Analysis of mixture stratification effects on unstrained laminar flames
Introduction
Stratified combustion can help to meet the strict emission requirements for internal combustion engines and gas turbines. Stratification can occur implicitly due to insufficient premixing of the fuel and the oxidiser or explicitly through multiple injections or staged combustion. Compared to homogeneous premixed combustion, the equivalence ratio stratification affects the local flame properties and may enhance flame propagation and lower emissions [1]. To study stratified combustion, experiments [2], [3] and numerical methods [4], [5], [6], [7], [8], [9] have been applied. A detailed overview is provided in the review paper by Lipatnikov [1].
The flame may either propagate normal to the equivalence ratio gradient, which is typically observed in triple-flame structures [10], or burn in gradient direction, which has been investigated for its influences on flammability limits [11], flame propagation speeds [5] and flame structure for various fuels including hydrogen [9], methane [12] and iso-octane [13]. The flame propagation in the direction of the equivalence ratio gradient can be examined in one dimension by analysing oscillation responses. The unsteady response of strained flames to velocity [4] and mixture composition [6] oscillations have been documented before, but the response of the flame propagation under unstrained conditions in one dimension is yet to be analysed in detail and the present paper addresses this void in the existing literature.
Industrially relevant stratified flames are usually fuel-lean, and the flame may burn into an even leaner mixture, which is referred to as back-supported (BS) combustion, or into a richer mixture closer to stoichiometry, which may be called front supported (FS). A fuel-lean BS flame propagates faster and has a thinner reaction zone [7] than a homogeneous mixture flame at the same (local) equivalence ratio. This is due to the relatively high heat and (radical) species concentrations on the burnt side of the flame diffusing into the leaner preheat zone. As a result, the flammability limits of a flame in a stratified mixture can be extended [5].
Studying stratified mixture combustion in one-dimension is not trivial, as many solvers do not support unsteady solutions, and the resulting computational cost is surprisingly high. A common method to analyse stratification effects is to artificially constrain the flame to the stratification layer by introducing compressible strain in a counter-flow configuration [8], which limits such studies to (strongly) strained cases. This is problematic for studies where the equivalence ratio gradients occur more naturally, without imposed straining (e.g. in the Darmstadt and Cambridge Flames [2], [3]). Another method allows for the flame to propagate through an equivalence ratio gradient [5], but, diffusion diminishes the equivalence ratio gradients, yielding a rapid reduction of the stratification effect.
The present work takes a different approach which, albeit computationally expensive, avoids these problems: a sinusoidal equivalence ratio field is generated by suitable unsteady inflow conditions, and a premixed stratified flame is initialised to freely propagate through the mixture. This work aims to (i) analyse the unsteady response of an unstrained flame to mixture stratification oscillations, to (ii) test the influences of the diffusivity assumptions and the reaction mechanisms on the results, and to (iii) propose and demonstrate the usefulness of an alternative consumption speed definition that is applicable for stratified flames.
Section snippets
Computational setup
To prepare the work presented here, the numerical experiment by Cruz et al. [5] was reproduced first to ensure the quality of our numerical techniques. A very good agreement was obtained, which is not shown here for the sake of brevity, however, can be found in the supplementary material.
One-dimensional unsteady unstrained laminar flames propagating through a sinusoidal methane-air mixture at 300 K in atmospheric conditions are calculated, as sketched in Fig. 1. The upper and lower limits of
Definitions and equations
A stratification indicator is considered as the equivalence ratio wavelength over the laminar flame thickness. A value of λ ≈ 3 is typical of the stratification layer of the well-known Cambridge turbulent stratified flame experiment [3]. Similarly, a stratification thickness δS is introduced as . The stratification indicator λ is estimated a priori, whereas the stratification thickness δS provides a posteriori quantification of the stratification strength.
The local
Stratified flame dynamics using equal diffusivities
The density-weighted displacement speed and its components defined in Eq. (7) are presented in Fig. 2: due to the stratification, these quantities are no longer unique function of equivalence ratio: they ‘orbit’ around the homogeneous mixture solutions, indicating a hysteresis. The area within the orbit or the deviation from the homogeneous mixture flame depends on the wavelength and amplitude of the stratification. The direction of the orbit (circular arrow in each figure) indicates
Stratified flame dynamics using mixture-average diffusivities
The effects of the variable diffusivity assumption on stratified flames have been investigated for turbulent flames by DNS in two [23] and three-dimensions [24] and by experiments [25]. A detailed study of the effect of variable diffusivity on flame chemistry has been performed by Hilbert et al. [26]. These analyses suggest that it is worthwhile to investigate the cases investigated here again for variable diffusivities to examine their influences on the results. The local mixture fraction now
Implications for consumption speed and flame thickness
Simulations of combustion often require dimension reduction techniques, e.g. [20]. For example, the popular reduction method tabulation techniques relies on an accurate description of flame propagation speed and thermal flame thickness. In this section, the popular methods to compute the flame speed and thickness and the consequences of neglecting stratification effects on the results will be discussed. Once again, we stress that this analysis is only carried out for flames propagating towards
Conclusions
Detailed one-dimensional simulations of unsteady unstrained laminar flames subjected to both linear and sinusoidal equivalence ratio perturbations have been conducted. Greater amplitudes and shorter wavelengths of sinusoidal equivalence ratio variations led to stronger stratification effects and subsequently gave rise to greater phase shifts between the equivalence ratio oscillations and the responses of species concentrations and flame properties. However, in the absence of compressible
Acknowledgments
The authors gratefully acknowledge the financial support by DFG (Proj. No.: 393710272, KE 1751/13-1) and EPSRC (EP/R029369/1) and the compute time on magnitUDE, Duisburg (DFG INST 20876/209-1 FUGG) and on ARCHER, Edinburgh (e305 – UKCTRF).
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