Simulation of a shock wave interaction with a bounded inhomogeneous gas–particle layer using the hybrid large-particle method
Keywords:hybrid large-particle method, inhomogeneous gas–particle layer, shock wave, relaxation, asymptotically exact solution
The problems of shock wave interaction with a bounded layer of gas suspension is studied in the case when a square-section inhomogeneity of reduced or increased density is situated inside this layer. The hybrid large-particle method of the second-order approximation in space and time is used for calculations. The numerical correctness of discontinuous solutions, in particular jumps of porosity, is confirmed by comparison with the asymptotically exact profiles of the mixture density. Analytical dependences of shock wave attenuation by a gas suspension layer are given. Shock-wave structures in two-dimensional regions and the effect of relaxation processes on them are analyzed.
- B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Amer. Math. Soc., Providence, 1982).
- G. A. Ruev, B. L. Rozhdestvenskii, V. M. Fomin, and N. N. Yanenko, “Conservation Laws for Systems of Equations for Two-Phase Media,” Dokl. Akad. Nauk SSSR 254 (2), 288-293 (1980) [Sov. Math. Dokl. 22 (2), 352-357 (1980)].
- L. Huilin, D. Gidaspow, J. Bouillard, and L. Wentie, “Hydrodynamic Simulation of Gas-Solid Flow in a Riser Using Kinetic Theory of Granular Flow,” Chem. Eng. J. 95 (1-3), 1-13 (2003).
- K. N. Volkov, V. N. Emelyanov, A. G. Karpenko, and I. V. Teterina, “Simulation of Unsteady Gas-Particle Flow Induced by the Shock-Wave Interaction with a Particle Layer,” Vychisl. Metody Programm. 21 (1), 96-114 (2020).
- R. Saurel and R. Abgrall, “A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows,” J. Comput. Phys. 150 (2), 425-467 (1999).
- R. Abgrall and R. Saurel, “Discrete Equations for Physical and Numerical Compressible Multiphase Mixtures,” J. Comput. Phys. 186 (2), 361-396 (2003).
- S. A. Tokareva and E. F. Toro, “HLLC-Type Riemann Solver for the Baer-Nunziato Equations of Compressible Two-Phase Flow,” J. Comput. Phys. 229 (10), 3573-3604 (2010).
- R. Jackson, “The Mechanics of Fluidized Beds. I: The Stability of the State of Uniform Fluidization,” Trans. Inst. Chem. Eng. 41, 13-21 (1963).
- G. Rudinger and A. Chang, “Analysis of Non-Steady Two-Phase Flow,” Phys. Fluid 7, 1747-1754 (1964).
- R. I. Nigmatulin, Fundamentals of the Mechanics of Heterogeneous Media (Nauka, Moscow, 1978) [in Russian].
- C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy, “Kinetic Theories for Granular Flow: Inelastic Particles in Couette Flow and Slightly Inelastic Particles in a General Flowfield,” J. Fluid Mech. 140, 223-256 (1984).
- J. Ding and D. Gidaspow, “A Bubbling Fluidization Model Using Kinetic Theory of Granular Flow,” AIChE J. 36 (4), 523-538 (1990).
- A. Boemer, H. Qi, and U. Renz, “Eulerian Simulation of Bubble Formation at a Jet in a Two-Dimensional Fluidized Bed,” Int. J. Multiph. Flow 23 (5), 927-944 (1997).
- M. A. Gol’dshtik, “Elementary Theory of the Boiling Layer,” Zh. Prikl. Mekh. Tekh. Fiz., No. 6, 106-112 (1972) [J. Appl. Mech. Tech. Phys. 13 (6), 851-856 (1972)].
- D. V. Sadin, “Behavior of the Unsteady Jet of a Mixture of a Pressurized Gas and Dispersed Particles Discharged from a Circular Duct into the Atmosphere,” Zh. Prikl. Mekh. Tekh. Fiz., 40 (1), 151-157 (1999) [J. Appl. Mech. Tech. Phys. 40 (1), 130-135 (1999)].
- D. V. Sadin, S. D. Lyubarskii, and Yu. A. Gravchenko, “Features of an Underexpanded Pulsed Impact Gas-Dispersed Jet with a High Particle Concentration,” Zh. Tekh. Fiz. 87 (1), 22-26 (2017) [Tech. Phys. 62 (1), 18-23 (2017)].
- D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions (Academic Press, New York, 1994).
- A. Goldshtein and M. Shapiro, “Mechanics of Collisional Motion of Granular Materials. Part 1. General Hydrodynamic Equations,” J. Fluid Mech. 282, 75-114 (1995).
- R. W. Lyczkowski, D. Gidaspow, C. W. Solbrig, and E. D. Hughes, “Characteristics and Stability Analyses of Transient One-Dimensional Two-Phase Flow Equations and Their Finite Difference Approximations,” Nucl. Sci. Eng. 66 (3), 378-396 (1978).
- L. A. Klebanov, A. E. Kroshilin, B. I. Nigmatulin, and R. I. Nigmatulin, “On the Hyperbolicity, Stability and Correctness of the Cauchy Problem for the System of Equations of Two-Speed Motion of Two-Phase Media,” Prikl. Mat. Mech. 46 (1), 83-95 (1982) [J. Appl. Math. Mech. 46 (1), 66-74 (1982)].
- D. A. Drew, “Mathematical Modelling of Two-Phase Flow,” Ann. Rev. Fluid Mech. 15, 261-291 (1983).
- V. S. Surov, “Hyperbolic Models in the Mechanics of Heterogeneous Media,” Zh. Vychisl. Mat. Mat. Fiz. 54 (1), 139-148 (2014) [Comput. Math. Math. Phys. 54 (1), 148-157 (2014)].
- M. Hantke, C. Matern, and G. Warnecke, “Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model,” in Theory, Numerics and Applications of Hyperbolic Problems I (Springer, Cham, 2018), Vol. 236, pp. 665-675.
- D. V. Sadin, “On the Convergence of a Certain Class of Difference Schemes for the Equations of Unsteady Gas Motion in a Disperse Medium,” Zh. Vychisl. Mat. Mat. Fiz. 38 (9), 1572-1577 (1998) [Comput. Math. Math. Phys. 38 (9), 1508–1513 (1998)].
- D. V. Sadin, “A Modified Large-Particle Method for Calculating Unsteady Gas Flows in a Porous Medium,” Zh. Vychisl. Mat. Mat. Fiz. 36 (10), 158-164 (1996) [Comput. Math. Math. Phys. 36 (10), 1453-1458 (1996)].
- D. V. Sadin, “A Method for Computing Heterogeneous Wave Flows with Intense Phase Interaction,” Zh. Vychisl. Mat. Mat. Fiz. 38 (6), 1033-1039 (1998) [Comput. Math. Math. Phys. 38 (6), 987-993 (1998)].
- L. Gascón and J. M. Corberán, “Construction of Second-Order TVD Schemes for Nonhomogeneous Hyperbolic Conservation Laws,” J. Comput. Phys. 172 (1), 261-297 (2001).
- Y. Xing and C.-W. Shu, “High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms,” J. Sci. Comput. 27 (1-3), 477-494 (2006).
- R. Saurel, O. Le Métayer, J. Massoni, and S. Gavrilyuk, “Shock Jump Relations for Multiphase Mixtures with Stiff Mechanical Relaxation,” Shock Waves 16 (3), 209-232 (2007).
- D. V. Sadin, “TVD Scheme for Stiff Problems of Wave Dynamics of Heterogeneous Media of Nonhyperbolic Nonconservative Type,” Zh. Vychisl. Mat. Mat. Fiz. 56 (12), 2098-2109 (2016) [Comput. Math. Math. Phys. 56 (12), 2068-2978 (2016)].
- D. V. Sadin, “On Stiff Systems of Partial Differential Equations for Motion of Heterogeneous Media,” Mat. Model. 14 (11), 43-53 (2002).
- D. V. Sadin, “Stiffness Problem in Modeling Wave Flows of Heterogeneous Media with a Three-Temperature Scheme of Interphase Heat and Mass Transfer,” Zh. Prikl. Mekh. Tekh. Fiz. 43 (2), 136-141 (2002) [J. Appl. Mech. Tech. Phys. 43 (2), 286-290 (2002)].
- V. M. Bojko, V. P. Kiselev, S. P. Kiselev, et al., “Interaction of a Shock Wave with a Cloud of Particles,” Fiz. Goreniya Vzryva 32 (2), 86-99 (1996) [Combust. Explos. Shock Waves 32 (2), 191-203 (1996)].
- S. L. Davis, T. B. Dittmann, G. B. Jacobs, and W. S. Don, “Dispersion of a Cloud of Particles by a Moving Shock: Effects of the Shape, Angle of Rotation, and Aspect Ratio,” Zh. Prikl. Mekh. Tekh. Fiz. 54 (6), 45-59 (2013) [J. Appl. Mech. Tech. Phys. 54 (6), 900-912 (2013)].
- D. A. Tukmakov, “Numerical Study of Intense Shock Waves in Dusty Media with a Homogeneous and Two-Component Carrier Phase,” Comput. Issled. Model. 12 (1), 141-154 (2020).
- D. V. Sadin and V. A. Davidchuk, “Interaction of a Plane Shock Wave with Regions of Varying Shape and Density in a Finely Divided Gas Suspension,” Inzh. Fiz. Zh. 93 (2), 489-498 (2020) [J. Eng. Phys. Thermophys. 93 (2), 474-483 (2020)].
- K. N. Volkov, V. N. Emelyanov, A. G. Karpenko, and I. V. Teterina, “Two-Dimensional Effects on the Interaction of a Shock Wave with a Cloud of Particles,” Vychisl. Metody Programm. 21, 207-224 (2020).
- R. I. Nigmatulin, Dynamics of Multiphase Media (Nauka, Moscow, 1987; Hemisphere, New York, 1990).
- D. V. Sadin, “A Modification of the Large-Particle Method to a Scheme Having the Second Order of Accuracy in Space and Time for Shockwave Flows in a Gas Suspension,” Vestn. Yuzhn. Ural. Gos. Univ. Ser. Mat. Model. Programm. 12 (2), 112-122 (2019).
- R. B. Christensen, Godunov Methods on a Staggered Mesh — An Improved Artificial Viscosity , Preprint UCRL-JC-105269 (Lawrence Livermore Nat. Lab., Livermore, 1990).
- C. Hirsch, Numerical Computation of Internal and External Flows. Vol. 2: Computational Methods for Inviscid and Viscous Flows (Wiley, New York, 1990).
- D. V. Sadin, Fundamentals of the Theory of Modeling Wave Heterogeneous Processes (Mozhaisky Military Space Academy, St. Petersburg, 2000) [in Russian].
- D. V. Sadin, “Stiff Problems of a Two-Phase Flow with a Complex Wave Structure,” Fiz.-Khim. Kinetika Gaz Din. 15 (4) (2014).
http://chemphys.edu.ru/issues/2014-15-4/articles/243/. Cited December 6, 2020.
How to Cite
Copyright (c) 2021 Numerical methods and programming
This work is licensed under a Creative Commons Attribution 4.0 International License.