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

  • Ryan W. Houim (a1) and Elaine S. Oran (a1)

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.

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Corresponding author

Email address for correspondence: rhouim@umd.edu

References

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Abgrall, R. & Karni, S. 2010 A comment on the computation of non-conservative products. J. Comput. Phys. 229 (8), 27592763.
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), 151185.
Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.
ASC FLASH Center 2012 Flash User’s Guide. University of Chicago.
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), 861889.
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), 405452.
Benkiewicz, K. & Hayashi, A. K. 2002 Aluminum dust ignition behind reflected shock wave: two-dimensional simulations. Fluid Dyn. Res. 30 (5), 269292.
Billet, G. & Abgrall, R. 2003 An adaptive shock-capturing algorithm for solving unsteady reactive flows. Comput. Fluids 32 (10), 14731495.
Brilliantov, N. V. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.
Capecelatro, J. & Desjardins, O. 2013 An Euler–Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 131.
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), 840873.
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), 14071413.
Colella, P. & Woodward, P. R. 1984 The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54 (1), 174201.
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.
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, 266292.
Dacombe, P., Pourkashanian, M., Williams, A. & Yap, L. 1999 Combustion-induced fragmentation behavior of isolated coal particles. Fuel 78 (15), 18471857.
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), 113121.
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.
Einfeldt, B., Munz, C. D., Roe, P. L. & Sjögreen, B. 1991 On Godunov-type methods near low densities. J. Comput. Phys. 92 (2), 273295.
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), 179187.
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), 578588.
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), 104113.
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), 29032917.
Gidaspow, D. 1994 Multiphase Flow and Fluidization. Academic.
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.
Grinstein, F. F., Margolin, L. G. & Rider, W. J. 2007 Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press.
Gunn, D. J. 1978 Transfer of heat of mass to particles in fixed and fluidized beds. Intl J. Heat Mass Transfer 21, 467476.
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomena. J. Fluid Mech. 134, 401430.
Harten, A., Lax, P. D. & van Leer, B. 1983 On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1), 3561.
Helland, E., Occelli, R. & Tadrist, L. 2000 Numerical study of cluster formation in a gas–particle circulating fluidized bed. Powder Technol. 110 (3), 210221.
Horio, M. & Kuroki, H. 1994 Three-dimensional flow visualization of dilutely dispersed solids in bubbling and circulating fluidized beds. Chem. Engng Sci. 49 (15), 24132421.
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), 85278553.
Houim, R. W. & Oran, E. S. 2015a Numerical simulation of dilute and dense layered coal-dust explosions. Proc. Combust. Inst. 35 (2), 20832090.
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, 214222.
Huang, K., Wu, H., Yu, H. & Yan, D. 2011 Cures for numerical shock instability in HLLC solver. Intl J. Numer. Meth. Fluids 65, 10261038.
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), 14311448.
Ishii, M. & Hibiki, T. 2006 Thermo-Fluid Dynamics of Two-Phase Flow. Springer.
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), 229236.
Johnson, P. C. & Jackson, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 6793.
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727730.
Kamenetsky, V., Goldshtein, A., Shapiro, M. & Degani, D. 2000 Evolution of a shock wave in a granular gas. Phys. Fluids 12, 3036.
Karni, S. & Hernández-Dueñas, G. 2010 A hybrid algorithm for the Baer–Nunziato model using the Riemann invariants. J. Sci. Comput. 45 (1–3), 382403.
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), 196207.
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), 547555.
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, 223229.
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), 570615.
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, 229263.
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.
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), 433448.
LeVeque, R. J. 2004 The dynamics of pressureless dust clouds and delta waves. J. Hyperbolic Diff. Equ. 1 (2), 315327.
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, 184194.
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), 10261044.
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), 10131025.
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.
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), 23452356.
Liou, M.-S. 1996 A sequel to AUSM: $\text{AUSM}^{+}$ . J. Comput. Phys. 129 (2), 364382.
Liou, M. S. 2006 A sequel to AUSM, Part II: AUSM $^{+}$ -up for all speeds. J. Comput. Phys. 214 (1), 137170.
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), 310316.
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, 223256.
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), 330354.
Markatos, N. C. 1986 Modelling of two-phase transient flow and combustion of granular propellants. Intl J. Multiphase Flow 12 (6), 913933.
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), 10371053.
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), 270289.
Miura, H. & Glass, I. I. 1982 On a dusty-gas shock tube. Proc. R. Soc. Lond. A 382 (1783), 373388.
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.
Nigmatulin, R. I. 1990 Dynamics of Multiphase Media, vol. 1. Taylor & Francis.
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.
Nussbaum, J., Helluy, P., Hérard, J.-M. & Carriére, A. 2006 Numerical simulations of gas–particle flows with combustion. Flow Turbul. Combust. 76, 403417.
Oran, E. S. & Gamezo, V. N. 2007 Origins of the deflagration-to-detonation transition in gas-phase combustion. Combust. Flame 148 (1–2), 447.
Pandolfi, M. & D’Ambrosio, D. 2001 Numerical instabilities in upwind methods: analysis and cures for the ‘carbuncle’ phenomenon. J. Comput. Phys. 166 (2), 271301.
Parmar, M., Haselbacher, A. & Balachandar, S. 2010 Improved drag correlation for spheres and application to shock-tube experiments. AIAA J. 48 (6), 12731276.
Pelanti, M. & LeVeque, R. J. 2006 High-resolution finite volume methods for dusty gas jets and plumes. SIAM J. Sci. Comput. 24, 13351360.
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), 9951011.
Poludnenko, A. Y. & Oran, E. S. 2011 The interaction of high-speed turbulence with flames: turbulent flame speed. Combust. Flame 158 (2), 301326.
Porterie, B. & Loraud, J. C. 1994 An investigation of interior ballistics ignition phase. Shock Waves 4, 8193.
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), 2945.
Saito, T., Marumoto, M. & Takayama, K. 2003 Numerical investigations of shock waves in gas–particle mixtures. Shock Waves 13 (4), 299322.
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), 229242.
Saurel, R. & Abgrall, R. 1999 A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (2), 425467.
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), 279292.
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), 490526.
Serna, S. & Marquina, A. 2005 Capturing shock waves in inelastic granular gases. J. Comput. Phys. 209 (2), 787795.
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), 469491.
Srivastava, A. & Sundaresan, S. 2003 Analysis of a frictional–kinetic model for gas–particle flow. Powder Technol. 129 (1), 7285.
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.
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), 737758.
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), 384397.
Thornber, B., Mosedale, A. & Drikakis, D. 2007 On the implicit large eddy simulations of homogeneous decaying turbulence. J. Comput. Phys. 226 (2), 12021929.
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), 48734894.
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. 472481. Springer.
Toro, E. F. 1999 Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn. Springer.
Toro, E. F., Spruce, M. & Speares, W. 1994 Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4 (1), 2534.
Van der Weele, K. 2008 Granular gas dynamics: how Maxwell’s demon rules in a non-equilibrium system. Contemp. Phys. 49 (3), 157178.
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), 10351051.
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), 15071517.
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.
Wilson, L. 1980 Relationships between pressure, volatile content and ejecta velocity in three types of volcanic explosion. J. Volcanol. Geotherm. Res. 8 (2), 297313.
Zèmerli, C.2013 Continuum mechanical modeling of dry granular systems: from dilute flow to solid-like behavior. PhD thesis, Technical University of Kaiserslautern.
Zhang, D. Z. 2005 Evolution of enduring contacts and stress relaxation in a dense granular medium. Phys. Rev. E 71, 041303.
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), 121142.
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), 528532.
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), 306314.
Zimmermann, S. & Taghipour, F. 2005 CFD modeling of the hydrodynamics and reaction kinetics of FCC fluidized-bed reactors. Ind. Engng Chem. Res. 44 (26), 98189827.
Zydak, P. & Klemens, R. 2007 Modelling of dust lifting process behind propagating shock wave. J. Loss Prev. Process. Ind. 20 (4), 417426.
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