J. K. Bechtold

The dependence of the Markstein length on stoichiometry

Both theory and experiment predict a linear relationship between flame speed and stretch, and the sensitivity of this dependence is given in terms of the Markstein number. In this note the dependence of Markstein number on mixture strength is explicitly determined. Results are presented for hydrogen-air, hydrocarbon-air, and alcohol-air mixtures over a range of equivalence ratio.

Hydrodynamic theory of premixed flames: Effects of stoichiometry, variable transport coefficients and arbitrary reaction orders

Based on a hydrodynamic length, which is typically larger than the nominal flame thickness, a premixed flame can be viewed as a surface of density discontinuity, advected and distorted by the flow. The velocities and the pressure suffer abrupt changes across the flame front that consist of Rankine-Hugoniot jump conditions, to leading order, with corrections of the order of the flame thickness that account for transverse fluxes and accumulation. To complete the formulation, expressions for the flame temperature and propagation speed, which vary along the flame as a result of local non-uniformities in the flow field and of flame front curvature, are derived. Unlike previous studies that assumed a mixture consisting of a single deficient reactant, the present study uses a two-reactant scheme and thus considers mixtures whose compositions vary from lean to rich conditions. Furthermore, non-unity and general reaction orders are considered in an attempt to mimic a wider range of reaction mechanisms and, to better represent actual experimental conditions, all transport coefficients are allowed to depend arbitrarily on temperature. The present model, expressed in a coordinate-free form, is valid for flames of arbitrary shape propagating in general fluid flows, either laminar or turbulent.

Wrinkling of spherically expanding flames

The onset of instability and subsequent development of cells on spherically expanding flames is examined theoretically. The model used accounts for both hydronamic and diffusive-thermal effects and, in contrast to earlier theories, is valid for variable transport properties over a wide range of equivalence ratios. The analysis yields predictions for a number of flame properties, including growth rate of small disturbances, critical flame size for the instability onset, cell size beyond the threshold, and an estimate of the speed of the developing turbulent flame. It is shown that results using the more realistic temperature-dependent transport coefficients are more commensurate with experimental data concerning the critical conditions, that is, flame size or Peclet number, at the transition from one burning regime to another.

Analysis of thermal ignition in the supersonic mixing layer

Ignition in a laminar supersonic mixing layer between two parallel streams of initially separated reactants is studied both numerically and through the use of large activation energy asymptotics. The asymptotic analysis provides a description of ignition characteristics over the entire range of system parameters. In particular, it is demonstrated that, for small values of viscous heating, the ignition distance scales approximately linearly with the freestream Mach number, whereas for large viscous heating it decreases rapidly due to the temperature-sensitive nature of the reaction rate. This indicates the potential of using local flow retardation to enhance ignition rather than relying solely on external heating. The asymptotic analysis further identifies several distinct ignition situations, yielding results that compare well with those obtained from the full numerical calculation. The effects of flow nonsimilarity are also assessed and are found to be more prominent for the mixing layer flow in comparison to the flat-plate configuration studied previously.

Effects of stoichiometry on stretched premixed flames

Abstract Within the framework of slowly varying flames we derive an equation for the burning rate of a flame in a strained flow field under near-stoichiometric conditions. Our expression exhibits a nonlinear dependence of flame speed on the strain rate and depends on the mixture’s equivalence ratio as well as two distinct Lewis numbers, corresponding to the fuel and oxidant. For a strained flame, the rate at which a given reactant reaches the reaction zone is strongly affected by its molecular diffusivity. We demonstrate that it is possible for the reactant which is initially in excess to be entirely consumed by the reaction, while the initially deficient reactant leaks through. This is shown to have important implications on the extinction characteristics of the flame. We calculate burning velocities using parameter values typical of several hydrocarbon–air and hydrogen–air mixtures and show that our predictions are in good agreement with experimental results.

Weakly stretched premixed flames in oscillating flows

The response of a premixed flame in stagnation-point flow with an imposed oscillating strain rate has been examined. This configuration is of fundamental interest and has potential application to turbulent combustion modelling. Of interest are flames which stand well clear of the front stagnation point of a bluff body. Under these conditions the flame can be treated as a surface of density discontinuity. A detailed solution is constructed in the burned and unburned gas regions and includes the flame response to the imposed fluctuations as well as the resulting displacement of the incident flow. Our analysis accounts for the full coupling between the flame and the underlying flow field and, unlike most previous studies, is not restricted to small-amplitude oscillations.

Nonlinear hydrodynamic stability and spinning deflagration of liquid propellants

Considered is a dynamic model of liquid propellant combustion that generalizes the classical model due to Landau. In appropriate parameter regimes, this model exhibits the well-known phenomenon of hydrodynamic (Landau) instability, in which steady planar deflagration is unstable to steady, but nonplanar (cellular), modes of combustion. In a cylindrical geometry, there exist special values of the radius such that the basic solution simultaneously loses stability to two cellular modes at critical values of a parameter that determines the sensitivity of the reaction rate to the local pressure field. A nonlinear stability analysis in the neighborhood of certain members of this sequence of double eigenvalues then leads to the prediction, by either one or the other of two methods, of secondary and tertiary branching of orbitally stable, spinning deflagration waves.

On the response of spherical premixed flames under rotation

We investigate the influence of flow rate and rotation on the structure and response of burner-stabilized spherical premixed flames. Asymptotic methods are employed in which we exploit the limit of large activation energy to resolve the reaction zone structure, and a perturbation analysis is carried out for small rates of rotation. We first construct the leading order solution, which describes a stationary (nonrotating) spherical flame. This configuration is analyzed to determine extinction characteristics, and the various mechanisms for stabilization of curved flames are also identified. Rotation is then included as a perturbation, and closed form solutions of the appropriate governing equations are obtained. We find that a secondary flow is induced inward toward the burner poles and outward from the equator. This in turn deforms the flame into a pancake shape that may be flattened either at the poles or the equator, depending on the combined effects of Lewis number, flame stretch, and ambient temperature. We also present results for the related problem of a rotating monopropellant droplet in the Appendix.

Response of spherical diffusion flames under rotation with general Lewis numbers

The structure and extinction of diffusion flames generated by a porous spherical burner or a fuel droplet in response to rotational motion were investigated through perturbation analysis, with emphasis on the effects of non-unity Lewis numbers (Le) for both fuel and oxidizer. The analysis shows that the rotational motion induces a secondary flow that distorts the otherwise spherical flame into a pancake shape. The flame temperature is also affected, such that the flame becomes more susceptible to extinction either at the poles or the equator, depending on the combined effects of Lewis numbers, flame stretch through local flow non-uniformity and ambient oxidizer concentration. The results reported here will help guide planned microgravity experiments.

Some new results on Markstein number predictions

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.