Combustion waves in a model with chain branching reaction and their stability

In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of nonadiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the nonadiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero and a standing combustion wave is formed as the extinction condition is reached. Prospects of further work are also discussed.Comment: pages 32, figures 2

Numerical methods for the travelling wave solutions in reaction-diffusion equations

In this work we consider how shooting and relaxation methods can be used to investigate propagating waves solutions of PDEs. Particular attention is paid to overcoming some of the numerical difficulties. The linear stability of these solutions are analyzed by using the Evans function approach. As an illustration, we shall apply the above methods to an autocatalytic reaction involving two diffusing chemicals in one spatial dimension. Prospects of further work are also discussed

Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number

A numerical shooting method for performing linear stability analyses of travelling waves is described and applied to the problem of freely propagating planar premixed flames. Previous linear stability analyses of premixed flames either employ high activation temperature asymptotics or have been performed numerically with finite activation temperature, but either for unit Lewis numbers (which ignores thermal-diffusive effects) or in the limit of small heat release (which ignores hydrodynamic effects). In this paper the full reactive Navier-Stokes equations are used with arbitrary values of the parameters (activation temperature, Lewis number, heat of reaction, Prandtl number), for which both thermal-diffusive and hydrodynamic effects on the instability, and their interactions, are taken into account. Comparisons are made with previous asymptotic and numerical results. For Lewis numbers very close to or above unity, for which hydrodynamic effects caused by thermal expansion are the dominant destablizing mechanism, it is shown that slowly varying flame analyses give qualitatively good but quantitatively poor predictions, and also that the stability is insensitive to the activation temperature. However, for Lewis numbers sufficiently below unity for which thermal-diffusive effects play a major role, the stability of the flame becomes very sensitive to the activation temperature. Indeed, unphysically high activation temperatures are required for the high activation temperature analysis to give quantitatively good predictions at such low Lewis numbers. It is also shown that state-insensitive viscosity has a small destabilizing effect on the cellular instability at low Lewis numbers

Mechanisms performance and pressure dependence of hydrogen/air burner-stabilized flames

The kinetic mechanism of hydrogen combustion is the most investigated combustion system. This is due to extreme importance of the mechanism for combustion processes, i.e. it is present as a sub-mechanism in all mechanisms for hydrocarbon combustion systems. Therefore, detailed aspects of hydrogen flames are still under active investigations, e.g. under elevated pressure, under conditions of different heat losses intensities and local equivalence ratios etc. For this purpose, the burner stabilized flame configuration is an efficient tool to study different aspects of chemical kinetics by varying the stand-off distance, pressure, temperature of the burner and mixture compositions. In the present work, a flat porous plug burner flame configuration is revisited. A hydrogen/air combustion system is considered with detailed molecular transport including thermo-diffusion and with 8 different chemical reaction mechanisms. Detailed numerical investigations are performed to single out the role of chemical kinetics on the loss of stability and on the dynamics of the flame oscillations. As a main outcome, it was found/demonstrated that the results of critical values, e.g. critical mass flow rate, weighted frequency of oscillations and blow-off velocity, with increasing the pressure scatter almost randomly. Thus, these parameters can be considered as independent and can be used to improve and to validate the mechanisms of chemical kinetics for the unsteady dynamics

Analysing combustion waves in a model with chain branching

We analyse the travelling wave solutions in an adiabatic model with two-step chain branching reaction mechanism. We show that the behaviour of the combustion waves are similar to the case of the corresponding nonadiabatic one-step reaction, namely there is residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. We also analyse how the speed of the travelling wave solutions and the residual amount of fuel left behind the fuel change, as control parameters are varied. References V. V. Gubernov, G. N. Mercer, H. S. Sidhu and R. O. Weber, Evans function stability of nonadiabatic combustion waves, Proc. R. Soc. Lond. A, 460, 2004, 2415--2435.doi:10.1098/rspa.2004.1285 G. Joulin, A. Linan, G. S. S. Ludford, N. Peters and C. Schmidt-Laine, Flames with chain-branching/chain-breaking kinetics, SIAM J. Appl. Math., 45, 1985, 420--434. A. Linan, A theoretical analysis of premixed flame propagation with an isothermal chain-branching reaction Insituto Nacional de Technica Aerospacial ``Esteban Terradas'' (Madrid), USAFOSR Contract No. E00AR68-0031, Technical Report No. 1, 1971. A. Makino, Fundamental aspects of the heterogeneous flame in the self propagating hightemperature synthesis (SHS) process, Prog. Energy Combust. Sci., 27, 2001, 1--74. A. G. Merzhanov and E. N. Rumanov, Physics of reaction waves, Rev. Mod. Phys., 71, 1999, 1173--1211. doi:10.1103/RevModPhys.71.1173 H. Pitsch, and M. Bollig, 1994, FlameMaster, A Computer Code for Homogeneous and One-Dimensional Laminar Flame Calculations, RWTHAachen, Institut fur Technische Mechanik, 1994. A. L Sanchez, G. Balakrishnan, A. Linan and F. A. Williams, Relationships between bifurcation and numerical analyses for ignition of hydrogen-air diffusion flames, Combust. Flame, 105, 1996, 569--590. A. L. Sanchez, A. Lepinette, M. Bolling, A. Linan, and B. Lazaro, 2000, The reduced kinetic description of lean premixed combustion, Combust. Flame, 123, 2000, 436--464. K. Seshadri, N. Peters and F. A. Williams, 1994, Asymptotic analyses of stoichiometric and lean hydrogen-air flames, Combust. Flame, 96, 1994, 407--427. P. L. Simon, S. Kallidasis and S. K.Scott, Inhibition of flame propagation by an endothermic reaction, IMA J. Appl. Math., 68, 2003, 537--562. doi:10.1093/imamat/68.5.537 J. K. Bechtold and C. K. Law, The structure of premixed methane-air flames with large activation energy, Combust. Flame, 97, 1994, 317--338. doi:10.1016/0010-2180(94)90024-8 J. Warnatz, U. Maas and R. W. Dibble, Combustion: physical and chemical fundamentals, modelling and simulation, experiments, pollutant formation, Springer, Berlin, 1996. R. O. Weber, G. N. Mercer, H. S. Sidhu and B. F. Gray, Combustion waves for gases ($Le = 1$) and solids ($Le \rightarrow 1$), Proc. R. Soc. Lond. A, 453, 1997, 1105--1118. doi:10.1098/rspa.1997.0062 C. K. Westbrook and F. Dryer, Simplified reaction mechanisms for the oxidation of Hydrocarbon Fuels in Flames, Combust. Sci. Tech., 27, 1981, 31--43. Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M Makhviladze, The mathematical theory of combustion and explosions Consultants Bureau, New York, 1985. J. W. Dold, R. O. Weber, R. W. Thatcher and A. A. Shah, Flame Ball With Thermally Sensitive Intermediate Kinetics Combust., Combust. Theory Mod., 7, 2003, 175--203. J. W. Dold and R. O. Weber, Reactive-Diffusive stability of planar flames with modified Zeldovich--Linan kinetics. In: F. J. Higuera, J. Jime'nez and J. M. Vega (Eds), Simplicity, Rigor and Relevance in Fluid Mechanics. A volume in honor of Amable Linan, CIMNE (Barcelona), 2004. V. V. Gubernov, G. N. Mercer, H. S. Sidhu and R. O. Weber, Evans function stability of combustion waves. SIAM J. Appl. Math., 63, 2003, 1259--1275. doi:10.1137/S003613990140024