Acta mathematica scientia, Series B >
THE GENERALIZED RIEMANN PROBLEM FOR A SCALAR NONCONVEX COMBUSTION MODEL—THE PERTURBATION ON#br# INITIAL BINDING ENERGY
Received date: 2010-06-26
Online published: 2012-05-20
Supported by
Supported by NUAA Research Funding (NS2011001), NUAA'S Scientific Fund for the Introduction of Qualified Personal, NSFC grant 10971130, Shanghai Leading Academic Discipline Project J50101, and Shanghai Municipal Education Commission of Scientific Research Innovation Project 112284.
In this article, we study the generalized Riemann problem for a scalar non-convex Chapman-Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane. We focus our attention to the perturbation on initial binding energy. The solutions are obtained constructively under the entropy conditions. It can be found that the solutions are essentially different from the corresponding Riemann solutions for some cases. Especially, two important phenomena are observed: the transition from detonation to deflagration followed by a shock, which appears in the numerical simulations [7, 27]; the transition from deflagration to detonation (DDT), which is one of the core problems in gas dynamic combustion.
PAN Li-Jun , SHENG Wan-Cheng . THE GENERALIZED RIEMANN PROBLEM FOR A SCALAR NONCONVEX COMBUSTION MODEL—THE PERTURBATION ON#br# INITIAL BINDING ENERGY[J]. Acta mathematica scientia, Series B, 2012 , 32(3) : 1262 -1280 . DOI: 10.1016/S0252-9602(12)60098-8
[1] Bao W, Jin S. The random projection method for hyperbolic conservation laws with stiff reaction terms. J Comput Phys, 2000, 163: 216–248
[2] Bdzil JB, Stewart DS. The dynamics of detonation in explosive systems. Annu Rev Fluid Mech, 2007, 39: 263–292
[3] Chang T, Hsiao L. The Riemann problem and interaction of waves in gas dynamics. Pitman Monoger, Surveys Pure Appl Math, Longman, Essex, 1989: 41
[4] Chorin A J. Random choice methods with application to reacting gas flow. J Comp Phys, 1977, 25: 253
[5] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience, 1948
[6] Fickett W. Detonation in miniature. Amer J Phys, 1979, 47: 1050–1059
[7] Geßner T. Dynamic Mesh Adaption for Supersonic Combustion Waves Modeled with Detailed Reaction Mechanisms [Dissertation]. Fakultäat f¨ur Mathematik, Albert-Ludwigs-Universität Freiburg, 2000
[8] Levy A. On Majda’s model for dynamic combustion. Comm Partial Differential Equations, 1992, 17: 657–698
[9] Li T. On the initiation problem for a combustion model. J Differential Equations, 1994, 112: 351–373
[10] Li T. On the Riemann problem for a combustion model. SIAM J Math Anal, 1993, 24: 59–75
[11] Li T T, Yu W C. Boundary Value Problem for Quasilinear Hyperbolic Systems. Duke University Mathematics Series V, 1985
[12] Li J Q, Zhang P. The transition from Zeldovich-von Neumann-D¨oring to Chapman-Jouget theories for a nonconvex scalar combustion model. SIAM J Math Anal, 2003, 34: 675–699
[13] Liu T P, Ying L A. Nonlinear stability of strong detonation for a viscous combustion model. SIAM J Math Anal, 1996, 26: 519–528
[14] Liu T P, Zhang T. A scalar combustion model. Arch Ration Mech Anal, 1991, 114: 297–312
[15] Majda A. A qualitative model for dynamic combustion. SIAM J Appl Math, 1981, 41: 70–93
[16] Oleinik O A, Kalashnikov A C. A class of discontinuous solutions for first order quasilinear equatons//Proceedings of the meeting on differential equations. Jarevon (in Russian), 1960: 133–137
[17] Dears F W, Salinger G L. Thermodynamics, kinetic theory and statistical thermodynamics. Addison-Wesley, 1975
[18] Pan L J, Sheng W C. The generalized Riemann problem for a scalar combustion model—the perturbation on initial binding energy. J P D E, 2008, 21: 335–346
[19] Pan L J, Sheng W C. The scalar Zeldovich–Von Neumann–D¨oring combustion model: (I)Interactions of shock and detonation. Nonlinear Analysis: Real World Applications, 2009, 10: 483–493
[20] Pan L J, Sheng W C. The scalar Zeldovich–Von Neumann–D¨oring combustion model: (II)Interactions of shock and deflagration. Nonlinear Analysis: Real World Applications, 2009, 10: 449–457
[21] Sheng W C, Sun M N, Zhang T. The generalized Riemann problem for a scalar nonconvex Chapman-Jouguet combustion model. SIAM J Appl Math, 2007, 68: 544–561
[22] Sheng W C, Zhang T. Structural stability of solutions to the Riemann problem for a scalar CJ nonconvex combustion model. Discrete Contin Dyn Syst, 2009, 25(2): 651–667
[23] Sun M N, Sheng W C. The ignition problem for a scalar nonconvex combustion model. J D E, 2006, 231: 673–692
[24] Sun M N, Sheng WC. The generalized Riemann problem for a scalar Chapman-Jouguet combustion model. Z angew Math Phys, 2009, 60(2): 271–283
[25] Teng Z H, Chorin A J, Liu T P. Riemann problems for reacting gas with application to transition. SIAM J Appl Math, 1982, 42: 964
[26] William C D. The detonation of explosives. Scientific American, 1987, 256(5): 98–104
[27] Xu S, Aslam T, Stewart D S. High resolution numerical simulation of ideal and non-ideal compressible reacting flows with embedded internal boundaries. Combust Theory Modeling, 1997, 1: 113–142
[28] Ying L, Teng Z. Riemann problem for a reaction and convection hyperbolic system. Approx Theory Appl, 1984, 1: 95–122
[29] Zhang X T, Ying L. Dependence of qualitative behavior of the numerical solutions on the ignition temperature for a combustion model. Journal of Computational Mathematics, 2005, 23: 337–350
[30] Zhang P, Zhang T. The Riemann problem for scalar CJ-combustion model without convexity. Discrete Contin Dynam Systems, 1995, 1: 195–206
[31] Zhang T, Zheng Y X. Two-dimensional Riemann problem for a single conservation law. Trans of AMS, 1989, 132: 589–619
[32] Zhang T, Zheng Y X. Riemann problem for gas dynamic combustion. J Differential Equations, 1989, 77: 203–230
/
| 〈 |
|
〉 |