Theory of premixed-flame propagation in large-scale turbulence

Abstract

A statistical theory is developed for the structure and propagation velocity of premixed flames in turbulent flows with scales large compared with the laminar flame thickness. The analysis, free of usual closure assumptions, involves a regular perturbation for small values of the ratio of laminar flame thickness to turbulence scale, termed the scale ratio ε, and a singular perturbation for large values of the non-dimensional activation temperature β. Any effects of the flame on the flow are considered to be given. In this initial study, molecular coefficients for diffusion of heat and reactants are set equal. The results identify convective-diffusive and reactive-diffusive zones in the flame and predict thickening of the flame by turbulence through streamwise displacement of the reactive-diffusive zone. Profiles for intensities of temperature fluctuations and for streamwise turbulent transport are obtained. A fundamental quantity occurring in the analysis is the longitudinal displacement of the reactive-diffusive zone in an Eulerian frame by turbulent fluctuations, and to first order in the scale ratio this equals the longitudinal displacement of fluid elements in an Eulerian frame by turbulent fluctuations, herein termed simply the Eulerian displacement. To first order in the scale ratio it is found that, if the Eulerian displacement experiences the same type of statistical non-stationarity as the corresponding Lagrangian displacement, then the diffusion approximation is valid for streamwise turbulent transport but the turbulent flame thickens as time increases, while if the Eulerian displacement is statistically stationary then the diffusion approximation necessitates a negative coefficient of diffusion in part of the flame but the flame thickness remains constant. By carrying the analysis to second order in the scale ratio it is shown that the turbulent-flame speed exceeds the laminar-flame speed by an amount proportional to the mean square of the transverse gradient of the Eulerian displacement. This result can be understood from the mechanistic viewpoint of a wrinkled laminar flame in terms of the increase in flame area produced by turbulence. Thus the theory provides a precise statistical quantification of the model of the wrinkled laminar flame for describing structures of turbulent flames.

Front propagation in laminar flows

“…u = 0, one expects that the front propagates with an average limiting speed V f . A problem of primary interest is to determine the dependence of V f on the properties of the velocity field u [10,11]. In this article we consider front propagation in simple laminar flows (shear flow and systems with cellular structures)…”

“…We will provide a detailed analysis of these two regimes, highlighting their differences. In the context of the thin front regime we can mention, among the many contributions, the works on the G-equation approximation and its relation with the "geometrical optics" regime [19,20], the work on turbulent flows [10,11] and the numerical study of front propagation in synthetic turbulence [23].…”

Bulk Burning Rate in¶Passive–Reactive Diffusion

“…In the physics literature one can find a number of models and approaches that yield different predictions -relations between the turbulent intensity and the burning rate ( [8,21,20,41]). These results are usually obtained using heuristic models and physical reasoning.…”

Flame propagation in a small-scale parallel flow

“…To O( −2 ) we find T 0ηη = 0 which, when used with (11), implies that T 0 must be independent of η, T 0 = T 0 (t, ζ ).…”

“…This is relevant in the so-called flamelet regime [10] and has received significant attention and confirmation in the literature. An example of analytical contributions relevant to this regime is the often cited Clavin-Williams formula derived for flames propagating in largescale, low-intensity turbulence [11]. For a detailed recent account on turbulent combustion, see [12] and references therein.…”