Turbulent flame speeds of non-equidiffusive ($Le \neq 1$) mixtures: a generalized scaling with Markstein/Lewis-number effects, the mechanism of stretch-affected local extinction, and a diffusivity-aware combustion regime diagram

For expanding turbulent premixed flames, the normalized turbulent flame speed follows a $Re_{T,f}^{1/2}$ scaling (turbulent Reynolds number based on the mean flame radius and the thermal diffusivity) only at fixed Markstein number, and the Markstein-diffusivity-based collapse proposed in the source work is valid only for positive-Markstein-number ($Le>1$) flames. Wu, Saha, Chaudhuri and Law (2015), measuring C4 to C8 n-alkane/air flames at 1 to 5 atm in a fan-stirred vessel, found that the turbulent flame speeds of stoichiometric and rich mixtures with Lewis numbers close to or below unity assume similar values, that the corresponding lean ($Le>1$) flames exhibit a strong propensity to stretch-induced local extinction, and that the averaged turbulent flame speed decreases monotonically with fuel molecular weight for lean mixtures. These findings, together with the restriction of the Markstein-diffusivity-based collapse to positive-Markstein flames, leave four linked open questions: (i) a generalized turbulent flame-speed scaling that incorporates thermo-diffusive effects and collapses data for both $Le<1$ and $Le>1$ mixtures; (ii) the physical mechanism by which $Le>1$ controls the onset and spatial extent of stretch-induced local extinction, and the role of subsequent re-ignition; (iii) a turbulent combustion regime diagram that integrates molecular diffusivity and separates extinction-prone from extinction-free conditions; and (iv) the confirmation or refinement of the reported monotonic decrease of averaged turbulent flame speed with fuel molecular weight for lean mixtures. A decade of follow-up work has produced partial empirical correlations and unity-Lewis-number theory but no resolution of any of these items; the problem remains open.

Background

A premixed flame propagates through a combustible mixture at the laminar flame speed $S_L$, set by the balance of heat release, chemical kinetics, and molecular transport. In a turbulent flow, wrinkling and distortion of the flame front by eddies increase the burning rate, and the resulting turbulent flame speed $S_T$ (the effective propagation or consumption rate of the turbulent flame brush) is the central quantity of premixed turbulent combustion: it is required by essentially every combustion model used for engines, gas turbines, and safety analyses. Classical arguments due to Damkohler identify a turbulent Reynolds number as the controlling group; for outwardly expanding spherical flames ignited in near-isotropic turbulence inside a fan-stirred, constant-pressure vessel, experiments showed that $S_T/S_L$ scales as $Re_{T,f}^{1/2}$, where $Re_{T,f}$ uses the mean flame radius as length scale and the thermal diffusivity as transport property. The turbulent flame speed of an expanding flame grows with flame radius (wrinkling develops with scale) without reaching a plateau, so $S_T$ must be defined at a stated radius window, and different measurement definitions (displacement, consumption, or schlieren-based — schlieren being optical density-gradient imaging) coexist in the literature.

Two concepts from laminar flame physics govern how the mixture's transport properties enter. First, flame stretch (the combined effect of strain and curvature on a flame surface) changes the local flame speed; the linear response coefficient is the Markstein length $L_M$, normalizing which by the flame thickness gives the dimensionless Markstein number $Mk$, and the Markstein diffusivity $D_M = S_L L_M$ acts as the diffusivity that dissipates small-scale flame-surface fluctuations. Second, the Lewis number $Le$, the ratio of thermal diffusivity to mass diffusivity of the deficient reactant, measures non-equidiffusion (differential diffusion): lean hydrogen and hydrogen-blended mixtures have $Le<1$ (thermo-diffusive instability, local burning-rate enhancement at convex flame segments), while lean heavy hydrocarbons have $Le>1$ (local burning-rate reduction and, at sufficient stretch, local extinction, meaning the formation of holes in the flame sheet that may subsequently re-ignite). Chaudhuri, Wu and Law (2013) showed experimentally and theoretically that replacing the thermal diffusivity by the Markstein diffusivity in the Reynolds number definition, giving $Re_{T,M}$, collapses normalized turbulent flame-speed data fuel-independently as $Re_{T,M}^{1/2}$ - but only for positive Markstein number flames, i.e. $Le>1$ mixtures, because the Markstein diffusivity is ill-defined or negative otherwise. An alternative theoretical route, the leading-points concept of Venkateswaran et al. (2015), attributes turbulent propagation to the most advanced, negatively stretched flame elements and correlates turbulent consumption speed with the stretched laminar flame speed, again without achieving a pressure- and fuel-independent collapse.

Within this framework, Wu, Saha, Chaudhuri and Law (2015) measured propagating turbulent flames of n-butane through n-octane (C4 to C8 n-alkanes), at pressures from 1 to 5 atm, motivated by the known laminar 'fuel similarity' of these heavy n-alkanes (nearly identical $S_L$ and transport properties at fixed equivalence ratio). They found that the $Re_{T,f}^{1/2}$ scaling collapses their data only at fixed Markstein number; that lean ($Le>1$) flames exhibit a strong propensity to stretch-induced local extinction; and that the averaged turbulent flame speed decreases monotonically with fuel molecular weight for lean mixtures. Standard turbulent combustion regime diagrams (the Borghi-Peters diagram) classify flame-turbulence interaction by ratios such as $u'/S_L$ and the ratio of integral scale to flame thickness, with boundaries set by the Karlovitz number (the ratio of the chemical time to the smallest-eddy turnover time) and the Damkohler number (the ratio of turbulent to chemical time scales), but they contain no molecular-diffusivity (Lewis or Markstein number) coordinate and do not separate extinction-prone from extinction-free conditions. Reviews (Driscoll et al. 2020) confirm that thermo-diffusive effects remain significant even in highly turbulent flames, so the omission is material. The open questions left by the source paper's findings therefore target the missing thermo-diffusive physics of turbulent flame propagation: a scaling valid across both signs of $Le-1$, the extinction mechanism for $Le>1$, and a regime description that incorporates diffusivity.

Problem Statement

Establish how thermo-diffusive effects enter the turbulent propagation of premixed flames, generalizing the $Re_{T,f}^{1/2}$ scaling of expanding turbulent flames - which holds only at fixed Markstein number - to a relation $S_T/S_L = f(Re_{T,f}, Le)$ (equivalently parameterized by the Markstein number $Mk$) that collapses experimental data for both $Le<1$ and $Le>1$ mixtures across fuels, pressures, and turbulence intensities. Within this common objective, four linked questions arise from the findings of the source work (Wu et al. 2015):

1. Derive or establish a generalized turbulent flame-speed scaling relation incorporating thermo-diffusive effects (through the Markstein or Lewis number), reducing to the known $Re_{T,M}^{1/2}$ collapse for positive-Markstein-number flames and valid for negative-Markstein-number ($Le<1$) mixtures as well, with demonstrated collapse of the expanding-flame data spanning both classes onto a common curve within their reported experimental scatter.
2. Identify the physical mechanism by which $Le>1$ controls the onset and the spatial extent of stretch-induced local extinction in turbulent flames, and the role of subsequent re-ignition of extinguished segments.
3. Construct a turbulent combustion regime diagram that incorporates molecular diffusivity (Lewis or Markstein number), distinguishing extinction-prone from extinction-free conditions.
4. Confirm, refute, or refine the reported monotonic decrease of the averaged turbulent flame speed with fuel molecular weight for lean n-alkane/air mixtures.

The four questions are jointly posed within this common objective: item 1 is the central scaling question, and items 2-4 are the companion questions.

References

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