Sivashinsky equation for corrugated flames in the large-wrinkle limit

Abstract

Sivashinsky's [Acta Astron. 4, 1177 (1977)] nonlinear integrodifferential equation for the shape of corrugated one-dimensional flames is ultimately reducible to a 2N -body problem, involving the 2N complex poles of the flame slope. Thual, Frisch, and Hénon [J. Phys. (France) 46, 1485 (1985)] derived singular linear integral equations for the pole density in the limit of large steady wrinkles (N>>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

“…In this way, we reproduce the results obtained in [20] for −1 ≤ c ≤ 0, and explain and fix the pathologies of (3) for c > 0. The method is valid whatever the sign of c, and generalizes results on wrinkled fronts with two unlike crests per wavelength, which was so far available for the MS equation only [19]. In all the configurations envisaged, we compare the analytical predictions with numerical determinations of the discrete B k and of φ.…”

“…When N 1, corresponding to large wrinkles, the spacing between consecutive poles scales like B N /N , enabling one to adopt a continuous approximation of the equilibrium conditions, and rewrite them as integral equations for pole densities [17]. As first shown for c = 0 in [19], and confirmed when c < 0 [20], a key step to describe the structure of large steady wrinkles is to first solve the simpler case of isolated crests ; these have infinite wavelengths and poles at Z k = iB k in arbitrary number 2N .…”

Premixed-flame shapes and polynomials

“…Equation (1) exhibits a number of remarkable features, most notably the existence of poledecompositions whereby the search for ( , ) t x  is converted to a 2N -body problem for the complex poles of the front slope [10,11]; see §2. Thanks to this property one can: (i) Explain the formation of front arches joined by sharper crests whose mergers ultimately produce the widest admissible steady cell; (ii) Access the latter's arc-length vs. wavelength curve [12], which yields the effective flame speed; (iii) Solve stability issues [13,14] without the effect of spurious noises hampering the non-self-adjoint linearized dynamics [15]; (iv) Compute pole density and front shapes for isolated crests, and then for periodic cells [11,16], if 1 N  ; (v) Study stretched crests [17]; (vi) Set up tools to study extensions of (1) that incorporate higher orders of the 1   expansion, at least in the large-N limit ( [18,19] and Refs. therein).…”

“…This choice was already made when drawing Fig.1 [16,23] whatever the value of  is: it is indeed known [23] that  mainly encodes information about the lowest zeros, whence it disappeared from (18).…”

Wrinkled flames and geometrical stretch

“…In the stretch-free case, the uppermost pole altitude increases with the number N of pole pairs, and the typical difference Equation (3.4) is solved by a Fourier method like in [21] (see Appendix A). In terms of an…”