Stability of attached two-dimensional diffusion flame leading edges

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Stability of attached two-dimensional diffusion flameleading edgesIndrek S. Wichman a & Robert Vance aa Department of Mechanical Engineering , MichiganState University , East Lansing, Michigan, USAPublished online: 17 Sep 2010.

To cite this article: Indrek S. Wichman & Robert Vance (2003) Stability of attachedtwo-dimensional diffusion flame leading edges, Combustion Science and Technology,175:10, 1807-1834, DOI: 10.1080/713713117

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STABILITYOFATTACHEDTWO-DIMENSIONALDIFFUSIONFLAMELEADINGEDGES

INDREKS.WICHMAN*ANDROBERT VANCE

Department of Mechanical Engineering,Michigan State University,East Lansing, Michigan, USA

This article examines the stability of a diffusion flame whose leading edge is

located near the cold wall to which it attaches. Such leading edges appear in

spreading flames, in burner and jet flames, and in flames burning over het-

erogeneous propellants. The flame model is idealized in order to facilitate

analysis and interpretation. The method of analysis follows the examination

of a one-dimensional diffusion flame by Vance et al. (2001, Combust. Theory

Model., 5, 147). First, a basic state attached-flame problem is defined, which is

subjected to small perturbations. The linearized disturbance equations are

solved numerically to determine their eigenvalues, which indicate either sta-

bility or instability in response to infinitesimal perturbations of the basic state.

Many of the one-dimensional model results of Vance et al. (2001) carry over

to the two-dimensional case examined here. It is determined that the leading

edge dictates the behavior of the remainder of the flame. In addition, the

oscillation of the flame leading edge during both flame retreat and flame

advance toward the stable position is examined.

Keywords: diffusion flame, stability, analysis, leading edge, triple flame

Received 19 August 2002; accepted 15 April 2003.

The authors express their gratitude for research support provided by NASA Micro-

gravity Combustion Research Contract #NCC3-662. The contract monitor was Dr. Fletcher

Miller. Mr. Stefauns Tanaya helped with many of the figures.

*Address correspondence to [email protected]

Combust. Sci. andTech., 175:1807^1834, 2003

Copyright#Taylor & Francis Inc.

ISSN: 0010-2203 print/1563-521X online

DOI: 10.1080/00102200390230873

1807

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BACKGROUND

Almost all diffusion flames burn in the presence and under the influence

of nearby solid- or condensed-phase surfaces and boundaries. For

burners, the surfaces guide the flow and provide the means for mixing the

reactants and securing the needed flame attachment. In flame spread, the

surfaces guide the flow and also provide the fuel for the flame. For het-

erogeneous propellant combustion, the fuel and oxidizer may be supplied

in the form of gaseous discharges from the pyrolyzing and regressing

propellant surface.

An important issue for wall-attached diffusion flames is their

strength or vigor, which is measured by their ability to survive when

prevailing conditions are altered. These survival criteria characterize the

robustness of the flames and indicate whether or not they can be used (in

a design context) for applications.

A quantifiable theoretical measure of the robustness and survivability

of flames exists in the form of a stability analysis. The amount of lit-

erature on stability theory in fluid mechanics is as large as it is small for

combustion. One reason for the paucity of diffusion flame stability

studies is the lack of experimental measurements to which any competing

theories might be compared. In fluid mechanics, by contrast, numerous

meticulous and detailed experiments have been performed, such as the

Schubauer and Skramstad experiments and others described by

Schlichting and Gersten (2000, chap. 15). The lack of diffusion flame

stability experiments is explained by the comparative difficulty of the

experimental setup and the measurement procedure itself, which often

requires sophisticated diagnostics and the tracking of formidable quan-

tities, such as a particular chemical species (e.g., OH� or CHþ ). Another

complication in earth-based experimentation is the buoyancy-induced

flow of hot gases, which often serves to alter the stability behavior of the

flame. Because buoyancy is a severe complication in ordinary nonreactive

fluid mechanical stability analyses, its influence in combustion calcula-

tions proves to be especially challenging and difficult. Additional for-

midable complications include chemical reaction kinetics and

thermophysical property variations.

Our purpose in this article will be to examine the stability of a dif-

fusion flame that is attached to a surface (as in flame spread, or pro-

pellant burning, or burner-attached flames in engines, and so on) and in

which the two-dimensional nature of flame attachment cannot be

1808 I. S. WICHMAN AND R. VANCE

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ignored. Our approach will be to construct a simplified model, which can

be examined using methods that have been previously established by

Vance et al. (2001) and Cheatham and Matalon (1996, 2000) for simpler

one-dimensional diffusion flames. Our model, because it ignores buoy-

ancy-induced flow, can be considered a microgravity combustion stability

analysis.

A review of the previous literature of one-dimensional diffusion

flame stability analysis is given by Vance et al. (2001). There are, how-

ever, several recent works that have a bearing on this research, such as the

articles by Cheatham and Matalon (1996, 2000) and Mills and Matalon

(1998). Cheatham and Matalon (2000) outline the entire three-

dimensional stability problem and describe the full stability calculation in

detail. The model problem that they examined was a one-dimensional

diffusion flame like the one considered by Vance at al. (2001), where cold

oxidizer and fuel surfaces are located on the two sides of the one-

dimensional planar flame. Other relevant studies are those of Kim (1997),

Kim and Lee (1999), and Sohn et al. (1999), where asymptotics were used

to develop simplified stability equations that were solved analytically in

certain cases. Sohn et al. (1999) employed numerical computation instead

of asymptotic methods. Nevertheless, asymptotic calculations are in fact

needed for a fuller understanding of the numerical treatments, because

they predict trends and indicate which model features are or are not

important. Additional studies that influence this work are Wichman and

Ramadan (1998), Wichman et al. (1999), and Wichman (1999). In these

articles, the flame attachment problem was studied and features of

attached diffusion flames were examined, including liftoff (Wichman and

Ramadan, 1998), heat fluxes to nearby surfaces (Wichman et al., 1999),

and the structure of the flame leading edge (Wichman, 1999).

Fundamental differences were observed between (1) flames that propa-

gated freely in stratified gas mixtures, and (2) flames that were ‘‘held’’ in

place by cold, adjacent fixtures, such as a burner lip (Wichman and

Ramadan, 1998).

The variant of the attached flame considered here is one in which the

lower cold wall is solid and the cold porous sidewalls upstream and down-

stream of the flame edge allow a convective flow through the domain

from oxidizer to fuel side (see Figure 1a). In this sense the model quali-

tatively resembles opposed-flow flame (and flamelet) spread (Wichman,

1992; Wichman et al., 2003) with blowing oxidizer and diffusing fuel (see

Figure 1b). It also resembles diffusion flame attachment at the top of a

STABILITY OF DIFFUSION FLAME LEADING EDGES 1809

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burner for which the burner is wide and the fuel and oxidizer ‘‘wrap

around’’ the exit edge and meet at the flame. The model of flame

attachment studied herein also strongly resembles flame attachment to,

and spread over, heterogeneous propellants, where adjacent clusters of

oxidizer and fuel produce surface flames that are attached in a manner

similar to flame spread shown in Figure 1b. The reader may consult

Lengelle et al. (2000; see their Figure 28) and Parr and Hanson-Parr

(2000; see their Figures 5, 7, 11, 12, 14, and 15). The former article

provides theoretical discussions and the latter describes detailed diag-

nostics and provides numerous photographs.

It is possible to obtain a representative model for slot burners by

changing the convective term to model flowing reactants along the in-

flame coordinate (i.e., the y-direction in Figure 1a). This configuration

produces another type of flame leading edge that in some cases will more

nearly resemble a triple flame structure (Wichman and Ramadan, 1998).

The present model, however, more closely resembles the one-dimensional

Figure 1a. Schematic of two-dimensional edge-flame model with boundary conditions. Note

the correspondence of the vertical walls with the ‘‘walls’’ a–a and b–b drawn in Figure 1b.


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model studied by Vance et al. (2001) with the cold inert lower surface of

Figure 1a adding the multidimensionality.

The only previous studies of the stability of multidimensional dif-

fusion flames (to the knowledge of the authors) are by Buckmaster and

colleagues (Buckmaster, 1997; Buckmaster and Zhang, 1999a; Buckmaster

et al., 2000). There the influence of the leading edge was described by

formulating a step change in the sidewall boundary condition (BC; e.g.,

one of the vertical walls in Figure 1a). Some of our results presented in

the section ‘‘Flame Behavior during Oscillations’’ closely resemble those

of Buckmaster and Zhang (1999a).

Previous studies that examined stability processes as inherent fea-

tures of flame-surface interaction have appeared principally in the field of

flame spread over liquid fuels (Kim et al., 1998; Schiller et al., 1996). The

fundamental mechanisms differ from the stripped-down model problem

considered here because the liquid phase adds multiple complications to

the instability such as thermocapillary flow (Garcia-Parra et al., 1996),

in-depth bulk motion, and vortex formation. It has been speculated that

the use of wide channels might lead to morphological flame front

instabilities, which might distort the two-dimensional character of the

flame front. If thermocapillary flows under some conditions control the

Figure 1b. Diagram of opposed-flow flame (and flamelet) spread over a gasifying fuel sur-

face. Note the convective influx of oxidizer from upstream across a–a (corresponding to

the boundary x¼�1 in Figure 1(a) and the primarily diffusive influx from the gasifying sur-

face across b–b (corresponding to the boundary at x¼ þ 1 in Figure 1a. The fuel surface

temperature is higher than the ambient by the amount needed to gasify it (i.e., Tn>To)

but this feature is ignored in the model of Figure 1a. The characteristic flamelet shape is

shown as the dashed line. Photographs of flame spread (first three frames) and flamelet

spread (final five frames) are shown in Figure 6.


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propagation of the flame then these morphological instabilities should

likewise appear first in the liquid phase.

Lingens et al. (1996) used experimental data to model the flow field of

a burner-attached flame. The stability analysis used a one-dimensional

model in a radial coordinate but was applied at various locations above

the burner. Their model takes on an approximate form of multi-

dimensionality. The experiments, combined with the stability analysis,

‘‘give proof of the connection between the inflectional velocity profile on

the lean side of the reaction zone, and global instability characteristics.’’

Numerical work has been done with two-dimensional models to identify

cellular flames for Le< 1, and flame oscillations for Le> 1 (Buckmaster

and Zhang, 1999a).

MODEL

The governing equations are similar to those found by Vance et al.

(2001), with the exception of a second spatial coordinate (i.e., y). In

nondimensional form these equations are

@T

@tþ Pe

@T

@x¼ H2Tþ w ð1aÞ

Le@yi@t

þ PeLe@yi@x

¼ H2yi � w i ¼ O;F ð1bÞ

where H2 ð�Þ ¼ @2 ð�Þ=@x2 þ @2 ð�Þ=@y2. Equations (1a) and (1b) are

coupled through the nonlinear reaction term w (see Nomenclature). A

single value of Le is employed for both reactants.

The boundary conditions (BCs) for Eqs. (1a) and (1b) are

T¼ To;yF ¼ 0;yO ¼ 1; at x¼�1 and T¼ To;yF ¼ 1;yO ¼ 0; at x¼þ1

ð2aÞ

T ¼ To; @yF=@y ¼ 0; @yO=@y ¼ 0; at y ¼ 0 and@ð�Þ@y

¼ 0 at y ¼ 1

ð2bÞHere the symbol (�) denotes the quantities T; yF; and yO. The last

BC expresses the postulate that the far-field flame does not change with


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coordinate y and that the flame is one-dimensional there. Computational

limitations restrict the number of nodes available; hence, this condition of

one-dimensionality cannot be applied at y ¼ 1 but instead is applied at

the rather closer location y¼ h¼ 3. As our subsequent calculations show,

this feature produces a continuous spectrum of eigenvalues in the limit as

h! 1, and it adds a measure of stability to the flame. Furthermore, we

assume for simplicity that the fuel and oxidizer inflow temperatures both

equal a reference temperature To, which is much smaller than the flame

temperature Tf and of the same order of magnitude as the gasification

temperature for the fuel beneath the flame in Figure 1b. This is the reason

why the model of Figure 1a does not employ a higher temperature (say

T1) at the y ¼ 0 surface than at the two sidewalls. For simplicity, we use

T ¼ To at all of the cold surfaces. The gradient conditions on the fuel and

oxidizer at the lower wall ( y ¼ 0) express the fact that this wall is inert

and no fuel or oxidizer is produced there.

STABILITYANALYSIS

The stability analysis conducted here is identical to that performed by

Vance et al. (2001). Into Eqs. (1) we write each dependent variable

(T; yO; yF) as the combination of a steady (basic state) solution and

a perturbation, which may depend on the time, namely, jðx; y; tÞ¼ �jjðx; yÞ þ j0ðx; y; tÞ. The basic state is subtracted out of the resulting

equations and BCs, and the nonlinear reaction terms are linearized to

produce the following set of linear partial differential equations for the

j’s:

@j0T

@tþ Pe

@j0T

@x¼ H2j0

T þ w 0 ð3aÞ

Le@j0

i

@tþ PeLe

@j0i

@x¼ H2j0

i � w0 i ¼ O;F ð3bÞ

We note that Eq. (3b) for oxidizer and fuel can be combined to

eliminate w0, namely,

Le@j0

Z

@tþ PeLe

@j0Z

@x¼ H2j0

Z ð3cÞ


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where j0Z ¼ j0

F � j0O and H2 ð�Þ ¼ @2 ð�Þ=@x2 þ @2 ð�Þ=@y2. The linear-

ized reactivity perturbation function w0 is defined in the Nomenclature.

We note that the general solution of Eqs. (3a), (3b), and (3c) consists of a

solution of one homogeneous equation, Eq. (3c), and two inhomoge-

neous equations, Eq. (3a) and one form of Eq. (3b). In the special case

Le¼ 1, however, it is possible to define another perturbation function

(for example, j0E ¼ j0

T þ j0F) that is linearly independent of j0

Z and

also satisfies Eq. (3c). Thus, when Le¼ 1 the equation set contains two

homogeneous equations and only one inhomogeneous equation. We

note, for thoroughness, that all of the BCs for these perturbation functions

are homogeneous, since the basic state also subtracts out of the BCs.

The solution of the linear Eqs. (3) is achieved by writing

j0j ¼ cjðx; yÞ expðstÞ, where subscript j¼T, O, Z (or T, F, Z) when

Le 6¼ 1 and j¼ (T, E, Z) when Le ¼ 1. This procedure yields the following

equations for cj:

LesciþPeLe@ci

@x¼H2ci�Da

�yyO�yyFTacT

�TT2þ �yyOcFþ �yyFcO

� �

� expð�Ta= �TTÞ i¼O;F ð4aÞ

scTþPe@cT

@x¼H2cT�D

�yyO�yyFTacT

�T2T2þ �yyOcFþ �yyFcO

� �expð�Ta= �TTÞ ð4bÞ

LescZþPeLe@cZ

@x¼H2cZ ð4cÞ

In the case that Le ¼ 1, we replace Eq. (4b) with

scE þ PeLe@cE

@x¼ H2cE ð4dÞ

The BCs on these equations are the homogeneous versions of the BCs in

Eqs. (2a) and (2b). Equations (4c) and (4d) can be solved analytically for

their eigenvalues sn; n ¼ 0; 1; 2; . . . ;1. The other equations must be

solved numerically for the remaining eigenvalues. We note that since Eqs.

(4c) and (4d) do not depend upon Da, the eigenvalues produced from

these equations will also not depend upon Da. Only the eigenvalues

arising from Eqs. (4a) and (4b) will depend on Da. When the s’s are realand negative the flame is stable and all infinitesimal perturbations to the

original steady-state two-dimensional attached diffusion flame die out.


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When even one of the s’s is positive, the flame is unstable to infinitesimal

perturbations, which then grow exponentially in time. If s is imaginary

the disturbances produce oscillatory behavior and, depending on the sign

of Re(s), these oscillations may either grow or decay.

We proceed to outline the analytical calculation of the eigenvalues

of Eq. (4c). Let cðx; yÞ ¼ XðxÞYðyÞ and apply the BCs cð�1; yÞ ¼cð1; yÞ ¼ cðx; 0Þ ¼ cðx; hÞ ¼ 0 to find the following: (1) ck;lðx; yÞ ¼expðPeLe x=2Þ sinðkpy=hÞ sinðlpxÞ with k; l ¼ 0; 1; 2; . . . ;1 and the

eigenvalues sk;l ¼ �½ðPe=2Þ2Leþ ðkp=hÞ2=Leþ ðlpÞ2=Le� for the odd

eigenfunctions (in coordinate x); (2) ck;lðx; yÞ ¼ expðPeLe x=2Þsinðkpy=hÞ cos½ðlþ 1

2Þpx� with k; l ¼ 0; 1; 2; . . . ;1 and the eigenvalues

sk;l ¼ �½ðPe=2Þ2Leþ ðkp=hÞ2=Leþ p2=4Leþ lðlþ 1Þp2=Le� with k; l ¼0; 1; 2; . . . ;1 for the even eigenfunctions. When Le ¼ 1 these eigenvalues

and eigenfunctions have multiplicity two (i.e., they should each be

counted twice). The smallest value of jsk;lj with nonzero (i.e., nontrivial)

ck;l occurs for the case of even eigenfunctions with k ¼ 1; l ¼ 0 giving

s1;0 ¼ �½ðPe=2Þ2Leþ ðp=hÞ2=Leþ p2=4Le�. The next-smallest value of

jsk;lj occurs for the case of odd eigenfunctions with k ¼ l ¼ 1, giving

s1;1 ¼ �½ðPe=2Þ2Leþ ðp=hÞ2=Leþ ðpÞ2=Le�. These eigenvalues jsk;lj canbe arranged as sn; n ¼ 0; 1; 2; . . . ;1, where jsnþ1j � jsnj, so that as the

eigenvalues become larger in magnitude (i.e., more negative) their influ-

ence on the potential instability of the diffusion flame becomes ever more

negligible. In practice, we will calculate usually only the first three

eigenvalues, s1; s2; s3; which should adequately characterize the response

of the flame to infinitesimal perturbations.

In the case Le ¼ 1 it is usually true that s1, the smallest or least

negative or potentially positive eigenvalue, arises from the computation

of the more complicated Eqs. (4a) and (4b), which contain the linearized

reaction term. Thus, under most circumstances it is the numerically

computed eigenvalue whose behavior characterizes the flame response.

This eigenvalue computation must be performed over the entire domain.

For computational purposes a relatively coarse grid was used and, hence,

some loss of accuracy near the turning point (see Figure 2, point A) is

unavoidable. A finite difference algorithm was used employing a 92� 92

uniformly spaced mesh. All of the eigenvalues, including those of Eqs.

(4c) and (4d), were obtained by employing standard LAPACK sub-

routines designed for large, banded matrices. The coarse grid cannot

produce smooth, continuous eigenvalue curves, as in the one-dimensional

model of Vance et al. (2001). Nevertheless, the results appear to represent


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a plausible qualitative picture of the flame behavior near the upper

turning point of the S curve.

There are, however, several differences between the two- and one-

dimensional models. One of these is that for the two-dimensional case

the domain size on the vertical direction (h) directly influences the

eigenvalues, as seen by the term ðp=hÞ2=Le, which appears in both s1;0and s1;1. Such domain-size-dependent terms of course did not appear

in the eigenvalues of the one-dimensional model. In addition, three

points must be mentioned: (1) When h ! 1 the eigenvalue spectrum

becomes continuous because the difference between two successive

eigenvalues jskþ1;l � sk;lj ! 0 as h ! 1. Thus, there is a size depen-

dence of the stability on the location of the upper ‘‘insulated’’

boundary. (2) The finite displacement h of the ‘‘insulated’’ boundary

contributes to the stabilization of the attached flame because this

contribution further decreases the eigenvalues sk;l. One thus expects,

based on the earlier simple analytical eigenvalue calculation, that the

larger h, the less stable is the flame. This qualifier may lose importance

in the face of the available numerical evidence that it is really the first

eigenvalue s1; which is computed numerically from Eqs. (4), that

Figure 2. Qualitative plot of one-dimensional nonpremixed flame temperature versus

Damkohler number, Da. Three distinct branches are traditionally represented. The upper

branch (above A) represents steady burning, the middle branch is unstable, and the lower

branch corresponds to the ignition branch. In actual plots the lower turn may appear at

values of Da many orders of magnitude higher than Da at the upper turn.


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determines the stability, and the higher eigenvalues sn; n ¼ 2; 3; . . . ;1only enter into consideration if and when the principal eigenvalue s1exchanges places with one of the sn; n ¼ 2; 3; . . . ;1 (e.g., if s1 becomes

more negative than s2 so that s2 is now the leading eigenvalue).

(3) The crossflow Pe appears to add stability by diminishing the sn.But the subsequent decrements to the sn do not contain any additional

Pe dependence, they contain only Le and h dependence.

Comparisons of the two-dimensional model to the one-dimensional

model are made for S-curve evaluations. We recall that the S-curve plots

graph the maximum temperature versus the Damkohler number, and in a

one-dimensional domain (like that of Vance et al., 2001) there is no

ambiguity: the maximum temperature is simply the highest flame tem-

perature between the two parallel walls. In a two-dimensional domain,

however, the selection of the location of maximum temperature is unclear

(e.g., along isocontours of mixture fraction Z that intersect the quench

point, at heights above the lower cold wall where the maximum flame

temperature had attained to within a certain chosen fraction of its one-

dimensional value, etc.). We opted for the least ambiguous definition and

chose to use the maximum temperature in the entire domain. This value

usually occurred at the top of the domain ðy ¼ hÞ at the flame location,

where the flame had essentially become one-dimensional. The value of Da

at the turning point is, just as in the one-dimensional case, simply the

value that produces the drop to an extinction (‘‘slow-burning’’) state

along the bottom of the S.

Eigenvalue Results For LeJ1

The case Le¼ 1 is illustrated in Figure 3. Here s1 represents the leading

eigenvalue, which is negative along the entire upper branch and turns

positive (indicating instability) at the turning point. The second two

eigenvalues, which are degenerate because Le¼ 1, are therefore inde-

pendent of Da and are always negative. From the preceding discussion,

they should have the values s2¼ s3¼ s1,0¼�[(Pe=2)2Leþ (p=h)2=Leþp2=4Le]¼�3.626524. The numerical value, however, is somewhat

smaller in magnitude, close to s2¼ s3¼ –2.75. This approximately 33%

error apparently arises from computation with a very coarse mesh, which

does not produce smooth curves in the vicinity of the upper turning point

on the S curve. Furthermore, the solutions along the middle branch are

difficult to obtain and produce ragged-looking curves. Nevertheless, the


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eigenvalue behavior for this case is qualitatively the same as the one-

dimensional model (see Figure 3 of Vance et al., 2001).

The picture becomes slightly more complicated for Le< 1. According

to Vance et al. (2001, Figure 6), and according to our discussion con-

cerning multiplicity of eigenvalues, there was only one constant eigen-

value among the three leading eigenvalues s1, s2, and s3. It is the

behavior of what is labeled the third eigenvalue, s3, that changes as Ledeviates from unity. We see from Figure 4 that s3 departs from the

constant value s2 by bending lower as the solution travels along the upper

branch; it asymptotically approaches the constant s2 as the middle

branch is traversed from the upper turning point toward the lower

turning point. This behavior is also seen when Le< 1 (slightly) in the one-

dimensional model (Vance et al., 2001).

The first qualitative difference with the one-dimensional model

appears when Le is further reduced. If Le is too small, s3 will not

asymptotically approach s2 but will increase past this value as the middle

branch is traversed (see Figure 5). Although this difference does not

Figure 3. Eigenvalue distribution showing the variation of Real(s) with Da for the case

Le¼ 1, Pe¼ 0.5, Ta¼ 4, f¼ 1.0. Similar distributions are seen for the one-dimensional

model of Vance et al. (2001) when Le¼ 1.


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change the physical behavior of the system, which still exhibits stable

solutions along the entire upper branch and unstable solutions along the

middle branch (with no oscillations), one begins to notice the influence of

the flame edge and the cold boundary.

As noted by Vance et al. (2001) and other (see, e.g., Cheatham and

Matalon, 2000), when Le� 1 the instability manifests itself in the form of

cellular flames. The relative size of these ‘‘flamelets’’ is determined in part

by the critical wave number for which the ‘‘new’’ leading eigenvalues

become zero. The cellular flame computation is next briefly outlined and

explained.

The essence of the method is to ‘‘relieve’’ the tendency for instability

by introducing an additional, available spatial degree of freedom. The

previously two-dimensional unstable attached diffusion flame (DF) then

becomes a stablized three-dimensional flame (the additional degree of

freedom is the newly introduced z-coordinate) with a characteristic

‘‘wavelength’’ in the z-coordinate. This wavelength may be interpreted,

with some physical latitude, as the characteristic size of the flamelets that

Figure 4. Eigenvalue distribution showing the variation of Real(s) with Da for Le¼ 0.9,

Pe¼ 0.3, Ta¼ 4, f¼ 1.0. Here s1 becomes positive at the upper turning point, indicating

instability along the middle branch. The third eigenvalue approaches s2 asymptotically

along the middle branch.


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form in the z-direction. This interpretation supports available physical

observations (Olson, 1997; Wichman et al., 2003), which demonstrate

that stable spreading flames near condensed fuel surfaces can, under

suitable conditions, break apart into separate flame fragments called

flamelets. These flamelets continue to propagate over the surface as small,

separate, isolated units. The precise reasons for the breakup are not

perfectly clear, but experiments have shown that they occur under near-

limit conditions of low oxidizer mass fraction, low opposed velocity, and

high heat losses. Figure 6 shows the breakup of a spreading flame in the

form of a photographic sequence. In this experiment the oxidizer mass

fraction of the free stream is held fixed (YOO¼ 0.233). The opposed flow

speed (directed from the bottom to the top of each photograph in Figure 7)

is approximately 5 cm=s and the sample width is 10 cm. A cross section of

each flamelet would resemble the two-dimensional flamelet slice shown in

Figure 1b as the dashed line.

To perform the cellular flame calculation we start by using Eqs. (1)

and (2) withH2ð�Þ ¼ @2ð�Þ=@x2 þ @2ð�Þ=@y2 þ @2ð�Þ[email protected] then introduce

for the dependent variables (T, yO, yF) the functions jðx; y; z; tÞ ¼

Figure 5. The variation of Real(s) with Da for the case Le¼ 0.5, Pe¼ 0.5, Ta¼ 4.0, f¼ 1.0.

When Le is decreased below a certain value, s3 crosses s2 and continues its monotonic

decrease. This behavior is not seen in the one-dimensional model of Vance et al. (2001).

The leading eigenvalue is not shown.


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Figure 6. Flame spread instabilities in a drop test in the NASA zero-g 5-s facility. At t*¼ 0 s

the flame is a continuous front across the cellulose sheet (0.001 in. thick, 10 cm wide). The air

opposed flow velocity is 7 cm=s. Wavelengths are visible at t*¼ 1.43 and 2.27 s into the drop,

and development into full flamelets occurs at t*¼ 3.46 and 4.13 s, followed by recombina-

tion at t*¼ 4.76 and 4.99 s. The drop ends at t*¼ 5.2 s. The characteristic flamelet major axis

is � 2–4 cm.


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�jjðx; yÞ þ j0ðx; y; z; tÞ (note that the basic state is still two-dimensional

but the perturbation is now three-dimensional), we subtract out the

basic state as before, and we introduce the functions j0j ¼ cjðx; yÞ

expðstþ iKzÞ. The result is to produce Eqs. (4a)–(4d) with the term

Le s on the left-hand side (LHS) of Eq. (4a) replaced by Le sþ K2

and the term s on the LHS of Eq. (4b) replaced by sþ K2. In the

preceding eigenvalue computations, for purposes of illustration with the

simple case with Le¼ 1, we have snew¼ sold�K 2 so that when s1 is

positive the contribution of K2 introduces stability with wavenumber

Kcrit¼ffipsold (i.e., wavelength l¼ const� [2p=

ffipsold]). The general

numerical eigenvalue computation is, of course, unchanged. For the one-

dimensional model of Vance et al. (2001) it was found that the critical

wavenumber Kcrit was not greatly altered by the different possible values

of Le or Pe. This is seen also for the two-dimensional model in Figure 7.

Here various Le and Pe combinations are presented along with the one-

dimensional result for Le¼ 1, Pe¼ 0, and Ta¼ 4.0. The two-dimensional

model produces critical wave numbers that are of the same order of

magnitude as those predicted with the one-dimensional model. Figure 7

Figure 7. Critical wave numbers for several different configurations. It is seen that (1) Kcrit is

not greatly affected by changes in Le or Pe and (2) the order of magnitude of Kcrit is the

same for the one- and two-dimensional models.


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also indicates that the critical wave number is only slightly altered

by thermal-diffusive effects (as quantified through changes in Le) or

hydrodynamic considerations (as quantified through changes in Pe). In

other words, Kcrit appears to be a physically robust quantity.

The characteristic flamelet size is calculated from Kcrit by writing

lflamelet¼KcritL. The domain size in the direction transverse to the flame

(i.e, in the x-direction in Figure 1a) can be chosen as the appropriate

length scale L*. It makes more sense, however, to follow Vance et al.

(2001) and Vance (2001) by employing the ‘‘characteristic near-limit

length’’ taken from the turning point (i.e., point A in Figure 2) value

of the Damkohler number, namely, L*¼ffip[(a*=B*(PeDaturn pt.=YOOÞ].

Use of Daturn pt.� 5.5Eþ 06 (see the discussion in Conclusions),

a* � 1.7 cm2=s, YOO � 0.233, B*¼ 2.1Eþ 08 s�1 and Pe� 1 gives

L*� 0.43 cm. Using Kcrit � 0.2 from Figure 7 gives l*flamelet� 2 cm,

which is in good agreement with the experimentally observed flamelet

sizes. This agreement is illustrated in Figure 6, which shows the breakup

between 1.43 and 3.46 s of the flame front into a flamelet front. The

wavelength of the disturbances at 1.46 s is approximately 2 cm, and

the characteristic flamelet size is O(2–4 cm) along the major flamelet axis.

The choice of parameters used in the evaluation of l*flamelet is explained

briefly in the Appendix.

Eigenvalue Results For Le d1

In the previous section it was shown that flame instabilities for Le < 1

result in cellular flames. When Le > 1 cellular flames are not observed,

flame oscillations can occur that either decay to stable burning or con-

tinue to grow, resulting in flame quenching. This behavior was described

by Vance et al. (2001), Cheatham and Matalon (2000), and Sohn et al.

(1999) for the one-dimensional flame. If Le is large enough the upper

branch solutions exhibit pure perturbation growth or oscillation growth

or decay. This behavior is described by the eigenvalue response examined

in the following for the two-dimensional model. Vance et al. (2001)

restricted the analysis to low Pe because the eigenvalue behavior became

more complicated as Pe increased (and large Pe was not the focus of their

work). Here, the restriction of low Pe is used because of the relatively

coarse grid needed to perform the stability calculations.

Small deviations from Le¼ 1 are studied and shown in Figure 8.

They were also discussed in detail by Vance et al. (2001; see their Figures


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4 and 5) and will not be described here. If one compares the nature of

eigenvalue behavior for Figure 8 to that of Figure 4 of Vance et al. (2001),

many similarities arise. Primarily, the entire upper branch is stable with

decaying oscillations present over a small portion. Note that Figure 8

does not extend as far as Figure 4 of Vance et al. (2001) along the upper

branch, but the behavior is qualitatively similar. That is, the oscillations

will persist until the complex conjugate pair s1,2 crosses s3 and s3 in turn

becomes the leading eigenvalue.

There exists a critical value of Le in the one-dimensional model for

which larger values produce upper branch instabilities. This is the case

also for the edge-flame model. Figure 9 shows the upper branch results

for Le¼ 2.0 and, as can be seen, the complex conjugate pair extends into

the positive region and then breaks apart. This behavior indicates that

not only do unstable oscillations exist but also pure perturbation growth

exists. This is the same behavior seen in Figure 5 of Vance et al. (2001).

Both Figures 8 and 9 represent eigenvalue responses for the case of zero

convection (Pe¼ 0). The characteristics of the flame behavior do not

Figure 8. The variation of Real(s) with Da for the case Le¼ 1.1, Ta¼ 4.0, Pe¼ 0.0, f¼ 1.0.

Stable oscillations exist on the upper branch while the entire middle branch is unstable. The

regions within the circle represent eigenvalues along the middle branch.


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change appreciably with changes in Pe when Le < 1. To identify any

effect of small Pe for Le > 1 the eigenvalue behavior was obtained for

Le¼ 1.2 with Pe¼ 0.0 and 0.5. These results appear in Figure 10. It is

seen that the small changes in Pe do not alter the behavior of the

eigenvalues, with the exception that the curve is shifted toward higher Da.

This is consistent with the Le < 1 results of the previous section.

FLAMEBEHAVIORDURINGOSCILLATIONS

For the case when Le >1 we examined in some detail the oscillations of

the flame leading edge. Buckmaster and colleagues (Buckmaster and

Zhang, 1999b; Buckmaster et al., 2000) made similar observations. They

demonstrated, using a qualitative model of DF leading edges (LEs), that

the flame LE structure changes during retreat and advance. When the

flame retreats from the attachment location its LE resembles the

‘‘flame nub’’ described by Wichman (1999): such LEs are found for

Figure 9. The variation of Real(s) with Da for the case Le¼ 2.0, Pe¼ 0.0, Ta¼ 4.0, f¼ 1.0.

The upper branch eigenvalues indicate regions of pure perturbation growth as well as oscil-

lation growth leading to instability. The oscillations cease to exist when s3 becomes the

leading eigenvalue.


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nonisenthalpic heat-losing DFs. When the flame advances toward the

attachment point in its return, the LE resembles a triple flame propa-

gating through a stratified premixed mixture. In the time that the LE

retreated, the fuel and oxidizer in front of the LE had additional time to

mix; it is this mixture into which the oscillating LE now propagates.

Figures 11a and 11b show our results for our model for the case

Le¼ 2, Ta¼ 4, and Da¼ 7.5 E þ 6. Figure 11a shows the oscillation

pattern of the LE over time. In Figure 11b the shape of the LE is shown at

two time instants (t¼ 3 and 4.5), one for retreat and the other for

advance. It is seen that the retreating LE is more nublike, whereas the

advancing flame LE is more like a triple flame. This calculation

demonstrates that the ‘‘premixed flame wings’’ are a characteristic feature

of propagating diffusion flames. Retreating flames, however, and

attached diffusion flames near cold surfaces (from which, or through

which, there is at most a low flow of reactant, as in ordinary flame spread

across a condensed fuel) show an LE whose structure is visibly simpler

(because there are no premixed flame wings) but whose method of

Figure 10. The case Le¼ 1.2, Ta¼ 4.0, f¼ 1.0. Shown are the three leading eigenvalues for

different Pe values. Changes in Pe have only a small influence on the flame behavior, which

is consistent with Le< 1 (Vance et al., 2001).


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‘‘survival’’ is much more difficult to understand. The LE apparently

consists of a location that allows the fuel and oxidizer mass fractions to

approach ‘‘large’’ values even as the temperature is not high. Neverthe-

less, the multiplication of the Arrhenius exponential by the mass fraction

terms (which are O[1]) produces a large reaction rate there, in fact, at least

one to two orders of magnitude larger than in the trailing DF.

CONCLUSIONS

A two-dimensional model of an edge flame was examined to identify the

validity of a simpler one-dimensional model. This work examined both

the subunity Le regime whose instability is exhibited as cellular flame

behavior (see Figure 6), and the Le> 1 regime, which has the char-

acteristic of producing oscillatory behavior.

The matter of whether a flame that attaches to a lower cold wall (base

of Figure 1a) is more or less robust than one existing in a suspended one-

dimensional state between walls of oxidizer and fuel (as is the flame

Figure 11a. The case Le¼ 2, Ta¼ 4, Da¼ 7.5 Eþ 6, and f¼ 1.0. Shown is the oscillation

pattern of the LE over time.


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section toward the top of Figure 1a) is answered by comparing the values

of Da at the turning point of the upper S curve. If the flame is less robust

this value of Da will be larger. It can be stated without exception for all

cases studied that the values of Da for the two-dimensional flame are

always larger (which is discussed in detail following). The cold lower wall

in a certain sense weakens the one-dimensional flame, making it sus-

ceptible to extinction. Nevertheless, it is generally true that near the cold

wall the flame LE has a very high reactivity, which serves to stabilize it.

Combining these two conditions of a weakened flame having a high

reactivity suggests an analogy with the strong yet brittle solid, which has

high strength (high reactivity) but past a certain threshold fractures

Figure 11b. The case Le¼ 2, Ta¼ 4, f¼ 1.0, and Da¼ 7.5 Eþ 6 as in Figure 11a. Here the

shape of the LE is shown at two time instants (t¼ 3 and 4.5), the former for retreat and the

latter for advance. The retreating LE is blunt, whereas the advancing flame LE has wings.

Premixed flame wings are a characteristic feature of propagating DFs. Retreating flames

show a visually simple LE without appreciable wings and whose method of survival is much

more difficult to understand.


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readily. (Extinction for two-dimensions occurs at higher values of Da,

i.e., more readily, than in one-dimension).

For the cases examined in this work, the average Da value at

extinction was Daext� 5.5 Eþ 06 with a maximum value of 5.88 Eþ 06

and a minimum of 5 Eþ 06, whereas for the 1-D model Vance et al.

(2001) found Daext� 1.72 Eþ 06, with a maximum of 1.72 Eþ 06 and a

minimum of 1.715 Eþ 06. The ratio of the two- and one-dimensional

values gives Daext,2D=Daext,1D 3: the two-dimensional flame extin-

guishes at the turning point at three times the value of Da for the one-

dimensional case. Since Da/L2 (we will take L to be nondimensional

since only ratios will be considered here) this result implies that

L2D=L1D 31=2 1.7. The characteristic length over which heat losses

occur in the two-dimensional case is nearly twice that of the characteristic

length for heat losses in the one-dimensional case. If we take the ratios of

the total lengths of the side walls in both cases, for the two-dimensional

case it is 2(2)þ 2(3)¼ 10 units, whereas we may take it as 2(3)¼ 6 (in the

vertical direction we take the dimension as 3), whose ratio is 1.67, close to

1.7. In another way we may take L1D¼ 2 as the nondimensional

separation distance between the two sides and L2D¼ [22þ 32]1=2¼ 3.61

(the diagonal of the area for the two-dimensional case), giving

L2D=L1D 1.8. A more physically based means for estimating char-

acteristic lengths is to estimate the integrated ratio of the wall heat losses

to the integrated heat release, expressed in nondimensional form as

L2D

L1D ð

R_qq00 dA=

R_qq000 dVÞ2D

ðR_qq00 dA=

R_qq000 dVÞ1D

ð5Þ

where the symbols are defined in the Nomenclature. The nondimensional

heat transfer to the surfaces occurs purely by conduction so thatR_qq00 dA / 2ðh � sÞ=l along the vertical wall and

R_qq00 dA / ðl � sÞ=h along

the horizontal lower wall. Note that we use s to describe the non-

dimensional depth coordinate perpendicular to the plane. For the volume

integral we useR_qq000 dV / ðh � l � sÞ=l2 � s in both cases. The result is

L2D=L1D 1þ (l=h)2=2, giving L2D=L1D 1.2, which is slightly lower

than the numerical result. Nevertheless, order-of-magnitude agreement is

achieved.

It was demonstrated that both of the Le regimes studied behave

similarly to the one-dimensional model. A more exacting numerical

analysis, that is, a more refined mesh, will be needed to more accurately


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predict the precise eigenvalue response. This will have to wait until more

powerful computers are available. However, the present study should

provide strong support for the qualitative validity of previous and future

work on one-dimensional models. It is clear the basic characteristics and

qualitative behavior of flames can be delineated with only the most

skeletal of equations and the bare minimum of spatial dimensions (1).

This is an encouraging result for theoretical research.

Interesting agreement was produced with the models by Buckmaster

and colleagues (Buckmaster and Zhang, 1999b; Buckmaster et al., 2000)

for the shape and nature of the flame LE as it oscillated back and forth

near the attachment point. The winglike triple flame structure is char-

acteristic of diffusion flames advancing into gas mixtures; the more

nublike structure is characteristic of attached, retreating flames where

permanent heat losses are a dominant feature (Wichman, 1999). We

attained this behavior by carrying out full numerical simulations in a

physically realistic configuration (Figure 11b).

NOMENCLATURE

A dimensionless area

A* preexponential factor [(gmol=cm3)1�m�ns�1]

B* dimensional preexponential factor (1=s)

cp Specific heat (kJ=kg-k)

D* mass diffusivity (m2=s)

Da Damkohler number (ratio of flow time to chemical reaction time),

Da¼B*YOOL*2=a*Pe

E* activation energy (kJ=kmol-K)

h nondimensional length characterizing the height of the two

vertical walls

K wave number

k* rate constant [(gmol=cm3)1�m�ns�1]

l nondimensional length between the two vertical walls (l¼ 2)

L* characteristic length used to nondimensionalize the coordinates

x*,y*,z* (m)

Le Lewis number, Le¼ l*=r*cp*D*

MWi molecular weight of species i (g=gmol)

Pe Peclet number, Pe¼ u*L*=a*_qq00 nondimensional heat flux (W=m2)


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_qq000 nondimensional heat release rate per unit volume (W=m3)

R* universal gas constant (kJ=kmol-K)

s nondimensional depth of domain in direction perpendicular to

x,y-plane

T* temperature (K)

T nondimensional temperature, T¼T*=To*

Ta nondimensional activation temperature, Ta¼Ta*=To*

To* normalization temperature, To*¼ q*YOO=n cp*t nondimensional time, t¼ t*=to*

to* normalization time, to*¼L*2=a* (s)

u* velocity (m=s)

V dimensionless volume

w* reactivity, w*¼ r*B*YOYFexp (�Ta*=T*) (kg=m3-s)

w nondimensional reactivity, w¼DayOyF exp(�Ta=T)

w0 linearized reactivity perturbation,

w0 ¼Da �yyo�yyF expð�Ta= �TTÞ½f0o=�yyo þ f0

F=�yyF þ f0TTa= �TT

2�x,y,z nondimensional spatial coordinates defined in Figure 1a,

x¼ x*=L*, etc.

Yi mass fraction of species i, i¼O, F

YFF mass fraction of fuel in the fuel stream

YOO mass fraction of oxidizer in the oxidizer stream

yO normalized mass fraction of oxidizer, yO¼YO=YOO

yF normalized mass fraction of fuel, yF¼YF=(YOO=n)Z mixture fraction, Z¼ (yFþ 1�yO)=2

Greek Letters

a* thermal diffusivity, a*¼ l*=r*cp* (m2=s)

l* disturbance wavelength (m); also thermal conductivity

(kJ=m-s-K)

n mass-based stoichiometric coefficient for the irreversible chemical

reaction Fþ nO! (1þ n)Pr* density (g=cm3)

s temporal growth eigenvalue (dimensionless)

f global stoichiometric coefficient, f¼ nYFF=YOO

c disturbance function amplitude

Subscripts

a activation

c,crit critical


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E second coupling function for the case Le¼ 1, from the definitions

j0E¼j0

T þ j0F or j0

E¼j0T þ j0

O

F fuel

f flame

i species (i¼O, F)

O oxidizer

o reference (ambient)

T temperature disturbance

Z mixture fraction disturbance, from j0Z ¼ j0

F�j0O

1D one-dimensional model

2D two-dimensional model

Superscripts

— (overbar) basic state quantity0 disturbance quantity

* dimensional quantity

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APPENDIX

Here we explain the parameter choices for the wave number calculation

in the section ‘Eigenvalue Results for Le� 1.’ For our one-step chemical

reaction Fþ nO ! (1þ n)P between a hydrocarbon fuel and air we write

d [CxHy]*=dt*¼�k*[CxHy]*m[O2]

*n, k*¼ A*exp(�Ta*=T*). The values

of A*, m, n, and Ta* are given by Turns (2000, Table 5.1), and we choose

for evaluation the fuels C2H6, C3H8, C4H10, C5H12, C6H14, C7H16,

C8H18, C2H4, C3H6, and C6H6. We write the concentration (gmol=cm3)

[Xi]¼ r*Yi=MWi to find

dYCxHy=dt ¼ �BYm

CxHyYnO2 expð�T

a=TÞ

where B*¼A*(p*)rmþ n�1MW1�mCxHy=MWn

O2 has units s�1. For our

ten fuels the average value of B*¼ 2.1 Eþ 08 when r*¼ 3.4 E � 04 cm3=s

(i.e., air at T *� 1000 K, an average of the flame temperature 1800 K and

room temperature 300K). We use the value a*¼ 1.7 cm2=s for air (also

evaluated at T *� 1000K) and we write YOO¼ 0.233. With Pe� 1 we

obtain L*� 2 cm. We note that the ten hydrocarbon fuels produced B*

values that deviated no more than 10% from the mean, except for C2H4

(3.373 Eþ 08), C3H6 (1.077 Eþ 08), and C6H6 (1.014 Eþ 08).


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Stability of attached two-dimensional diffusion flame leading edges