On-Shell Description of Unsteady Flames

TL;DR

This paper develops a non-perturbative, on-shell integral framework for describing unsteady premixed flames with arbitrary gas expansion in two dimensions, unifying and extending existing flame dynamics theories.

Contribution

It introduces an implicit integral representation and explicit vortex component expression that eliminate the need to solve bulk flow equations, providing a unified approach to flame front dynamics.

Findings

Derives a system of integro-differential equations for flame front and velocity.

Recovers known linear and nonlinear flame instability equations as special cases.

Provides a new analytical tool for studying unsteady flame behavior.

Abstract

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.