Premixed flame shapes and polynomials

TL;DR

This paper investigates the shapes of premixed flames using pole decomposition methods applied to the Michelson-Sivashinsky equation, revealing polynomial structures that accurately describe steady crest shapes for certain conditions.

Contribution

It introduces a polynomial-based approach linked to Meixner Pollaczek recurrences for modeling flame crest shapes, extending understanding of nonlinear flame dynamics.

Findings

Polynomials closely follow Meixner Pollaczek recurrence for steady shapes

Accurate crest shapes obtained for N>=3 poles

Explicit shapes for finite-N periodic flames remain elusive

Abstract

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.