Pierre Cambray

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 moderately-forced premixed flames

By means of spectral, numerical integrations of a forced evolution equation (Michelson-Sivashinsky type) for the front shape we qualitatively study how potentially unstable premixed flames react when moderately excited by incoming velocity fluctuations. We also perform parallel calculations on the associated model-equation of “passive” propagation (Kardar-Parisi-Zhang type). Both models are nonlinear. Harmonic shear flows of various intensities, frequencies and wavelengths are successively used as forcings. Next, a Doppler-like frequency-shift, due to enhanced propagation through eddies of fixed extend, also is accounted for. In each case, the flowfield disturbances induced by flame wrinkling—the very mechanism of hydrodynamic instability—turns out to be a very important ingredient of flame forced dynamics In particular, retaining the induced flowfield leads to a very steep increase in forcing-aided flame speed S T with the intensity u′ of forcing, at least when u′ is small enough compared to the laminar burning velocity S L . More generally, in all our numerical experiments the “passive” model noticeably under-estimates S T /S L when u′ ∼ S L or less, even if the exciting field contains eddies with unlike sizes.

Mean evolution of wrinkle wavelengths in a model of weakly-turbulent premixed flame

Abstract New informations on weakly-turbulent premixed flames in the cusped-flame regime are obtained upon studying a Michelson-Sivashinsky equation whose solution (the instantaneous flame shape) is excited by the influence of a weak, pseudo-random additive forcing, that is meant to mimic incoming velocity fluctuations. Attention is focused on the time-dependent mean spacing lc{t) between crests, in cases where the front is flat before forcing is switched on. Despite the complexity of each individual run, a surprisingly simple history Ac(t) always emerges upon averaging lc(t) over many realizations of the forcing function, especially in the limit of weak forcings: a noise-triggered linear stage is followed by a stage of crest-coalescence during which dAJdt is a constant and the direct role of forcing is minor; then, rather abruptly, an equilibrium stage is reached where Ac(t) acquires a time-independent value because the crest implants induced by the noise statistically equilibrate the coalescences. Impro...

Coalescence Problems in the Theory of Expanding Wrinkled Premixed Flames

Abstract In the framework of a Michelson-Sivashinsky (MS) evolution equation for the front shape we study cylindrically-expanding premixed flames, focusing attention on the spontaneous dynamics of collections of finite-amplitude wrinkles To begin with, we consider sharp crests described by pole-decomposed solutions to the MS equation, for which the flame dynamics is reduced to a finite set of Complex ODEs. The latter are simplified in the limit of large flame radii to result in a restricted N-body problem (with long-range interactions) for the real angular locations of the wrinkle crests. The corresponding statistical problem of coalescence/expansion competition is treated approximately, by a mean-field method, and yields analytical predictions for the cell-size distributions vs. time. Comparisons with spectral integrations of the MS equation and with a simulation of the restricted N-body problem reveal fair agreements Next we consider initial conditions outside the previous class, yet again representing...

Temperature-Lags and Radiative-Transfer in Particle-Laden Gaseous Flames Part II : Unsteady Propagations

We study the dynamics of particle-laden flames, the steady propagation regimes of which have been studied in Part I (Joulin, 1986). Attention is focused on the non-steady couplings between radiative-preheating and conductive losses. Non-linear evolution equations are obtained for the instantaneous burning speed. In the case of a simplified model of radiative-transfer and of conductive exchanges between the phases, the stable regimes are selected, if any. When no stable regime exists, strong relaxation oscillations in burning speed are exhibited. In cases of bi-stability, transitions-induced by head-on flame/flame radiative interactions—between stable regimes are shown. Finally, hints are given to generalize the results to more accurate modellings of radiative transfers, chemistry, and conductive exchanges.

Nonlinear Dynamics of Wrinkled Premixed Flames and Related Statistical Problems

The flames propagating through premixed gaseous reactants are surface-like interfaces, that consume the former at a local curvature-dependent burning speed, and heat them. The Michelson-Sivashinsky (MS) equation and its variants well describe how the fluid-mechanical (Landau-Darrieus) instability competes with curvature and geometrical (Huygens) non-linearity in the overall flame shape dynamics. For nearly-straight or nearly-circular fronts, the dynamics translates into N-body problems for complex singularities. Approximate mean-field treatments yield predictions on the statistics of spontaneous wrinkling, that are checked against numerical integrations of MS equations : mean wrinkle-wavelength evolutions, etc... The paper next addresses how flames respond to weak broad-banded forcing, e.g. residual turbulence. Open questions relating to 3-D flames, or to the influence of a large tangential velocity, are finally evoked.

Stoichiometry Effects in the Point-Source Initiation of Lean Flames of a Light Fuel

Abstract We numerically study a non-linear, integro-differential equation which has recently been obtained in the context of analytical theories of flame initiation. It describes the dynamics of a spherical flame kernel, the growth of which is triggered by a time-dependent, localized source of heal in a mixture of a light fuel, a heavier oxidizer and a diluent in large excess; actually, this equation pertains to situations in which the concentrations of both reactants are small at the reaction zone (nearly-stoichiometric burning). By performing go-no-go numerical experiments, we obtain the critical energy for successful initiation of an expanding, autonomous flame and we show how it depends on the energy-input duration and shape, the mixture initial equivalence ratio, and the diffusive properties of both reactants. We show that lean mixtures can provisionally lead to locally-rich burning as the flame kernel grows or shrinks, as a consequence of differential diffusion effects. For mixtures characterized by...

Mean cell wavelengths of wrinkled premixed flames in weak gravity fields: Spontaneous evolutions

Starting from a Rakib–Sivashinsky type of equation for the local amplitude of wrinkling of premixed flames propagating upward or downward, we numerically study how the ensemble-averaged cell wavelengths (crest-to-crest, or from spectral density of wrinkling) evolve from random initial conditions, when the gravity is weak. Tentative theories yield simple scaling laws that are in good agreement with the numerical results; yet some lack of universality suggests that such laws are merely accurate and not exact. Open problems, e.g. relating to noise, are finally evoked.