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Thermofluid Mechanics (Heat and Mass Transfer): Autoignition in Mixing Flows

In the aftermath of the advent of the Homogeneous Charge Compression Ignition (HCCI) engine and Lean Premixed Pre-vaporized (LPP) gas turbine concepts and the increasingly strict emission regulations for diesel engines, the immediate practical needs concerning autoignition in the presence of strong turbulence and mixture inhomogeneities served as a renewed driving force for research in this field, with inhomogeneities making their first, reluctant appearance. Initially, the 'conventional' approach towards inhomogeneous autoignition resulted directly from the pre-existing work concerning autoignition delay times in the various homogeneous configurations, on account of the considered increased importance of the chemistry. An argument was put forward that attempted to explain inhomogeneous autoignition once again in terms of a residence time and envisaged the possibility of separating the total autoignition delay time into a fluid-mechanical 'mixing time' (set by turbulent mixing) and a 'chemical delay time' (set by the chemistry as before), the latter of which was believed to be the most important, or 'controlling', parameter. According to this idea, the initially inert inhomogeneous mixture will need some 'mixing time' before becoming a reactive homogeneous mixture, after which it will need a further (homogeneous) 'chemical delay time' to autoignite. As a consequence of this simple decoupling much of the early literature is filled with studies that report only on an 'autoignition delay time' in:

1. A wide variety of unsteady/transient flow configurations such as shock tubes, jet stirred reactors, constant volume vessels or other diesel-like combustion chamber environments, rapid compression machines, etc., in which:

(a) Mixture inhomogeneities were not present.
(b) Mixture inhomogeneities and turbulence were present, but could not be properly characterized due to the chosen nature of the flow and/or were until recently considered unimportant.

2. Continuous flow reactors/configurations that (potentially) allowed measurements of turbulence quantities and characterization of the inhomogeneities, but in which:

(a) The mixture inhomogeneities were purposefully minimized (for example by promoting intense turbulent mixing between the fuel and oxidizer early on and in the minimum time possible by specially designed injection schemes) in order to study other effects.
(b) The mixture inhomogeneities were present, but whose effect was overlooked/not measured, or were present, were measured but whose effect was investigated only qualitatively.
(c) The presence of inhomogeneities was linked with the two-phase physics of liquid fuel spray evaporation in complex mixing fields, such as the steady and unsteady/transient cross-streamwise (perpendicular) injection of a fuel spray into a heated air stream.

Even in the work mentioned in the final point, where the turbulent flow field was relevant and inspected (such as measuring the turbulence intensity) together with the measurements of autoignition, direct measurements of the mixing fields were not made and a consensus regarding the effects of turbulent mixing cannot be reached since various investigators diametrically disagreed on the true effects of turbulence on their results. Some have found that the influence of turbulence was insignificant, whereas others have concluded that autoignition delay times were increased by strong turbulence, at least for a certain temperature range.

Currently, the autoignition of a turbulent mixing flow is a problem of great fundamental importance and practical interest. The further development of combustors for HCCI engines and LPP gas turbines, in terms of improved performance and efficiency and reduced emissions, can be significantly aided by a better understanding and ability to predict the phenomenon of autoignition in the presence of considerable fluctuations of velocity, composition and temperature, whose effect must be understood. In parallel, turbulent autoigniting flows in which the chemical and fluid-mechanical timescales are of the same order (such as in the current experiments), are a theoretically challenging problem due to the non-linear coupling between complex chemistry and turbulent mixing that is interesting to explore. The phenomenon (and hence any successful theory) of inhomogeneous, turbulent autoignition involves the dynamic interplay between several underlying mechanisms, such as:

1. The turbulent, fluid-mechanical mixing (mass transfer) processes that bring together the oxidizer and fuel, but also any important intermediate species together in the radical pool,

2. The chemistry of the slow pre-ignition reactions between the reactants and intermediate species, and,

3. The thermodynamical fate of the heat (heat transfer) released from these reactions, leading to thermal runaway in localized sites called 'autoignition kernels'.

With each one carrying equal importance, the convenient assumptions of scale-separation cannot be applied and so it can be said, that turbulent inhomogeneous autoignition truly stands on the boundaries where the disciplines of chemistry, fluid mechanics and thermodynamics meet. These phenomena can be complicated and immensely diverse, highlighting the many possibilities for non-linear interaction between the equally important aspects. Measurements of the phenomena that are the subject of this study can serve as excellent test-beds for testing both Computational Fluid Dynamics (CFD) and chemistry models, since they are by definition the outcome of de rigeur turbulence-chemistry interactions.

The major advances in this field have come from all aforementioned disciplines. They have been borne out of experimental, theoretical and computational approaches and by considering a wide variety of configurations: homogeneous and inhomogeneous, gaseous fuels and evaporating sprays, simple and complex chemistries, constant volume/pressure vessels and continuous flows; laminar and turbulent, two- and three-dimensional. In each of these configurations it has been possible to isolate and treat different aspects of the phenomenon, from the chemistry to the mixing patterns of passive scalars in turbulent flows, but also to progressively examine more complex cases of autoignition in a turbulent reacting flow. For a brief presentation of the latest theories and observations of autoignition, it is necessary to go back to the simple case of the constant volume or pressure vessel (see also the Autoignition Tutorial) and this is done below.

Homogeneous Autoignition:

The simplest relevant analysis is the treatment of the thermal explosion of a homogenous mixture in a closed vessel (bomb). In this, a uniform combustible mixture undergoes either an isochoric or isobaric process of autoignition, assuming no heat losses and one-step Arrhenius chemistry described by an exponential reaction rate term proportional to exp{-T_act/T}. The numerator is a constant termed the 'activation temperature' and is related to the 'activation energy' of the reaction via T_act = E_act/R^o, with R^o the Universal Gas Constant. Given this theoretical, one-step exothermic chemistry model, with finite reaction rate, autoignition always occurs after an 'autoignition delay time' (τ_IGN), during which the rate of change of all scalars is exponential. This is important since it reveals that initially the evolution of the chemistry is very slow and the temperature (and reactant concentrations) can be idealized as being approximately constant, whereas at later times autoignition has a strongly self-accelerating character, termed 'thermal runaway'. Denoting by the subscripts 'fu' the fuel and 'ox' the oxidizer and by the superscript '0' initial conditions, the main results for the isobaric autoignition process are that τ_IGN:

1. Decreases very quickly with increasing initial temperature and is inversely proportional to the initial density, and thus, the initial pressure.

2. Is minimized when Y_fu⁰ = Y_ox⁰ = 0.5 and increases for leaner or richer initial mixture compositions. Note that Arrhenius (unlike real) chemistry does not set lean/rich composition limits outside which reaction is not possible.

For complete combustion the temperature in the vessel always reaches the adiabatical flame temperature after autoignition. The next step in complexity is to allow for a relaxation of the adiabatic constraint. The main result concerns the importance of the balance between:

1. The heat release from the proceeding exothermic chemical reaction, and,

2. The heat losses from the system through the walls of the container.

and is a fundamental consideration for autoignition in general. Depending on the magnitude of the heat losses relative to the non-linear generation of heat due to the chemical reaction, the following scenarios are possible:

1. For low heat losses, the generation of heat is larger than heat loss from the vessel. Autoignition will always occur as in the case of no heat losses, yet the autoignition delay time increases relative to the adiabatic case and the eventual temperature is now lower than the adiabatic flame temperature.

2. For excessive heat losses, a stable, slow reacting, or 'cool-flame', solution is found. The heat generation and loss terms balance and thermal runaway does not occur. The vessel keeps reacting, at very low rate, yet the heat of the reaction is lost as soon as it is generated. The temperature rise is small and the system no longer experiences an exponential explosion as before.

The analytical analyses mentioned so far have been concerned with stagnant, homogeneous mixtures and the simple, one-step chemistry of thermal explosions and as such, cannot give insight to the effects of turbulence, inhomogeneities or real chemistry on autoignition. In order to understand the individual processes that constitute a complex phenomenon such as this and their possible interactions, it is common practice to begin by observing and studying isolated aspects of the overall phenomena where possible. The insight gained in each of the more elementary treatments is always instructive and could lead in some ways to the understanding of the more sophisticated ones. Nevertheless, it is generally dangerous to extrapolate results and conclusions outside the bounds of the physical processes for which they were reached and it does not always follow that knowledge from the simplified cases can be applied blindly to the more complex ones.

Qualitatively different phenomena of homogeneous autoignition, that rely on deviations from the high activation energy (or temperature) single-step Arrhenius description of thermal explosions, have traditionally dominated autoignition studies. In these studies emphasis rests on the study of the slow chemical paths and shifting explosion limits of the so-called 'multiple ignitions' and 'cool flames', similar to what was done above for hydrogen and methane. These phenomena are observed at relatively lower temperatures and are associated with weaker luminescence, lower heat release and negative temperature coefficients. In the experimental work that is the subject of this thesis, real chemistry effects are of course present as a matter of fact. Attention is placed on the investigation of the effects of turbulent mixing on this chemistry.

The phenomenon of turbulent inhomogeneous autoignition is dominated by turbulent scalar mixing, set up by the non-uniformities and turbulent flow. In the governing equations for the turbulent fluctuations of species and temperature, the only source terms come from the chemistry, with turbulent transport terms playing a 'balancing' role. Seen otherwise, scalar fluctuations can be 'transported', but not created by turbulence. For the autoignition of a homogeneous mixture in the absence of temperature fluctuations, the fluctuations of the velocity field cannot give rise to scalar fluctuations. Hence, this case is of little fundamental importance. In the turbulent autoignition of homogeneous mixtures, even the inclusion of temperature fluctuations is of reduced fundamental importance. In the absence of turbulent mixing, which is by definition non-existent for a homogeneous mixture, an important part of the turbulent transport processes is not treated. A true understanding of turbulent autoignition in the presence of mixture inhomogeneities should come from dealing with the full, turbulent inhomogeneous situation. For this reason, the experiments presented in this thesis were designed to physically model the case in which fluctuations exist in the velocity and all scalars, with (length and time) scales chosen to be similar to those of the chemistry and thus of autoignition itself. Knowledge concerning the effects of scalar mixing on autoignition has come from the theoretical treatment of autoignition in laminar inhomogeneous flows and more recently from experiments, DNS and modelling of autoignition in turbulent inhomogeneous flows. The following two sections are a brief introduction to inhomogeneous autoignition in laminar and turbulent flows respectively.

Laminar Inhomogeneous Autoignition:

Laminar inhomogeneous autoignition includes the classical problems of the laminar (most commonly planar) co-flowing mixing layer and strained counterflow. Due to its simplicity, the former has been approached mostly from an analytical and theoretical point of view. On the other hand, the autoignition of non-uniform counterflow mixing layers has been explored analytically, experimentally and with simulations. In both cases the character of the solution depends on the pre-ignition chemistry, which can, in general, give rise to a thermal or chain-branched explosion. Many investigators have tended to treat these two as completely different processes, based on the fundamental difference of the dynamic build-up before the explosion. Specifically, thermal explosions exhibit thermal runaway at a finite time, whereas for chain-branched explosions radical concentrations evolve exponentially and mathematically approach infinity only at infinite times.

Laminar Co-flowing Mixing Layers:

The first autoignition problem to be investigated was that of two, initially separated, streams of fuel and oxidizer downstream of a splitter-plate in a planar (two-dimensional) flow geometry, with a one-step Arrhenius chemistry model and high energy asymptotics. This was extended to take into account the wake formed by the shedding of the boundary layers on the walls of the splitter-plate into the merging region of the main flow. In both cases the results revealed that initially the reactants mix in a nearly 'frozen' flow and reaction is slow, with low heat release and temperature rise. This is always overtaken by a sudden thermal-runaway behaviour that clearly identifies the location of autoignition.

The laminar, inhomogeneous mixing layer is mathematically a parabolic, or initial-value, problem and can be understood in the following way. The initial conditions of the problem, together with the mixture fraction, or normalized conserved scalar (ξ) at a point in the flow, completely determine the state of mixing. For a constant temperature domain, as the mixing layer evolves from an initial state in which the fuel and oxidizer were separated, the probability density function (pdf) of mixture fraction spreads from initial delta functions to occupy a range of values in mixture fraction space. Seen differently, as the fuel and oxidizer mix, fluid particles with a range of different mixture compositions appear. Each fluid particle can be roughly thought of as a homogeneous reactor, reacting based on its composition, in a similar way to that described for a homogeneous mixture in a closed vessel in the Autoignition Tutorial. Then, in terms of the mixture fraction field, the conditional reaction rates are a function of mixture fraction. Fluid particles with the 'right' composition will have the maximum reaction rate allowable by the temperature and will autoignite first, since they will be associated with the shortest τ_IGN. These are the 'most reactive' conditions and are associated with the most reactive mixture fraction, ξ_MR. For one-step chemistry and a constant temperature throughout the domain ξ_MR = 0.5, while ξ_MR will shift to leaner values as T_ox increases relative to T_fu.

A very important graph for inhomogeneous autoignition that was arrived at by the treatment of the laminar mixing layer is the curve of temperature, or heat release, versus Damköhler number. The non-dimensional Damköhler number, Da = τ_res/τ_chem, is the ratio of a representative physical timescale, such as a residence time (τ_res), to a representative chemical timescale (τ_chem). The figure below with the sharp rise indicating thermal runaway at τ_IGN (and corresponding Da_IGN) is characteristic of autoignition. The lower left portion of the curve corresponds to the slowly reacting state prior to autoignition, when τ_res are much shorter than the large timescales of slow chemistry. As the chemistry becomes faster or τ_res increases, so does Da until the autoignition point is reached. For Da values greater than this, explosive ignition leads to fully-fledged combustion with its associated high temperatures, e.g. the adiabatic flame temperature for complete combustion.

Indirectly, this figure also demonstrates that as long as there is no restriction to τ_res, no excessive heat losses from the system and the chemistry allows the reaction to proceed, there is no theoretical reason for autoignition not to occur, because, as with the homogeneous bomb, the reaction rate is always finite and any heat released from the reaction will inevitably lead to an increase in temperature and an acceleration in the reaction rate.

In addition, the same figure has important implications for the turbulent case. The chemical reactions that take place at high temperatures, e.g. in flames, are nearly always fast compared to all turbulent timescales, whereas at the low temperatures of low Da, the chemistry is usually slow relative to the turbulence. Hence, in both cases the length and timescales of the chemistry and turbulence may be separated. However, just prior to autoignition, the chemical timescales are of the same order as those of the turbulence and any assumptions of scale-separation are no longer valid.

In recent years chain-branching autoignition in the co-flow mixing layer has also been approached analytically. As previously mentioned, hydrogen chemistry has been very much at the centre of attention, due to its fundamental role in hydrocarbon combustion, relative simplicity and interesting behaviour near the crossover temperature (T_c), at which the rate of the main three-body, exothermic chain-terminating reaction H + O₂ + M ---> HO₂ + M is balanced by the rate-controlling chain-branching reaction H + O₂ ---> OH + O. At temperatures higher than T_c the recombination reactions responsible for the heat release are overtaken by the branching reactions due to the lower T_act and pre-exponential (frequency) factor of the former. Hence, initially, the mixing layer achieves a thermally-frozen state. A short chain-initiating region is followed by a long autocatalytic region of weakly exothermic chain-branching in which the radical-pool builds up exponentially before the point of autoignition. It has also been determined that initial radical build-up cannot be guaranteed unless chain-initiating steps such as H₂ + O₂ ---> 2OH are taken into account, even though these reactions are slow. The effect of the wake from the boundary layers of the separating plate has been treated, with similar conclusions.

Laminar Counterflows:

Steady and un-steady counterflows of cold fuel and hot oxidizer have been extensively studied both analytically and numerically. Analytic, asymptotic approaches include ones for large activation energy thermal autoignition and others for chain-branching autoignition. Generally, the literature, as with the co-flow mixing layer, focuses on hydrogen chemistry. Yet, unlike for the co-flows, numerous experimental investigations have been performed, for a variety of hydrocarbon fuels and hydrogen and over a wide range of conditions. Another situation that has been considered is that of a premixed fuel-oxidizer stream being ignited by an inert, opposed hot nitrogen flow.

The counterflow, or strained mixing layer, is a boundary-value problem and as such can be described by an elliptic set of partial differential equations. Conveniently, for the velocity field diffusion can be neglected. A potential (inviscid) flow stream-function transformation can be used, based on an aerodynamic strain rate (γ), such that ψ = γxy; effectively u_x = γx and u_y = -γy. For the species in steady counterflow mixing layers diffusion is balanced by species advection that brings the reactants together in the mixing zone. Mathematically for species σ:

u_x dY_σ/dx + u_y dY_σ/dy - D_σ(d²Y_σ/dx² + d²Y_σ/dy²) = ω_σ/ρ

with D_σ its molecular diffusivity (in m²/s) and ω_σ it reaction rate per unit volume (in kg/m³s). It becomes straightforward to define two characteristic timescales; one for the fluid-mechanics (left-hand-side) and sometimes called the 'diffusive time', and, one for the chemistry (right-hand-side). A Damköhler number can be defined as before, based on the ratio of these scales and it can be seen, directly from the definition of γ, that Da is proportional to γ^-1. Going a step further, an equivalent Da can also be defined from the inverse of the scalar dissipation rate (χ) since in this flow, γ is proportional to χ (but depends also on the position across the flamelet) according to χ = γ/π exp{z^-2}, with z a non-dimensional length. Seen otherwise, γ acts as a 'scaling parameter' for the conditional χ, χ|ξ = γ/π exp{-2[erfc^-1(2ξ)]²}, where erfc^-1 is the inverse of erfc.

Solutions can be obtained of the form of an S-shaped curve, as illustrated in the figure below. Careful consideration of the ignition regime at lower Da (higher γ and χ) leads to the conclusion that critical straining conditions exist, beyond which a low temperature, frozen solution is not possible and a diffusion flame temperature exists instead. This means that autoignition (i.e. transition to flame in this context) can occur only if Da is above a certain value, whereby the mixing rate is below a critical value. The autoignition delay time, τ_IGN, was found to increase with increasing mixing rate. Theory, numerical simulations and experiments have indicated, that indeed, autoignition is not possible for χ > χ_crit. This result has important implications for the effects of turbulence. If extrapolation were possible, it would have meant that autoignition would be expected to occur later if the mixing rate was increased and possibly not at all. This would seem contrary to empirical knowledge of autoignition, for example in engines, where it is empirically known that 'faster mixing' in fact results in earlier autoignition. Actually, the discrepancy is not as clear as it seems and partly arises due to the fact that empirical knowledge of autoignition based on 'mixing' and 'chemical' times can be misleading.

More detailed treatment of the hydrogen autoignition chemistry in the laminar counterflow configuration was possible by computational and experimental/computational investigations. Various hydrocarbons such as methane, ethylene, ethane, propene, propane, butane and n-heptane have also been investigated, with numeric studies and experiments/simulations. These studies have provided an excellent understanding of autoignition chemistry in the presence of strain rate and have revealed the existence of complex thermokinetic/diffusive effects based on which reduced chemical kinetic mechanisms have been developed.

In laminar inhomogeneous autoignition the existence of critical conditions, in terms of a strain-rate, scalar dissipation rate or Damköhler number, has been verified. The observed influence of the non-uniform flow on the emergence of autoignition highlights the fact that, on a fundamental level, autoignition does not follow directly from expectations based on the homogeneous case.

Turbulent Inhomogeneous Flows:

An appreciation of the fundamental issues of turbulent autoignition must come from truly dealing with the turbulent case. Theoretical approaches are extremely difficult because of the complex chemical kinetics of the slower chemistry of autoignition, the turbulent closure problem and the coupling of the chemistry and turbulence. Experimentally there has been a lack of interest in treating this problem, perhaps inhibited by the difficulty of performing well-characterized measurements in the 'hostile', and immensely sensitive to conditions, turbulent autoignition environment. It is well known that the underlying physical and chemical processes of this phenomenon and their mutual cross-interference is sensitive to initial conditions. As a result, there is currently little data available concerning the effect of the local turbulence character on autoignition. It is only recently, with the increase in residence times, τ_res and consequently Da associated with the low emission LPP industrial turbines and the increased interest in controlling the autoignition timing in HCCI engines, that this issue has become relevant. Improved understanding of turbulent autoignition has come from Direct Numerical Simulations (DNS), even though there are the issues of the low Reynolds Numbers that have been examined and/or the simplified description of the chemistry. Moreover, and crucially, the validity of the DNS results has not been experimentally verified.

Stagnant Mixing Layers:

Knowledge in this field has been recently expanded with the help of DNS, which have amply emphasized the importance of the mixture fraction field in determining the location of autoignition. Current, 'state of the art', understanding of the autoignition of an inhomogeneous mixture, in a turbulent flow, stemmed from the observation of autoignition in isotropic, homogeneous, decaying two-dimensional turbulence in a shear-less mixing layer with a simple chemistry model. Re-examination of these results in terms of the original observations concerning the laminar mixing layer lead to the conclusions that:

1. Autoignition always occurred at a well-defined mixture fraction ξ_MR, termed the most reactive mixture fraction. The value of ξ_MR could be (approximately) determined a priory from knowledge of the initial conditions and the chemistry. At any instance, there were many such possible locations in the layer.

2. The fluid particles that eventually autoignited were the ones associated with small gradients of the mixture fraction ξ, i.e. experienced lower values of conditional scalar dissipation rate, χ|ξ = ξ_MR. Plots of the reaction rate (or even the 'reactedness' B_auto; i.e. the non-dimensional temperature increment from the initial condition) versus χ|ξ = ξ_MR were demonstrated to be very well correlated as far as the first autoignition location was concerned.

Since then, evidence to support these findings has come from simulations with different codes and with detailed chemical mechanisms for hydrogen and a more reliable four-step reduced mechanism for n-heptane. An extension has also been made to three-dimensional simulations with complex hydrocarbon chemistry, agreeing fully with the earlier results.

It is significant to mention that, according to the latest understanding, a residence time until autoignition cannot fully define the problem in the absence of information on the mixture fraction field. It is surprising that even today, the number of autoignition experiments that are being performed in turbulent flows for which the mixture fraction field has been directly characterized, or at least for which an attempt has been made to estimate the mixing quantities based on accurate turbulence measurements, are very scarce indeed. In our work here the importance of information on the mixture fraction is further justified, viewed in the light of experimental results concerning the various phenomena that have been observed in the apparatus.

A point is being reached, whereby the location of autoignition can be reasonably predicted, but these predictions can only be confirmed by experiment. On the other hand, the effect of the turbulence character, length and timescales on the magnitude and randomness of τ_IGN is far from understood. For the purposes of the current study, the exploration of the sensitivity of autoignition to the aforementioned parameters, is best attempted in conditions in which the chemical timescales are of the same order, or close to the order, of the fluid-mechanical/turbulent timescales; whence the direct effect of turbulent mixing is most significant.

Turbulent Counterflows:

Recently, experiments of autoignition in the counterflow configuration have been extended to turbulent counterflows of hydrogen. These experiments and successful Probability Density Function (PDF) modelling have produced the very interesting result that, as with the laminar counterflows, depending on the temperature of the air stream autoignition may not occur at all (at least during the residence time available, as determined by the mean bulk strain rate).

The existence of critical conditions in the case of the counterflow is an interesting outcome, in that the DNS have not shown that such a limit exists for autoignition in turbulent mixing layers. This is not altogether surprising, since the decaying turbulence and hence χ meant that high values of this variable could not be sustained for long, eventually leading to right (in the sense of low χ|ξ = ξ_MR) conditions for autoignition. Going back to the laminar mixing layers, both co-flow and counterflow, τ_res is completely determined by χ and vice versa. For example, for equal velocity everywhere, the cross-streamwise profile of χ in the co-flow is identical for equal τ_res and independent of the bulk flow velocity. Yet for the co-flow, unlike the counterflow:

1. τ_res (and hence Da) can always be made arbitrarily long, and,

2. χ always decays downstream, rather than remaining constant as the reactants flow radially outwards,

such that the solution will always jump to the burning temperature (as in the S-Curves shown above in the figures).

The novelty of the turbulent case is that χ does not completely determine τ_res. For the counterflow it is known that the criticality holds, but then the allowable τ_res is not arbitrary and it is relevant to approach the problem in terms of a consistently high bulk χ that can be achieved in practice by having high bulk strain rates. On the other hand, for the co-flow τ_res can still be arbitrarily long. This means that it is possible to have a 'hypothetical' situation in which χ is sustained above a critical value (for all τ_res) such that autoignition fails (for any τ_res). Physically this would mean that the flow always has the ability to diffuse heat and radicals away from the reaction zone, promoting heat losses and depleting the radical pool and thus precluding autoignition.

This possibility of critical conditions is interesting and has been theoretically demonstrated, albeit for artificially sustained (constant) high levels of dissipation with zero-dimensional (0-D) Conditional Moment Closure (CMC). Nevertheless, it must be said that in parabolic flows, χ will always decay with increasing τ_res in the absence of sources, such as droplets (as is the situation that has been investigated in this work). Furthermore, the CMC approach was a modelling-based effort to predict autoignition in these flows and hence, the subsequent conclusions are subject to an uncertainty arising from the simplifications made for closure. Experimentally such an outcome has not been explored for parabolic flows and so the question is as yet open.

For the turbulent counterflow it was also shown experimentally, that increased turbulence in the air stream resulted in a higher critical temperature necessary for autoignition, suggesting a delaying effect of turbulence on the pre-ignition reactions. This finding is in subtle contrast to the DNS, that have shown that turbulence may accelerate autoignition. This discrepancy is most interesting and so the reasons behind the DNS finding will be reflected on briefly. It has been stated above that the DNS results have revealed that, locally, autoignition occurs at ξ_MR and at regions with low values of χ|ξ = ξ_MR. The simulations did not last very long relative to the turbulence turnover time and also had to resolve the fuel-air interface. Hence, the autoignition time was found to depend strongly on the initial condition (i.e. the initial value of χ), with the turbulence affecting autoignition time only insofar as it affected the emergence of the lowest value of the conditional χ|ξ = ξ_MR. Therefore, autoignition was promoted by fast mixing, due to the earlier emergence of well-mixed ξ_MR spots. It becomes essential to experimentally clarify this disparity and to validate or not the theoretical results from DNS regarding turbulent autoignition in parabolic flows.

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