Premixed flame shapes and polynomials

The nonlinear nonlocal Michelson-Sivashinsky equation for isolated crests of unstable flames is studied, using pole-decompositions as starting point. Polynomials encoding the numerically computed 2N flame-slope poles, and auxiliary ones, are found to closely follow a Meixner Pollaczek recurrence; accurate steady crest shapes ensue for N>=3. Squeezed crests ruled by a discretized Burgers equation involve the same polynomials. Such explicit approximate shape still lack for finite-N pole-decomposed periodic flames, despite another empirical recurrence.Comment: Accepted for publication in Physica D :Nonlinear Phenomen

Flame Wrinkles From The Zhdanov-Trubnikov Equation

International audienceThe Zhdanov-Trubnikov equation describing wrinkled premixed flames is studied, using pole decomposi tions as starting points. Its one-parameter (−1 0 (over-stabilisation) such analytical solutions can yield accurate flame shapes for 0 < c < 0.6. Open problems are invoked

Wrinkled flames and geometrical stretch

Localized wrinkles of thin premixed flames subject to hydrodynamic instability and geometrical stretch of uniform intensity (S) are studied. A stretch-affected nonlinear and nonlocal equation, derived from an inhomogeneous Michelson-Sivashinsky equation, is used as a starting point, and pole decompositions are used as a tool. Analytical and numerical descriptions of isolated (centered or multicrested) wrinkles with steady shapes (in a frame) and various amplitudes are provided; their number increases rapidly with 1/S > 0. A large constantS > 0 weakens or suppresses all localized wrinkles (the larger the wrinkles, the easier the suppression), whereasS < 0 strengthens them; oscillations of S further restrict their existence domain. Self-similar evolutions of unstable many-crested patterns are obtained. A link between stretch, nonlinearity, and instability with the cutoff size of the wrinkles in turbulent flames is suggested. Open problems are evoked

Shapes and speeds of forced premixed flames

Steady premixed flames subjected to space-periodic steady forcing are studied via inhomogeneous Michelson-Sivashinsky (MS) and then Burgers equations. For both, the flame slope is posited to comprise contributions from complex poles to locate, and from a base-slope profile chosen in three classes (pairs of cotangents, single-sine functions or sums thereof). Base-slope-dependent equations for the pole locations, along with formal expressions for the wrinkling-induced flame-speed increment and the forcing function, are obtained on excluding movable singularities from the latter. Besides exact few-pole cases, integral equations that rule the pole-density for large wrinkles are solved analytically. Closed-form flame-slope and forcing-function profiles ensue, along with flame-speed increment vs forcing-intensity curves; numerical checks are provided. The Darrieus-Landau instability mechanism allows MS flame speeds to initially grow with forcing intensity much faster than those of identically forced Burgers fronts; only the fractional difference in speed increments slowly decays at intense forcing, which numerical (spectral) timewise integrations also confirm. Generalizations and open problems are evoked.Comment: Revised version submitted to Phys. Rev.

On-Shell Description of Unsteady Flames

The problem of non-perturbative description of unsteady premixed flames with arbitrary gas expansion is solved in the two-dimensional case. Considering the flame as a surface of discontinuity with arbitrary local burning rate and gas velocity jumps given on it, we show that the front dynamics can be determined without having to solve the flow equations in the bulk. On the basis of the Thomson circulation theorem, an implicit integral representation of the gas velocity downstream is constructed. It is then simplified by a successive stripping of the potential contributions to obtain an explicit expression for the vortex component near the flame front. We prove that the unknown potential component is left bounded and divergence-free by this procedure, and hence can be eliminated using the dispersion relation for its on-shell value (i.e., the value along the flame front). The resulting system of integro-differential equations relates the on-shell fuel velocity and the front position. As limiting cases, these equations contain all theoretical results on flame dynamics established so far, including the linear equation describing the Darrieus-Landau instability of planar flames, and the nonlinear Sivashinsky-Clavin equation for flames with weak gas expansion.Comment: 21 pages, 3 figures; extended discussion of causality, new references adde

The Sivashinsky equation for corrugated flames in the large-wrinkle limit

Sivashinsky's (1977) nonlinear integro-differential equation for the shape of corrugated 1-dimensional flames is ultimately reducible to a 2N-body problem, involving the 2N complex poles of the flame slope. Thual, Frisch & Henon (1985) derived singular linear integral equations for the pole density in the limit of large steady wrinkles $(N \gg 1)$, which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise. Here we solve those analytically for isolated crests, next for monocoalesced then bicoalesced periodic flame patterns, whatever the (large-) amplitudes involved. We compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole-decomposition approach. Good agreement is obtained, even for moderately large Ns. The results are extended to give hints as to the dynamics of supplementary poles. Open problems are evoked

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation

Using pole decompositions as starting points, the one parameter (-1 =< c < 1) nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of premixed gaseous flames is studied in the large-wrinkle limit. The singular integral equations for pole densities are closely related to those satisfied by the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They can be solved via the introduction of complex resolvents and the use of complex analysis. We retrieve results obtained recently for -1 =< c =< 0, and we explain and cure their pathologies when they are continued naively to 0 < c < 1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the shapes of steady isolated flame crests, and then bicoalesced periodic fronts. These theoretical results fully agree with numerical resolutions. Open problems are evoked.Comment: v2: 29 pages, 6 figures, some typos correcte

On the hydrodynamic stability of curved premixed flames

We propose a non-linear, model equation describing the dynamics of finite amplitude disturbances superimposed to a two-dimensional, weakly unstable, flame tip of parabolic shape. By showing that solutions of this equation admit a pole decomposition, we illustrate how the local curvature effects, non-linearity and the geometry-induced flame stretch compete with the hydrodynamic instability. Cases of stability, of metastability or leading to « sidecusping » are exhibited. For spatially-periodic disturbances, a non-linear analog to Zel'dovich et al.'s criterion (C. S. T. 24 (1980)) is obtained. The appearance of steady tip-splitting is also shown to be non-generic in the class of pole-decomposable solutions.On propose une équation non linéaire modèle qui décrit la dynamique de perturbations d'amplitude finie superposées à une flamme faiblement instable et de forme parabolique. Montrant que des solutions admettent une décomposition en pôles, on illustre comment les effets de courbure locaux, la non-linéarité, et l'étirement de la flamme dû à la géométrie rivalisent avec l'instabilité hydrodynamique. Des situations stables, métastables ou conduisant à des structurations latérales sont mises en évidence. Dans le cas de perturbations spatialement périodiques, un analogue non linéaire du critère de Zel'dovich et al. (C.S.T. 24 (1980)) est obtenu. On montre aussi que les dédoublements symétriques et permanents du sommet de la flamme sont non génériques dans la classe de solutions envisagée