zhou, W., Zhao, C. S., Duan, L. B., Qu, C. R. & Chen, X. P. 2011 Two-dimensional computational fluid dynamics simulation of coal combustion in a circulating fluidized bed combustor. Chem. Engng J. 166 (1), 306–314.Google Scholar
Zimmermann, S. & Taghipour, F. 2005 CFD modeling of the hydrodynamics and reaction kinetics of FCC fluidized-bed reactors. Ind. Engng Chem. Res. 44 (26), 9818–9827.Google Scholar
Zydak, P. & Klemens, R. 2007 Modelling of dust lifting process behind propagating shock wave. J. Loss Prev. Process. Ind. 20 (4), 417–426.Google Scholar