Consistent definitions of “Flame Displacement Speed” and “Markstein Length” for premixed flame propagation

Abstract The definition of the flame displacement speed (FDS), often used to characterize the dynamical properties of premixed flames, is generally ambiguous because, except for a steadily propagating planar flame, the mass flow rate through the combustion region varies with distance through the flame and one is therefore faced with the difficulty of choosing a proper iso-surface to represent the flame surface. A directly related issue is the determination of the proportionality coefficient in the linear flame speed-flame stretch relation of weakly-stretched flames, known as the Markstein length, which depends strongly on the location inside the flame zone where it is measured or calculated. The objective of the present study is to identify an iso-surface and thereby a definition of the FDS that is well conditioned and less prone to uncertainties, and a consistent and unambiguous expression for the Markstein length. With a selected isotherm to represent the flame surface, the two most common definitions of the FDS are based either on the energy equation with the temperature as the progress variable, or on the kinematic characteristics of the surface (the propagation speed relative to the flow). In this study we examine the spherical flame geometry, a setup that provides an independent determination of the FDS that is not contingent upon an arbitrary selection of the flame surface and thus permits a proper evaluation of the two FDS definitions. A large number of simulations of premixed spherical propane/air flames with equivalence ratio ranging from 0.8 to 1.4 were carried out at various temperatures and pressures using both global single-step and detailed reaction schemes. Outwardly propagating spherical (or cylindrical) flames and inwardly propagating stationary spherical flames were examined. The dependence of the flame speed and flame temperature on stretch, and the corresponding Markstein length were identified for different isotherms selected to represent the flame surface, and the results were carefully compared to the asymptotic theory of weakly-stretched flames. The excellent agreement between theory and simulations provides a clear explanation and quantification of the differences found between the trends in the flame speed-flame stretch relation and the corresponding Markstein lengths, exhibited when the FDS was calculated based on an isotherm in the burned or unburned sides of the flame. We show that the proper isotherm for the evaluation of the FDS which is well-conditioned and properly accounts for the physics must be sufficiently close to the burned side of the flame. This choice is less prone to uncertainty, as the slope of the flame speed-flame stretch relation when reaching the burned side of the flame becomes less dependent on the selected iso-value. On the other hand, the choice of the fresh combustible mixture temperature as a reference location for the calculation of the FDS, and the corresponding unburned Markstein length, is ill conditioned and should be avoided.