Two-phase shock-tube problems and numerical methods of solution. (English) Zbl 1203.76117
Summary: The study involves the flow of compressible gas in a porous bed: We are interested in exploring the solution to a shock-tube problem that also includes a discontinuous jump in the porosity of the bed. The averaging process that is used to derive the governing system of equations introduces terms that prevent the equations taking a standard conservation form. The purpose of this work is to explore how the porosity jump effects the solution to the Riemann problem and to assess whether standard conservative and nonconservative numerical methods can provide the correct solution to the Riemann problem. The study illustrates serious shortfalls in the numerical solution to these problems using standard techniques. By exploring the exact solution, the author suggests how the standard numerical methods can be modified to provide the correct solution to these systems.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76T99 | Multiphase and multicomponent flows |
Keywords:
Nonconservative hyperbolic equations; Multi-phase compressible flow; Combustion; Conservative numerical methods; Riemann problemsReferences:
| [1] | Andrianov, N.; Warnecke, G., On the solution to the Riemann problem for the compresisble flow in a duct, SIAM Journal on Applied Mathematics, 64, 878-901 (2004) · Zbl 1065.35191 |
| [2] | Andrianov, N.; Warnecke, G., The Riemann problem for the Baer and Nunziato model for two-phase model, Journal of Computational Physics, 195, 2, 434-464 (2004) · Zbl 1115.76414 |
| [3] | Asay, B. W.; Son, S. F.; Bdzil, J. B., The role of gas permeation in convective burning, International Journal of Multiphase Flow, 22, 5, 923-952 (1995) · Zbl 1135.76350 |
| [4] | Baer, M. R.; Nunziato, J. W., A two-phase mixture theory for the deflagation-to-detonation transition (DDT) in reactive granular materials, Journal of Multiphase Flow, 12, 6, 861-889 (1986) · Zbl 0609.76114 |
| [5] | M.R. Baer, J.W. Nunziato, P.F. Embid, Deflagration-to-Detonation Transition in Reactive Granular Materials, Progress in Astronautics and Aeronautics Series, vol. 66, 1979, pp. 481-512; M.R. Baer, J.W. Nunziato, P.F. Embid, Deflagration-to-Detonation Transition in Reactive Granular Materials, Progress in Astronautics and Aeronautics Series, vol. 66, 1979, pp. 481-512 |
| [6] | Bale, D. S.; LeVeque, R. J.; Mitran, S.; Rossmanith, J. A., A wave propagation method for conservation laws and balance laws with spatially-varying flux functions, SIAM Journal of Scientific Computing, 24, 3, 955-978 (2002) · Zbl 1034.65068 |
| [7] | J.B. Bdzil, S.F. Son, Engineering models of deflagration-to-detonation transition, Technical Report LA-12794-MS, UC-741, Los Alamos, Los Alamos National Laboratories, NM 87545, 1995; J.B. Bdzil, S.F. Son, Engineering models of deflagration-to-detonation transition, Technical Report LA-12794-MS, UC-741, Los Alamos, Los Alamos National Laboratories, NM 87545, 1995 |
| [8] | Gough, P. S.; Zwarts, F. J., Modeling heterogenous two-phase flow, AIAA Journal, 17, 1, 17-25 (1979) · Zbl 0392.76093 |
| [9] | Greenberg, J. M.; Leroux, A. Y.; Baraille, R.; Noussair, A., Analysis and approximation of conservative laws with source terms, SIAM Journal of Numerical Analysis, 34, 5, 1980-2007 (1997) · Zbl 0888.65100 |
| [10] | A.W. Horst, G.E. Keller, P.S. Gough, New directions in multiphase flow interior ballistic modeling, Technical Report BRL-TR-3102, Ballistics Research Laboratory, Aberdeen Proving Ground, MD, 1990; A.W. Horst, G.E. Keller, P.S. Gough, New directions in multiphase flow interior ballistic modeling, Technical Report BRL-TR-3102, Ballistics Research Laboratory, Aberdeen Proving Ground, MD, 1990 |
| [11] | Lowe, C. A.; Clarke, J. F., Aspects of solid propellant combustion, Philosophical Transactions of the Royal Society, Combustion Science Thematic Volume, 357, 1764, 3639-3654 (1999) · Zbl 0965.80012 |
| [12] | Sainsaulieu, L., Finite volume approximation of two-phase-fluid flows based on an approximate Roe-type Riemann solver, Journal of Computational Physics, 121, 1-28 (1994) · Zbl 0834.76070 |
| [13] | Saurel, R.; Abgrall, R., A multiphase godunov method for compressible multifluid and multiphase flows, Journal of Computational Physics, 150, 425-467 (1998) · Zbl 0937.76053 |
| [14] | Toro, E. F., Riemann-problem based techniques for computing reactive two-phase flows, (Dervieux; Larrouturou, Lecture Notes in Physics, Numerical combustion, vol. 351 (1989), Springer: Springer Berlin) · Zbl 0747.76027 |
| [15] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Springer: Springer Heidelberg · Zbl 0888.76001 |
| [16] | Toro, E. F.; Sviglia, A., PRICE: Primitive centred schemes for hyperbolic systems, International Journal for Numerical Methods in Fluids, 42, 1263-1291 (2003) · Zbl 1078.76566 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
