Effect of thermal expansion on the linear stability of planar premixed flames for a simple chain-branching model: The high activation energy asymptotic limit

The linear stability of freely propagating, adiabatic, planar premixed ames is investigated in the context of a simple chain-branching chemistry model consisting of a chain-branching reaction step and a completion reaction step. The role of chain-branching is governed by a crossover temperature. Hydrodynamic effects, induced by thermal expansion, are taken into account and the results compared and contrasted with those from a previous purely thermal-di�usive constant density linear stability study. It is shown that when thermal expansion is properly accounted for, a region of stable ames predicted by the constant density model disappears, and instead the ame is unstable to a long-wavelength cellular instability. For a pulsating mode, however, thermal expansion is shown to have only a weak e�ect on the critical fuel Lewis number required for instability. These e�ects of thermal expansion on the two-step chain-branching ame are shown to be qualitatively similar to those on the standard one-step reaction model. Indeed, as found by constant density studies, in the limit that the chain-branching crossover temperature tends to the adiabatic ame temperature, the two-step model can be described to leading order by the one-step model with a suitably de�ned e�ective activation energy

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

Nonlinear cellular dynamics of the idealized detonation model: Regular cells

High-resolution numerical simulations of cellular detonations are performed using a parallelized adaptive grid solver, in the case where the channel width is very wide. In particular, the nonlinear response of a weakly unstable ZND detonation to two-dimensional perturbations is studied in the context of the idealized one-step chemistry model. For random perturbations, cells appear with a characteristic size in good agreement with that corresponding to the maximum growth rate from a linear stability analysis. However, the cells then grow and equilibrate at a larger size. It is also shown that the linear analysis predicts well the ratio of cell lengths to cell widths of the fully developed cells. The evolutionary dynamics of the growth are nonetheless quite slow, in that the detonation needs to run of the order of 1000 reaction lengths before the final size and equilibrium state is reached. For sinusoidal perturbations, it is found that there is a large band of wavelengths/cell sizes which can propagate over very long distances (~1000 reaction lengths). By perturbing the fully developed cells of each wavelength, it is found that smaller cells in this range are unstable to symmetry breaking, which again results in cellular growth to a larger final size. However, a range of larger cell sizes appear to be nonlinearly stable. As a result it is found that the final cell size of the model is non-unique, even for such a weakly unstable, regular cell case. Indeed, in the case studied, the equilibrium cell size varies by 100% with different initial conditions. Numerical dependencies of the cellular dynamics are also examined

The differential diffusion effect of the intermediate species on the stability of premixed flames propagating in microchannels

The propagation of premixed flames in adiabatic and non-catalytic planar microchannels subject to an assisted or opposed Poiseuille flow is considered. The diffusive-thermal model and the well-known two-step chain-branching kinetics are used in order to investigate the role of the differential diffusion of the intermediate species on the spatial and temporal flame stability. This numerical study successfully compares steady-state and time-dependent computations to the linear stability analysis of the problem. Results show that for fuel Lewis numbers less than unity, LeF 1, flames propagating in adiabatic channels suffer from oscillatory instabilities. The Poiseuille flow stabilises the flame and the effect of LeZ is opposite to that found for LeF < 1. Small values of LeZ further destabilise the flame to oscillating or pulsating instabilities.This research was supported by the Spanish Ministerio de Ciencia e Innovación (MICINN) [Projects #ENE2011-27686-C02-01, #ENE2012-33213]; the Comunidad de Madrid [Project #S2009/ENE-1597, CONSOLIDER CSD2010-00011].Publicad

Bifurcation of pulsation instability in one-dimensional H2−O2 detonation with detailed reaction mechanism

Classical modes of one-dimensional (1D) detonation characterized by a simplified reaction model are reproduced by using a real chemical kinetics for the H2−O2 system with argon dilution. As Ar dilution is varied, the bifurcation points of pulsating instability are identified and a formed bifurcation diagram is compared with that obtained by the one-step reaction model. Eventually, the numerical results demonstrate that, for real detonations with detailed chemistry, the criterion of Ng et al. works well on prediction of the 1D detonation instability. Furthermore, the detonability limits are found respectively at low and high Ar dilutions. Above the high Ar dilution limit, detonations decays to the minimum level where long autoignition time and small heat release rate make reestablishment impossible for both 1D and 2D simulations. However, below the low Ar dilution limit, a 1D detonation cannot be sustained due to high instability, while the corresponding cellular detonation can propagate sustainably due to the role of transverse instability

Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves

The nonlinear stability of cylindrically and spherically expanding detonation waves is investigated using numerical simulations for both directly (blast) initiated detonations and cases where the simulations are initialized by placing quasi-steady solutions corresponding to different initial shock radii onto the grid. First, high-resolution one-dimensional (axially or radially symmetric) simulations of pulsating detonations are performed. Emphasis is on comparing with the predictions of a recent one-dimensional linear stability analysis of weakly curved detonation waves. The simulations show that, in agreement with the linear analysis, increasing curvature has a rapid destabilizing effect on detonation waves. The initial size and growth rate of the pulsation amplitude decreases as the radius where the detonation first forms increases. The pulsations may reach a saturated nonlinear behaviour as the amplitude grows, such that the subsequent evolution is independent of the initial conditions. As the wave expands outwards towards higher (and hence more stable) radii, the nature of the saturated nonlinear dynamics evolves to that of more stable behaviour (e.g. the amplitude of the saturated nonlinear oscillation decreases, or for sufficiently unstable cases, the oscillations evolve from multi-mode to period-doubled to limit-cycle-type behaviour). For parameter regimes where the planar detonation is stable, the linear stability prediction of the neutrally stable curvature gives a good prediction of the location of the maximum amplitude (provided the stability boundary is reached before the oscillations saturate) and of the critical radius of formation above which no oscillations are seen. The linear analysis also predicts very well the dependence of the period on the radius, even in the saturated nonlinear regimes. Secondly, preliminary two-dimensional numerical simulations of expanding cellular detonations are performed, but it is shown that resolved and accurate calculations of the cellular dynamics are currently computationally prohibitive, even with a dynamically adaptive numerical scheme

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

Combustion theory and modeling

In honor of the fiftieth anniversary of the Combustion Institute, we are asked to assess accomplishments of theory in combustion over the past fifty years and prospects for the future. The title of our article is chosen to emphasize that development of theory necessarily goes hand-in-hand with specification of a model. Good conceptual models underlie successful mathematical theories. Models and theories are discussed here for deflagrations, detonations, diffusion flames, ignition, propellant combustion, and turbulent combustion. In many of these areas, the genesis of mathematical theories occurred during the past fifty years, and in all of them significant advances are anticipated in the future. Increasing interaction between theory and computation will aid this progress. We hope that, although certainly not complete in topical coverage or reference citation, the presentation may suggest useful directions for future research in combustion theory