Shapes and speeds of forced premixed flames

TL;DR

This paper analyzes the behavior of steady premixed flames under space-periodic forcing using mathematical models, deriving analytical solutions for flame speed and shape, and comparing the dynamics of Michelson-Sivashinsky and Burgers equations.

Contribution

It introduces a novel analytical framework for understanding forced premixed flames via pole dynamics and integral equations, extending previous models with explicit solutions and numerical validation.

Findings

Flame speed increases rapidly with forcing intensity initially.

Michelson-Sivashinsky flames grow faster than Burgers flames under forcing.

Speed difference decays slowly at high forcing intensities.

Abstract

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.