Towards the Distributed Burning Regime in
Turbulent Premixed Flames
Claremont Road, Newcastle-Upon-Tyne, NE1 7RU, UK
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory,
MS50A-3100, 1 Cyclotron Road, Berkeley, CA 94720, USA
25th June 2018
Three-dimensional direct numerical simulations of canonical statistically-steady statistically-planar turbulent flames have been used in an attempt to produce distributed burning in lean methane and hydrogen flames. Dilatation across the flame means that extremely large Karlovitz numbers are required; even at the extreme levels of turbulence studied (up to a Karlovitz number of 8767) distributed burning was only achieved in the hydrogen case. In this case, turbulence was found to broaden the reaction zone by around two orders of magnitude, and thermodiffusive effects (typically present for lean hydrogen flames) were not observed. In the preheat zone, the species compositions differ considerably from those of one-dimensional flames based a number of simplified transport models (mixture-averaged, unity Lewis number, and a turbulent eddy viscosity model). The behaviour is a characteristic of turbulence dominating non-unity Lewis number species transport, and the distinct limit is again attributed to dilatation and its effect on the turbulence, and suggests a temperature-dependent eddy viscosity model is required. Peak local reaction rates are found to be lower in the distributed case than in the lower Karlovitz cases but higher than in the laminar flame, which is attributed to effects that arise from the modified fuel-temperature distribution that results from turbulent mixing dominating low Lewis number thermodiffusive effects. Finally, approaches to achieve distributed burning at experimentally-realisable conditions are discussed.
The distributed burning regime of turbulent premixed flames represents the limiting case where flame propagation is driven by turbulent mixing rather than molecular diffusion (Aspden et al., 2011a), and corresponds to the small-scale turbulence limit (Peters, 2000; Damköhler, 1940). The feature that distinguishes burning in the distributed mode is that turbulent eddies comparable with the reaction zone thickness can mix faster than the species can react chemically. Experimentally, there is some evidence of what will be referred to here as “transitionally-distributed” burning, particularly in high-speed piloted lean-to-stoichiometric methane jet flames; see, for example, Dunn et al. (2010), Zhou et al. (2017), Wabel et al. (2018), and the references therein. Previous numerical studies of transitionally-distributed flames include Poludnenko and Oran (2010), Aspden et al. (2011a), Savard and Blanquart (2015), Nilsson et al. (2018), Wang et al. (2018), and the references therein. All of these studies have demonstrated flames with preheat zones that were significantly broadened by turbulence, but none have shown significant broadening of the actual reaction zone.
Distributed burning has been observed numerically in an astrophysical context (Aspden et al., 2008a), where a single-step reaction model was used to represent thermonuclear fusion in a type Ia supernova flame. Subsequently, Aspden et al. (2010) demonstrated scaling laws for distributed supernova flames following Damköhler (1940) and Peters (2000), and introduced the so-called “-flame” regime; the combination of large Karlovitz and Damköhler numbers gives rise to flames simultaneously in the small-scale and large-scale limits, resulting in such a large range of turbulent scales that the flame burns in the distributed mode, but the larger scales are unable to mix before the flame burns. These supernova studies used an idealised configuration capable of subjecting the flame to arbitrary levels of turbulence favourable for distributed burning (the Reynolds number in a supernova can be in excess of and the Mach number around ); realisable conditions for distributed burning in terrestrial chemical flames are yet to be established.
In the present paper, lean premixed methane and hydrogen flames have been simulated with extreme levels of turbulence (rms velocity fluctuations exceeding four hundred times the laminar flame speed). At the highest turbulence levels, the hydrogen flame has been found to present substantial broadening of the reaction zone, but even with such intense turbulence, the methane flame did not. Despite the abstracted configuration and unrealistic conditions, this hydrogen flame represents the transition to distributed burning expected of a terrestrial chemical flame if suitable conditions can be contrived. Phenomenological observations are first presented, followed by consideration of global consumption speeds, flame thickening, and conditional means of heat release and species mass fractions. The paper concludes with a discussion of the distributed burning regime and potential conditions required to realise distributed burning experimentally.
2 Simulation Details
2.1 Numerical solver
The numerical solver used here is based on the well-established low Mach number formulation of the reacting flow equations (Day and Bell, 2000; Nonaka et al., 2012). The fluid is treated as a mixture of perfect gases, and a mixture-averaged model is assumed for diffusive transport. A source term is used in the momentum equation to establish and maintain turbulence with the desired properties (Aspden et al., 2008b). The chemical kinetics and transport are modelled using the hydrogen mechanism of Li et al. (2004) consisting of 9 species with 21 fundamental reactions, and the GRIMech 3.0 methane mechanism (Frenklach et al., 1995) with the nitrogen reactions removed, resulting in 35 species and 217 reactions. These evolution equations are supplemented by CHEMKIN-compatible databases for thermodynamic quantities, and transport properties computed using EGLIB (Ern and Giovangigli, 1996).
2.2 Simulation configuration
Following our previous studies (e.g. Aspden et al., 2011a, 2015, 2016), a canonical periodic-box configuration was used, where a lean premixed flame was allowed to propagate through maintained zero-mean homogeneous isotropic turbulence. All simulations were run at atmospheric conditions in a high aspect ratio domain, with periodic lateral boundary conditions, a free-slip base and outflow at the top. Lean premixed hydrogen (equivalence ratio , Lewis number ) and methane (, ) were considered. The freely-propagating hydrogen flame speed and thickness are and (Aspden et al., 2011b); the laminar methane flame speed and thickness are and . It was shown in Aspden et al. (2008b) that the forcing approach gives approximately 10 integral length scales across the domain width. In all cases the length scale ratio is , consistent with our previous studies (e.g. Aspden et al. (2015) and Aspden (2017)). Three further Karlovitz () numbers have been considered , 974, and 8767, which correspond to velocity ratios , 98.3, and 425, respectively, where is the rms velocity fluctuation. A conventional regime diagram is shown in figure 1.
It should be stressed that these simulations are numerical experiments, and this set of values is not realisable experimentally; in particular, the low Mach number approximation is exploited here to preclude strong compressibility effects and prevent any potential detonation (e.g. Poludnenko and Oran, 2010). However, these simulations represent the transition towards the limiting case of flame propagation driven by turbulent mixing, i.e. distributed burning, and are of significant interest and relevance to the transition away from the thin reaction zone with increasing levels of turbulence. Possible steps to realise distributed burning experimentally are discuss in section 4.
Most simulations were conducted with a domain size of discretised on a grid of 1921921536 computational cells; the hydrogen case with the highest Ka required a larger domain (; 1921922304 cells) to accomodate the growth of the flame brush. This resolution corresponds to 19.2 cells across a thermal thickness, which is more than sufficient to resolve these chemical mechanisms (Aspden et al., 2011a, 2016). At such high turbulence levels, the Kolmogorov length scale is between 44 and 544 times smaller than the thermal thickness of the flame; it is not fully-resolved on this grid. The effective Kolmogorov length scale (see Aspden et al. (2008b) for details) can be evaluated for this solver, and was found to range between 38 and 67 times smaller than the thermal thickness, which indicates that the turbulence that interacts with the flame (i.e. at the flame scale) is sufficiently well-resolved (and is maintained by the momentum source term through the inertial cascade). Moreover, as previously argued in Aspden (2017), turbulence is strongly affected by dilatation, therefore, close to the flame, the Kolmogorov length is not the value that would be expected from the classical cascade; scales not represented on the grid are inconsequential, and resolving them would be a waste of computational effort.
3.1 Flame response overview
A general overview of flame response to high Ka turbulence is presented in figure 2, which depicts slices of fuel mass fraction, temperature, fuel consumption rate (FCR), and heat release rate (HR), normalised by the corresponding laminar values (note for for hydrogen, the FCR and HR are normalised by ten times the laminar value to account for the enhanced reaction rates due to the thermodiffusive instability).
Methane at appears to be similar to moderate Ka (see the case in Aspden et al., 2016); the flame surface is convoluted but smooth, generally similar to the laminar flame, with a decrease in reaction rates correlated with high positive curvature, which was attributed to atomic hydrogen diffusion (Aspden et al., 2016). The temperature field shows clear evidence of turbulent mixing, but restricted to the preheat region.
Methane at continues the trend of increased turbulent mixing in the preheat region, but the reaction rates appear to be changing; while not immediately apparent in the image, the peak value exceeds the laminar flame values in places, and there appears to be greater variability along the flame surface, which is becoming more convoluted – again, this can be attributed to atomic hydrogen diffusion (Aspden et al., 2016).
Methane at shows different behaviour than the lower Ka cases; in addition to the mixing observed in the preheat region, there are indications in the temperature field that there is an onset of turbulent mixing in the post-flame region. Interestingly, the reaction rates appear slightly increased in places (see magenta/white regions; an enlarged image is shown in figure 3), but do not show any broadening.
Hydrogen at presents significant thermodiffusively-unstable behaviour (similar to that observed in Aspden et al., 2011a, 2015); turbulence exaggerates the thermodiffusive instability, creating small-scale structures with higher curvature than at lower Ka, resulting in more intense burning over a broad flame brush (localised heat release rates in excess of fifteen times the laminar value are observed; shown by the white regions). The decorrelation between fuel consumption and heat release rates (reported by Aspden et al. (2015) and attributed to atomic hydrogen diffusion Aspden (2017)) is present, further indicating persistence of preferential diffusion at this Ka. Super-adiabatic temperatures still exist in the near post flame region. The temperature field presents limited turbulent mixing; the thermodiffusive instability leads to a resistance to turbulent mixing.
At , the hydrogen flame presents the first evidence of a change in behaviour; the thermodiffusively unstable structures at lower Ka are no longer observed, reaction rates have generally decreased, and the temperature field shows significant evidence of turbulent mixing in the preheat region.
At the highest Ka, the hydrogen flame presents significantly different behaviour to all of the other cases; there is now substantial turbulent mixing throughout the flame, there is no evidence of the thermodiffusive instability, the reaction rates are substantially lower than the other two hydrogen cases, and distributed relatively smoothly across a region that is over ten thermal thicknesses across. This behaviour is consistent with the distributed supernova flame presented in Aspden et al. (2008a); figure 3 reinforces this similarity comparing the reaction rates from the distributed supernova flame with hydrogen fuel consumption rate and heat release (normalised by four times the laminar values rather than ten). The peak reaction rates in the hydrogen case are lower than those of the moderate turbulent cases, but remain higher than in the one-dimensional laminar flame.
3.2 Turbulent flame speeds
Turbulent flame speeds are shown as a function of Ka in figure 4 (vertical lines denote the full range of values attained after reaching a statistically-steady state); time histories are presented as supplementary material (note that at such extreme levels of turbulence, tens of integral length eddy turnover times are required for the flame to reach a statistically-steady state, and over 100 for the highest cases). Power laws derived in analogy with Damköhler (1940) for the distributed limit of turbulence-driven mixing are shown as dashed lines (see also Peters (2000) or Aspden et al. (2010)). Predicting a turbulent flame speed , where , is a turbulent diffusion for some constant and (turbulent) chemical time scale (both to be determined), three equivalent expressions can be derived
where the constant of proportionality in all three cases is ; note the latter can be written as for . It is as yet unclear why both hydrogen and methane flames appear to follow this scaling law (in a regime where it is not intended to apply) followed by an apparent transition, after which the flame speeds may again follow the scaling law, especially since distributed burning behaviour is only observed for hydrogen at .
3.3 Thickening factor
A local thickening factor was previously defined (Aspden et al., 2016), analogously to thermal thickness, as the ratio of the conditional means of temperature gradients as
where the normalisation by the conditional mean at is used (in preference to the laminar profile) to account for the thermodiffusive instability in the hydrogen flames. Previously (Aspden et al., 2016), the methane flames were found to be broadened in the preheat region, unlike the hydrogen flames, which were found to become progressively thinner with increasing Ka due to generation of more highly-curved flame surface by turbulence thereby enhancing the thermodiffusive effects. The thickening factor in the present flames are compared in figure 5, where the methane flames are broadened further in the preheat region with increasing Ka, but seem to get slightly thinner at in the post-flame region (the vertical dashed line indicates the location of peak HR); the thinning trend in hydrogen is reversed in the preheat region at higher Ka, but remain thinner in the post-flame region. This thinning in the post-flame region may be due to the normalisation, where the temperature profile flame (as for the laminar flame) has a long tail at high temperatures (i.e. small gradients), resulting in a smaller due a modified post-flame structure at high Ka.
3.4 Reaction rates
The response of heat release rate to turbulence is considered using conditional means in figure 6, including profiles for the laminar flame (red), unity Le laminar flame (blue) and (following Aspden et al. (2016)) laminar flames supplemented by a constant turbulent diffusion term (magenta; dashed lines correspond to different magnitudes, with the limiting case as a solid line). At low-to-moderate Ka, the methane profile is close to the laminar profile, but at the highest Ka the profile appears to have shifted toward higher temperatures, with a slight increase in peak magnitude (consistent with figure 2). A more pronounced response is observed in the hydrogen flames; at low-to-moderate Ka, there is substantial heat release rate in the preheat region (due to the thermodiffusive instability exaggerated by moderate turbulence), whereas at higher Ka the distribution is narrower, with a peak that again has shifted to higher temperatures, consistent with the peak temperature in the one-dimensional profiles.
3.5 Species mass fraction distributions
For the methane flames, the turbulent response of conditional means of species mass fractions remain consistent with the classification set out in Aspden et al. (2016); figure 7 presents conditional means of mass fractions for H and HO at (conditional means for all species and Ka are provided as supplementary material). These species in particular present a strong response in the preheat region, specifically, turbulence dominates species diffusion at high Ka. The standard deviation decreases with Ka, and the distribution not only transitions from that comparable with the laminar profile (red) towards the unity Lewis number profile (blue), but also appears to reach a state that is distinct from all three of the one-dimensional profiles (red, blue, and magenta).
A similar response is found in the hydrogen flames; figure 8 presents conditional means of species mass fraction for H and HO at . The response of H is characteristic of O and OH (again, all species are presented as supplementary material), where the enhanced radical pool at low-to-moderate temperatures observed at lower Ka (see Aspden et al., 2015, and supplementary material) is suppressed at . A strong response of HO is observed in the preheat region, and again appears to tend to a limit distinct from the three one-dimensional profiles. The standard deviation is also significantly reduced.
4 Discussion and Conclusions
Direct numerical simulations at extreme levels of turbulence have shown a transition to distributed burning in lean premixed hydrogen flames. The phenomenology of the transition is similar to that reported in an astrophysical context (Aspden et al., 2008a); there are no indications that suggest other fuels should not undergo a similar transition at sufficiently intense turbulence levels. Unrealistically-high turbulence conditions were required to observe the transition, which is argued to be primarily due to dilatation and the higher density ratio between unburned and burned conditions (the density ratio is less than two for the supernova case, about four for hydrogen, and over six for methane).
The key behaviour characteristic of distributed burning not previously observed in chemical flames is the signficant broadening of the reaction zone by turbulence; here, the heat release region was found to be approximately fifteen thermal thicknesses across, an increase of two orders of magnitude. Thermodiffusive effects were found to be suppressed, if not eliminated, with the peak reaction rates falling from about fifteen times the peak laminar value to about four times. Distributed reaction rates that are higher than the corresponding laminar flame is different than the response found in the supernova flame, where the peak reaction rates were found to be much smaller. This can be attributed to global Lewis number by considering the fuel-temperature distribution, as discussed in Aspden et al. (2011c). Hydrogen has a low Lewis number, and so the fuel-temperature distribution of the laminar flame lies below the linear mixing distribution. Strong turbulent mixing gives an effective unity Lewis number for all species (species and enthalpy are advected together in packets) so the distributed flame profile is close to linear – fuel concentrations are higher at the same temperature, so reaction rates and heat release increase. Conversely, for the supernova, the Lewis number is large, so the laminar flame profile lies above the linear mixing distribution, turbulent mixing results in fuel concentrations that are lower at the same temperature, giving lower reaction rates.
Turbulent flame speeds were found to follow scaling laws at low-to-moderate Karlovtiz numbers (despite not satisfying the assumptions made), followed by an apparent transition. Even at the exteme turbulence levels considered, there was insufficient evidence to draw solid conclusions about behaviour in the distributed burning regime, and will require significant further work to demonstrate the behaviour observed in the supernova flames (Aspden et al., 2010).
The species distribution tends towards a limit that is distinct from the one-dimensional profiles (laminar, unity Le, and turbulent diffusive limit), with a standard deviation of almost zero; Aspden et al. (2016) suggested that a temperature-dependent diffusion coefficient is required to account for dilatation through the flame. A change in distribution was observed for species that experience low-temperature activity (e.g. HO), which is argued to result from molecular diffusion of mobile species such as H and/or H being overcome by turbulent mixing.
The conditions required here for distributed burning are unphysical and unrealisable in practice, but there are steps that could be taken that may lead to distributed burning. Reducing the density ratio (e.g. by preheating) will reduce the impact of dilatation on turbulence so that it can survive further into the flame. Given that the strength of the turbulence that interacts at the flame scale depends solely on Ka, fuel conditions that minimise (e.g. by running lean) will also reduce the critical Ka required for transition. Despite observing distributed burning in the hydrogen flame, the response of reaction rates suggests that high-Le flames (at the same density ratio) might transition at a lower critical Ka.
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