A reacting shock in a spherically symmetric gas. (English) Zbl 1448.76098
Summary: In this paper, we model the propagation of a reacting shock wave in a spherically symmetric combustible gas. Approximate solutions are found using a spectral method which builds the necessary shock jump conditions into the infinite series, and the problem is then solved in the spectral space. We present some numerical results, and it is then shown that the system may develop a secondary imploding shock front behind the primary shock. By application of a small amount of artificial diffusion, the spectral method is able to resolve this behaviour. The accuracies of these numerical solutions are verified by comparing them with a Godunov scheme.
MSC:
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76V05 | Reaction effects in flows |
| 80A25 | Combustion |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
References:
| [1] | Sal’nikov IY (1949) Contribution to the theory of the periodic homogeneous chemical reactions. Zh Fiz Khim 23:258-272 |
| [2] | Gray BF, Roberts MJ, Gray P (1988) Analysis of chemical kinetic systems over the entire parameter space I. The Sal’nikov thermokinetic oscillator. Proc R Soc Lond A 416(1851):391-402 · Zbl 0635.34027 · doi:10.1098/rspa.1988.0040 |
| [3] | Gray BF, Roberts MJ, Gray P (1988) Analysis of chemical kinetic systems over the entire parameter space II. Isothermal oscillators. Proc R Soc Lond A 416(1851):403-424 · Zbl 0635.34028 · doi:10.1098/rspa.1988.0041 |
| [4] | Coppersthwaite DP, Griffiths JF, Gray BF (1991) Oscillations in the hydrogen + chlorine reaction: experimental measurements and numerical simulation. J Phys Chem 95(18):6961-6967 · doi:10.1021/j100171a043 |
| [5] | YaB Zeldovich, Barenblatt GI, Librovich VB, Makhviladze GM (1985) The mathematical theory of combustion and explosions. Springer, Berlin |
| [6] | Paul RA, Forbes LK (2015) Travelling waves and oscillations in Sal’nikov’s reaction in a compressible gas. ANZIAM J 56:233-247 · Zbl 1312.76056 · doi:10.1017/S1446181114000479 |
| [7] | Forbes LK, Derrick W (2001) A combustion wave of permanent form in a compressible gas. ANZIAM J 43(1):35-58 · Zbl 1065.80505 · doi:10.1017/S144618110001141X |
| [8] | Rankine WJM (1870) On the thermodynamic theory of waves of finite longitudinal disturbances. Philos Trans R Soc Lond A 160:277-288 · JFM 02.0832.01 · doi:10.1098/rstl.1870.0015 |
| [9] | Hugoniot H (1889) Sur la propagation du mouvement dans les corps et spcialement dans les gaz parfaits. J Ecole Polytechnique 58:1-125 · JFM 21.0958.01 |
| [10] | Whitham GB (1974) Linear and nonlinear waves. Wiley, New York · Zbl 0373.76001 |
| [11] | Paul RA, Forbes LK (2016) Combustion waves in Sal’nikov’s reaction scheme in a spherically symmetric gas. J Eng Math 101(1):29-45 · Zbl 1360.80012 · doi:10.1007/s10665-016-9843-0 |
| [12] | Anderson JD (2003) Modern compressible flow with historical perspective. McGraw-Hill, New York |
| [13] | Williams FA (1965) Combustion theory: the fundamental theory of chemically reacting flow systems. Addison-Wesley Publishing Company, Reading |
| [14] | Liepmann HW, Roshko A (1957) Elements of gasdynamics. Wiley, New York · Zbl 0078.39901 |
| [15] | Forbes LK, Krzysik OA (2017) Shock-jump conditions in a general medium: weak-solution approach. Shock Waves 27:457-466 · doi:10.1007/s00193-016-0695-3 |
| [16] | Bourlioux A, Majda AJ (1995) Theoretical and numerical structure of unstable detonations. Philos Trans R Soc A 350(1692):29-68 · Zbl 0820.76091 |
| [17] | Courant R, Friedrichs KO (1948) Supersonic flow and shock waves. Springer, New York · Zbl 0041.11302 |
| [18] | Arfken G (1985) Mathematical methods for physicists. Academic Press, London · Zbl 0135.42304 |
| [19] | Godunov SK (1959) A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik 89:271-306 · Zbl 0171.46204 |
| [20] | Roe PL (1981) Approximate Riemannn solvers, parameter vectors, and difference schemes. J Comput Phys 43(2):357-732 · Zbl 0474.65066 · doi:10.1016/0021-9991(81)90128-5 |
| [21] | Toro EF (2009) Riemannn solvers and numerical methods for fluid dynamics. Springer, Berlin · Zbl 1227.76006 · doi:10.1007/b79761 |
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.
