DOI: https://doi.org/10.26089/NumMet.v21r109

Simulation of unsteady gas-particle flow induced by the shock-wave interaction with a particle layer

Authors

  • K.N. Volkov
  • V.N. Emelyanov
  • A.G. Karpenko
  • I.V. Teterina

Keywords:

two-phase flow
numerical simulation
shock wave
particle
concentration

Abstract

A numerical simulation of the unsteady gas-particle flow arising from the shock-wave interaction with a layer of inert particles is performed based on a continuum model. Each phase is described by a set of equations describing the conservation laws of mass, momentum and energy. The interphase interaction is taken into account using source terms in the momentum and energy equations. The governing equations for the gas and dispersed phases are of a hyperbolic type, they can be written in a conservative form and can be solved with a Godunov-type numerical method. A third order Runge-Kutta method is used to discretize the governing equations in time. The proposed model allows one to calculate a wide range of gas-particle flow regimes occurring when the volume concentration of the dispersed phase varies. The closure of the mathematical model and some details of numerical model implementation are discussed. The shock-wave flow structure as well as the space-time dependencies of particle concentration and other flow parameters are presented.

New computational technologies

Downloads

Published

2020-03-05

Issue

Vol. 21 (2020): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biographies

K.N. Volkov

D.F. Ustinov Baltic State Technical University «Voenmekh»
Faculty of Rocket and Space Engineering
• Leading Researcher

V.N. Emelyanov

D.F. Ustinov Baltic State Technical University «Voenmekh»,
Faculty of Rocket and Space Engineering
• Professor

A.G. Karpenko

St Petersburg University,
Mathematics and Mechanics Faculty
• Associate Professor

I.V. Teterina

D.F. Ustinov Baltic State Technical University «Voenmekh»,
Faculty of Rocket and Space Engineering
• Associate Professor

References

  1. K. N. Volkov and V. N. Emel’yanov, Flows of Gas with Particles (Fizmatlit, Moscow, 2008) [in Russian].
  2. Kh. A. Rakhmatulin, “Fundamentals of the Gasdynamics of Interpenetrating Motions of Continuous Media,” Prikl. Mat. Mekh. 20 (2), 184-195 (1956).
  3. A. N. Kraiko and L. E. Sternin, “Theory of Flows of a Two-Velocity Continuous Medium Containing Solid or Liquid Particles,” Prikl. Mat. Mekh. 29 (3), 418-429 (1965) [J. Appl. Math. Mech. 29 (3), 482-496 (1965)].
  4. H. Staedtke, G. Franchello, B. Worth, et al., “Advanced Three-Dimensional Two-Phase Flow Simulation Tools for Application to Reactor Safety (ASTAR),” Nucl. Eng. Des. 235 (2-4), 379-400 (2005).
  5. R. I. Nigmatulin, Dynamics of Multiphase Media (Nauka, Moscow, 1987; Hemisphere, New York, 1990).
  6. A. N. Kraiko, “On Correctness of the Cauchy Problem for a Two-Fluid Model of a Gas Flow Containing Particles,” Prikl. Mat. Mekh. 46 (3), 420-428 (1982) [J. Appl. Math. Mech. 46 (3), 327-333 (1982)].
  7. A. N. Osiptsov, “Investigation of Regions of Unbounded Growth of the Particle Concentration in Disperse Flows,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 46-52 (1984) [Fluid Dyn. 19 (3), 378-385 (1984)].
  8. M. R. Baer and J. W. Nunziato, “A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials,” Int. J. Multiph. Flow 12 (6), 861-889 (1986).
  9. Y.-Y. Niu, “Numerical Approximations of a Compressible Two Fluid Model by the Advection Upwind Splitting Method,” Int. J. Numer. Methods Fluids 36 (3), 351-371 (2001).
  10. P. Embid and M. Baer, “Mathematical Analysis of a Two-Phase Continuum Mixture Theory,” Continuum Mech. Therm. 4, 279-312 (1992).
  11. E. F. Toro, “Riemann-Problem-Based Techniques for Computing Reactive Two-Phase Flows,” in Lecture Notes in Physics (Springer, Heidelberg, 1989), Vol. 351, pp. 472-481.
  12. I. Toumi, “An Upwind Numerical Method for Two-Fluid Two-Phase Flow Models,” Nucl. Sci. Eng. 123 (2), 147-168 (1996).
  13. C.-H. Chang and M.-S. Liou, “A Robust and Accurate Approach to Computing Compressible Multiphase Flow: Stratified Flow Model and AUSM^+-up Scheme,” J. Comput. Phys. 225 (1), 840-873 (2007).
  14. N. Andrianov and G. Warnecke, “The Riemann Problem for the Baer-Nunziato Model of Two-Phase Flows,” J. Comput. Phys. 195 (2), 434-464 (2004).
  15. D. W. Schwendeman, C. W. Wahle, and A. K. Kapila, “The Riemann Problem and a High-Resolution Godunov Method for a Model of Compressible Two-Phase Flow,” J. Comput. Phys. 212 (2), 490-526 (2006).
  16. V. Deledicque and M. V. Papalexandris, “An Exact Riemann Solver for Compressible Two-Phase Flow Models Containing Non-conservative Products,” J. Comput. Phys. 222 (1), 217-245 (2007).
  17. S. Karni and G. Hernández-Dueñas, “A Hybrid Algorithm for the Baer-Nunziato Model Using the Riemann Invariants,” J. Sci. Comput. 45, 382-403 (2010).
  18. C. A. Lowe, “Two-Phase Shock-Tube Problems and Numerical Methods of Solution,” J. Comput. Phys. 204 (2), 598-632 (2005).
  19. C. Parés, “Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework,” SIAM J. Numer. Anal. 44 (1), 300-321 (2006).
  20. S. Rhebergen, O. Bokhove, and J. J. W. van der Vegt, “Discontinuous Galerkin Finite Element Methods for Hyperbolic Nonconservative Partial Differential Equations,” J. Comput. Phys. 227 (3), 1887-1922 (2008).
  21. 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).
  22. R. V. R. Pandya and F. Mashayek, “Two-Fluid Large-Eddy Simulation Approach for Particle-Laden Turbulent Flows,” Int. J. Heat Mass Tran. 45 (24), 4753-4759 (2002).
  23. B. Shotorban, “Preliminary Assessment of Two-Fluid Model for Direct Numerical Simulation of Particle-Laden Flows,” AIAA J. 49 (2), 438-443 (2011).
  24. L. I. Zaichik, O. Simonin, and V. M. Alipchenkov, “An Eulerian Approach for Large Eddy Simulation of Particle Transport in Turbulent Flows,” J. Turbul. 10 (9), 1-21 (2009).
  25. B. Shotorban and S. Balachandar, “Two-Fluid Approach for Direct Numerical Simulation of Particle-Laden Turbulent Flows at Small Stokes Numbers,” Phys. Rev. E 79 (2009).
    doi 10.1103/PhysRevE.79.056703
  26. B. Shotorban, G. B. Jacobs, O. Ortiz, and Q. Truong, “An Eulerian Model for Particles Nonisothermally Carried by a Compressible Fluid,” Int. J. Heat Mass Tran. 65, 845-854 (2013).
  27. F. Mashayek and R. V. R. Pandya, “Analytical Description of Particle/Droplet-Laden Turbulent Flows,” Prog. Energy Combust. Sci. 29 (4), 329-378 (2003).
  28. B. Vreman, B. Geurts, and H. Kuerten, “Realizability Conditions for the Turbulent Stress Tensor in Large-Eddy Simulation,” J. Fluid Mech. 278, 351-362 (1994).
  29. L. Y. M. Gicquel, P. Givi, F. A. Jaberi, and S. B. Pope, “Velocity Filtered Density Function for Large Eddy Simulation of Turbulent Flows,” Phys. Fluids 14 (3), 1196-1213 (2002).
  30. T. Saito, “Numerical Analysis of Dusty-Gas Flows,” J. Comput. Phys. 176 (1), 129-144 (2002).
  31. F. Laurent, M. Massot, and P. Villedieu, “Eulerian Multi-Fluid Modeling for the Numerical Simulation of Coalescence in Polydisperse Dense Liquid Sprays,” J. Comput. Phys. 194 (2), 505-543 (2004).
  32. D. Kah, F. Laurent, M. Massot, and S. Jay, “A High Order Moment Method Simulating Evaporation and Advection of a Polydisperse Liquid Spray,” J. Comput. Phys. 231 (2), 394-422 (2012).
  33. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, New York, 2002).
  34. P. V. Bulat and K. N. Volkov, “One-Dimensional Gas Dynamics Problems and Their Solution Based on High-Resolution Finite Difference Schemes,” Nauchno-Tekhn. Vestn. Inform. Tekhnol. Mekhan. Optiki 15 (4), 731-740 (2015).
  35. J. D. Regele, J. Rabinovitch, T. Colonius, and G. Blanquart, “Numerical Modeling and Analysis of Early Shock Wave Interactions with a Dense Particle Cloud,” AIAA Paper (2012).
    doi 10.2514/6.2012-3161