Guy Joulin

Linear Stability Analysis of Nonadiabatic Flames: Diffusional-Thermal Model

Abstract The method of matched asymptotic expansions, in terms of a suitably reduced activation energy, is applied to investigate the effects of heat losses on linear stability of a planar flame, which is governed by a one-step irreversible Arrhenius reaction. The density change associated with the heat release is neglected in order to eliminate the Landau hydrodynamic instability. Attention is focused on diffusional-thermal instability mechanisms. The dispersion relation is obtained in terms of the diffusive properties of the limiting reactant and of the heat-loss intensity. For a given loss intensity-less than the critical value leading to extinction-the two steady planar regimes have different stability properties: (1) the “slow” regimes (which do not reduce to the adiabatic one when the heat-loss intensity goes to zero) are shown to be always unstable; and (2) for the “fast” regimes (which include and generalize the adiabatic one) cellular structures are predicted to occur when the limiting component is sufficiently light. If the limiting component is moderately light, the “fast” regimes are stable and unstructured in nearly adiabatic conditions; however, they are destabilized by an increase of heat losses and must exhibit cells before the extinction limit is reached. Similarly, for mixtures involving a realistically heavy limiting component, our analysis predicts the appearance of transverse travelling waves near the extinction limit.

The structure and stability of nonadiabatic flame balls

Abstract Recent experiments in microgravity suggest the possibility of stationary spherical premixed flames (flame balls) in which the only fluxes are diffusional. We construct stationary solutions of this nature, starting with simple model equations and using activation energy asymptotics. Sufficiently large volumetric heat losses quench the flame, and for heat losses less than the quenching value there are two possible solutions, a small flame, and a large flame. For vanishing heat loss the small solution is identical to one constructed by Zeldovich, and is known to be unstable, whereas the large solution is characterized by a flame of infinite radius. We examine the linear stability of these stationary solutions, and show that all small flames are unstable to one-dimensional (radial) perturbations. Large flames are unstable to three-dimensional perturbations, but only if they have a radius greater than some critical value. Thus there is a band of large flames, lying between the quenching point and unstable flames, that are stable.

Spherical flame initiation: Theory versus experiments for lean propaneair mixtures

Abstract The initiation of a spherical premixed flame is analyzed from the starting point of view of a high activation energy asymptotic and then investigated both numerically and experimentally. As a result of asymptotic analysis the ignition is expected to be controlled, at least for lean mixture of heavy fuel, by the existence of an unstable equilibrium spherical radius R c . This critical radius is calculated as a function of the various physicochemical parameters together with a characteristic evolution time t c of the flame and a reference energy E c for flame initiation. These reference quantities are shown to give a sound basis for a numerical solution of the spherical flame initiation. In addition the dynamical behavior of the flame front as well as the initiation energy-energy deposit time duration that we compute are in complete accordance with the initiation mechanism conjectured from the asymptotic result. Such a conjecture is also confirmed by the experimental test that we performed on lean propaneair mixtures, where the flame initiation is triggered by an electric spark. Moreover it is found that, at least for moderate pressure, experimental data and numerical results are in good agreement, both qualitatively and quantitatively.

The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses

Abstract A theory of spherical premixed flames in which the only transport processes are diffusion and radiation (“flame balls”) is extended to include the effects of heat loss from the far-field (unburned gas). Using matched asymptotic expansions for large activation energy, stationary solutions are constructed and an evolution equation for the radial motion of the flame is derived. Linear stability analyses are performed for both one-dimensional and three-dimensional perturbations. It is shown that when far-fieldlosses are included, the stability properties are qualitatively changed. A smaller range of conditions produces flames that are stable to one-dimensional disturbances and, in some cases, the linear growth rates are found to have imaginary components. Numerical intergation of the evolution equation reveals oscillations (but not limit cycles), in agreement with a bifurcation analysis. The minimum flame-ball radius for which three-dimensional instabilities will occur is shown to depend only on the near-field losses, and becomes arbitrarily large as these losses are reduced.

On stability of premixed flames in stagnation - Point flow

Abstract An equation is derived which describes the evolution of a corrugated flame front stabilized in stagnation-point flow φτ+4▿4φ+ ▿2φ+½(▿φ)2+α(ηφ)η = 0 It is shown that, if blowing is sufficiently strong, the corrugations disappear and a plane flame results. The problem turns out to be rather unusual in the sense that the phenomena cannot be fully described by means of the classical linear stability analysis.

Point-Source Initiation of Lean Spherical Flames of Light Reactants: An Asymptotic Theory

Abstract We study the dynamics of small spherical flame kernels whose evolution is triggered in a lean mixture of a light mobile reactant by a time-dependent point- source of energy. For simplicity, the analysis is conducted in the framework of a one-reactant flame model, with a one-step Arrhenius overall kinetics. Using the method of matched asymptotic expansion for large activation energies, we show that: The concentration and temperature fields split into quasi-steady regions, inside and around the flame kernel, and an unsteady far-field; in both regions convection effects may be neglected. Match ing the near - and far-fields furnishes a parameter-free, non-linear evolution equation for the flame radius ; in addition to memory effects of diffusive origin, it includes explicitly the functional form of the chemical rate and the instan taneous power of the point source of energy we used as ignition device. Through a numerical integration of this equation, flame front trajectories and critical energies are...

On The Response of Premixed Flames to Time-Dependent Stretch and Curvature

Abstract In the framework of a constant-density model, we study the linear response of a premixed flame to large scale, but time dependent, curvature and stretch of given strengths and frequency. It is analytically suggested that: i) as the frequency of forcing increases above the reciprocal transit time across the flame, the local instantaneous burning speed gets less and less sensitive to hydrodynamical stretch. ii) the influence of differential diffusion of heat and deficient reactant is milder and milder when the high-frequency limit is approached, so that the Lewis-number-effects tend to disappear. The predicted trends are put in perspective with the results of recent measurements on acoustically destabilized flames.

On a Tentative, Approximate Evolution Equation for Markedly Wrinkled Premixed Flames

Abstract Synthesizing several analytical results on weakly wrinkled premixed flames, we suggest a simple, phenomenological extension of the Michelson-Sivashinsky evolution equation. It possesses an infinite number of closed-form solutions, the mathematical relevance of which is checked through comparisons with the results oraccurate numerical, spectral integrations. Unexpectedly enough, the proposed equation seems to be quantitatively compatible with the few pre-existing, independant results on markedly wrinkled steady fronts.

On model evolution equations for the whole surface of three-dimensional expanding wrinkled premixed flames

Borrowing the structure of building blocks from potential hydrodynamics, we set up model evolution equations to mimic moderately wrinkled three-dimensional flames expanding in gaseous premixtures. These evolution equations incorporate a hydrodynamic instability, local curvature effects, a Huygens-type nonlinearity, and can cope with broad-banded forcing whenever needed. Pseudo-spectral integrations in the Legendre–Fourier basis yield evolutions of the whole front that are in striking qualitative agreement with experiments on free or weakly forced propagations. Our results are robust against educated changes in the modelling. Provided an accurate evolution equation is available, this approach can simulate expanding flames which are quite a bit larger than what DNS can currently handle, at least in the chosen configurations.

The effect of stirring on the limits of thermal explosion

It is analysed theoretically how the stirring of an exothermically reacting fluid layer affects its thermal explosion limits. Analytical and numerical analyses reveal that, in accordance with the intuitive expectations, the short-scale stirring makes the thermal explosion more difficult through the increased heat transfer to the boundaries. However, under the long-scale stirring, promoting formation of hot spots, transition to the explosion may be facilitated rather than hampered.