Some new results on Markstein number predictions

Abstract

The flame speed of a premised flame is known to depend on curvature and local flow conditions, i.e. flame stretch. For flames in unconfined environments. both theory and experiment predict for weakly-stretched fla.mes a 1inea.r relationship bet.ween flame speed and stretch, and the sensitivity of this dependence is given in terms of the Markstein number. In this paper we first review known theoretical results concerning Markstein numbers and present some new results showing an explicit dependence on equivalence ratio and variable transport properties. We also present new theoretical predictions regarding stretch effects on premixed flames in enclosed vessels. Specifically, we derive an expression for flame speed in a constant volume vessel. The inherent unsteadiness associated with the increasing pressure gives rise to a more complicated expression for flame speed. W’e find that flame propagation is strongly influenced by geometry, and the corresponding “Markstein” numbers must be interpreted differently from the constant-pressure case. We use our model to examine the propagation of a spherical flame in a. closed vessel and relate our results to recent experimental measurements. *Associate Professor ‘Professor, Associate Fellow AIAA Copyright @ZOO0 by hslatalon. Published by the American Institute of Aeronautics and Astronautics, Inc with pernlission. All rights reserved. Introduction Much progress in the theoretical underst.anding of premixed flame dynamics has been based on esa.mining the problem on two separate length scales (cf. Matalon and Matkowsky’ ). One length scale, LD = Z3Dth/S~, characterizes the thermal thickness of the flame: here Z)th is the thermal diffusivity of the mixture and 5’~ is the laminar flame speed. The other length scale, L, characterizes the flame shape: for example it is associated with the wavelengt.11 of wrinkles that, develop on the flame front or nit11 the geometrical dimensions of the vessel ivithin ~vhich the flame propagates. Typically LD 10-2c~n. namely much smaller than L. Viewed on the hydrodynamic length scale. the flame may be regarded as a surface of density discont,inuity, advect,ed and distorted by the flow. Tlrc flow field is determined by a global analysis Ivherr the hydrodynamic equations must be solved subject to jump relations across the flame and appropriate conditions along the boundary of the domain. The flame sheet is characterized by t\vo parameters which vary along its surface and in time: the flame temperature Tf, and its propagation speed .S’f . In general, Tj determines t.hc dcnsity drop across the flame front while S’f determines its evolution relat.ive to t,he fresh unburned gas and they differ from the adia.batic flame temperat.urc T0 and the laminar flame speed SL. Thus, flame speed and temperature are influenced by local nonuniformities in the flow field and by the flame front cur va.ture, the combined effects which are kno\vn as flame stretch.