## Abstract

This article describes the macroscopic and microscopic features of flames spreading over solid-fuel surfaces by examining and comparing three models. The first model examines ignition and flame spread over a solid-fuel surface using a two-dimensional numerical simulation code. This model employs variable density, variable thermophysical properties and one-step global finite-rate chemistry. The second model, a macroscopic ‘field’ model, is solved in terms of the mixture fraction (*Z*) and total enthalpy (*H*) functions. Comparisons are made with numerical predictions for primitive quantities: temperature, species distributions and velocity fields; and derived quantities: heat flux, mass flux, mixture fraction, enthalpy function and flame stretch rate. The third model yields a ‘localized’ flame structure description near the flame attachment point. Theoretical formulas are produced for the quenching distance, the leading edge heat flux, and the flame structure, as characterized by reactivity, temperature field and species distributions. The analytical predictions are compared with numerical simulations to derive flame microstructure scaling parameters.

## 1. Background and motivation

Opposed-flow flame spread over fuel surfaces is a classical cyclical feedback mechanism. The flame heats the surface, which decomposes to gaseous volatiles that are transported by diffusion and convection into the surrounding gas. A locally combustible mixture of fuel and oxidant is then formed. The mixture ignites to release thermal energy from the flame, which is transported to the solid fuel, beginning the cycle again.

In opposed-flow flame spread, the dominant variation of the physical variables usually occurs in the direction of the downstream flow along the sample surface (*x*) and in the direction perpendicular to the decomposing sample surface (*y*). Flame-spread experiments are often conducted in devices that arrange the fuel surfaces such that the *x* and *y* directions contain the only variations; this requires that the length scale associated with the *z*-wise sample width be smaller than the length scale required for *z*-wise variations in the flame structure. It is becoming increasingly obvious, however, that a standard two-dimensional description of the flame spread process is often inadequate and that more detailed models are needed.

Experimental research by Olson *et al*. (2006, 2009) has shown that complicated burning patterns occur when ‘near-limit’ forms of flame spread are analysed. Such near-limit behaviours are generated by devising conditions of large flame heat loss to the surroundings, by reducing the oxygen content of the air inflow or by reducing the air inflow rate. These types of flame spread are encountered in reduced oxygen environments, in narrow channels or confined spaces, or under microgravity (mg) conditions (in which there is no buoyant inflow of oxidant). However, it is also known that many other influences, such as fuel thickness, fuel homogeneity and heterogeneity, fuel regression pattern, melt layer (or char layer) formation, volatile transport in the gas or fluid-dynamical flow pattern near the surface, will influence the flame structure and the flame pattern in complicated ways. Most real materials regress, liquefy or form spotty regions with various degrees of charring and liquefaction. In some cases, they consist of highly heterogeneous mixtures such as propellants. Large-scale fires usually spread over heterogeneous materials (forest, brush, etc.), and this is one of the reasons why aerial photographs often show irregular shapes for large-scale fire fronts. The complicated nature of flame spread over these heterogeneous materials suggests that statistical descriptions may be appropriate.

The exact reasons of why complicated flame structures appear on solid-fuel surfaces are not presently known. They may arise from an instability mechanism, in which the original two-dimensional problem becomes unstable to small perturbations in the third coordinate (*z*) direction along the advancing flame front. In order to examine this question, it would be necessary to perform a full three-dimensional stability analysis of the governing differential equations.

Our objectives are not so comprehensive. In order to ask relevant, informed questions concerning flame-front behaviour and pattern formation, we felt it was first necessary to understand the two-dimensional flame-spread mechanism in detail. Once such an understanding of the scope, nature and peculiarities of the two-dimensional model are established, it will become easier to place the three-dimensional model in context and to more carefully and accurately perform a complete three-dimensional analysis. This article constitutes a concentrated attempt to synthesize the existing theoretical and conceptual understanding of the classical two-dimensional flame-spread problem.

In order to achieve this goal, it is imperative to characterize two outstanding features of flame spread. The first outstanding feature is the *flame macrostructure*, comprising the heat and mass transfer behaviours in the gas, in the solid and at the interface during flame spread. A model of this feature of flame spread should produce an explanation of the global structure of attached flames. The predicted basic structure should resemble, in its macroscopic features, the macrostructure observed in numerical simulations. The second outstanding feature is the *flame microstructure*, which includes attachment characteristics; local reactivity; local temperature, enthalpy and species fields; reaction-rate contours for the quenching flame front; and characteristic time and length scales. A third outstanding feature of flame spread, not addressed explicitly in this article, examines the fundamental reason of why real flame spread is actually a three-dimensional process. This requires performing a three-dimensional analysis, which can only be presently accomplished numerically. It also requires studying the causes of the three-dimensional flame formation from initially two-dimensional plane flames. We expect that the reason will be found in the very nature of the solutions produced by the two-dimensional models described here.

The mathematical and numerical models examined in this article attempt, in a preliminary way, to address the first two outstanding features of flame spread discussed above, namely its macrostructure and microstructure. Two detailed numerical models of two-dimensional flame spread are examined in conjunction with two analytical models, one describing the macroscopic features, and the other describing the microscopic features of flame spread. The theoretical predictions for a highly simplified model are shown to be in excellent qualitative agreement with the numerical computations for a detailed computational model, suggesting that the physical features of such models are characterized by the simplified theory.

## 2. Literature review and discussion

There is extensive literature on the problem of flame spread over solid-fuel surfaces. For convenience, we shall describe these in the context of pre-1995 and post-1995 research. Pre-1995 reviews by Fernandez-Pello & Hirano (1983) and Wichman (1992) have emphasized the basic physical concepts, the construction of meaningful and accurate correlations of the experimental data and the importance of gaining a thorough understanding of the scope and limitations of the various theoretical models. Great advances were made in the application of basic theory for the development of accurate, comprehensive correlations, later successfully used to quantify flame-spread behaviour in a broad range of flame-spread regimes (Wichman 1992). By contrast, the work of the past decade (post-1995) has been dominated by numerical simulations in which detailed equations and boundary conditions are computationally solved in order to describe or simulate the complex, interactive problem of flame spread over fuel surfaces. For detailed numerical models, the reader may consult, for example, the work of Nakamura *et al*. (2002, 2006), Lina & Chen (1999) and Zheng *et al*. (2001). Several comprehensive numerical codes have been written to describe large fires in structures such as rooms, hallways, buildings, ships and aircraft. Possibly, the best-known of the simulation codes is the fire dynamics simulator (FDS) developed by Dr K. McGrattan and his colleagues at the National Institute of Standards. The FDS, built on work begun in the early- to mid-1980s by Dr H. Baum and Dr R. Rehm, is a large, adaptable code that can be used to simulate fire growth and spread, and may thus be considered as a fire-safety tool. Information about this code can be found at the website www.fire.nist.gov/fds/ in the web-based user’s guidelines, and in written form in McGrattan *et al*. (2008).

Numerical models, regardless of their degree of sophistication, generally produce impressive graphics that visually resemble actual spreading flames (fires), whether over vertical walls or horizontal surfaces, whether they be large fires or small flames or whether their spread is wind-aided or wind-opposed. The beguiling power of numerical simulation methods, arising partly from the impetus for their rapid use in engineering applications and partly from the blurring of the distinction between basic and applied research, has seemingly eclipsed the need to stand athwart this juggernaut, calling for a brief pause while examining simulation veracity. It is necessary, but not sufficient, that the simulations resemble actual fires; they must also be capable of making accurate quantitative predictions.

From the time that numerical simulations became the accepted norm in fire spread, little research has been performed to reconcile the ever-more detailed numerical models with the theory that preceded and underlaid them. The advantage of a theoretical model lies both in the production of a functional relationship (in the person of a formula) and a quantitative correlation between two or more parameters. A theoretical solution also provides a third benefit, namely a field in space (e.g. Cartesian *x*, *y*, *z* coordinates) that can be compared in an exhaustive manner with a simulation. This is very unlike many experimental evaluations, wherein the measurements are generally local. The examination of the theoretical field should provide clues for the interpretation of its simulation counterparts. If the theoretical model is accurate, and if it successfully retains essential physics, it should predict quantities such as inflection points, singularities, maxima and minima. These same features, in less distinct form1 must also appear in the simulations (,Aris 1999).

The goal of this article is to examine the theoretical content of a simplified model of flame spread in an attempt to determine which simulation features it can predict. It will be shown that the simplified theory produces functional field dependencies that appear (in diluted form) in the simulations. In addition, the simplified model predictions lend themselves to clear and cogent explanations, which assists greatly in the examination (and explanation) of the numerical results. Clarification of the nature of the model solutions can only help as additional details are added to numerical models and the quantity of information to be processed increases. If means are not available for the interpretation of the results (or if there is no means available for determining whether the additional information is superfluous, or wrong), progress cannot be said to have occurred. Without guidance provided by correct theory, scientific advances cannot be made (Truesdell 1984).

## 3. Model formulation

In this section, we formulate two two-dimensional mathematical models of ignition and flame spread over cellulosic materials. The only difference between the models is the treatment of the solid phase.

In order to simulate radiant ignition, an external heat flux is applied to a narrow segment of the surface. This segment of the surface eventually decomposes, expelling fuel volatiles into the gas where they mix and react with the inflowing oxidant. The ignition kernel grows into a spreading flame that consumes the solid fuel as it propagates against the oxidant inflow. Note that the nascent flame can also spread in the downstream direction from the ignition kernel, this is wind-aided or co-current flame spread. The ignition source is removed once the flame begins to spread.

The initially quiescent ambient temperature gas is described using the following restrictions: flow is laminar; viscous dissipation and compressive work are neglected; the gas-phase chemical reaction is one-step, second-order, irreversible and exothermic; thermal and flow properties in the gas vary with temperature; and the gas is ideal. For the solid, heat of pyrolysis is neglected; thermal conductivity and heat capacity are linear functions of temperature; thermal swelling, shrinkage and surface regression are neglected; volatiles leave the solid instantaneously; and volatile solid degradation products flow only towards the heated (top) surface because of the heightened porosity of the burning solid layer.

We now write the governing equations for the general numerical model.

### (a) Gas-phase governing equations

The equations for conservation of mass, streamwise (*x*) and transverse (*y*) momentum, energy and species (fuel and oxidant) are
3.1
3.2
3.3
3.4
3.5
where *i*=fuel (F), oxidizer (O) for equation (3.5). The ideal gas equation is *p*=*ρ*_{g}*RT*. The reader should note that the definitions of model variables discussed in this and subsequent sections are given in table 1 of the electronic supplementary material (ESM).

### (b) Condensed-phase governing equations

The condensed-phase equation for conservation of energy is 3.6 Species conservation requires careful examination of the solid degradation process.

### (c) Decomposition models

Two models of solid-phase degradation will be analysed.

#### (i) Cellulosic solid phase with multi-step decomposition reaction

The Broido–Shafizadeh scheme, describes the thermal degradation of a cellulosic fuel into gaseous volatiles and residual char. Upon suffering external heating, the virgin cellulose forms an ‘intermediate’ or ‘active’ cellulose, whose chemical composition resembles virgin cellulose. Although its existence was a subject of debate (see Villermaus *et al*. (1986) and especially Varhegyi *et al*. (1994) and references therein), the experiments of Lede *et al*. (2002) have verified the presence of active cellulose. This compound, once formed, decomposes through a multi-step temperature-dependent rate mechanism into volatiles and char, see figure 1. The char breaks down to ash, whereas the volatiles (consisting of light, reactive species) diffuse through the degraded cellulosic matrix into the gas, forming the hydrocarbon species that mix and react, as gaseous fuels, with the oxidant. The gas-phase chemical reaction produces the spreading flame (figure 2).

The Broido–Shafizadeh scheme was proposed by Dr A. Broido (Broido & Weinstein 1971; Broido & Nelson 1975; Broido 1976). It has been described in detail and in the review article by Shafizadeh (1984). It was also analysed theoretically and numerically (Wichman & Oladipo 1993, 1994; Wichman & Melaaen 1994). The scheme is based on extensive experimental research using small, thin samples of cellulose at constant, elevated test temperatures. It captures important features of the char–volatile competition (DiBlasi 1992, 1994; Wichman & Oladipo 1993, 1994; Wichman & Melaaen 1994). Other models may produce char and volatiles in fixed proportions. The Broido–Shafizadeh model produces them in variable proportions, depending on the temperature and the heating rate.

The cellulose decomposition processes is described by the rate equations
3.7a
3.7b
3.7c
3.7d
where , *i*=cell, char, volatiles. Equation (3.7*a*–*d*) is solved subject to the initial conditions *y*_{cell}(0)=1, *y*_{active}(0)=*y*_{char}(0)=*y*_{volatiles}(0)=0 and the solid-fuel sample is initially virgin cellulose, without any active cellulose, char or volatiles. During decomposition, mass conservation requires *y*_{cell}+*y*_{active}+*y*_{char}+*y*_{volatiles}=1. The chemical rate constants in equation (3.7*a*–*d*) are given in table 2 of the ESM. During decomposition, the average density of the fuel sample changes as the mass fractions of cellulose, active cellulose, char and volatiles change. The cellulose, active cellulose and char are solid-phase constituents. The volatiles constitute the gas-phase fuel species.

During degradation, the sample specific heat and thermal conductivity can be approximated by
3.8a
and
3.8b
respectively. In our model, the volatiles leave the solid fuel by instantaneous transport, not through a transport mechanism involving fuel sample porosity, raised internal gas pressure and tortuous internal flow paths (Slattery 1999). Based on this restriction, the two terms *y*_{volatiles}*c*_{p volatiles} and *y*_{volatiles}*k*_{ volatiles} in equation (3.8*a*,*b*) are ignored, implying that the volatile products of solid-phase degradation do not contribute to the average specific heat and thermal conductivity of the sample. The physical properties in equation (3.8*a*,*b*), along with equation (3.6) couple the Broido scheme equation (3.7a–*d*) with the gas-phase equations (3.1)–(3.5).

#### (ii) Cellulosic solid phase with one-step decomposition reaction

The second solid-phase decomposition model is a one-step scheme. Here, the decomposition of the solid into volatiles and residual char occurs via
3.9
where , *M* is the total sample mass at any instant of time and *M*_{p}=*M*−*M*_{char} is the mass of the remaining solid. The final char mass *M*_{char} is treated as a constant with a specific numerical value. This reaction behaves as though the char were present as an inert pre-existing component of the solid, which remains behind once the volatiles have vanished from the solid matrix. Unlike the three-step Broido–Shafizadeh model, there is no competition between volatilization and charring.

### (d) Initial and boundary conditions; solution procedure

In the gas phase, the temperature is initially ambient and there is no fuel in the gas since the heat flux has not been started. There is no solid-phase degradation. The initial velocity is zero. The gas phase initial conditions are 3.10 In the solid phase, the initial conditions are 3.11

For the far-field boundary conditions, we impose an isothermal inflow condition at *x*=0, 0<*y*<*l*_{ gy}, where *l* denotes the linear dimension of the computational domain. At the outlet boundary *x*=*l*_{ x}, 0<*y*<*l*_{ gy}, the zero gradient (‘weak’) condition is applied. A weak condition is also imposed at the upper boundary *y*=*l*_{ gy}, 0<*x*<*l*_{ x}. An isothermal boundary condition is applied at the lower wall *y*=−*l*_{ sy}, 0<*x*<*l*_{ x}, which is essentially the bottom of the solid sample. Thus, the far-field conditions are
3.12a
3.12b
3.12c
3.12d
3.12e
3.12f
At the interface between the gas and solid, we impose the no-slip condition for the streamwise flow, and standard continuity of heat flux and species fluxes
3.13
The mass flux from the solid is mathematically calculated for both types of solid as the integral across the full solid thickness of the time rate of change of the volatile mass per unit volume, *viz*.
3.14
The initial boundary value problem is solved by starting the flow at time *t*=0, along with the incident heat flux impinging on the surface at the prescribed ignition spot. The surface heats up and decomposes, producing volatiles (gaseous fuel, F) that enter the gas and mix with the inflowing air. The solid fuel must produce a sufficient quantity of fuel vapour. For a suitable range of gas-phase parameters, heating of the ignition spot eventually ignites the gas mixture. If the gas-phase pre-exponential factor is too small, the gas-phase reaction will never ignite. Some trial-and-error is required. The ‘normal’ parameter values used in most hydrocarbon combustion simulations suffice. Parameter values are given in table 2 of the ESM.

The numerical analyses were performed using a domain consisting of a Cartesian gas-phase channel and the condensed phase. The computation uses the iterative alternate direction implicit procedure. The gas-phase flow field is calculated using a SIMPLEC-based scheme employing under-relaxation techniques. The numerical grid cells are rectangular. The grid at the upper gas–solid interface has the smallest size, Δ*y*=0.05 mm. Other grid sizes along both positive and negative *y* directions increase in algebraic order as Δ*y*, *r*Δ*y*, *r*^{2}Δ*y*, etc. (*r*=1.07). Along the streamwise direction, we retain Δ*x*=0.1 mm. A staggered grid is employed. Scalar quantities are evaluated at the cell centre, velocity components at the cell faces. The two-dimensional computational domain is 2×3.16 cm, with grid dimension 200×160.

## 4. Theoretical model

In this section, we shall first consider the large-scale (global) structure of the solution in terms of *macro* field variables (§4*a*), then the small-scale structure of the solution in terms of the *micro*variables (§4*b*). Two macro variables are the mixture fraction, *Z*, and the total enthalpy function, *H*. Micro variables include local quantities such as the flame-quenching distance. The goals are to characterize a simplified flame-spread problem and to compare the accuracy of the predictions with the numerical solutions of equations (3.1)–(3.14).

It will be shown that the mathematical formulas produce accurate qualitative agreement with the numerical solutions, even though the latter constitute far more detailed and presumably precise models of the spread process.

### (a) Macrostructure of a spreading diffusion flame

Here, we employ the mixture function variable2*Z*=(1−*Z*_{f})*y*_{F}+*Z*_{f}(1−*y*_{O}) and the total enthalpy variable3*H*=*τ*+*y*_{O}+*y*_{F}−1 and we write the following three coupled governing equations:
4.1
where
is the non-dimensional convective/diffusive differential operator. In equation (4.1), *D* is the Damköhler number, *ϕ* is the stoichiometry parameter, is the dimensionless reaction-rate function and *ξ*=*x*/*L*, *η*=*y*/*L*, where *L*=*α*/*u* is the ratio of the gas-phase thermal diffusivity and the constant streamwise (Oseen) gas opposed-flow velocity. The fuel and oxidant mass fractions are given in terms of *Z*, *H* and *τ* in the following form:
and
The simplified gas-phase conservation equation (4.1) (compare with equations (3.1)–(3.5)) was derived by assuming an Oseen flow with characteristic velocity everywhere unity in the streamwise (*ξ*) direction and zero in the transverse (*η*) direction. The thermophysical properties were assumed constant and the Lewis number was assumed to be unity. The chemical reaction is one-step and irreversible, *F*+ν*O*→(1+ν)*P* on a mass basis. The reaction-rate function is linear in the oxidant (*y*_{O}) and fuel (*y*_{F}) normalized mass fractions.

All three equations must be solved simultaneously since they are coupled through the boundary conditions, by the definitions of *Z* and *H* (which both contain *y*_{F} and *y*_{O}) and by the chemical reaction rate function
Nevertheless, means exist to simplify the problem so that a reduction to two homogeneous equations *L*(*Z*)=0, *L*(*H*)=0 is possible.

#### (i) Solution for *Z*

The solution for *Z* is coupled to the condensed phase through the gasification zone, as in droplet combustion analyses (Williams 1985). We restrict our theoretical analysis to the ‘simple vaporizing solid’ (Wichman 1992) where surface gasification occurs when the solid-fuel surface temperature is near the gasification temperature. This constitutes the simplest possible solid-phase model, which is simpler (by far) than the one-step or the Broido–Shafizadeh models of §3. The equilibrium partial pressure of gasified fuel is given by
where *P*_{v} is the partial pressure of the volatilized fuel when the surface temperature *T*_{s} is lower than *T*_{v}. In the limit as , gasification occurs across a step front that we choose to locate—without loss of generality—at the origin. Since the partial pressure is proportional to the mass fraction,
4.2

Our second principal restriction is to consider infinite-rate chemistry. Thus, there can be no fuel on the (upstream) oxidant side of the flame and vice versa. Upstream of the flame attachment point, there is no gasification of the surface () so that, from equation (3.13), we have
This leads to the upstream interface boundary condition
Also, there is no gasification between *ξ*=−*Δ* and the origin. Thus,
leading to
For *ξ*>0, we have *y*_{F}=*y*_{Fv} and *y*_{O}=0, giving *Z*=*Z*_{v}=(1−*Z*_{f})*y*_{Fv}+*Z*_{f}.

In the far upstream and transverse fields, *y*_{F}=0 and *y*_{O}=1, so that *Z*=0. In the far downstream, we impose the ‘weak’ boundary conditions
These boundary conditions are illustrated in figure 3. The standoff distance separates the attachment point (*ξ*=−*Δ*) from the gasification front (*ξ*=0).

The mathematical solution for *Z* is (Carrier *et al*. 1966)
4.3
Deductions can be made from equation (4.3) for various quantities.

#### Flame shape and location; flame fuel, oxidant and heat fluxes

The flame sheet attaches at *η*=0, *ξ*=−*Δ*, giving *Z*_{f}=*Z*_{v}erfc√*Δ*, indicating that *Z*_{f}<*Z*_{v}, as expected. When there is no standoff separating the flame attachment point and the gasification point, we find that *Z*_{f}→*Z*_{v} from below. The flame sheet is located along the locus *Z*(*ξ*,*η*)=*Z*_{f}, giving, from equation (4.3), *η*=2Δ[1+*ξ*/Δ]^{1/2} for . The unit normal vector perpendicular to the flame sheet is therefore given by
4.4
which points from the fuel side toward the oxidant side of the flame sheet. Hence,
where
is a function that monotonically increases from μ(0)=0 to . Since the fuel and oxidant sides of the flame have *y*_{O}=0 and *y*_{F}=0, respectively, then, from the definition of *Z*, we obtain directly
4.5
Since the unit normal vector points toward the fuel side of increasing *Z*, the oxidant flux is positive while the fuel flux is negative. As *Δ*→0,
respectively. As , these gradients vanish, indicating the occurrence of downstream diffusion flame broadening.

The heat flux liberated at the flame is calculated by forming the product
where *Q* is the heat released per unit mass of oxidant consumed, approximately 13.8 MJ kg^{−1} O_{2} for most hydrocarbon fuels, while *ρ*_{O}, *D*_{O}, and *L* are the oxidant density, mass diffusivity, far-field mass fraction and characteristic gas-phase length. Using this expression for *Q*, the flame heat flux is
4.6
The total heat flux per unit width to the surface is defined as the integral
which yields
4.7
This reduces to as *Δ*→0. The analysis predicts an approximate square-root dependence on distance downstream, starting from the point of surface gasification.

#### Mass flux from the gasification surface

From equation (3.13), the fuel and oxidant fluxes from the gasifying surface can be written as
and
which simplify to
and
respectively, when we define as the non-dimensional mass flux into the gas. These two boundary conditions can be combined to give
4.8
which states that the mass flux profile can be deduced from the mathematical solution for *Z*. Use of equation (4.3) yields
4.9
which will later be compared with the numerical computations from the one- and three-step solid-fuel decomposition models formulated in §3*c*. This derivation of *M*(*ξ*) arises solely from the boundary conditions at the non-regressing interface coupled to the gas fuel and oxidant species fields. However, several qualitatively realistic predictions are made. First, there is a high mass efflux near the gasification front. Second, a significant diminishment of the mass flux occurs with increased downstream distance; equation (4.9) predicts an inverse square-root decay. Third, the integrated mass flux should increase with the square root of distance from the origin gasification front.

#### Distribution of fuel and oxidant mass fractions along the fuel surface

Upstream of the flame attachment point along sector ν_{1} (figure 3) we have, according to the definition of *Z* and using *y*_{F}=0, the result *Z*=*Z*_{f} (1−*y*_{Os}), so that *y*_{Fs}=0 and *y*_{Os}=(1−*Z*(*ξ*,0)/*Z*_{f} along the surface. At the attachment point, we have *Z*=*Z*_{f}, so that *y*_{Fs}=*y*_{Os}=0 there. Along sector ν_{2} downstream of the attachment point, we have *y*_{Os}=0 and *Z*(*ξ*,0)=(1−*Z*_{f})*y*_{Fs}+*Z*_{f}, so that *y*_{Fs}=(*Z*(*ξ*,0)−*Z*_{f})/(1−*Z*_{f}). Along ν_{3}, we have *y*_{Os}=0 and *y*_{Fs}=*y*_{Fv}=(*Z*_{v}−*Z*_{f})/(1−*Z*_{f}). Using the solution given by equation (3.3), along ν_{1} the condition on *y*_{Os} is given by *y*_{Os}=1−*Z*(*ξ*,0)/*Z*_{f}=1−*Z*_{v}[erfc√|*ξ*|]/*Z*_{f}=1−[erfc√|*ξ*|]/erfc√Δ. Expanding this formula in a Taylor’s series in the vicinity of *ξ*=*Δ* for small values of ϵ=|*ξ*|−*Δ* gives
4.10a
Similarly, along ν_{2}, we write
Expanding in a Taylor’s series in the vicinity of *ξ*=*Δ* for small values of *δ*=*ξ*+*Δ* yields
4.10b
These distributions are illustrated in figure 4*a*, showing that *y*_{Os} is concave downward with negative curvature near the flame attachment point, whereas *y*_{Fs} is concave upward with positive curvature. Based on these profile shapes, one would expect more oxygen transport across the attachment front than fuel leakage. The numerical results computed and discussed in §5 are shown for comparison in figure 4*b*.

#### (ii) Solution for *H*

The enthalpy function *H* is coupled to the condensed phase and to the gas-phase thermal and species fields. Consequently, it is not possible to solve for *H* until these other solutions themselves are known. Decoupling can be achieved only if the boundary conditions can be simplified, thereby allowing an approximate form of *H* to be deduced. We follow this latter approach, keeping in mind that our primary question is: how accurate, qualitatively and quantitatively, is this approximate solution for *H*? We can answer this question only by comparing the approximate solution with the detailed numerical solution.

The elliptic differential equation for *H* given in equation (4.1) requires boundary conditions at all surfaces. In the far field (), we have *y*_{O}=1,*y*_{F}=*τ*=0, so that *H*=0 there. Far downstream, *∂**H*/*∂**ξ*=0.

Along ν_{1} (figure 3), *y*_{Fs}(*ξ*)=0,*y*_{Os}(*ξ*)=1−*Z*(*ξ*,0)/*Z*_{f} and *τ*(*ξ*)=*τ*_{s}(*ξ*,0). Thus,
Since both *τ*_{s} and *Z*_{s} vanish far upstream, *H*_{s}→0 along the far-upstream interface. At the attachment point *ξ*=−*Δ*, we have *Z*_{s}=*Z*_{f} and *τ*_{s}=*τ*_{s}(−*Δ*), giving
Since *τ*_{s}(−*Δ*) is known, the value of *H*_{s}(−*Δ*) is uniquely determined.

Along ν_{2}, *y*_{Os}(*ξ*)=0,*y*_{Fs}(*ξ*)=(*Z*_{s}(*ξ*,0)−*Z*_{f})/(1−*Z*_{f}), *τ*=*τ*_{s}(*ξ*) and
which is clearly less negative than *H* at the attachment point, *H*_{s}(−*Δ*).

On ν_{3}, we have *H*_{s}(*ξ*)=*τ*_{v}−(1−*Z*_{v})/(1−*Z*_{f}).

Since the solid surface heated by the flame acts as an enthalpy sink to the gas, we expect *H*_{s}<0 there. We therefore expect that *τ*_{v}<(1−*Z*_{v})/(1−*Z*_{f}).

Assembling all of these boundary conditions into a single function suggests that the *H* should resemble the distribution shown in figure 5. Upstream, *H*=0. *H* then decreases to a minimum at the flame attachment point (where enthalpy losses to the surface are largest) and subsequently increases to a constant, negative value in the gasification region. The entire surface is an enthalpy sink to the gas phase, with the greatest sink located directly under the point of flame attachment.

Figure 6 shows the solution for the boundary value problem for *H* defined by equation (4.1). To obtain a numerical solution, it was assumed that the surface distribution of the temperature functionally resembled the surface distribution of the mixture fraction:4 thus, *τ*_{s}∼*τ*_{v}erfc[−*ξ*]^{1/2}=*τ*_{v} (*Z*(*ξ*,0)/*Z*_{v}). Along ν_{1} upstream of *ξ*=−*Δ*, this leads to *H*={*τ*_{v}/*Z*_{v}−1/*Z*_{f})*Z*(*ξ*,0). Along ν_{2} and ν_{3} downstream of *ξ*=−*Δ*, this leads to *H*={*τ*_{v}/*Z*_{v}−1/*Z*_{f})*Z*(*ξ*,0)−1/(1−*Z*_{f}). On ν_{1} and ν_{2}, we put *Z*(*ξ*,0)=*Z*_{v}erfc(−*ξ*)^{1/2} and on ν_{3}, we write *Z*(*ξ*,0)=*Z*_{v}. The numerical parameter values used to compute figure 6 are *Z*_{f}=0.1, *Z*_{v}=0.2, *τ*_{v}=0.2 and *Δ*=0.1. In figure 6*a*,*b*, the minimum of *H* is at *ξ*=−*Δ*, *H* increases to zero in the upstream far field and to a constant negative value in the downstream far field, in accordance with figure 5. It will be recalled that this solution is valid only for the theoretical model constant-property Oseen-flow case. The kink in the *H*-profile upstream of the minimum is caused by the numerical use of a finite domain. If that domain were extended upstream indefinitely, the *H*-profile would rise without any points of inflection to zero in this region.

Near the attachment point *ξ*=−*Δ*=−0.1, the *H*-contour of figure 6*b* can be considered to be approximately symmetric. As a consequence of that symmetry, the model configuration examined in Wichman *et al*. (1999) becomes applicable and relevant. Thus, the model problem examined here is amenable to qualitative description in its region of strongest variation by the model solved in Wichman *et al*. (1999). In that model, the extinguished gaseous diffusion flame produced an enthalpy sink region that *symmetrically* traversed both sides (fuel and oxidant) of the attached diffusion flame. We note also that, very near the attachment point, the value of *H* is smallest (i.e. most negative), as suggested by the discussion of the boundary conditions illustrated in figure 5.

Later in this article (conclusion (v)), the surface temperature distribution *τ*_{s}(*ξ*) will be compared with the assumed error-function profile.

#### (iii) Flame-stretch calculation

For the two-dimensional flame, the formula for the flame stretch takes the simple form *X*=d*v*_{s}/d*s*, where coordinate *s* traverses the flame arc and is the tangential component of the gas velocity, where *α*(*s*) is the angle between the streamwise velocity *u*(*s*) and the local unit normal to the flame sheet. The flame stretch becomes
4.11
The stretch is evaluated at each point *s* along the flame contour to determine where, exactly, the stretch is largest. The Oseen flow has *u*=1 and therefore . Along the flame sheet, the calculations above equation (4.4) yield the flame shape *η*=2*Δ*[1+*ξ*/*Δ*]^{1/2}, whereby, along the flame arc, d*s*=[d*ξ*^{2}+*d**η*^{2}]^{1/2}=[(2+*ξ*/*Δ*)/(1+*ξ*/*Δ*)]^{1/2} d*ξ*. Thus, in equation (4.11), d*α*/d*s*=[(1+*ξ*/*Δ*)/(2+*ξ*/*Δ*)]^{1/2} *d**α*/d*ξ*. From the definition of the flame unit normal vector, one may write
whence
Using all of these results in equation (4.11) produces the flame-stretch formula
4.12
The flame stretch is zero at *ξ*/*Δ*=−1 and at . It has a single maximum value given by *X*=0.182/*Δ* at *ξ*/*Δ*=−1/3. Figure 7 shows *X* for the case *Δ*=0.1. The stretch increases linearly for small *ξ*/*Δ* and decreases in proportion to (*ξ**Δ*)^{−5/2} for large *ξ*/*Δ*.

The Oseen flame model produces a maximum flame-stretch rate slightly upstream of the surface pyrolysis region, one-third of the distance from the pyrolysis front to the (fictitious) flame attachment point. Figure 7 compares the theoretically predicted calculation (figure 7*a*) with the numerical results (figure 7*b*) from the full simulation of the equations of §3 (for the Broido–Shafizadeh model). These curves are qualitatively similar, with maxima located very close to the flame attachment point.

This maximum stretch rate must be resisted by the flame if it is to avoid blowoff and extinction to continue to spread. The resistance to stretch is accomplished in two ways. One is through a local increase of the flame reactivity. The other way by which high stretch is resisted (which is not modelled in this article) is by the process of flame tunnelling, where the spreading flame, faced with a high-speed forced-opposed-inflow of oxidant, burrows into the surface to form a parabolic ‘trough’ that allows the flame to spread, albeit slowly, against the high-speed-opposed inflow. This mechanism is analogous to high-speed flameholding.

### (b) Microstructure of a spreading diffusion flame

The flame microstructure enters into consideration when one examines closely the spreading diffusion flame near its upstream leading edge. Here, the length scales become very small in comparison with the remainder of the flame, being of the order of millimetres to fractions of millimetres rather than the macroscale of centimetres or tens of centimetres. The ‘micro’ length scales characterize quantities such as the quenching distance from the wall and the reaction-zone size. These length scales allow the calculation of quantities such as the heat flux to the solid surface and the characteristic widths of heat and mass efflux zones. Additional scalings exist for the reaction-zone intensity and magnitude of the flame heat release. This permits calculation of the heat flux to the sample surface near the flame leading edge. The goal is to produce heat and mass flux distributions more realistic than those given by global equations (4.6)–(4.9).

The procedure for flame microstructure calculations is different to that of the macrostructure. Here, it is imperative that the reaction zone be of finite (not infinitesimal) size. This constraint preserves at least one reaction-zone length scale, which, for a wall-quenched flame, measures streamwise distance near the flame tip. This length scale will be different from the diffusion flame thickness, which scales with the inverse Zeldovich number *β*^{−1} (Linan 1974).

Another length scale that must be preserved is the quenching distance. Since this is a difficult quantity to calculate, even for a quiescent gas in an idealized geometry (Wichman *et al*. 1999), here the influences of streamwise convection are neglected and the quenching characteristics of the flame tip are calculated in the absence of these real effects. Such calculations have been performed previously by Wichman & Ramadan (1998), Wichman *et al*. (1999) and Wichman (1999). For the flame leading edge (tip) characteristic length scale (figure 1), we have
4.13
where the dimensionless multiplicative constant is of order unity. As the flame temperature increases, the characteristic flame width decreases through both factors in equation (4.13). For the heat flux to the sample surface beneath the flame leading edge, we have
4.14
The constant is given as 0.572 in Vance & Wichman (2000). Although the spreading flame here is not globally symmetric (*Z*_{f}≪1/2), it is true that, near the point of flame attachment, the spreading flame resembles the symmetric model of Vance & Wichman (2000); see also the discussion in §4*a*(ii) above. In equation (4.14), the coordinate *δ*=*ξ*+Δ measures non-dimensional distance from the point beneath the flame leading edge, and *r*_{q} is the quench distance measured from beneath the flame leading edge to the point of maximum flame reactivity. The quenching distance between the flame tip and the sample surface is given by (Vance & Wichman 2000)
4.15

A similar deduction can be made from a different approach in which a preflame transport-layer theory is used to describe the evolution of a diffusion flame leading edge as it approaches a cold wall from the gas side (Wichman 1999). Wichman (1999) showed that a reduced Damköhler number *d* varied between zero and unity: when *d*→0, the flame was ‘free’ and isenthalpic; as *d*→1, the flame attached to the surface (non-isenthalpic), whereby the transport layer reduced to the quenching distance, yielding equation (4.15). In the theories leading to equation (4.15), the factor ‘const.’ is a function of *Z*_{f}. Near the flame leading edge, equations (4.14) and (4.15) give *q*∼const. (*D*/*β*^{3})^{1/2}; a similar expression was derived in Wichman *et al*. (1997). This quantity scales the heat flux to the solid surface beneath the flame tip. Another scaling, which is related to the *q* scaling, examines the variation of the heat flux with streamwise distance from the flame tip. Here, the distance near flame attachment *δ*=*ξ*+*Δ* is scaled with *l*_{ r} of equation (4.13). The heat flux should therefore be scaled with and the distance from *ξ*=−*Δ* is scaled with *l*_{ r}.

In fact, all quantities that vary in the streamwise direction near the flame attachment point must be scaled with *l*_{ r}. These include the surface and gas-phase distributions of *y*_{F} and *y*_{O}, *Z* and *H*. The latter varies most near *ξ*=−*Δ* (figures 5 and 6). In particular, the mixing of the reactants near the flame leading edge and in the quench layer should resemble the distribution described in Wichman & Ramadan (1998), where an onion-shaped intermeshing set of isochors indicated diffusive mixing. Without this mixing, the flame-tip structure, consisting of a ‘terminated’ gas-phase flame with its point of extremely high reactivity5 located almost exactly at its termination point, would not be possible. The chemical reactivity function in the reaction zone scales as
4.16
where *l*_{ q}*l*_{ r} represents the cross-sectional area of the diffusion flame leading edge. The reaction order is *m*=*n*=1. The dimensionless reactivity function should have a maximum value near unity; the length-scale product *l*_{ q}*l*_{ r}, as well as the use of order unity normalization mass fractions *Y*_{OO} and *Y*_{FF}, play important roles in the attainment of this maximum value.

It was deduced in Wichman *et al*. (1999) that, between the surface and the flame leading edge, the temperature variation is concave upwards. Thus, the temperature rate of increase is lowest near the surface and highest near the flame tip. This profile (i) maximizes reactant consumption by assuring a high temperature and reactivity at the flame and (ii) minimizes heat losses by the flame tip, since conduction to the wall is reduced.

The non-dimensional mass flux can be normalized by the quantity *Z*_{f}/*r*_{q}(1−*Z*_{f})=(*r*_{q}*ϕ*)^{−1}. The latter is a flame ‘microstructure’ parameter, whereas the former non-dimensionalization for *M* was deduced from its macrostructure. The relevant non-dimensional quantity is therefore *m*(*ξ*)≡*Mr*_{q}*ϕ*∼const. *M**ϕ*(*β*^{3}/*D*)^{1/2}=const. 2/(*π**ξ*)^{1/2}*ϕ*(*β*^{3}/*D*)^{1/2}. If we define an average value of *m*(*ξ*) as and we use *Z*_{f}∼*O*(10^{−1}), *β*^{3}/*D*∼*O*(10^{2}/10^{4})∼*O*(10^{−2}), we obtain

## 5. Numerical simulations and comparisons

In the numerical analysis, the initial conditions are zero flow and ambient air over an unheated sample. At *t*=0, a flux of heat is applied to the surface between 0.015*m*<*x*<0.016*m* until the gas above the surface ignites and a flame grows from the ignited flame kernel. The nascent flame propagation speed is unsteady while it grows. Once the flame size is fixed, its spread rate is constant. This steady flame-spread regime will be compared, with the steady theoretical solutions of §4. The quantities examined are temperature, species, velocity, density, mixture fraction, excess enthalpy and reaction-rate fields, the heat flux to the surface, the mass flux from the surface, and the integrated heat and mass fluxes to (from) the surface.

It must be re-emphasized that the theoretical model assumes that all thermophysical and transport parameters are fixed, whereas the numerical model assumes that they vary with temperature and composition. Certain differences between the predictions must necessarily be manifested.

In the numerical analysis (§3), two types of solid materials were considered, the three-step (Broido–Shafizadeh) decomposing solid with a competitive char/volatiles sub-mechanism and the one-step decomposing (cellulosic) solid. The decomposition waves propagate into the virgin solid while leaving behind char. In this feature, the numerical model that describes flame spread over a ‘realistic’ solid fuel is different from the simplified theoretical model that describes flame spread over an ‘ideal vaporizing solid’ (Wichman 1992). In fact, the comparison was conducted with these differences in mind since it was desired to determine how ‘realistic’ the ideal vaporizing solid model is.

Shown in figures 8 and 9 are the *H*- and *Z*-distributions at *t*=12 s after ignition. Here, steady propagation has been established. The *H*-distribution resembles figure 6*a*, even though the numerically computed profiles are slightly more complicated in appearance near the diffusion flame leading edge. The constant-*H* contours in figure 6*a* have a maximum of one inflection point (in fact, only those constant-*H* contours originating very close to the minimum of *H*), whereas some of the contours in figure 8 (with *H*=−0.244, −0.341, −0.537, etc.) clearly have two inflection points. In figure 8, the second inflection point arises from the upward displacement of the *H*-contour towards larger *y* as *x* increases in the streamwise direction. At the surface, the distributions of the *H*-contours are qualitatively identical to those illustrated in figure 5 and calculated in figure 6*b*, suggesting that the theoretical arguments of §4 concerning the functional form of the boundary conditions are valid. In the gas, the variable-property full-flow models produce complicated *H*-contours near the flame arc, some closed and decreasing in all directions outward, while others develop inflection points.

Shown in figure 10*a* is the volatile mass flux from the sample surface into the gas, displaying the beginnings of the ‘dipole’ gas distribution of *y*_{F} when the flame burns out in the near-downstream region. The upstream peak is approximately double the downstream peak, but the latter is wider by the same amount. The integrated distributions are expected to be approximately identical in these two regions, as demonstrated in figure 10*b*. Note that the integrated distribution does not follow a *ξ*^{1/2} (or rather, *x*^{1/2}) functional form, but instead it lags the theoretical distribution near the first peak and, because of the appearance of a second peak, exceeds the theoretical distribution. Equation (4.8) reproduces the functional form and magnitude of the computed theoretical *M*-profiles: this profile was used in the boundary conditions of the numerical model of §3. The implication, however, is that, regardless of how a particular *Z*-profile is obtained, equation (4.8) can be used to deduce the qualitative surface mass flux distribution. For the idealized mathematical model of §4, it yields equation (4.9), which is singular at the origin.

Shown in figure 11*a* is the numerically computed heat flux distribution, *q*(*ξ*). The integrated distribution resembles the theoretical *ξ*^{−1/2}-profile of §4, as shown in figure 11*b*. This result is obtained despite the fact that the theoretical profile was deduced for an ‘ideal vaporizing solid’, whereas the numerical computation was generated for a charring cellulosic material. Absent from the *q*-profile of figure 11*a* is a second, downstream local peak. As in the simplified theory of equation (4.6), the heat flux profile decreases monotonically. There are some differences from the simplified model. From figure 11*b*, it is seen that the theoretical integrated flux commences abruptly at *x*∼−0.004 m upstream of the origin, whereas the numerical profile begins to increase gradually from approximately −0.006 m upstream of the origin. This numerical increment is produced by upstream heat transfer from the lifted flame leading edge. Note how truly small this increment is. Another difference arises downstream, where the two curves diverge because the rate at which the numerical *q*(*x*)-profile diminishes is smaller than the theoretical *q*(*x*)∼*x*^{−1/2} rate. The cause is attributed to the additional heat flux contribution to *q*(*x*). This contribution is absent from the theoretical flame, which has no trailing edge.

Note that the theoretical model predicts that the heat flux peak precedes the first mass flux peak. This is observed in the numerical simulations.

Shown in figure 12*a*,*b* are surface distributions of various computed global quantities. In figure 12*a*, the surface distribution of *H* produces the negative minimum at the flame ‘attachment’ point. Also shown is the concomitant increase to zero of *H* in the upstream as well as the local *H*-maximum in the fuel pyrolysis zone downstream of the flame leading edge. All features are shared by the theoretical model (figure 6*b*), although the theoretical curves are smoother, its extremes larger (particularly the minimum, which is also narrower) and pyrolysis zones much flatter. Qualitatively, figures 6*b* and 12*a* are identical.

We now consider the diffusion flame microstructure near the flame leading edge, as described by the theoretical formulas of §4*b*. We focus our attention on the characteristic length scales *l*_{ r} and *r*_{q}.

Several examinations of *l*_{ r} and *r*_{q} were carried out by varying the reaction parameters *A*_{g} and *E*_{g} from ±2.5 per cent to ±20 per cent. We calculate the ratio *r*_{q}/*r*_{qBASE}, where *r*_{qBASE}, is the base case flame-quenching distance calculated from the parameters in table 2 of the ESM. Shown in figure 13 are the numerical and theoretical profiles of *r*_{q}/*r*_{qBASE} versus *E*_{q}/*E*_{qBASE}. The curves show an identical trend (monotonic increase). In figure 14, we plot
5.1
versus *A*_{q}/*A*_{qBASE} and *E*_{q}/*E*_{qBASE}. Neither profile yields order-unity flat lines. The constant in equation (4.15) is not constant in the numerical model. It likely depends on global stoichiometry (*ϕ* or *Z*_{f}).

## 6. Conclusions

Possible extensions and generalizations of this research include: (i) issues related to variations of the Lewis and Schmidt numbers (which are constant in this article). These parameters affect edge flame structure in attached and lifted burner flames. Their influence on spreading flames should also therefore be investigated. (ii) Examination of flame-spread response to variable stretch. Although this has been carried out theoretically in the early 1980s (see Wichman 1983), a numerical study that examines this flow-field influence is yet to appear.

Several conclusions can be drawn. These are listed below.

(i) The theoretical constant-coefficient infinite reaction-rate Oseen-flow model produces accurate qualitative functional dependences for quantities ranging from surface oxidant and fuel distributions, to surface total enthalpy function distributions, and even to integrated heat and mass flux distributions. Equations (4.8) and (4.9), which predict an inverse square-root dependence on streamwise distance along the pyrolysis front from a boundary condition (equation (4.8)) without solving a solid-phase problem, stand as the most potentially unphysical of the results. Nevertheless, as the computations show, this function produces a strong qualitative resemblance to the numerical solution of figure 10*a*. The integrated mass flux of figure 10*b* is also a reasonable approximation to the integrated inverse square-root distribution.

(ii) Near the point of flame attachment, the enthalpy ‘well’ of figure 6*b* is reasonably symmetric, making viable the theoretical analysis of simplified models of flame leading edge structure based on that fact (and on the symmetric boundary condition for *H* there). The same qualitative dependence is observed in the numerical solutions (figure 12*a*,*b*) although in less stark form. Simplified models of flame leading edge structure can elucidate numerous details that simulations of full flame spread and flame-attachment problems cannot. Examples of such simulations for attached and lifted jet diffusion flames can be found in Takahashi & Katta (2002, 2003). Despite the accuracy of these numerical computations, they cannot quantify relationships or predict the functional dependence of quantities that a model can.

(iii) In the discussion of figure 8, it was observed that the field contours of figure 8 possessed up to two inflection points (e.g. the contour of *H*=−0.244), whereas the contours of figure 6*a* possessed at most one inflection point. The second inflection point arises from the change in curvature of the downstream *H*=−0.244 contour (for example) as it rises, falls and rises once again. In figure 6*b*, by contrast, a similar *H*-contour will rise and then fall; the subsequent rise does not occur. Why is this? Partly because of the weak boundary condition *∂*(·)/*∂**x*=0 imposed downstream. If the zero-downstream gradient condition were to be relaxed for the numerical model (or at least imposed very far downstream) there would doubtless appear a rise of the *H*-contours there. Another important difference is that the downstream heat sink for the theoretical model persists indefinitely, whereas for the numerical model, the flame terminates and the heat sink essentially disappears. As the values of *H* along the surface rise towards zero, the low-*H* contours are forced further outward into the gas.

(iv) In figure 8, it is seen that the contour *H*=−0.244 after emerging from the upstream interface first rises and then, after reaching a maximum *y*-value of approximately 0.005 m, drops backward towards the surface while also moving upstream. Thus, this contour intersects at the vertical line *x*=−0.0075 m thrice. The total enthalpy distributions for the full numerical solution are more complicated than for the simplified Oseen model as we compare figures 6*a* and 8. When the diffusion flame is lifted from the surface (figure 8), the distribution of *H* produces contours near the quenching flame like the *H*=−0.244 contour that double back. Part of this may arise from the increase of streamwise convection as we move vertically further from the surface, which sweeps the outward *H*-contours further downstream. By contrast, in figure 6*b*, the Oseen model produces uniform streamwise convection independent of transverse position. This constitutes part of the explanation. What is clear is that another part of the explanation cannot be finite-rate chemical kinetics. In one model (the theoretical model), kinetics were infinite-rate, whereas in the other (the numerical model), they were finite-rate. Nevertheless, in both cases, the diffusion flame contour crosses (intersects) the contours of constant *H* in a similar manner. Thus, a second part of the explanation must reside in simple thermal expansion, which was permitted in the numerical model, but forbidden in the theoretical simulation. Thermal expansion of the hot gases produces part of the downstream ‘thrust’ of the *H*-contours.

(v) In the theoretical model, it was postulated, for expedience and simplicity, that the upstream surface distributions of *Z* and *τ* were functionally identical, with *τ*_{s}∼*τ*_{v}erfc[−*ξ*]^{1/2} and *Z*_{s}∼*Z*_{v}erfc[−*ξ*]^{1/2}. Shown in figure 15 is the numerical *Z* profile, and also the numerical *τ*/*Z*-profile along the interface. If these profiles were functionally identical, the ratio would always be constant. The fact that the ratio *τ*/*Z* is not constant upstream of the flame leading edge at *x*=0.005 m indicates that these profiles are not functionally identical in the region upstream of the flame leading edge. Nevertheless, both *Z*- and *τ*-profiles monotonically increase; the latter more slowly in the far upstream and more rapidly in the near upstream. Qualitatively, the *Z*- and *τ*-profiles are identical. Nothing is lost of the qualitative description of the flame attachment process by employing the approximate profile. The exact is preferable, but the error in the former does not alter the nature of the results.

(vi) Figures 13 and 14 show that the theoretically derived flame microstructure parameters are reasonably represented by the numerical simulations. The ratio *r*_{q}/*r*_{qBASE} shown in figure 14 correlates better with the pre-exponential ratio *A*_{g}/*A*_{gBASE} than it does with the activation energy ratio *E*_{g}/*E*_{gBASE} in figure 13. The latter is an extremely sensitive parameter. Although the agreement in figure 13 is not exemplary, the trends and the predicted ranges of values are the same.

(vii) In the research presented here, one remarkable fact has gone thus far unmentioned, namely the locations of the reactivity and stretch-rate maxima coincide. A possible explanation for this fact is that the flame, which self-optimizes for survival, must oppose through enhanced chemical energy release any attempt to quench it through the mechanisms of heat loss and flame weakening by physical stretching. For this reason, the flame must reinforce itself at the point of greatest assault and hence the location of maximum reactivity coincides almost identically in the streamwise direction with the location of the maximum heat loss (which occurs at *ξ*=−*Δ*; figure 5) and in the transverse direction with the maximum gas-phase flow-stretch rate at the surface of the flame. This fact, apparent for wall-attached flame spread over combustible solid fuels, does not coincide with the reality (via computer simulation) of pure gas-phase turbulent flame fronts, where the reactivity maxima and the flame-stretch maxima are nowhere near being coincident (Yaldizli 2008). It is beyond the scope of this article to explain the latter phenomenon, but well within its scope to point out the completely opposite nature of its predictions to the results deduced here.

## Acknowledgements

The authors wish to thank Dr Sandra L. Olson and Dr Fletcher J. Miller for their advice and encouragement and to express their gratitude to the Microgravity Combustion Branch at NASA Glenn Research Center for funding this project under contract no. NCC3-662. I.S.W. would like to acknowledge generous support during the writing of this article from the MSU Strategic Partnership Grants (SPG) programme.

## Footnotes

↵† Present address: NSK Corporation, 4200 Goss Road Ann Arbor, MI 48105, USA

↵1 Only theoretical models can predict singularities, which are inaccessible to computational algorithms. As a rule, the mathematical severity of extreme solution behaviour is far lower in simulations. Thus, they are less stark and consequently more difficult to interpret than skeletal theoretical models.

↵2

*Z*_{f}=(1+*ϕ*)^{−1}is the value of*Z*at the theoretical flame front, where*y*_{O}=*y*_{F}=0.↵3 The total enthalpy is defined as the sum of the thermal (sensible) and chemical enthalpies. When the total enthalpy is constant, its value is unity everywhere in the gas. In the definition given here, the quantity ‘1’ is subtracted from this total enthalpy variable, and thus it becomes an enthalpy defect function in the vicinity of the solid-fuel surface. The flame loses enthalpy to the solid while heating and decomposing it into volatile gases (fuel); the loss of gas-phase enthalpy to the solid is manifested in the creation of local regions in the gas and near the solid where

*H*<0.↵4 This assumption can only be justified by the precedent of previous analytical solutions for the upstream surface temperature, in all of which (Wichman 1992; and references therein) it increases monotonically between and

*ξ*=−|*Δ*|. Even if the fundamental functional*form*of the assumed surface temperature profile is different, the qualitative similarity of the resultant*H*and*Z*profiles will not be affected. It will be shown later that the surface temperature distribution is quantitatively different from, but qualitatively identical to, the assumed profile.↵5 The reactivity at the flame leading edge near a cold wall is of the order of at least 10× that in the downstream one-dimensional diffusion flame.

- Received March 25, 2009.
- Accepted June 17, 2009.

- © 2009 The Royal Society