## Abstract

Numerical methods have been used to examine the effects of (a) stretch alone, and (b) a combination of stretch and radiative loss, on the properties and extinction limits of methane–air flames near the lean flammability limit. Two axisymmetric opposed flow configurations were examined: (i) a single flame, unburnt-to-burnt (UTB) system in which fresh reactant is opposed by a stream of its own combustion products at the unburnt temperature, and (ii) a symmetric unburnt-to-unburnt (UTU) configuration where twin flames are supported back to back, one on each side of the stagnation plane. The maximum temperatures achieved in the UTB system are always away from the stagnation plane. For a fixed sufficiently sub-adiabatic product stream temperature, increasing flame stretch or gaseous radiative emissivity, or a combination of both, will augment downstream conductive heat loss, leading to a reduction in *T*_{max} and eventually to an abrupt extinction if the loss rate is sufficiently large. The UTU system is more complex, and offers the additional possibility of purely stretch-induced extinctions where the flames are forced together back-to-back so that radiative loss is restricted to upstream of the maximum temperature. Extinction in these cases occurs by straightforward truncation of the hot sides of the reaction zones. At sufficiently low stretch, near and at the standard flammability limit, radiative loss makes a major contribution to the overall extinction mechanism in both configurations.

The detailed effects of flame stretch on extinction behaviour depend on the diffusion characteristics within the near-limit mixtures, in particular the Lewis number, Le, of the deficient component. The effect of high stretch is always to attenuate the composition range of flammability. However, for Le<1 this range is extended at low to moderate stretch, particularly in the UTU situations where downstream radiative loss is not present at extinction. Lewis number effects for a global methane–air chemistry, and with assumed Le≥1, are discussed in the light of numerical results previously presented by Ju *et al*. (Ju *et al*. 1998 *Combust. Flame* **113**, 603–614).

## 1. Introduction

It has long been recognized that, without restrictions on the length of the computational domain, adiabatic flame models for strictly unidimensional flows are unable to predict the abrupt composition flammability limits exhibited by premixtures of fuels with oxidizers. The need for such restriction requires in turn the existence of some mechanism which limits the residence time in the high temperature region of the flame. By far the most likely such mechanisms are radiative loss (Zel'dovich 1941; Mayer 1957; Spalding 1957), flame stretch, or some combination of the two. Strong support for the radiative loss hypothesis has been provided by the extensive experimental and theoretical studies of spherically expanding near-limit flames under micro-gravity conditions by Ronney (1985, 1988*a*,*b*), Ronney & Wachman (1985), Abbud-Madrid & Ronney (1990) and Farmer & Ronney (1990); from numerical studies of such flames by Lakshmisha *et al*. (1988, 1990), Sibulkin & Frendi (1990) and Frendi & Sibulkin (1991); and further from a study of unstrained planar flames by Law & Egolfopoulos (1992). Detailed effects of radiation models, including effects of re-absorption, have more recently been studied by Ju *et al*. (1998*b*).

Dixon-Lewis (1996, part I) numerically examined the combined effects of stretch and non-radiative upstream heat loss on the properties of a laminar, stoichiometric methane–air flame in the symmetric, twin-flame, unburnt-to-unburnt (UTU) opposed flow configuration. Introductory material relating to opposed flow configurations and associated extinction limits is included in that paper, and will not be repeated here. The upstream heat loss in the specific instance was by thermal conduction to each of the supply nozzles, which were maintained at ambient temperature. Radiative loss was not considered. Curves of flame properties, for example the maximum temperature achieved, versus stretch rate formed a series of isolae, each having one high- and one low-stretch extinction limit.

Driven by the probable direct relevance of radiative loss to flammability limits, the present paper uses the opposed flow configurations shown in figure 1 to investigate the combined effects of stretch and radiative loss on the properties and extinction behaviour of methane–air flames near the lean flammability limit. Following descriptions in §2 of some further computational details, §§3–5 describe numerical studies of these combined effects in both the twin-flame UTU configuration (§§3 and 5), and an unburnt-to-burnt (UTB) configuration in which fresh combustible mixture is opposed by its own equilibrium combustion products at a defined burnt boundary temperature (§§3 and 4). The latter configuration is the more relevant to flammability limits as normally encountered, and in such circumstances the primary requirement for abrupt extinction is for the product temperature to be sufficiently sub-adiabatic (Libby & Williams 1982, 1983, 1984; Libby *et al*. 1983; Darabiha *et al*. 1986, 1988; Dixon-Lewis 1988, 1990; Rogg 1988). In the presence of radiative exchange with surroundings at such a product temperature, both increased emissivity and increasing flame stretch will augment the heat loss from the flame by steepening the temperature gradient on the downstream (product) side of the maximum temperature. Radiative loss also occurs, of course, on the upstream side of the maximum.

The UTU situation is more complex, and can lead to a further occurrence of twin-flame extinctions at constant composition, where the flames impinge directly on each other back-to-back and there is no downstream heat loss. Dependent on the Lewis number of the deficient component (Lewis number is defined throughout as the thermal diffusivity of the overall mixture divided by the mass diffusivity of the deficient component; Le≺1 for lean methane–air) small stretch intensities may somewhat extend the composition range of flammability in both configurations. The UTU behaviour of lean methane–air has previously been examined by Sung & Law (1996), Buckmaster (1997) and Ju *et al*. (1997, 1998*a*). The present investigation of the UTB behaviour has led to a much greater understanding than hitherto of the relative magnitudes of the interactions between chemistry, stretch, heat loss and Lewis number effects in the overall system. The results further show that the stretch/radiative loss mechanisms are able satisfactorily to account for the observed lean flammability limits of methane–air and the properties of the limit flames, at least within the precision of the detailed reaction mechanism and rate parameters employed.

## 2. Computational matters

### (a) Equations to be solved

Referring to figure 1, we consider the general case of coaxial opposed flow nozzles separated from each other by a fixed distance. Nozzle exit velocities and compositions, at temperature *T*_{u}, are assumed uniform in both configurations, so that ignition results in flat flames that lie in the radial plane of coordinates. The properties of such flames are functions of axial distance alone: the systems become quasi-unidimensional.

The steady-state continuity equations that govern such a system may be derived directly from the Navier–Stokes equations (Kee *et al*. 1988; Dixon-lewis 1990). For fixed composition and boundary conditions the derivation implies an eigenvalue radial pressure curvature *J* which may also be expressed as a global radial velocity gradient *a*_{e}=(−*J*/*ρ*_{e})^{1/2}, where *ρ*_{e} is the unburnt free stream density. With use of *a*_{e} to characterize the specific flame, and non-dimensionalization of the density-weighted distance , the axial velocity *u*, and the strain rate *F* by(2.1)(2.2)(2.3)the conservation equations governing the steady-state system become(2.4)

(2.5)

(2.6)

(2.7)Here the subscript ‘e’ refers to the unburnt free stream, *C*=(*ρμ*)/(*ρμ*)_{e}, *μ* is the viscosity and *h* the specific enthalpy of the *N*-component mixture, *σ*_{i}=*Y*_{i}/*m*_{i} where *Y*_{i} is the mass fraction and *m*_{i} the molecular mass of the *i*th component, and *R*_{i} is its net molar rate of chemical formation per unit volume. Expressions for the density *ρ*, the specific enthalpy *h*, the diffusive fluxes *j*_{i} and the energy transport fluxes *Q*_{D} and *Q*_{T} are given by Dixon-Lewis (1996). Molecular interactions for the associated multicomponent calculations were represented by the Lennard–Jones (12 : 6) potential, with the use of force constants given by Dixon-Lewis & Islam (1983).

The term *q*_{h} in equation (2.6) represents a volumetric rate of receipt of energy by radiative exchange with the surroundings. On the assumptions that the flames are optically thin and that Kirchhoff's law applies, such rates are calculated by use of equation (2.8):(2.8)where *σ* is now the Stefan–Boltzmann constant, *T*_{s} is a radiative ‘sink’ temperature, *p*_{i} is the partial pressure of the *i*th component, and *α*_{i,T} is its mean absorption coefficient at temperature *T*. CO_{2} and H_{2}O are assumed to be the sole emitters, and the mean absorption coefficients for these are expressed as functions of temperature by equations (2.9) and (2.10):(2.9)(2.10)where *T*^{*}=*T*/1000. The equations give a close fit to the curves given by Hubbard & Tien (1978) in the range 250–2000 K.

The situations considered in the present paper have *T*_{s}=*T*_{u}=295 K. *T*_{b} in the UTB configuration takes the same value.

### (b) Computational domain and boundary conditions

Solution of the equations was by the implicit finite domain procedure described by Dixon-Lewis (1996). Discretization into mass intervals was performed on a new independent variable *ω*, defined so that *η*=*η*_{L}*ω*, where *L* is the total length of the computational domain. The cell boundary between nodes at *ω*_{j} and *ω*_{j+1} was located at (1/2)(*ω*_{j}+*ω*_{j+1}). On the assumption of linear variation of *F*, *T*, *σ*_{i}, *R*_{i}/*ρ* and *q*_{h}/*ρ* with *ω* between nodes, equations (2.4)–(2.7) may be integrated analytically across individual cells to give, for the whole system, a large set of algebraic equations of the form(2.11)where the arguments are the solution vectors. and *L* are related by(2.12)In what follows, the origin of the axial distance coordinate *x* is always situated at the stagnation plane. By definition, the axial velocity at this position is also zero, in both the UTU and UTB configurations.

The stagnation plane also forms the burnt boundary of the symmetric UTU configuration, for which the boundary conditions thus become(2.13)Both the UTU and UTB systems will have the same unburnt free stream boundary conditions, and the UTB product boundary condition will be of similar form. Since only radiative losses are to be considered, the computational domain must be long enough to avoid diffusive losses at any of these outer boundaries. Considerable variations in length of the domain are to be expected at low stretch rates, and consistency of treatment throughout the whole investigation is maintained by use of the potential flow condition at all unburnt boundaries, and at the UTB product boundary. With the proviso of a sufficiently long computational domain, the unburnt boundary conditions at a defined stretch rate *a*_{e} become, for *both* configurations:(2.14)where the *G*_{i,u} are the mass fractions of the respective components in the input stream.

The UTB computational domain, again of total length *L*, has regions on both sides of the stagnation plane, with lengths *L*_{1} and *L*_{2}, respectively. After setting *ω*=0 at the outer product boundary and *ω*=0.5 at the stagnation plane, the internal boundary conditions *x*=0 and are introduced at the latter position to give, for the two locations,(2.15)

(2.16)Here the *G*_{i,b} are the mass fractions in the product stream. Modifications of the solution method to accommodate the internal boundary conditions are straightforward.

### (c) Solution of equations. Adaptive gridding

As has frequently been remarked by others, full Newton solutions of the discretized conservation equations for compositions near flammability limits are much more difficult than for near-stoichiometric flames, with the domain of convergence becoming so small that it is necessary for success to input an already virtually completely converged solution. Such a solution is determined by application of an iterative process (see Part I) which introduces a timestepping procedure into the discretization process. Even so, the achievement of full Newton steady-state solutions was here restricted to the UTU configuration. The overall strategy employed was initially to obtain a series of full Newton solutions for a methane–air equivalence ratio *ϕ*=0.57 in the adiabatic UTU system, for a discrete series of moderately separated stretch rates ranging from zero to just short of the extinction limit, and then to determine the limit itself by use of the continuation procedure which used the stagnation plane temperature as the independent variable (cf. Dixon-Lewis 1996). Central difference formulations were used for final solutions, and the latter were used in turn as starting profiles for the examination of effects of radiative loss in both the UTU and UTB configurations. Full Newton solutions were obtained in all the UTU situations.

Because of the practical constraints noted above, UTB extinction limits at fixed compositions were computed by careful use only of the timestepping procedure to determine, at a series of specified stretch rates, the occurrence or otherwise of convergence into a steady state. Alternatively, one notes that a composition extinction limit at fixed stretch rate can be determined in an analogous manner.

For a fixed position of the stagnation plane, the timestepping procedure for approach to a steady-state solution will involve changes both in profile shapes and in their positions within the computational domain. Efficient computation then demands periodic adaptation of the grid structure so that maximum node concentrations coincide always with regions of maximum physical and chemical activity. The basic features of the equidistribution procedure designed to do this are described in Part I. Having regard to problems encountered by Carter *et al*. (1982) in an earlier attempt to model hydrogen–air flames near the rich flammability limit, two additional grid constraints were applied here:

the total number of nodes are increased until there are always at least twenty to thirty included in the region of chemical production of H atoms; and

the temperature interval between nodes on the unburnt side of this zone does not exceed 25 K.

With suitable further constraints on the maximum *ω*-interval size in low activity regions of weakly stretched flames, the procedure worked well for those UTU situations where the maximum temperature was at the stagnation plane. It did not behave so satisfactorily for those UTU and UTB situations, with radiative loss, where the maximum temperature occurred away from the stagnation plane. The problem was overcome by rather heavily populating this maximum temperature region initially, and then, during regridding, retaining in fixed positions all those nodes lying between the burnt boundary and, inclusively, one node on the unburnt side of the maximum temperature. Satisfactory node distribution was then obtained by application of the standard equidistribution procedure to the remaining, unburnt side of *T*_{max}.

The major processes occurring on the burnt side of *T*_{max} are the final stages of the combustion of carbon monoxide, the decay of the radical pool, and radiative loss. The profiles there do not change rapidly, and any necessary adjustments to the grid, including those to the overall length of the computational domain, were made by hand at longer intervals. To maintain precision of representation of the radiative loss, temperature intervals between successive nodes were restricted to a maximum of 35–40 K at higher equivalence ratios, and 25 K otherwise. Near *T*_{max} the intervals were 1 K or less. They were only allowed to increase to the larger values gradually, and well onto the burnt side.

The flame profiles computed here are based on totals of some 140–160 nodes (UTU) and 200–220 nodes (UTB).

### (d) Reaction mechanism, rate parameters and computation of burnt gas profiles

The reaction mechanism employed, consisting of forty two reversible elementary steps and associated rate parameters, is that used by Dixon-Lewis & Islam (1983), and again presented in Part I. In that it includes only a truncated series of C_{2} reactions, and no reactions of CH and CH_{2}, it is not a complete methane–air mechanism. It nevertheless contains all the essential features for examination of the macroscopic properties of the fuel-lean flames.

The results in §4 will show that the CH_{4}→CO oxidation in the UTB flames is more or less completed in the main flame zone, and that the methane and the intermediate products of that stage of the combustion have virtually disappeared very soon after the maximum temperature is reached. To simplify the computation and to avoid numerical noise in the bulk of the subsequent CO oxidation/radiative cooling zone, the chemistry there was limited, after a short transition region, to the reactions concerned in the oxidation of hydrogen and carbon monoxide only. Specifically:

only the forward reactions of the minor pathway CH

_{3}→CH_{3}O→CH_{2}O are ever considered;when the residual methane had fallen below the ethane-forming dimerization reaction CH

_{3}+CH_{3}=C_{2}H_{6}was suppressed altogether, as also were all elementary steps (either forward or reverse) of the sequence CH_{4}→CH_{3}→…→CHO which run counter to the direction of oxidation.a vanishingly small value,

*σ*_{i}=10^{−16}, is imposed as a minimum for each species on both the burnt and unburnt sides of the system, with the further condition in each case that the*σ*_{i}vary monotonically. That is, if the condition is introduced by the computation at a specific node, it implies also a constant*σ*_{i}=10^{−16}for that species at all further nodes towards the free burnt or unburnt boundary. The chemistry is modified slightly to allow by-passing of the species concerned, and provision is made also for subsequent movement of the species into or out of the 10^{−16}region as necessary.limitation of the mechanism to the H

_{2}/CO oxidation reactions alone took place at*σ*_{CHO}≤10^{−16}. Between this position and the burnt boundary the value*σ*_{i}=10^{−16}was assigned to each of the intermediate species not directly retained. This procedure allowed the retention of a uniform block size for the Jacobian matrix throughout the flame.

## 3. Selected UTU and UTB flame profiles at a single equivalence ratio

The effects of (a) stretch alone, and (b) a combination of stretch and radiative loss on the strain rate, temperature and species profiles for an equivalence ratio *ϕ*=0.57, well inside the flammability limit, will be presented. We note here that in addition to defining a *global* stretch rate *a*_{e}, the eigenvalue *J* may be used to define a set of *local* stretch rates within a flame, by , and thus in turn a set of dimensionless local stretch rates . They are *not* the same as the dimensionless *strain* rates of equation (2.3), to which they are related by equation (2.5). The quantities *a* and *F* are analogues of the mass flux fraction and mass fraction in the chemical species equations. The profiles for both dimensionless quantities for the adiabatic UTU flame (*ϕ*=0.57, *T*_{u}=295 K) and for the UTU and UTB flames with radiative loss are illustrated in figure 2. It is interesting to note that although marked hysterysis effects occur across the flame zones, the strain rate on the burnt side of the UTB configuration adjusts almost instantaneously to the density changes there.

Figure 3 shows two effects of stretch alone in the adiabatic UTU configuration. The maximum temperature achieved, at the stagnation plane, rises from 1610 K for the unstretched flame to a maximum of 1632 K at a radial stretch rate *a*_{e}=40 s^{−1}, before falling to 1461 K at the extinction limit (*a*_{e}=229 s^{−1}). The initial increase in temperature is caused by preferential diffusive retention of the deficient, low molecular weight fuel within the region near the flow axis (Lewis number effect for Le<1). This same effect results in an increase of the space integral of the chemical rate of methane removal from 0.181 mol m^{−2} s^{−1} for the unstretched flame to 0.261 mol m^{−2} s^{−1} at *a*_{e}=222 s^{−1}, followed by a small decrease in the integral, to 0.252 mol m^{−2} s^{−1}, during the final approach to the limit. These space integrals, defined as , are the negatives of those shown in curve 2. They represent chemically induced flux changes across the flame.

For *a*_{e}=2.5 s^{−1}, *T*_{u}=295 K and *ϕ*=0.57, figure 4 shows computed temperature profiles for four different circumstances. Curve 1 refers to the adiabatic UTU system, for which *T*_{max}=1613 K, at the stagnation plane. The flames themselves are each separated from the stagnation plane by about 49 mm. Inclusion of realistic radiative loss in the same configuration causes each flame to move approximately 9 mm towards the stagnation plane (curve 2), and leads to maximum temperatures of 1578 K at distances of 39 mm. Between the maxima is a temperature well, with a minimum of 1052 K at the stagnation plane.

As the UTU stretch rate increases, the flames move towards the stagnation plane and the depth of the temperature well rapidly decreases. At *a*_{e}=20 s^{−1} maximum temperatures of 1582 K are achieved at 3.5 mm from the stagnation plane, at which the minimum is 1553 K. The temperature well disappears completely before *a*_{e}=167 s^{−1} is reached, and at 167 s^{−1} the maximum temperatures with and without radiative loss are 1544 and 1551 K, respectively, both now being at the stagnation plane. The respective space integrals for methane removal rate are 0.2461 and 0.2486 mol m^{−2} s^{−1}. The high stretch UTU extinction limit at 229 s^{−1} will thus not be appreciably changed by the radiative loss.

For situations where both the input and product streams are in thermal equilibrium with the surroundings at *T*_{s}, curves 3 and 4 show the corresponding UTB temperature profiles on the assumptions of infinitesimally small gaseous emissivities and realistic emissivities, respectively. The temperature and major species profiles for this last situation are further shown in figure 5. The computations draw attention to an interesting thermal diffusion effect in the immediate vicinity of the stagnation plane, where the gradient of the reciprocal temperature is high.

## 4. Extinction limits in the UTB configuration

As implied above, increasing the stretch rate in either of the UTB situations illustrated in figure 4 will lead to an abrupt extinction limit. For infinitesimally small emissivity this limit occurs at *a*_{e}=20.8 s^{−1}. With realistic emissivities it is reduced to 17.5 s^{−1}.

The pairs of curves 1 and 2 in figure 6 show the maximum temperatures achieved (open circles) and the space integrals (shaded circles) on approach to the respective limits. The curves are sections of the stable branches of the semi-isolae that, in the particular circumstances, relate the flame properties with stretch. On increasing the stretch rate from 2.5 s^{−1}, Lewis number effects (Le<1) cause the magnitude of each space integral to rise to some maximum value before decreasing again towards extinction. Further quantitative material will relate only to situations with realistic emissivities.

Table 1 lists selected properties of several stretched methane–air premixed flames at the UTB extinction limit. A feature of particular note is that whereas both *T*_{max} and the magnitude of the space integral decrease monotonically as the UTB stretch rate decreases, the equivalence ratio shows a minimum value at the small *a*_{e}=1 s^{−1}, below which reduction to zero stretch results in a small increase of the lean limit composition. The primary effect of increasing stretch, the attenuation of the flammable region, is ameliorated and briefly reversed by the Lewis number effect. The limit stretch rates and the corresponding space integrals for the whole range of compositions are further shown in figure 7.

The requirement for sufficiently sub-adiabatic burnt boundary conditions clearly implies an essential *primary* contribution of a downstream heat loss mechanism to the occurrence of all UTB (or single flame) extinction limits. In the absence of a downstream quenching surface the mechanism is provided by radiative exchange with the surroundings. Such radiative exchange may operate either alone (at the standard flammability limit where *a*_{e}=0) or in combination with flame stretch effects. In contrast with the primarily stretch-induced limits encountered specifically in UTU systems (see also below), the radical production zones in all the UTB flames, limit or otherwise, are separated from the maximum temperature by a zone of radical recombination. The situation is illustrated at low stretch rate by figure 8, which is a zoom showing selected features of the early part of the carbon monoxide oxidation zone in the flame of figure 5 (*a*_{e}=2.5 s^{−1}). At this stage the hydrocarbon fuel has almost completely disappeared by reaction upstream with the radicals H, O and OH which have diffused there (Dixon-lewis 1990, 1996). To clarify the role of flame stretch, it is instructive to compare the profiles of figure 8 with the similar profiles of figure 9, which belong to the flame at the extinction limit, *a*_{e}=17.55 s^{−1}, for the same composition. Figures 8 and 9 also each show on their abscissa the extent BZ of the zone of net production of radical pool, considered as *R*=(H+2O+OH+HO_{2}+CH_{3}O+CHO+CH_{3}+C_{2}H_{5}). The respective crossover temperatures to net radical production are virtually the same, at 1426 and 1425 K. Both occur slightly downstream of the position of maximum CH_{3}, and the residual methane mole fraction in both cases is exactly the same, at 1.6×10^{−4}. The influence of the chain termination reaction O+CH_{3}=CH_{2}O+H entails that must fall to a very low value before net branching can occur.

The maximum temperatures achieved in the two flames are 1578 and 1534 K, at distances of 39.2 and 3.87 mm, respectively, from the stagnation plane. Apart from these differences there is a remarkable similarity between the two sets of profiles, extending even to the result that the methane mole fractions of 8.6×10^{−7} (at *T*=1536 K) and 9.6×10^{−7} (at *T*=1520 K) on exit from the respective radical production zones differ by only just over ten percent. The significant effects of increasing stretch are to reduce *T*_{max}, and simultaneously to move its location nearer to the radical production region BZ. At high enough stretch rates these effects become sufficient to modify the structure of the production region itself, leading to a weaker radical pool and, necessarily, to reduced mass fluxes through the reaction zone if appreciable fuel breakthrough beyond *T*_{max} is to be avoided. Such reductions are not sustainable indefinitely in the face of the additional radiative loss introduced thereby, and extinction eventually occurs.

The space integrals for the flames of figures 8 and 9 are 0.172 and 0.160 mol m^{−2} s^{−1}, respectively.

## 5. Overall behaviour of UTU and UTB systems

The general features of lean methane–air UTU behaviour have already been outlined by, amongst others, Buckmaster (1997), Sung & Law (1996) and Ju *et al*. (1997, 1998*a*). Particularly significant at low stretch rates is an extension of the flammable region to compositions considerably leaner than the flammability limit as normally encountered. The situation is further illustrated in figure 10, in which the presently computed lean extinction limit compositions are plotted as functions of stretch rate, for both the UTU and UTB configurations. The flammable region in each case is to the right of the appropriate curve, ABCD or DCE.

At very low stretch rates, below about 1 s^{−1}, the UTB extinction limits and the appropriate UTU limits corresponding with section CD (to be discussed below) are more or less identical. At these extremely low stretch rates the flames in both systems are far from the stagnation plane, as shown in figure 4. Viewed macroscopically, both increasing stretch and increasing emissivity in the UTB system will steepen the temperature gradient, initiated by the low product stream temperature, that lies between the flame and the stagnation plane. Both therefore also increase downstream conductive loss from the flame. On the assumption of a sufficiently low radiative sink temperature *T*_{s}, it is this stretch/radiative loss combination which is responsible for the attenuation of the UTB flammable range above *a*_{e}≈1 s^{−1}. For fixed *T*_{s} and *T*_{b} UTB extinction limits at sufficiently low stretch are still critically dependent on emissivity: however, at higher stretch rates the quantitative influence of emissivity changes will become much diminished, as illustrated in figure 6.

The situation in the UTU flames is more complex. As the stretch rate is increased, so the flames are pushed closer together, the depth of the intermediate temperature well is reduced, and the well is eventually eliminated, as noted in §3. That is, downstream heat loss is attenuated and eventually removed. Finally, a high stretch, purely stretch-induced extinction limit is obtained which is close to or (for compositions that support fast enough flames) virtually identical with that for the adiabatic flame of the same composition. The only radiative loss to be considered is that *upstream* of the temperature maximum.

As the equivalence ratio is reduced along the line AB in figure 10, the flames at the high stretch UTU limits become slower, and upstream radiative loss begins to play a more prominent role in the extinction process. The sequence again leads eventually to a composition limit at B. However, the reduced loss compared with the UTB system leads to the result that UTU stretch-induced extinction limits occur at compositions far outside the standard flammability limit, or the slightly extended UTB limits.

The UTU extinction limits displayed in figure 10 are global results derived from an extensive series of stretched laminar premixed flame computations. For realistic radiative loss at fixed composition, the stability curve relating any steady state flame property with stretch rate forms an isola with high and low stretch extinction limits at points of vertical tangency. These limits separate branches of the isola that relate, respectively, to stable and unstable solutions. The maximum temperatures achieved in the UTU flames studied are used in figure 11 to define the stable branches of the isolae for several methane–air equivalence ratios up to *ϕ*=0.53. The high stretch limits F and the low stretch limits G form the branches AB and BC of figure 10. The individual flames at both these particular limits are separated by only very small distances, and the limits themselves will be referred to below as ‘twin-flame’ limits. The term ‘radiation limit’ normally applied to the limits at G has been discarded for semantic reasons. The participation of significant thermal loss is in any case an essential requirement for the low stretch closure of both the UTU and UTB isolae.

The computed temperature profiles at the limits F and G are shown, for *ϕ*=0.53, in figure 12. They are typical of both non-adiabatic and adiabatic stretched, twin premixed flame systems in that the maximum temperature is located *at* the stagnation plane. The *limit* flames are further distinguished from those at the single flame UTB limits, in that the net radical production zones, each somewhat truncated, sit back-to-back to form a single such zone that extends uninterrupted across the symmetry plane and shows its maximum radical concentrations there. In the absence of downstream heat loss it is this truncation that weakens the radical pool and eventually leads to extinction. Thus, although all the observed limits are dependent on both stretch and radiative loss, the twin flame limits at both F and G may be regarded as *primarily* stretch-induced, with the radiative loss restricted to upstream of *T*_{max}. The combination of Lewis number and radiative loss effects entails that the limit flames at G, having suffered the greater loss, are much cooler and slower than those at *F*. Further, the residence times in both cases are such that the flames are able to propagate with much lower *T*_{max} than at the standard flammability limit. Note also that the low stretch limit curve BC of figure 10 extrapolates into the main flammability region as a subset of ‘weak flame’ extinction limits, as already observed by Buckmaster (1997) and Ju *et al*. (1997). The curve ABC and its (dashed) extension are themselves part of a global isola that is the locus of the twin-flame limits at F and G, over the whole composition range.

To continue, examination of the full temperature profiles associated with all the flames of figure 10 further reveals the following.

That at the three compositions

*ϕ*=0.46, 0.48 and 0.50 none of the flames at any stretch rate suffers downstream radiative loss.*T*_{max}is always at the stagnation plane.At

*ϕ*=0.52 and 0.525 the same applies for global stretch rates outside the approximate range*a*_{e}=2.5–14 s^{−1}. Between these stretch rates, and dependent on both the composition and*a*_{e}, the*T*_{max}separate by up to about 9 mm. The additional downstream radiative loss on this account gives rise to the observed dips in the maximum temperature isolae. At these compositions the*T*_{max}are unable to depart from the stagnation plane by more that about 3.0 and 4.5 mm, for*ϕ*=0.52 and 0.525, respectively.The same overall behaviour persists right up to and beyond

*ϕ*=0.53, with one important addition. Commencing at a composition consistent with the line CD of figure 10, the isolae of figure 11 develop an upper lobe J which, for compositions at and inside the standard flammability limit, will reach out to zero stretch. Similarly to the main isolae, the lobes J have a stable upper and an unstable lower branch. Although the unstable branches for methane–air have not been formally computed, their approximate locations are denoted by the dashed lines in figure 11. More quantitative information is given by Ju*et al*. (1997). The flames along JK are hotter and stronger than those along the branch*GH*. They are separated by larger distances, and all suffer from both upstream and downstream radiative loss. On reducing the stretch rate the overall structure moves progressively towards that of a pair of single UTB flames.For a limited range of composition just

*outside*the standard flammability limit, the lobes J show low stretch limits at*M*. Though stretch dependent, these are essentially single flame limits with downstream radiative loss. Dependent on the relative positions of the limits at*M*and G, they may be either true extinction limits or they may be ‘jump’ limits down to the stable twin flame structures along*GH*. The composition range over which these effects occur corresponds approximately with the region in figure 7 between the turning point C and the standard flammability limit at D.It remains to consider the high stretch limit

*H*, also designated a jump limit by Ju*et al*. (1997). For*ϕ*=0.53, trajectories on figure 11 were examined, with potential flow boundary conditions, for the same nozzle velocity of 9.86 cm s^{−1}and a series of three nozzle separations: 37.0, 40.9 and 44.2 mm. The largest separation led to a steady state flame on branch*GH*, with*a*_{e}=2.8 s^{−1}. The intermediate separation led to uniform oscillations with a clockwise path around the left-hand trajectory near*H*. These apparent oscillations may be an artefact of the computation due to proximity to the turning point at*H*. On the other hand they do give the impression of a flame trying unsuccessfully to make the jump to branch*JK*.

The smallest of the three separations leads to a successful jump as shown by the right-hand trajectory near H (this starts below the *ϕ*=0.53 curve GH to which it is related). During the jump the flame separation increases to about 18 mm, and *a*_{e} from around 3.5 to about 4.5 s^{−1}. On moving from left to right along the curve GH at constant methane–air composition, the effect of increasing stretch is to increase the flame strength on two counts: first, diminution of the upstream radiative loss, and secondly, the Lewis number effect. The requirement for the jump is analogous to that for an ignition. The jump takes place at the stretch threshold where the flame has acquired sufficient resources to withstand the additional, downstream radiative loss involved—though this is mitigated to an extent by diminished upstream loss.

For the particular numerical situation studied, the jump limit at H is confined to equivalence ratios greater than *ϕ*=0.5265, while that at *M* is further confined to the interval 0.5265<*ϕ*<0.527 67. Within this interval the effect of mixture enrichment, which enlarges the isolae, is to move both limits towards lower stretch rates.

## 6. Correlation of UTB and UTU behaviour. Lewis number effects

Returning briefly to the UTB discussion of §4, we note that, for equivalence ratios between *ϕ*=0.5265 and the standard flammability limit at *ϕ*=0.527 67 (that is, between C and D in figure 7), curves of individual flame properties versus stretch rate will again form a series of complete isolae, with both high and low stretch extinction limits. At the standard flammability limit the low stretch extinction will be at *a*_{e}=0, and at compositions richer than this limit there will be only a semi-isola, with no low stretch extinction. In the present case (0.5265≤*ϕ*≤0.527 67) the last few lines of table 1 show *T*_{max} at both the high and low stretch UTB limits to be away from the stagnation plane. Both are true radiation-induced limits, in the sense that both depend critically on radiative and conductive loss effects downstream of *T*_{max} to weaken the radical pool.

Attention has already been drawn in §3 and elsewhere to the effect of sub-unity Lewis number in causing fuel enrichment in the reaction zones of stretched, fuel-lean methane–air flames. Such enrichment in turn causes enlargement of both the single and twin flame isolae. Decreasing Le will also cause the turning point C in figure 7 to move towards weaker compositions. Conversely, increased Le causes weakening of the mixture at the reaction zone, shrinkage of the isolae, and a movement of C towards the standard flammability limit at D. By definition, satisfaction of either of the conditions Le=1 or *a*_{e}=0 provides diffusional neutrality. It follows from figure 7 that the formation of a complete UTB isola is restricted to situations where Le<1. There are no low stretch UTB limits for Le≥1, and, locally within the flammable region, the high stretch limits of the semi-isolae will increase monotonically with equivalence ratio, from *a*_{e}=0 at the standard limit. The rate of increase will become smaller as Le increases, corresponding with reduced high stretch extinction rates at fixed composition.

UTU behaviour is more complex, and may involve close interaction between the UTB and twin flame isolae. Such close interaction is displayed by the lean methane–air isolae of figure 11 having *ϕ*>0.5265; and these may equally well be regarded either as twin flame isolae that exhibit a lobe, or as UTB isolae modified to fit the changed downstream conditions. On the latter view, the high stretch, radiation-induced single flame limit of the UTB isola is replaced by a stretch-induced twin flame limit in a composite isola. We note that such interaction is confined to those situations where, at least at one location, there is a satisfactory match of temperature and other properties between the two components of the composite isola.

Lewis number effects associated with lean UTU methane–air flames have been examined numerically by Ju *et al*. (1998*a*), at Le=0.67, 1.0, 1.25, 1.43 and 1.82, with use of a global one-step chemistry. Increasing Le at constant composition results in a progressive narrowing of the neck *HK* in figure 11 due to the shrinkage of both component isolae. At Le≥1 regions of separation of the composite isolae into their single- and twin-flame components appear at compositions inside the standard flammability range, and penetrate further towards higher equivalence ratios as Le increases. Mixture enrichment becomes necessary in order approximately to restore the *status quo.* The change in Le is thus to cause the section *ABC* of the twin-flame limit curve in figure 10 to move to the right relative to the standard limit and to the curve *DCE*, the position of which itself depends on Le. The movements may also of course be accompanied by some change of shape. The results presented by Ju *et al*. (1998*a*) indicate the approximate limiting equivalence ratios given in table 2. The jump limits are here the weakest equivalence ratios for which composite isolae are obtained, and for which transitions from twin- to single-flame behaviour (and occasionally vice versa) become possible solely by way of change of stretch intensity. The stretch rates at which the various limits occur depend on the details of the temperature responses of the constituent isolae to changes in Le. Table 2 includes also the computed response of the standard (*a*_{e}=0) composition limit to such changes.

We note from table 2 that, for Le=1 and above, the weakest jump limit is at a richer composition than the standard flammability limit, and that, for compositions between the two limits, this implies that the single flame (or pseudo-UTB) semi-isola is separated from the twin flame isola. Further, the single flames do not exist outside the standard limit, and only the twin-flame isolae may extend into the sub-limit region. The third and fourth lines of table 2 further indicate that, for Le≈1.2 and above, the twin flame isolae themselves become completely absorbed into the single flame region.

The immediate sub-unity Lewis number region has not been investigated numerically. Within this range progressive reduction of Le will occur without noticeable change of behaviour, *until* it leads eventually to coincidence of the weakest jump limit composition with the standard flammability limit. At this particular conjunction of Le and composition the single flame isola will have completely disappeared. There will remain only the composite isolae at richer mixtures and the twin flame isolae for weaker mixtures. Further reduction of Le will result in the formation of a progressively larger range of composite isolae, of which the upper lobes are parts of the UTB type, with low stretch extinction limits that increase with decreasing equivalence ratio. Above the weakest jump limit for the Lewis number concerned, the merging of single and twin flame isolae leads to a sequence of results as exemplified by the lobe *M* in figure 11. There is of course the restriction that the upper lobes will not exist at compositions beyond the equivalent of the turning point C in figure 7.

Yet another feature of the isolae at Le≥1 is that the low stretch limits corresponding to G in figure 11 can no longer be primarily stretch induced and reliant only on *upstream* radiative loss. The temperature maxima must presumably move away from the stagnation plane as the stretch rate is reduced, leading to supplementary downstream loss and a pair of primarily radiation-induced, single flame limits. As already noted, at Le≥1 there are no such low stretch radiation-induced limits associated with the isolae of the hotter and stronger single flames.

## 7. Conclusions

Premixed flame extinction limits depend strongly on the magnitudes of loss mechanisms produced by interaction of the flames with the surroundings. Independently of the flame chemistry, or indeed of specific fuels or oxidizers, flame propagation or extinction depends primarily on a delicate balance between, on the one hand, a chemical heat release process that is essentially localized near the maximum temperature achieved, and on the other, heat loss processes that occur across the whole flame. Within the flammable range the maximum temperature achieved, and in turn the rate of chemical reaction and the flame velocity, are determined by this balance. On account of the comparative localization of the flame chemistry, the loss processes assume greater importance within the balance as an extinction limit is approached and the maximum temperature becomes lower, until eventually, outside the limit, the balance can no longer be maintained. Looked at in another way, the maintenance demands a finite reaction zone thickness and flame velocity at the limit; and, for constant environmental conditions, a well-defined maximum temperature and limit composition. The limit composition is the minimum mole fraction of the deficient component that will allow achievement of a satisfactory maximum temperature for its own complete combustion in the time available. In this aspect of their behaviour, though not in a *detailed* mechanistic sense, freely propagating localized flame reaction zones may be likened to travelling thermal explosions.

In those flames, such as those of methane, hydrogen and other organic materials in air or oxygen, that proceed by way of a free radical chain mechanism, the chemical heat release may not itself be localized in the manner just indicated. Instead, as explained earlier for methane–air, most of the heat release occurs via low activation energy reactions of radicals that have diffused upstream from a high temperature region of net radical production. The overall heat release is nevertheless still initiated by this localized high temperature process that itself involves a further competition between rates of radical producing and radical removing elementary chemical reactions.

Flame stretch alone will not cause abrupt extinction of single adiabatic flames; and the level of agreement between the experimentally measured lean methane–air flammability limit and that predicted computationally for the single, radiatively cooled unstretched flame where the hot reaction zone is sandwiched between unburnt and burnt gas at 295 K confirms the essential role of radiative loss. The limit is additionally modified by flame stretch. Thus the following applies.

At very low stretch rates the UTB, single flame lean limit equivalence ratio for methane–air is reduced from

*ϕ*=0.527 67 at the standard limit, to*ϕ*=0.5265 at*a*_{e}≈1 s^{−1}. Further increased stretch then causes truncation of the flammable region. There is thus a small range of compositions outside the standard flammability limit where the response of a specific flame property such as*T*_{max}to stretch is represented by a complete isola with both high and low stretch extinction limits. Inside the standard limit the closed curve is replaced by a semi-isola that shows only a high stretch extinction. Despite their dependence on stretch, these extinctions are all primarily radiation-induced, with radiative loss both upstream and downstream of*T*_{max}. The existence of the*complete*isolae and the small extension of the flammable region at low stretch are consequences of Lewis number effects. They are observed only in situations where the Lewis number Le of the deficient component is less than unity.The lean methane–air limit composition is extended still further in the UTU or twin flame system, to

*ϕ*≈0.45 for the assumed chemistry and transport parameters. Between*ϕ*=0.45 and the UTB limit at*ϕ*=0.5265 the response of the flames to stretch is again represented by a series of simple isolae, with the difference from above that both the high and low stretch extinction limits are now stretch-induced. The flames are forced closely enough together for their reaction zones each to be truncated on the hot side, in a symmetrical manner. One notes from figure 10 that the stretch rate at*ϕ*=0.45, where the isola just disappears, is about 5 s^{−1}. In combination with larger Lewis number effects at the higher stretch rate, this much larger extension of the flammable range compared with the UTB configuration is a consequence of restriction of radiative loss to upstream of*T*_{max}.At compositions within the UTB flammable range the single and twin flame isolae will coexist, and in the case of lean methane–air (Le<1) will merge to form a complex isola with one high stretch extinction limit and two low stretch branches as shown in figure 11. The high stretch single flame limit becomes completely absorbed into the twin flame isola.

At Le≥1 no single flame can exist at compositions outside the standard flammability limit, but single flame semi-isolae will still exist inside the limit. They will exhibit only high stretch radiation-induced extinctions. Depending on conditions, merging with the twin flame isolae may or may not occur, as outlined briefly in §6.

Inasmuch as both can affect the balance between heat production and heat loss, Lewis number and configuration effects are extremely important determinants of the extinction behaviour of stretched, premixed, near-limit flames. As noted earlier, extension of the flammable range when Le<1 in the twin flame configuration is due to a combination of Lewis number effect with restriction of downstream heat loss. The second of these effects still operates to retain a (reduced) extension when Le=1, and for lean methane chemistry it is not until Le≈1.4 that the extension disappears.

## Acknowledgments

Parts of this work were carried out during visits by the author to the Engineering Department, University of Cambridge, and to the Max-Planck-Institut für Strömungsforschung, Göttingen. Invitations and the organization of these visits by Professor K.N.C. Bray F.R.S. and Dr B. Rogg (Cambridge) and Professor H. Gg. Wagner (Göttingen), and support from EPSRC and the Humboldt Foundation, are all gratefully acknowledged.

## Footnotes

↵† Present address: School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK.

- Received April 1, 2005.
- Accepted July 7, 2005.

- © 2005 The Royal Society