Stretch effects in counterflow and propagating spherical flames

Generalized expressions for the flame response to stretch rate variations were derived based on an integral analysis. Together with values of the laminar flame speeds, laminar flame thicknesses, and the one-step overall reaction order and activation energy determined from the computational results of the one-dimensional planar flame, these expressions for the stretched flames were then used to correlate the computational results of the spherical outwardly-propagating, spherical inwardlypropagating, and counterflow hydrogen/air and propane/air flames. Such correlations yielded the laminar flame speeds through linear extrapolation to zero stretch rate, the Markstein lengths representing the sensitivity of the flame response to stretch rate, and the flame Lewis number. Furthermore, it was shown that the extracted Markstein lengths and Lewis numbers from the three flame configurations are largely the same for given equivalence ratio and system pressure, and that these Lewis numbers also agree well with those predicted from the two-reactant flame theory of Joulin and Mitani. The feasibility of a priori determination of stretch effects on laminar premixed flames is emphasized. Introduction Since the late seventies, significant advance has been made on the theoretical description of stretched flames, and as such there now exists a reasonably mature qualitative understanding on the structure and response of laminar premixed flames to stretch rate variations. Specifically, it is well established that stretch effects can be manifested through aerodynamic straining, flame curvature, and flame motion, and that these influences are particularly strong in the presence of mixture nonequidiffusion because of the resulting modification of the flame temperature. Since these formulations invoke such simplifying assumptions as one-step overall reaction and constant transport properties, it is obviously not justifiable to conduct quantitative comparisons between the theoretical predictions and results obtained through either experimentation or computational simulation with detailed reaction and variable transport properties. The rational approach towards enabling the analytical results quantitatively useful is thus to extract certain bulk flame parameters characterizing the stretch expressions through comparisons between the analytical and experimental/computational results. Such an extraction would be especially meaningful if the bulk parameters extracted are shown to be applicable to flames subjected to diverse modes of stretching. As a first step towards such an approach, in Ref. 1 the response of the symmetrical counterflow methane/air and propane/air flames to strain rate and pressure variations were studied. The extraction was based on the analytical result that the sensitivity of the flame response to stretch rate is a function of several parameters characterizing the reference, one-dimensional unstretched adiabatic flame in the doubly-infinite limit. These include the laminar flame speed ($„) and flame thickness (Sf), which are directly determinable from the flame solution, the Zeldovich number (Ze) and hence the one-step overall activation energy (Ea), which can be extracted from the dependence of S^ on the adiabatic flame temperature (T^), and the mixture Lewis number (Le), which has been conventionally * Graduate Student ** Research Staff Member *** Robert H. Goddard Professor, Fellow AIAA Copyright © 1997 by C. K. Law. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. estimated for sufficiently off-stoichiometric mixtures from the freestream values of the mixture properties. It was, however, pointed out in Ref. 1 that since transport properties can be fairly strong functions of temperature and mixture composition, which vary significantly across the flame, Le is a global flame property and as such should be extracted from the flame response. Recognizing this concern, a systematic approach was identified through which a global Lewis number, for a given equivalence ratio, by comparing the analytical expression for the stretched flame response with results obtained from both computation and experiment. Furthermore, for sufficiently off-stoichiometric flames, these extracted Lewis numbers were shown to be close to those estimated from the freestream conditions, hence demonstrating the consistency and hence meaningfulness of such an approach. There are several focuses of the present study, all of which are directed towards the eventual establishment of quantitatively useful expressions of the stretched flame responses. The first aspect is a generalized integral formulation of stretched flames. In this regard we note that our previous integral formulation [2] was separately conducted for the stationary strained and curved flame with non-unity Lewis number, and for the propagating spherical flame with unity Lewis number. Furthermore, a linear temperature variation across the flame was assumed. We have now generalized such a formulation to describe the unsteady propagation of a curved flame in a strained flow field, allowing for non-unity Lewis number and a proper description of the temperature variation across the flame. We have chosen to use the integral instead of the activation energy asymptotic formulation for the extraction because the analysis utilizes the finite nature of the flame thickness, which can be readily identified from the experimental and computational flame structure. We haste to add, however, that although it is as yet not apparent to us, it is conceivable that a similar extraction procedure can be developed for the asymptotic results and we are indeed simultaneously pursuing this goal. The essential point to note is that as long as the extracted parameters are used only in the expressions from which they are defined, the predicted flame responses should not be affected. The second aspect of the present study is an investigation of the dynamics of outwardlyand inwardly-propagating spherical flames. Here we note that recent studies of laminar flames in general and stretched flames in particular have employed two prototypical flame configurations, namely the counterflow flame [3-11] and the propagating spherical flame [12-21]. Both these flames are configurationally simple, and hence are amenable to well-controlled experimentation as well as computational simulation. In the context of studies of stretch effects on the flame response, the counterflow flame simulates the influence of hydrodynamic straining while the spherical flame is associated with the flame front motion and flame curvature. It is therefore logical that we apply the extraction procedure previously developed for the counterflow flame to the spherical flame, both outwardlyand inwardly-propagating. If the extraction procedure is indeed consistent, then the extracted Lewis numbers should assume similar values because it is reasonable to expect that they are insensitive to the specific flame configuration. Furthermore, we shall also demonstrate the feasibility of using the propagating spherical flame to determine the laminar flame speed through extrapolation to zero stretch rate. Results obtained by using the present expression will be compared to those developed previously by others. The third aspect of the present study is the determination of the so-called Markstein length, which can be broadly interpreted as the sensitivity of the flame response to stretch rate variations. These values can be readily used in the simulation of stretched flame response. The study will be conducted for both hydrogen/air and propane/air flames because of their opposite nonequidiffusive behavior for the lean and rich flames. The study is computational because of the simplicity of the flame configuration and the fact that the kinetics of the hydrogen and propane oxidation are reasonably established as far as the prediction of the bulk flame parameters is concerned. In the next section the generalized integral formulation will be presented. This is followed by discussions on results for hydrogen/air and propane/air flames. The discussions on the hydrogen/air flames will be detailed, consisting of a section on the extraction of the various properties of the onedimensional unstretched flame, to be followed by three sections covering various aspects of flame responses and parameter extraction for the outwardly-propagating, inwardly-propagating, and counterflow flames respectively. The corresponding results for propane/air flames arc then presented and briefly discussed. Generalized Integral Analysis of Stretched Flames Since the present derivation is a generalization of that of Ref. 2, we only present the major differences between the two derivations. The schematic of the flow and flame configuration is shown in Fig. 1. Here the thin reaction zone has a surface area Af and an arbitrary curvature represented by the unit normal vector n pointed in the downstream direction. AT/U is the area of surface projected from Af onto the upstream boundary of the preheat zone, at a distance of 87 from the reaction zone. The interception of this boundary and a streamline defines another surface, with area Aj>st. Finally, projecting AT st onto the reaction zone surface yields Aj f, which differs from Aj>st because of the flame curvature. In this simple quasi-one-dimensional system, the differential conservation equations of mass, temperature (T) and species (Y) are respectively given by