Consumption and displacement speeds of stretched premixed flames - Theory and simulations

The flame displacement speed (FDS) and flame consumption speed (FCS) are commonly used in numerical studies to characterize flame dynamics. Although for a planar configuration they are both well defined and accurately represent the propagation speed of the flame into the combustible mixture, their definition in more general circumstances is ambiguous. The FDS and FCS are local quantities: the FDS is associated with the displacement of an arbitrarily selected iso-surface, and the FCS is an integrated quantity throughout a region that needs to be approximated, in a direction that is not always uniquely defined. The only unambiguously defined quantity is the global (volumetric) consumption rate obtained by integrating the rate of reactant consumption over the entire combustion volume. However, using it to determine the FCS requires a proper identification of the flame surface area which introduces uncertainty in the results. Indeed, numerical simulations show that combustion properties depend significantly on the choices made in the determination of the FDS and FCS. In order to utilize these quantities in a meaningful way, their limitations are explored by providing a detailed comparison between predictions of numerical simulations and theoretical expressions obtained for weakly-stretched flames. The theory is based on the assumption that the flame is thin relative to the representative hydrodynamic length scale and in this asymptotic limit both, the FDS (commonly referred to as the flame speed) and the FCS, are uniquely and unambiguously defined. Two configurations are examined in this paper: (i) spherical flames, unsteady expanding as well as stationary, where the flow is unidirectional and (ii) steadily propagating cusp-like flames (resulting from the Darrieus-Landau instability) whose structures are spatially varying and where the flow through the flame is nonuniform. The presented comparison validates the accuracy of the asymptotic expressions for the dependence of the FDS and FCS on stretch for one-step chemistry, and demonstrates that the theoretical predictions remain qualitatively, and to a large extent quantitatively valid for detailed chemistry for both, lean and rich flames.