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Premixed flame response to equivalence ratio perturbations
Shreekrishnaa; Santosh Hemchandraab; Tim Lieuwena
a
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA b Institute of
Aerodynamics, RWTH, Aachen, Germany
Online publication date: 04 October 2010
To cite this Article Shreekrishna, Hemchandra, Santosh and Lieuwen, Tim(2010) 'Premixed flame response to equivalence
ratio perturbations', Combustion Theory and Modelling, 14: 5, 681 — 714
To link to this Article: DOI: 10.1080/13647830.2010.502247
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Combustion Theory and Modelling
Vol. 14, No. 5, 2010, 681–714
Premixed flame response to equivalence ratio perturbations
Shreekrishna, Santosh Hemchandra† and Tim Lieuwen∗
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Downloaded By: [Georgia Technology Library] At: 17:38 20 April 2011
(Received 12 August 2009; final version received 10 May 2010)
This paper studies the heat-release oscillation response of premixed flames to oscillations in reactant stream fuel/air ratio. Prior analyses have studied this problem in the
linear regime and have shown that heat release dynamics are controlled by the superposition of three processes: flame speed, heat of reaction, and flame surface area oscillations.
Each contribution has somewhat different dynamics, leading to complex frequency and
mean fuel/air ratio dependencies. The present work extends these analyses to include
stretch and non quasi-steady effects on the linear flame dynamics, as well as analysis of
nonlinearities in flame response characteristics. Because the flame response is controlled
by a superposition of multiple processes, each with a highly nonlinear dependence upon
fuel/air ratio, the results are quite rich and the key nonlinearity mechanism varies with
mean fuel/air ratio, frequency, and amplitude of excitation. In the quasi-steady framework, two key mechanisms leading to heat-release saturation have been identified. The
first of these is the flame-kinematic mechanism, previously studied in the context of
premixed flame response to flow oscillations and recently highlighted by Birbaud et al.
(Combustion and Flame 154 (2008), 356–367). This mechanism arises due to fluctuations in flame position associated with the oscillations in flame speed. The second
mechanism is due to the intrinsically nonlinear dependence of flame speed and mixture
heat of reaction upon fuel/air ratio oscillations. This second mechanism is particularly
dominant at perturbation amplitudes that cause the instantaneous stoichiometry to oscillate between lean and rich values, thereby causing non-monotonic variation of local
flame speed and heat of reaction with equivalence ratio.
Keywords: flame-acoustic interactions; premixed flames; combustion instabilities;
equivalence ratio perturbations; flame transfer function
Nomenclature
hR
hRj
sL
sLc
sLd
sLj
q
r
∗
†
=
=
=
=
=
=
=
=
Heat of reaction
j th order sensitivity of heat of reaction to equivalence ratio
Flame speed
Consumption speed of the flame
Displacement speed of the flame
j th order sensitivity of flame speed to equivalence ratio
Instantaneous heat release
Radial coordinate
Corresponding author. Email: [email protected]
Present address: Institute of Aerodynamics, RWTH, Wuellnerstrasse 5a, Aachen 52062, Germany
ISSN: 1364-7830 print / 1741-3559 online
C 2010 Taylor & Francis
DOI: 10.1080/13647830.2010.502247
http://www.informaworld.com
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682
Shreekrishna et al.
t
z
A
DT
F
G
Lf
Ma
Mac
Mad
R
St
St2
Stδ
uo
α
β
δ
δ( )
ε
φ
φ̃
ρu
σc∗
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
ω
ξ
( )o
( )’
()
( )∗
=
=
=
=
=
=
Time variable
Axial coordinate
Instantaneous flame area
Average thermal diffusivity of the reactant mixture
Transfer function
Level set function
Length of the stationary flame
Markstein number
Consumption speed Markstein number
Displacement speed Markstein number
Burner radius
Strouhal number = ωLf /uo
Reduced Strouhal number = St β 2 + 1 /β 2
Non quasi-steadiness Strouhal number = ωδ/sLo
Mean flow velocity
β 2 /(1 + β 2 )
Flame aspect ratio = Lf /R
Thermal flame thickness
Dirac Delta function
Amplitude of the equivalence ratio perturbation
Equivalence ratio
Equivalent equivalence ratio
Reactant mixture density
Scaled Markstein length, Ma(δ/R)
1/2
β (1+β 2 )
Angular forcing frequency (rad s−1 )
Flame front coordinate
Stationary variable
Perturbed variable
Fourier transformed variable
Non-dimensional variable
1. Introduction
This paper describes the response of laminar premixed flames to perturbations in reactant
mixture equivalence ratio. This work is motivated by the problem of combustion instabilities, which causes significant problems in the operation of premixed combustion systems
[1–5]. During an instability, heat release fluctuations feed energy into one or more of the
acoustic modes of the system, causing high amplitude pressure and velocity oscillations,
thereby leading to combustion instability. These oscillations can result in poor system
performance and hardware damage.
Modeling these phenomena in order to develop rational mitigation approaches requires
an understanding of the various mechanisms that cause heat release oscillations in lean
premixed combustors. Significant among these are flame burning area fluctuations driven
by acoustic velocity oscillations [2,3] or convected, vortical structures [4–7], flame extinction and re-ignition, flame–wall interactions [8] and reactant mixture composition, i.e.
equivalence ratio fluctuations [9–13]. Studying this latter equivalence ratio mechanism is
the focus of this paper. Several studies have shown strong evidence for its significance
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Combustion Theory and Modelling
683
in exciting instabilities either by direct measurement of equivalence ratio oscillations during instabilities [14–17] or by comparing the dependence of instability characteristics on
geometry and operating conditions with correlations developed from theoretical analyses
[5,12,18].
From an analytical viewpoint, much insight into the phase response of the flame to such
perturbations can be obtained from a simple time delay analysis that treats the flame as a
concentrated source of heat release [19]. In general, however, flames are distributed axially
over a length scale where the mixture equivalence ratio can significantly vary. In other
words, they are convectively non-compact, although perhaps being acoustically compact.
The flame Strouhal number, St (= ωLf /Uo ), which equals the length of the mean flame
to the length scale of the imposed fuel/air ratio excitation, determines whether the flame
can be regarded as being a convectively compact or distributed source, and whether the
flame response is geometrically quasi-steady or non-quasi steady. This quasi-steady limit
has been discussed extensively by Polifke and Lawn [20], who presented a detailed analysis
of the general flame response characteristics in the low forcing frequencies (St → 0) limit.
There are a number of other publications that have considered the fuel/air ratio oscillation mechanism [19,21–28]. Several publications have reported experimental studies to
characterize this mechanism in greater detail [16,17,29–33]. Similarly, a number of computational studies, e.g., Flohr and co-authors [34,35], Angelberger et al. [36] and Polifke
et al. [37], have studied the excitation, transport, and resultant fluctuations in heat release
due to fuel/air ratio oscillations. In addition, several reduced order modeling analyses have
incorporated this mechanism into actual instability models [9,21,26,29,38].
It is known that the basic phenomenology of the flame response is controlled by a
superposition of three processes [28], shown schematically in Figure 1. This can be seen by
noting that the instantaneous global heat release rate of a premixed flame may be expressed
as the product of the local mass burning rate and the heat of reaction of the reactant mixture,
integrated over the entire flame surface area. Mathematically,
q (t) =
ρu sLc hR dA
(1)
flame
Figure 1. Fundamental processes controlling the heat release response of premixed flames to equivalence ratio oscillations. Routes labeled ‘S’ denote additional routes due to influence of flame stretch.
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684
Shreekrishna et al.
Equivalence ratio perturbations cause fluctuations in the local flame speed (route 2a) and
heat of reaction (route 1) along the flame surface. These fluctuations in the flame speed and
mixture heat of reaction then cause the local heat release rate to oscillate. This is a direct
route of influence. Additionally, flame speed variations also excite flame wrinkles that propagate along the flame. This leads to an oscillation in the burning area of the flame (route 2b),
thereby causing the net heat release rate to oscillate. This is an indirect route of influence. It
is to be noted that the indirect route of influence is also non-local; i.e. the flame area fluctuations at a given time and position are a convolution of the flame surface oscillations at all
upstream locations at earlier times. Due to oscillations caused in the flame shape because of
equivalence ratio perturbations, oscillations arise in the curvature of the flame front, which
can perturb the flame displacement and consumption speeds [39], thereby establishing
another route by which the flame speed fluctuates (route 2S). These fluctuations in flame
speed can then disturb the heat release directly (route 2Sa) or indirectly through burning area
fluctuations (route 2Sb). Note for the problem of interest, that stretch rate oscillations are
indirectly caused by equivalence ratio oscillations; i.e. equivalence ratio oscillations perturb
the displacement flame speed, which causes flame wrinkles, which lead to oscillations in
flame stretch, which can now perturb both the displacement and consumption speeds of the
flame.
This study generalizes prior work by considering the following effects. First, it considers
finite amplitude effects, i.e. the modification of the linear results as nonlinear effects become
significant. These analyses are carried out in the quasi-steady limit. Nonlinear effects are
considered by performing a third order perturbation analysis to understand the factors that
influence the initial onset of nonlinearity, and complementary computations of the fully
nonlinear G-equation, so as to capture the flame front dynamics and heat release saturation
at high excitation amplitudes. This nonlinear analysis is needed because the prediction of
instability amplitudes requires consideration of nonlinear processes that control the response
of the flame to equivalence ratio perturbations at large amplitudes of excitation [40]. Some
nonlinear effects have also been recently analyzed by Birbaud et al. [1], who show pocket
formation due to coalescing of neighboring branches of the flame as an important kinematic
nonlinearity mechanism.
Second, this study also addresses higher frequency characteristics of the flame response in the linear regime. We show that two processes grow in importance at higher
frequencies. The first is the non quasi-steady response of the internal flame structure. Non
quasi-steadiness here refers to “flame structural” non quasi-steadiness, i.e. where the period
of oscillations in the mixture composition are comparable to internal flame time scales,
such as the characteristic diffusion time in the preheat zone of the flame. This is characterized by a Strouhal number Stδ . This is different from “geometric” non quasi-steadiness,
which occurs when St∼O(1), where the period of oscillations in the mixture composition
are comparable to the characteristic response time of the entire flame. The second phenomenon that can potentially become important at higher frequencies is flame stretch. The
oscillating flame displacement speed creates flame wrinkles, whose radius of curvature
scales roughly as 1/St2 . This causes both the displacement and consumption flame speeds
to be modulated not only directly by the fuel/air ratio oscillation, but also indirectly by the
stretch effects associated with flame wrinkling due to the flame displacement speed oscillations, analogous to the two additional routes it provides for velocity-coupled flame response
[41–43].
Finally, this study also explicitly discusses the response of rich, premixed flames. The
rich regime response is motivated by interest in operating premixed systems in partial
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Combustion Theory and Modelling
685
Figure 2. Qualitative plot showing dependence of flame speed, sL , and heat of reaction, hR , dependence on fuel/air ratio, φ.
oxidation mode that can, for example, be used for co-production of synthesis gas [44]. The
rich flame response dynamics are quite different from that of lean flames for two reasons
that can be inferred from Figure 2, which plots the typical φ dependence of the flame speed
and heat of reaction.
First, the sensitivities of the mean flame speed to fuel/air ratio fluctuations of lean
and rich flames are opposite in sign; i.e. an increase in fuel/air ratio causes a flame speed
increase and decrease on the lean and rich side, respectively. Second, the heat release per
unit mass of reactant varies with fuel/air ratio on the lean side, but is nearly constant on the
rich side. Hence, there is negligible influence of the heat of reaction (hR ) term on the rich
flame response – this term plays an important role in the lean flame response, particularly
under low Strouhal number conditions.
Because these sL and hR transition regions do not occur at the same φ value, the
flame response has qualitatively different characteristics in the three stoichiometry regions,
illustrated schematically in Figure 2. Region I is the lean regime which has been explicitly
considered in prior studies. Region II is associated with the same sL sensitivity trend as that
of a lean flame, but near-zero hR sensitivity. Region III is associated with the opposite sL
sensitivity trend and near-zero hR sensitivity. Depending upon the specific flame chemistry
and reactant composition, the size of region II in φ space can vary; e.g., for fuels like
methane with reactants at STP, the nature of both flame speed and heat of reaction change
at φ ∼1.0 leading to a very narrow region II width of φ∼0.07. On the other hand, the
flame speed for 80%H2 /20%CO–air synthesis gas mixture peaks at values close to φ ∼ 1.8.
This leads to a much larger region II width of about φ ∼ 0.8.
The remainder of this paper is organized as follows. The analytical development and
details of the numerical method are presented in Section 2. Section 3 presents typical results
for a CH4 /air flame. Results are also provided to demonstrate the influence of flame stretch
and non quasi-steady effects on linear flame response, the key mechanisms of nonlinearity,
and the parameter regimes where these different mechanisms are dominant. Section 4
concludes with a summary of results and comments on issues that must be addressed in
future work.
686
2.
Shreekrishna et al.
Formulation and analysis
2.1. Basic outline
The analytical framework adopted to model the flame response closely follows that of
Markstein [45], Yang and Culick [46], Boyer and Quinard [47], and Fleifil et al. [2]. The
flame is assumed to consist of a thin sheet whose surface can be represented implicitly by
the zero contour of a two dimensional function G(r, z, t). The evolution of this contour can
then be tracked using the G-equation [45].
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∂G
+ U .∇G = sLd |∇G|
∂t
(2)
where U is the local flow velocity. This equation can be solved numerically to capture
complex flame front motions, such as cusp and pocket formation [1] or multi-valued flame
fronts. This will be discussed later in the section on numerical approach to follow.
To achieve analytical progress, the axial location of the flame is given by a function
ξ (r, t). Thus, we may express G in an explicit form as G = z − ξ (r, t). Substituting this
into equation (2), we obtain the following flame front tracking equation (assuming axisymmetry).
2 1/2
∂ξ
∂ξ
∂ξ
d
+ sL 1 +
=u−v
∂t
∂r
∂r
(3)
Introducing the non-dimensionalization scheme: r ∗ = r/R, z∗ = z/Lf and t ∗ = tuo /Lf
with R being chosen to be an appropriate flame holder length scale, equation (3) is written
as
∂ξ ∗
+
∂t ∗
sLd
sLo
sLo
uo
∗ 2 1/2
∗
u
v
∂ξ
2 ∂ξ
=
− β
1+β
∂r ∗
uo
uo
∂r ∗
(4)
The subscript ‘o’ denotes the value of the respective quantity evaluated at the mean equivalence ratio, φo , or the mean value as in the case of flow velocity. The ratio of the instantaneous
flame displacement speed to mean flame speed, sLd /sLo is influenced by equivalence ratio
perturbations and the stretch rate of the flame. For small amplitude perturbations, we may
write sLd /sLo as
∂ sLd /sLo φ ∂ sLd /sLo ∗
sLd
=1+
+
κ
sLo
∂ (φ/φo ) φo
∂κ ∗ o
(5)
o
where κ ∗ ’ is the flame curvature non-dimensionalized by the inverse of the burner radius,
(1/R). The coefficients of the perturbation terms on the RHS of equation (5) are the sensitivities of the flame speed to perturbations in equivalence ratio and stretch rate, respectively.
Further, the stretch sensitivity of the displacement speed may be expressed in terms of the
Markstein number [42] as
∂ sLd /sLo δ
= −Ma d
∗
∂κ
R
o
(6)
Combustion Theory and Modelling
687
φ
δ
sLd
= 1 + sL1 − Ma d κ ∗
sLo
φo
R
(7)
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leading to:
Equation (4) is then solved by prescribing the characteristics of the disturbance field. The
analysis neglects the density jump across the flame, which changes the character of the
approach flow field. This assumption is well understood as a necessary approximation to
achieve analytic progress in velocity-coupled analyses [41,42,48], but is also required here
for slightly more subtle reasons. Even if there are no velocity perturbations imposed upon
the flame, the oscillating fuel/air ratio disturbance generates flame wrinkles. These wrinkles
will necessarily excite velocity disturbances in the upstream and downstream flow due to
the temperature change across the flame.
The instantaneous heat release of the flame is calculated using equation (1). For fuel/air
ratio perturbation occurring at constant density, equation (1) can be rewritten in terms of
burning velocity magnitude and heat of reaction perturbations as,
A (t)
q(t)
=
+
qo
Ao
flame
sLc dA
+
sLo Ao
flame
hR dA
+
hRo Ao
flame
sLc hR dA
sLo hRo Ao
(8)
The first term on the RHS denotes the contribution to heat release fluctuation due to
oscillations in the net burning area of the flame. The second and third terms represent the
contributions from burning velocity and heat of reaction oscillations, respectively. Note that
the burning velocity used in equation (8) is the consumption speed, sLc , as opposed to the
displacement speed, sLd , used in the level set equation, equation (2); sLc and sLd are identical
for unstretched flames but may differ in the presence of flame stretch [39]. The fourth term
represents the nonlinear coupling between flame speed and heat of reaction oscillations.
These terms can be explicitly evaluated for a given flame surface geometry—e.g., the first
term may be evaluated for an axisymmetric conical flame as [48],
2
A(t)
=
1/2
Ao
1 + β2
1
r∗ 1 + β2
0
∂ξ ∗
∂r ∗
2 1/2
dr ∗
(9)
The heat release transfer function of the flame due to equivalence ratio fluctuations can
then be defined as
F =
q (ω)/qo
φ̂ /φo
(10)
where the numerator and denominator are respectively the heat release of the flame and
equivalence ratio perturbations at the flame base, evaluated at the excitation frequency.
In the quasi-steady case, the flame speed and heat of reaction terms are functions
of fuel/air ratio, stretch rate, fuel type, and operating condition. In the general unsteady
case, however, these quantities, particularly the flame speed, introduce additional dynamics
related to the flame structure so that the instantaneous flame speed is also a function of
frequency; see Lauvergne and Egolfopoulos [49] and Sankaran and Im [50]. The former
study analyzes these unsteady effects for a flat flame, and shows that the quasi-steady
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688
Shreekrishna et al.
assumption is valid when f sLo /δ or Stδ 2π . They show that the non quasi-steady
flame response is equal to the quasi-steady response of a flame that is excited by an
equivalent mixture fraction oscillation that is obtained by spatially averaging the mixture
fraction over the preheat zone thickness at each instant in time. This “effective equivalence
ratio oscillation” is decreased in magnitude and shifted in phase from that of instantaneous
fuel/air ratio oscillations just upstream of the preheat zone. Thereafter, modeling the non
quasi-steady flame dynamics closely follows the quasi-steady modeling approach.
In general, the flame speed response to stretch is also non quasi-steady; an equivalent
way of stating this, is that the Markstein number is a function of frequency in the frequency
domain representation of equation (6). This point has been discussed by Joulin [51] who
shows that, while the Markstein length for hydrodynamic strain sensitivity of the flame
decreases as 1/f 1/2 , the Markstein length for unsteady curvature induced stretch depends
very weakly on frequency and asymptotes to the same value (the thermal thickness of the
flame) at quasi-steady (f → 0) and high frequency (f → ∞) conditions. For the problem
of interest, the flame is only stretched by curvature, and as such, we assume constant Ma.
Finally, in comparing the finite amplitude response of lean and rich flames, it is important
to note that the definition of fuel/air ratio is intrinsically non-symmetric, i.e. the lean side
ranges from 0 to 1 while the rich ranges from 1 to infinity. The opposite behavior occurs for
the inverse of φ, λ =1/φ. Thus, for a given φ, a perturbation of ε results in a larger absolute
perturbation in φ on the rich side than on the lean. As such, response graphs plotted in the
next section show the rich flames exhibiting nonlinear behavior at lower ε values than lean
flames – this is partly due to the definition of ε used here. The opposite behavior would be
observed if ε were used to measure perturbation amplitude in terms of air/fuel ratio, λ.
2.2.
Perturbation analysis
This section presents the development of the perturbation solution for the evolution of
the flame surface. For the sake of illustration, we consider an axisymmetric conical flame
stabilized on a burner tube as shown schematically in Figure 3.
Figure 3. Schematic of the axisymmetric conical flame geometry.
Combustion Theory and Modelling
689
The velocity field is assumed to be axially uniform; the continuity equation then yields
a zero radial velocity. With this assumption, equation (4) becomes
∂ξ ∗
1
sd
+ L ∗
∂t
sLo 1 + β 2 1/2
1+β
2
∂ξ ∗
∂r ∗
2 1/2
=1
(11)
The non-dimensional flame speed, sLd /sLo depends upon flame curvature as described in
equation (5). The non-dimensional curvature is given by:
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∂ξ ∗ /∂r ∗
∂ 2 ξ ∗ /∂r ∗2
1
κ∗ = β +β ∗ 3/2
r 1 + β 2 (∂ξ ∗ /∂r ∗ )2 1/2
1 + β 2 (∂ξ ∗ /∂r ∗ )2
(12)
The two terms on the RHS of equation (12) account for axial and azimuthal curvature
respectively. The azimuthal curvature term is smaller than the axial term by a factor of the
Strouhal number (see Appendix A) and, as such, is negligible at higher Strouhal numbers
where stretch effects are significant. The one exception to this occurs in the vicinity of the
flame tip, r = 0 where, however, the contribution to the flame area is negligible. Hence, we
only consider axial curvature effects in the subsequent stretched flame analysis.
Next, the upstream equivalence ratio perturbations are assumed to be radially uniform,
harmonically oscillating, and are advected by the mean flow:
z
−t
φ = φo 1 + ε cos ω
uo
(13)
Employing the non-dimensional scheme described in the previous subsection, equation (13)
may be written in dimensionless form as:
φ = φo [1 + ε cos (St (z∗ − t ∗ ))]
(14)
This assumed form of equivalence ratio is an exact harmonic solution of the species conservation equations for a spatially uniform mean equivalence ratio and axial flow field,
neglecting axial diffusive effects. This latter assumption of neglecting axial diffusive ef1/2 sLo
β δ
T 1
fects can be shown to be reasonable if St ωD
1 or, equivalently, f 2π
Le
.
R
δ
u2o Le
Strictly speaking, this shows that a general analysis of higher frequency flame characteristics must include these effects. Neglecting diffusion of equivalence ratio disturbances is
consistent with retaining flame non quasi-steadiness and flame stretch, the two other higher
frequency effects that are considered here, for flames that satisfy β (δ/R)1/2 Le1/2 2π
and Le Ma 1, respectively. Clearly, however, incorporating all of these effects simultaneously is an area requiring future work, as these latter two inequalities significantly
constrict the parameter space of applicability.
The origin of co-ordinates is fixed at the center of the tube exit plane. The flame is
assumed to be attached at the burner lip. This yields the following boundary condition for ξ
ξ (t)|flame - holder = 0
(15)
In non-dimensional form, equation (15) becomes
ξ ∗ (1, t ∗ ) = 0
(16)
690
Shreekrishna et al.
where the flame holder length scale parameter, R, is chosen to be the burner tube radius.
The centerline boundary condition is due to symmetry, and is given by
∂ξ
(0, t) = 0
∂r
2.3.
(17)
Dynamics of the flame front
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For the quasi-steady flame, a perturbation analysis is performed to third order in the
excitation amplitude ε in order to capture the leading order nonlinear dynamics of the flame
analytically. More specifically, we expand the flame front position ξ (r, t) as,
ξ (r, t) = ξ0 (r) + εξ1 (r, t) + ε2 ξ2 (r, t) + ε3 ξ3 (r, t) + O ε4
(18)
Using the above in equation (11) and equation (16) and collecting terms of the same order
in ε yields evolution equations for each of the ξ i ’s. In order to make analytic progress, we
also assume that the flame thickness is small relative to the burner radius, i.e. δ/R 1. We
may first write an equation for the shape of the mean flame as follows.
∗ ∗
ξo (r ∗ ) = 1 − r ∗ − O e−r /σc
(19)
Here, σ ∗c is a scaled Markstein length non-dimensionalized by the burner radius, defined as
σc∗ = Ma d
1
δ
R β 1 + β 2 1/2
(20)
The mathematical details of the derivation of equation (19) are presented in detail in
Appendix B. Equation (19) shows that the correction to the mean flame
due to flame
shape
∗n
∗
∗
stretch is exponentially small in σ ∗c . This implies that lim
σ
exp
−r
/σ
c
c = 0, for all
∗
σc →0
positive integers, n. This result is quite helpful for asymptotic analysis in the small σ ∗c limit,
since this mean flame shape correction term does not enter the solution when expanded
in powers of σ ∗c ; i.e. it can be neglected to any order of σc∗ , except for the very small
region where r ∗ < σ ∗c and thus, the contribution to flame area is negligible. Furthermore,
as described earlier, azimuthal stretch can be neglected at high frequencies where flame
stretch is important. Under these assumptions, the evolution equation for ξ1 may be written
as follows.
∂ξ1∗
∂ 2ξ ∗
∂ξ ∗
− α 1∗ − ασc∗ ∗21 + sL1 cos(St (1 − r ∗ − t ∗ )) = 0
∗
∂t
∂r
∂r
(21)
The parameter α is given by the expression, α = β 2 / 1 + β 2 .
Also,
sLj
1 ∂ j sLd /sLo =
j ! ∂ (φ/φo )j φ/φo =1
;
hRj
1 ∂ j (hR /hRo ) =
j ! ∂ (φ/φo )j φ/φo =1
(22)
Combustion Theory and Modelling
691
are respectively the j th order sensitivities of flame speed and heat of reaction of the reactant
mixture to fluctuations in equivalence ratio. To linear order in excitation amplitude, these
solutions are still exact for the non quasi-steady case, with the sensitivity coefficients, sL1
and hR1 , simply being functions of frequency. However, the nonlinear corrections in this
perturbation analysis implicitly assume quasi-steady sensitivities of flame speed and heat
of reaction, and also neglect stretch effects. For the sake of brevity, the evolution equations
for ξ 2 and ξ 3 are presented in Appendix C.
These evolution equations can be solved to yield expressions for ξi∗ (r ∗ , t ∗ ). For the
sake of illustration, we present the solution for ξ1∗ (r ∗ , t ∗ ) in the absence of stretch (σc∗ =
0).
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ξ1∗ (r ∗ , t ∗ ) =
sL1
sin St (1 − r ∗ − t ∗ ) − sin (St/α) (1 − r ∗ − αt ∗ )
(1 − α) St
(23)
Equation (23) is very helpful in understanding the physics of flame front dynamics. This
solution explicitly contains two contributions to the linear dynamics of the flame surface
evolution. The first term within the brackets represents the effect of local non-uniformities
in the burning velocity due to the spatial and temporal oscillations in equivalence ratio. The
second term arises because of the fixed-anchor boundary condition, i.e. equation (16), that
the flame does not move at the burner lip, even though the flame speed is oscillating. In
physical terms, equation (23) shows that the flame front position is controlled by two sets
of waves that travel along the front: (i) waves generated at each point along the flame due to
spatial variations in flame speed and (ii) waves generated at the flame attachment point due
to the boundary condition, equation (16). Notice that the propagation velocities of these
two waves along the flame surface are different. The former travels with the mean flow
velocity (unity in the nondimensional case) and the latter with a non-dimensional velocity
1/α along the axis of the flame. Thus, these two waves interfere constructively at some
flame surface locations and destructively at others. This has a significant influence on the
characteristics of the heat release transfer function of the flame. This is similar to the result
obtained by Preetham and Lieuwen [48] who emphasized these superposition effects upon
the dynamics of flames subjected to excitation in flow velocity.
The corresponding solutions for ξ 2 and ξ 3 are presented in Hemchandra et al. [52].
2.3.1. Heat release response transfer functions
Quasi-steady response. We next consider the quasi-steady heat release transfer function.
The transfer function in equation (10) can be decomposed in a manner similar to that of
heat release in equation (8). To first order in excitation amplitude, the transfer function
for a stretch insensitive flame, Fo can be written as a sum of three contributions arising
from burning velocity oscillations, heat of reaction oscillations and flame area oscillations,
which may respectively be written as:
2
(1
(i
+
i
St
−
exp
St))
St 2
2
(1
(i
= hR1
+
i
St
−
exp
St))
St 2
2α
1 − α − exp (i St) + α exp (i St/α)
= sL1
1−α
St 2
Fo,sL = sL1
(24)
Fo,hR
(25)
Fo,A
(26)
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The same decomposition however, cannot be strictly performed in the non-linear regime.
Thus, following equation (8), we decompose the net transfer function in the nonlinear
regime as,
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F = FA + FsL −A + FhR −A + FsL −hR −A
(27)
Full expressions for the terms on the RHS may be found in Hemchandra et al. [52].
Next, consider the effects of flame stretch, which provides an additional route of flame
relaxation to high frequency disturbances. Hereafter, we assume that the flame displacement
speed and consumption speed are equal, and drop the superscripts ‘c’ and ‘d’ for notational
convenience. As described in Figure 1, flame stretch affects flame response by altering
the flame speed directly and the flame burning area indirectly. Its inclusion leads to the
following expressions valid for quasi-steady flames and weak flame stretch (σc∗ St → 0) to
leading order in σc∗ :
FA,c
iSt
2sL1
iα e − eiSt/α − (1 − α) St + O σc∗2 (28)
(1 − α) St
2 i αsL1 2
iSt
2 iSt/α
(1
−
α)
= Fo,A + σc∗
+
αe
−
(1
−
α
+
α
)e
(1 − α)2 St
(29)
+ O σc∗2
FsL,c = Fo,sL + σc∗
Some care must be exercised in analyzing the asymptotic dependencies of these expressions
at simultaneously low σc∗ and high Strouhal numbers. Analysis of the exact solution of
equation (21) shows that stretch influences the flame burning area term, FA , when
σc∗ St 2 ∼ 1
(30)
Atsuch Strouhal
numbers, the flame speed contribution to the stretch correction, FsL , is
∗1/2
smaller than the burning area contribution, FA . However, as the Strouhal number
O σc
further increases to satisfy
σc∗ St ∼ 1
(31)
both the burning area and flame speed terms become comparable in their contributions to
the total flame response. It is interesting to note that similar criteria were developed for
the effects of stretch on the velocity coupled flame response, see Preetham et al. [42] and
Wang et al. [43]. Equations (30) and (31) may respectively also be rewritten in terms of
dimensional frequency, flame speed, flame thickness and Markstein number, for tall flames
(β 1) as
f ∼
1 sLo β
2π δ Ma 1/2
1/2
δ
R
(32)
and
1 sLo
f ∼
2π δ
β2
Ma
(33)
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Finally, it is important to note that, in the linear regime (in excitation amplitude, ε), flame
stretch does not affect the heat of reaction route to heat release oscillations. Hence, the total
response of the flame under the influence of stretch may simply be expressed as
F = Fo,hR + FsL,c + FA,c
(34)
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Non quasi-steady analysis. We next account for non quasi-steady effects related to the
delayed response of the internal flame structure to equivalence ratio disturbances. Following
the modeling approach detailed by Lauvergne and Egolfopoulos [49], it can be shown that
in the linear approximation, for an instantaneous equivalence ratio oscillation given by
equation (13), the flame heat of reaction (hR ) and burning rate (sL ) respond to an equivalent
equivalence ratio whose instantaneous value in dimensionless form is:
Stδ
Stδ
∗
∗
cos
+ St (z − t )
φ̃ (z , t ) = φo 1 + ε sinc
2
2
∗
∗
(35)
where sinc(x) = sin(x)/x. Here, a second Strouhal number, Stδ is defined based on the
flame thickness and flame speed as Stδ = ωδ/sLo . This is simply the ratio of a characteristic
diffusion time in the preheat zone of the flame (τdiff ∼ δ/sLo ) to the characteristic time
associated with mixture composition fluctuations (τeq ∼ 2π /ω). As such, we may relate Stδ
and St as
St
α 1/2
=
(δ/R)
Stδ
(36)
On accounting for non quasi-steady effects in such a manner, the flame speed and heat of
reaction sensitivities are diminished and phase shifted by a non quasi-steady scaling factor
and may be expressed as [53]
nqs
sL1 = sL1 sinc (Stδ /2) exp (−iStδ /2)
(37)
nqs
hR1
(38)
= hR1 sinc (Stδ /2) exp (−iStδ /2)
The non quasi-steady transfer function, F nqs , is a relatively simple modification of the
quasi-steady transfer function, which may be expressed as
F nqs = g (Stδ ) F qs (St)
(39)
where the correction factor accounting for non quasi-steady phenomena is given by
g (Stδ ) = sinc (Stδ /2) exp (−iStδ /2)
(40)
By definition, non quasi-steady effects become important when τdiff ∼ τeq , which may be
written in terms of Stδ as
Stδ ∼ 2π
(41)
Comparing this with equation (32), it may be seen that for tall flames satisfying βMa −1/2 (δ/R)1/2 2π , non quasi-steady effects becomes important at smaller
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frequencies than those due to flame stretch. Conversely, stretch effects are important when
non quasi-steady effects are negligible if Ma −1 β 2 2π . Outside of these two inequalities,
however, the two are of comparable importance.
It is important to note that the manner of influence of non quasi-steadiness and flame
stretch effects on the global heat release response of the flame is different. While non
quasi-steadiness seems to affect all the three routes, viz., heat of reaction oscillations, flame
speed oscillations and burning area oscillations identically, flame stretch affects only the
latter two routes; in fact, these two routes are affected differently due to flame stretch.
Further, non quasi-steadiness scales the transfer function by the scaling factor, g, while
flame stretch corrects the unstretched quasi-steady transfer function.
In addition, it may be seen that the asymptotic high frequency dependence of the
stretched quasi-steady and unstretched non quasi-steady flame response gains differ by an
order of Strouhal number; i.e.:
Fo,c ∼ 1
St
(42)
Fo,nqs ∼ 12
St
2.4.
Numerical approach
We next discuss the numerical approach adopted to study the nonlinear heat release response
of a quasi-steady, unstretched flame. Formally, equation (2) is a non-conservative Hamilton–
Jacobi equation. This equation has the property that the nonlinear term, due to flame
propagation normal to itself, results in cusps, or discontinuities in derivative, and possible
topological changes (i.e. pocket formation) in the solution. The formation of pockets due to
merging of adjacent flame branches was recently emphasized as an important mechanism
of nonlinear flame response to fuel/air ratio oscillations by Birbaud et al. [1]. Hence robust
numerical schemes that can capture these effects without excessive smearing are required.
The solution domain is discretized using a uniform grid. The initial value for the Gfield was constructed from the assumed quiescent flame shape. This was done by defining
the value of G at each grid location to be the signed distance of that location from the
quiescent flame surface. The solution at later times was obtained using a low diffusion
Courant–Isaacson–Rees scheme with back and forth error compensation and correction
(BFECC) [54]. The G-field was reset to a distance function after each time step using the
re-initialization procedure described by Peng et al. [55].
A considerable reduction in computation time can be obtained by solving equation
(2) in only a narrow band around the actual flame location, rather than in the entire twodimensional domain. This was achieved by adopting the localization procedure introduced
by Peng et al. [55]. This band evolves in time as the flame moves or as pockets form and
burnout. These computations were performed using the general purpose level-set program
LSGEN2D developed by the authors [52].
As noted earlier, it is assumed that the flame remains attached at the burner lip. This
is achieved by setting G = 0 after every time step of the BFECC scheme at the points
corresponding to the burner tube. The velocity of these points is maintained to be identically
zero throughout the simulation. The instantaneous heat release of the flame is given by
equation (1). Following Smereka [56], equation (1) can be written using G as
(43)
2π rρu sL hR δ (G) |∇G| d
q(t) =
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where the integration is performed over the whole computational domain described earlier
and δ(G) is the Dirac-delta function. This integral is then evaluated at every sampling time
step, using the numerical technique described by Smereka [56]. The grid size (
r∗) for
all the above computations was fixed at 0.001 non-dimensional units in both directions.
The non-dimensional time-step was fixed at 0.1
r∗. These were chosen by successive
refinement of the grid until the temporal heat release variation changed by less than 5%.
Sufficient numbers of grid points were taken along the z-direction to ensure that all pockets
formed at the tip of the flame would burn out before being convected out of the grid. The
first three contributions to the total heat release on the RHS of equation (8) were obtained
independently using the same techniques described above.
These exact results were used to determine the accuracy of the third order perturbation
analysis. The domain in St2 − ε space where the magnitude and phase of the transfer
function can be determined within specified accuracies Em and Eφ respectively is defined
by
Fcomp (St2 ) − Fasymp (St2 ) ≤ Em ,
St2 (Em , Eφ , ε0 ) = min St2 : F
(St )
comp
2
Fcomp (St2 ) − Fasymp (St2 ) ≤ Eφ (44)
ε=ε0
The first term within the braces on the RHS gives the value of St2 for which the error in
magnitude prediction from the approximate solution obtained using asymptotics is bounded
byEm . The second term gives the value of St2 for which the error in phase prediction in the
asymptotics solution is bounded byEφ . The sizes of these regions depend on the assumed
burning velocity and heat of reaction dependencies on equivalence ratio (e.g., equations
(45) and (46)).
As will be shown later, two mechanisms contribute to nonlinearity in the flame response.
The first is due to flame sheet dynamics, as described by the G-equation, see equation
(2). The second is the nonlinearity of the quasi-steady flame speed and heat of reaction
dependence upon fuel/air ratio, as plotted qualitatively in Figure 2 and described in the
sensitivity derivatives in equation (22). Figure 4 shows the regions of specified accuracy of
Figure 4. Domain of applicability of asymptotic analysis (valid to within specified accuracy below
line) for (a) φo = 0.85 (lean), (b) φo = 1.28 (rich), β = 4.
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the asymptotic solutions for various values of Em and Eφ for two flames where φ o =0.85
and 1.28. It can be seen that there is an opposing influence of perturbation magnitude and
Strouhal number – i.e. the analysis is valid at larger fuel/air perturbation amplitudes at lower
Strouhal numbers. This is due to the effects of nonlinearity in the G-equation which grows
with amplitude and frequency, see Preetham and Lieuwen [48]. At low Strouhal numbers,
the analysis validity is limited by nonlinearities in the quasi-steady flame speed and heat of
reaction dependencies upon fuel/air ratio.
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3. Results and discussion
This section presents explicit results for a conical flame with aspect ratio, β = 4.0. The
investigated geometry is shown schematically in Figure 3. The following correlations for
the burning velocity magnitude and heat of reaction for a methane-air flame at STP are
assumed [28].
sL (φ) = Aφ B exp −C (φ − D)2 ;
A = 0.6079, B = −2.554,
C = 7.31, D = 1.230
hR (φ) =
3.1.
2.9125 × 106 min (1, φ)
1 + 0.05825φ
(45)
(46)
Linear, quasi-steady dynamics of stretch-insensitive flames
We begin with a brief discussion of the characteristics of the quasi-steady linear transfer
function for an unstretched flame. We refer the reader to Cho and Lieuwen [28] for a more
complete discussion of these linear dynamics and focus the discussion here primarily on
the manner in which the rich flame results are different from those of lean flames. In order
to make this comparison, results are presented for two mean equivalence ratios, φo = 0.85
and φ o = 1.28, which correspond to conditions where the flame speeds are identical, sL ∼
33 cm/s.
Figure 5 shows the variation of the phase and magnitude of the linear transfer function
with St2 = St/α for the two equivalence ratios. Also shown are the phase and magnitude
of the individual contributions to the total transfer function. First, note that both the phase
and the magnitude do not monotonically vary with St2 . This is due to the fact that the linear
flame response is determined by the net superposition of a boundary generated “wave” and
a local disturbance, as discussed in the previous section. Therefore the net flame response
depends on exactly how these waves superpose at different Strouhal numbers. It must also
be noted from equations (24) and (25) that the phase of the flame speed and the heat of
reaction contributions are identical in the linear limit. The linear transfer function for the
rich case is shown in Figure 5(b). Notice that the transfer function goes to a near zero value,
given by the heat of reaction sensitivity (hR1 ), at low values of St2 (see equation (52)).
This is in striking contrast to the corresponding lean case and is due to the fact that the
heat of reaction is a nearly constant function of equivalence ratio in the rich regime. This
means that in the linear regime, the heat release of a rich flame is relatively insensitive to
perturbations in equivalence ratio at low values of St2 .
Another difference between the two transfer functions is the presence of a zero response
in the rich case, e.g., at St2 ∼ 8.7. At this point, the oscillating flame speed and area
oscillation response exactly cancel each other. In the lean case, however, the node is not
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Figure 5. Linear transfer function for (a) φo = 0.85 (lean), (b) φo = 1.28 (rich), β = 4.
present. This is due to the fact that the lean case has an additional contribution to the total
transfer function, viz., the heat of reaction oscillations. The zero response does not occur in
the lean flame response because it consists of a superposition of three terms, whereas the rich
flame has only two major contributors. However, it must be noted that these characteristics
are strong functions of the sensitivities of flame speed and heat of reaction to equivalence
ratio. To understand this, consider flame responses at different mean equivalence ratios as
plotted in Figure 6.
Note, first, that the lean cases all start with a gain of nearly unity and the rich cases with
a gain of nearly zero at low Strouhal numbers. This is due to the fact that the flame response
is entirely controlled by the heat of reaction sensitivity hR1 in the quasi-steady case. All
Figure 6. Variation of the linear transfer function with St2 for different values of equivalence ratio,
β = 4.
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transfer functions then initially grow with increasing Strouhal number, because the burning
area and the fluctuating flame speed terms progressively come into phase with each other.
As the mean equivalence ratio is increased from an initial lean value, e.g., φ = 0.6,
the sL and hR sensitivities progressively decrease and stay nearly constant, respectively,
until φ∼1.06 where the sL and hR sensitivities vanish. Hence, the magnitude of the flame
response drops to a nearly zero response at φ∼1.06. This is due to the occurrence of the
flame speed maximum at this equivalence ratio. The heat of reaction is a very weak function
of equivalence ratio for φ o >1.0. Hence, the magnitude increases from a nearly zero value
with increasing St2 for rich mean equivalence ratios.
Even though the flame speeds at φo = 0.85 and φ o = 1.28 are identical, the magnitude
of the maximum gain is higher in the rich case due to the higher sL sensitivity at φ o =1.28.
Also to be noted is that the heat release response lags the excitation in the lean case, and
leads it in the rich case. This again is due to the fact that the linear sL sensitivity, sL1, changes
sign from positive to negative when φ o >1.06. This may be understood physically from
the fact that the burning area response is due to sL fluctuations that, in turn, are induced
by equivalence ratio oscillations. The change in sign of sL1 implies that an instantaneous
increase in equivalence ratio results in an increase and decrease in the instantaneous value
of sL on the lean and rich side, respectively. Therefore, given the same instantaneous
equivalence ratio increase, the corresponding instantaneous burning area decreases for a
lean mean equivalence ratio and increases for a rich mean equivalence ratio.
3.2. Linear, non quasi-steady dynamics of stretch-insensitive flames
This section presents typical results for the non quasi-steady response of the flame. These
effects are presented separate from stretch effects in order to illustrate their different
influences on the flame response.
Figure 7(a) compares the magnitude and phase of the linear total transfer function for
quasi-steady and non quasi-steady flame response. It may be seen that, as Strouhal number
increases, there is a marked departure of the non quasi-steady response from the quasisteady result. At higher frequencies, corresponding to St ∼ 60, the response is significantly
Figure 7. (a) Non-quasi-steady versus quasi-steady flame response for φo = 0.85, β = 4, δ = 0.1R.
(b) Non-quasi-steady correction factor.
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attenuated, while the quasi-steady response is non-zero. In fact, it may be observed from
Figure 7(b), that at Stδ ∼ π , there is already about 40% attenuation in the gain, and 90
degrees difference in phase with respect to the quasi-steady response.
To give a feel for typical numbers, consider a methane-air reactant mixture, establishing
a tall flame with β = 10. At 1 atm, the preheat zone thickness is about 1 mm. Thus, at
1 atm, the flame preheat zone diffusive processes become non quasi-steady at a frequency
f ∼ 400 Hz. At 10 atm, the preheat zone becomes non quasi-steady at f ∼ 4 kHz.
3.3. Linear, quasi-steady dynamics of stretch-affected flames
We next present results that quantify the effects of flame stretch on the quasi-steady flame
response. We begin by investigating the effect of flame stretch on the contributions due
to flame speed and burning area perturbations to the overall heat release for a lean flame
(φo = 0.85). Similar trends are seen in rich flames.
Figure 8. Effect of flame stretch on (a) flame speed contribution and (b) burning area contribution
to unsteady heat release for a lean CH4 /air flame, φ o = 0.85, β = 4, δ = 0.1R, Ma = 1.
Figure 8 plots the flame speed contribution of the overall flame response for an unstretched flame (solid curve) and the stretch correction to this contribution (dashed curve).
To understand the roles played by flame stretch effects and non quasi-steady effects,
Figure 9 plots the total linear response of the flame with and without stretch and with
non quasi-steadiness.
These plots show that the effect of stretch is primarily to smooth the undulations in the
unstretched gain results, while that of non quasi-steadiness is to rescale the gain.
3.4. Nonlinear, quasi-steady dynamics of stretch-insensitive flames
As the excitation amplitude/frequency is increased, the higher order contributions to the
transfer function become significant. Before presenting explicit results, it is useful to
first consider the various mechanisms for nonlinearity. These different mechanisms are
summarized in Figure 10, which plots parameter space boundaries where the various
physical mechanisms are dominant. These regions were calculated for the lean flame
corresponding to φ o = 0.85 (and is nearly the same for the rich case φ o = 1.28, hence not
shown).
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Figure 9. Comparison between unstretched, quasi-steady (solid blue), stretched, quasi-steady (dashdotted green) and unstretched, non quasi-steady (dashed red) global heat release responses of a conical
CH4 /air flame., β = 4.0, δ/R = 0.1, Ma=1: (a) φ o = 0.85, (b) φ o = 1.28.
Figure 10. Qualitative map illustrating regimes of dominance of various physical mechanisms at
φo = 0.85. Stδ = π . The solid lines denote quasi-steady boundaries. Dash-dot lines denote approximate non quasi-steady boundaries, obtained by substituting the frequency dependent φ̃ into the
quasi-steady boundary solution.
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The region labeled ‘Linear dynamics’ in Figure 10 corresponds to the region where
nonlinear corrections contribute less than 10% of the transfer function gain, and has the
characteristics described in sections 3.1, 3.2 and 3.3.
There are two basic processes causing nonlinearity in the flame response:
1. nonlinearities in burning area oscillation, due to the nonlinearities in flame kinematics
(term 1 in equation (8)) and
2. quasi-steady nonlinearities in the sL -φ and hR -φ relationships, as plotted qualitatively
in Figure 2 (terms 2-4 in equation (8)).
There is an additional complication, however, in the fact that the sL -φ nonlinearity has
both a direct and indirect influence on the heat release response through term 2 and term 1 in
equation (8), respectively. This indirect mechanism dominates the heat release nonlinearities
in the ‘sL -φ nonlinearity’ region in Figure 10. Physically, its origin may be explained
as follows. Flame surface motion is induced by flame speed fluctuations. The resulting
area fluctuations associated with this motion exhibit nonlinearity due to the intrinsically
nonlinear dynamics of flame propagation normal to itself. This latter ‘sL -φ nonlinearity’
dominates in the indicated region of the chart, due to the nonlinear dependence of the local
propagation velocity upon fuel/air ratio. This induces nonlinearities in the burning area
response.
The propagation of the flame normal to itself, as remarked above is the dominant
source of nonlinearity in flame area and overall heat release response in the region labeled
‘Kinematic Restoration’ [57, 58] in Figure 10. Larger amplitude fluctuations in flame position slope cause kinematic nonlinearities to correspondingly grow in significance. As St is
increased, equation (23) shows that the wavelength of the induced wrinkles on the flame surface is O(1/St). Thus, at high frequencies, propagation of the flame surface normal to itself
results in the rapid destruction of these wrinkles [59] causing the fluctuating flame surface
area to saturate. Kinematic restoration becomes important at higher frequencies merely
because higher frequencies provide short length scale wrinkles which can be destroyed
rapidly.
The boundary between these two regions indicated in the figure was determined from
the perturbation analysis by artificially setting the higher order flame speed sensitivities
(e.g., sL2 ) to zero. The only source of nonlinearity is then due to kinematic restoration.
The indicated boundary was then determined from the points where the nonlinear flame
contributions in the cases with and without the higher order sL sensitivity were within 10%
of each other.
We next consider the regime labeled “Stoichiometric cross-over mechanism”. This
nonlinearity is completely due to the second source of nonlinearity noted above, i.e. the
sL -φ and hR -φ nonlinearities. However, in this region, this mechanism dominates for all
Strouhal numbers and is due to the drastic change in sL and hR characteristics on the lean
and rich side of stoichiometric. As described in the introduction section, the equivalence
ratio space can be divided into three distinct regions (see Figure 2). For large excitation
amplitudes, the local equivalence ratio can instantaneously cross over from region I to
region II or region I to region III and vice versa. The trend in the variation of sL and hR
qualitatively changes when this cross-over occurs. For the sake of illustration, consider an
instantaneous variation of φ over an excitation cycle shown in Figure 11.
The instantaneous value of sL falls with decreasing φ over a portion of the excitation
cycle in the rich case, as opposed to rising further. Hence, if for some instantaneous
oscil
lation amplitude φ around some mean equivalence ratio φ o , if |φ − φo | > φsL ,max − φo ,
the trend of sL variation over one excitation cycle changes and causes a very abrupt saturation of the mass burning rate contribution to the total heat release, the second term in
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Figure 11. Variation of flame speed with equivalence ratio. The vertical line marks the equivalence
ratio for maximum sL . The arrows show the extent of variation of sL over one excitation cycle at ε =
0.25 in each case.
equation (8). Similarly, a sufficiently high excitation amplitude can result in significant
nonlinearities in the heat of reaction contributions in equation (8) if |φ − φo | > |1 − φo |.
The fact that differentiates this mechanism from the kinematic mechanisms is that even if
the flame area oscillation is linear, these alone can cause strong nonlinearities in the net
heat release. Fortunately, determining the excitation amplitude ε, when this mechanism
becomes significant is very straightforward, as it is simply the minimum of the absolute
difference in value between the mean equivalence ratio and the stoichiometry where the sL
and hR characteristics change abruptly; i.e.
εstoich =
1
min |1 − φo | , φsL max − φo φo
(47)
Henceforth this second non-linearity mechanism will be referred to as the “cross-over”
mechanism. Note that, in a quasi-steady sense, this mechanism is controlled purely by the
oscillation amplitude. Hence the boundary of the crossover region in Figure 10, where this
mechanism is dominant has no dependence on St2 . However, the fact that the flame speed
sensitivity to fuel/air ratio oscillations at high frequencies progressively diminishes due
to non quasi-steady effects, implies that, in reality, this boundary “bends” in St space, as
illustrated in the figure.
The final regime, labeled ‘Flammability cross-over”, is a special case of the “cross-over”
mechanism. For sufficiently high amplitudes, the equivalence ratio can instantaneously
assume values very close to or beyond the flammability limits of the fuel. This could
lead to local flame extinction over a part of the cycle, and presumably lead to burning
area saturation. However, a complete understanding of this region requires solution of the
conservation equations with finite rate chemistry, and is beyond the scope of the current
work. Moreover, due to spatial variation in the equivalence ratio, “holes” in the flame can
advance or retreat with their own associated edge flame dynamics [60].
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Define “cross-over” amplitude, as the minimum amplitude at which some form of
cross-over occurs:
⎞
⎛
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εcrossover =
1
⎜
min ⎝εstoich ,
φo
φo − φfl, lean , φfl, rich − φo
⎟
⎠
(48)
εf l =f lammability crossover amplitude
where φfl, lean and φfl, rich denote the dynamic lean and rich flammability limits, see Sankaran
and Im [50]. The mean equivalence ratio determines the type of cross-over that is first
encountered. For example, a CH4 /Air mixture at a mean equivalence ratio of 0.85 will
probably encounter the stoichiometric cross over mechanism prior to the lean flammability
limit mechanism (εstoich = 0.18, εf l = 0.41, at STP). However, for a mixture with a mean
equivalence ratio of 0.6, flammability cross-over probably occurs first (εstoich = 0.72, εf l =
0.09).
Consider further the behavior of the transfer function in the low St2 limit for the cases
where the asymptotic analyses detailed in the previous sections are valid. We have the
following results for the terms on the RHS of equation (27),
3 3
lim FA = − lim FsL −A = −sL1 − ε2 sL1
− 2sL1 sL2 + sL3
St2 →0
St2 →0
4
3
2
+ hR1 sL2
lim FsL −hR −A = ε2 hR2 sL1 − hR1 sL1
St2 →0
4
3 2
− hR1 sL2
lim FhR −A = hR1 + ε2 hR3 − hR2 sL1 + hR1 sL1
St2 →0
4
(49)
(50)
(51)
From equation (49), it can be seen that in the low St2 limit, the contributions due to the
burning area fluctuations and burning rate oscillations have the same absolute magnitude,
but opposite signs. This means that the contributions in this limit are exactly out of phase
and cancel each other. Physically, this may be reasoned as follows. Two lean flames with
the same fuel flow rate but different air-flow rates will have the same steady heat release
rate. Local variations in mass burning rate due to slow time scale perturbations in sL must
be balanced by the oscillations in the net burning area. As such, the low frequency limit for
the transfer function is given by:
3
lim F = hR1 + ε2 hR3
St2 →0
4
(52)
From this, it follows that in the limit of St2 → 0, the net flame response is purely dependent
on the sensitivities of the heat of reaction, hRj (see equation (22)).
With the preceding material as background, we next present results obtained from
numerical computations. Figure 12(a) and Figure 12(b) plot the variation of the magnitude
and phase of the total heat release transfer function with increasing excitation amplitudes
for the lean and the rich flames, respectively.
Notice first that the transfer function response for all excitation amplitudes tends toward
the linear value in both the lean and the rich flame cases as St2 →0. This is due to the low
frequency behavior of the transfer function explained in the previous paragraph. With
increasing St2 , the transfer function begins to deviate significantly from the linear value. As
such the slight deviation from the linear value at low amplitudes with increasing St2 can be
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704
Shreekrishna et al.
Figure 12. Variation of the gain and phase of the nonlinear transfer function, F , with Strouhal
number, (a) φo = 0.85 (lean), (b) φo = 1.28 (rich), β = 4.
ascribed to the manifestation of sL − φ nonlinearities in both cases. For a chosen amplitude,
with increasing St2 , the role of kinematic restoration as a means to destroy flame surface
area and cause heat release saturation becomes increasingly significant. As the excitation
amplitude is increased beyond ε = 0.18 in the lean case and ε = 0.15 in the rich case, the
crossover mechanism becomes dominant.
We now examine the converse scenario, i.e. the variation of heat release response with
excitation amplitude at a fixed value of St = 2π . Figure 13(a) and Figure 13(b) plot the
variation of the magnitude of the heat release response (not its transfer function, as in Figure
12) with increasing excitation amplitude for the lean and rich cases respectively.
Figure 13. Magnitude of individual contributions to the total heat release, q , (a) φo = 0.85 (lean),
(b) φo = 1.28 (rich), β = 4, for St2 = 6.68 (St = 2π ). The vertical dashed black line marks the
amplitude at which the instantaneous equivalence ratio begins to cross over into the rich/lean region
over a part of the excitation cycle. The dash-dot interpolations to zero amplitude are obtained using
corresponding expressions from asymptotic analysis [52]
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Combustion Theory and Modelling
705
Figure 14. Local burning area fluctuation magnitudes of a lean flame: φo = 0.85 (lean), β = 4,
St2 = 6.68 (St = 2π ). (a) Variation of normalized local burning area, ∂(A(r, t)/A◦ )/∂r ∗ with radial
location. (b) Variation of integrated burning area, A, with radial location.
The dashed vertical lines on both show the amplitude where the stoichiometric crossover mechanism is initiated. Overlaid are the magnitudes of the individual constituent
components (see equation (8)) of the total heat release response at the excitation frequency
in each case. First, notice that the amplitude of the burning area oscillation, A, varies
nonlinearly and non-monotonically, even with excitation amplitudes that are smaller than
εcrossover . This is a counterintuitive result as it shows that the absolute magnitude (i.e. not the
relative rate of increase) of A fluctuations decreases, with increasing ε. This result is due
to the spatially integrated character of the flame area. To better understand this, consider
Figure 14(a), which plots the spatial dependence of the local flame area fluctuation magnitude (defined as ∂ (A (r ∗ , t ∗ )/Ao ) /∂r ∗ ).
It can be seen from Figure 14(a) that the local flame area fluctuation magnitude exhibits
non-monotonic spatial dependence, but monotonically increases with ε at each position.
Additionally, different spatial locations contribute differently in terms of phase relative to
the flame base. The total magnitude of the area fluctuations is merely the magnitude of the
integral of the local flame area fluctuations over the flame surface area. Mathematically,
this amounts to a phasor addition. This phasor addition leads to a complex non-monotonic
dependence in the burning area, with the flame locations closer to the tip acting to effectively
reduce contributions from the parts of the flame closer to the base. This can be understood by
considering Figure 14(b), which plots the spatially integrated area fluctuation magnitudes
from the base of the flame to a given radial location. As such, the values at r=0, the flame
tip, indicate the magnitudes of the burning area fluctuations integrated over the entire flame.
At a given spatial location near the flame base, these curves monotonically increase with
perturbation amplitude. They deviate from each other near the flame tip, however, due to
cancellation associated with the amplitude dependent phase, leading to a net reduction in
the total flame area fluctuation magnitude at higher amplitudes of excitation.
Next, the trends of the ‘sL − A’ contributions to the total heat release occur because
of the various mechanisms that lead to area saturation. In the lean case, nonlinearities are
dominated by kinematic restoration, as the crossover mechanism sets in at a larger amplitude
than for rich flames, where kinematic restoration and crossover are both important at lower
excitation amplitudes. Hence, the ‘sL − A’ contribution saturates for rich flames. Although
706
Shreekrishna et al.
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not reproduced here, a similar saturation is seen for lean flames at higher amplitudes at
which cross-over occurs from the lean to the rich side. However, flammability cross-over
mechanism might become a potential competitor at such high amplitudes.
Finally, the ‘hR − A’ and ‘sL − hR − A’ contribution to the heat release magnitude is
negligible for rich flames since hR is fairly constant on the rich side and has negligible
sensitivity to equivalence ratio fluctuations.
At even larger excitation amplitudes, there is a competition between three processes –
crossover across the flame speed maximum, crossover across the flammability limits of the
fuel mixture and kinematic restoration. The dynamics in such a situation needs detailed
chemistry considerations and is not studied in this paper.
4. Conclusion
The key conclusions of this work are the following. First, the response of rich flames to
fuel/air ratio oscillations fundamentally differs from that of lean flames. This is due to the
difference between the heat of reaction and flame speed sensitivities on the rich and lean
sides.
Second, at higher Strouhal numbers, effects due to flame stretch and non quasi-steady
response of the flame structure because of a time lag due to internal flame processes
become significant. For tall flames such that βMa −1/2 (δ/R)1/2 2π , non quasi-steady
flame response occurs at lower frequencies than flame stretch. However, for moderately tall
flames, e.g., β ∼ 10, they become important at similar Strouhal numbers, but affect the
global flame response differently. Non quasi-steadiness of the flame structure introduces
a scaling factor and phase shift in the excitation amplitude which perturbs the flame, and
hence changes the gain and phase characteristics of the flame transfer functions with respect
to its quasi-steady counterparts. The transfer function gain drops O(1/St) more rapidly with
Strouhal number than the quasi-steady gain. On the contrary, stretch introduces a correction
to the unstretched quasi-steady transfer function and serves to smoothen out undulations in
the gain characteristics, without largely altering the asymptotic Strouhal number tendencies
of the transfer function gain.
Third, for both rich and lean flames, there are two mechanisms of nonlinearity in the
heat-release response of premixed flames. The first is due to nonlinearities in the flame speed
and heat of reaction dependence upon equivalence ratio. The second is due to the intrinsic
nonlinear property of premixed flames in that they propagate normally to themselves at
each point, the “kinematic restoration mechanism”. The first mechanism manifests itself in
two different ways, most prominently through the so called “cross-over” mechanism where
the instantaneous stoichiometry oscillates between lean and rich stoichiometries.
The examples shown in this paper were obtained for CH4 /Air flames where the width of
region II (see Figure 2) is relatively small. Other fuels, such as H2 /CO mixtures, have much
wider region II widths, so that the perturbation amplitude at which the cross-over mechanism
becomes important for the heat of reaction and flame speed sensitivity is quite different.
This could lead to much more complex amplitude dependence than in the examples shown
here.
Further study is needed in several areas. First, the current analysis does not consider
the combined effect of flow and equivalence ratio perturbations. In most practical systems,
equivalence ratio perturbations are caused by flow perturbations and, moreover, equivalence
ratio oscillations will also induce flow perturbations because of the temperature jump across
the flame. It has been shown by prior analyses that the flame response to flow perturbations
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Combustion Theory and Modelling
707
can be highly nonlinear [48, 61]. However, it is not clear as to when the effect of one
mechanism would dominate over the other.
Next, this work has demonstrated the existence of flame stretch and non quasi-steadiness
associated with diffusion processes in the flame preheat zone, as two important processes
at higher Strouhal numbers that act to reduce the gain of the flame response. Both these
process could potentially interact nonlinearly to further influence the gain of the flame
response. This matter requires further investigation.
Additionally, even for quasi-steady dynamics of flames in the absence of flame stretch,
this work has emphasized the role of the “flammability cross-over” mechanism. For systems
running very lean, it is more likely that they will encounter local extinction due to deviations
of the fuel/air ratio below the flammability limits, well before they begin crossing over into
the rich side. This will introduce local holes on the flame surface and the associated edge
flame dynamics. These flame edges advance or retreat at different points of the flame at
a velocity that is not equal to the laminar burning velocity [60]. It can be anticipated that
the perturbation amplitude where the instantaneous equivalence ratio passes through the
flammability limits will also be associated with significant changes in flame dynamics [50].
Consideration of these effects should also be a key focus for future studies.
Acknowledgements
This research was supported by the Gas Technology Institute, under a subcontract to the US Department of Energy, and the US-DOE and NSF under contracts DE-FG26-07NT43069 and CBET0551045, respectively (contract monitors: Dr Joseph Rabovitser, Dr Mark Freeman, and Dr Arvind
Atreya, respectively). Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors also wish to acknowledge Vishal Acharya, Dong-hyuk Shin and Jack Crawford
for many insightful discussions.
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Appendix A
Effect of azimuthal flame stretch
This appendix discusses the relative roles of axial and azimuthal stretch on the flame dynamics.
Retaining azimuthal stretch, the Fourier transform of the O(ε) correction to the mean flame surface
at the forcing Strouhal number ζ1 (r ∗ , St) = ξ̂1 is described by:
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σc∗
d 2 ζ1∗
+
dr ∗2
∗
∗
σc∗
dζ
iSt ∗
e−i(1−r )St
−
1
−
=
ζ
r∗
dr ∗
α
α
(53)
A comparison of azimuthal to axial stretch effects yields
∗
κazi
λ∗c sin ψ
σc∗ (1/r ∗ ) (∂ξ ∗ /∂r ∗ )
α 1
∼
∼
∼ ∗ ·
∗
κax
σc∗ (∂ 2 ξ ∗ /∂r ∗2 )
r∗
r St
(54)
Here, λc is the convective wavelength and ψ is the flame half-angle, given by cot ψ = α. This equation
shows that the influence of azimuthal stretch decreases as frequency increases, except for very small
r values. A representative result, which was obtained by solving equation (53) numerically, along
with the fixed-anchor and symmetry boundary conditions, equations (16) and (17) respectively, is
provided for the case of St = 10 in Figure 15.
This figure shows that the effect of azimuthal stretch is seen only at the flame tip and that over
the rest of the flame, there is no significant influence.
Figure 15. Variation of the response amplitude of perturbation of the flame front about the mean
flame for a conical flame with β = 4.0, Ma = 1, δ/R = 0.1 at a forcing Strouhal number of St = 10.
Combustion Theory and Modelling
711
Appendix B
Effect of flame stretch on the mean flame shape
This appendix considers the stretch-effected solution for the mean flame shape, obtained from equation
(11). The key objective of this section is to demonstrate that the stretch correction to the shape of the
mean flame is exponentially small everywhere except near the flame tip, r =0. Assume the following
relationship for flame speed [62]
sL
= 1 − δ∇ · n̂ + Ka (1 − Ma)
sLo
(55)
where n̂ is the local normal to the flame surface and Ka is the Karlovitz number may be expressed in
terms of the flame stretch rate kstr as
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Ka =
δ
kstr
sLo
(56)
Using equation (12) for the flame curvature, the dimensionless stationary flame equation is
∗
δ β
∂ 2ξ ∗
1 ∂ξ ∗
β2
+
+
∂r ∗2
r ∗ ∂r ∗
r∗
= 1+β
2
∂ξ ∗
∂r ∗
2 3/2 ∂ξ ∗
∂r ∗
3 1/2 1/2 1 + β 2 (∂ξ ∗ /∂r ∗ )2
− 1 + β2
1/2
1 + β 2 (∂ξ ∗ /∂r ∗ )2
− Ma o (1 + β 2 )1/2
(57)
Here, Ma o = 1 − Ma and δ ∗ = δ/R is the dimensionless flame thickness. The boundary conditions
for equation (57) are the anchor-fixed boundary condition at the base, equation (16) and the zero
slope at the tip boundary condition, equation (17). Rewrite these equations as [63]:
d (βξ ∗ )
=s
dr ∗
ds
f (s)
δ ∗ ∗ = V (s) − δ ∗ ∗
dr
r
(58)
Here,
V (s) = 1 + s
2 3/2
1/2 1/2
1 + s2
− 1 + β2
(1 + s 2 )1/2 − Ma o (1 + β 2 )1/2
f (s) = s + s 3
(59)
We now write the solution for the flame position as
βξ (r ∗ , δ ∗ ) = X (r ∗ , δ ∗ ) + x (ρ, δ ∗ ) − xmatch
s (r ∗ , δ ∗ ) = (r ∗ , δ ∗ ) + σ (ρ, δ ∗ )
(60)
where the stretched coordinate ρ = r ∗ /δ ∗ and xmatch is a constant used to match the inner solutions
(x, σ ) and outer solutions (X, ). Following standard matched asymptotics procedures in singular
perturbation theory [63–65], we note the outer solutions for the position and slope, which are the
O(1) terms in the expansion of X and in terms of δ ∗ , Xo and o are
o = −β; Xo = β (1 − r ∗ )
(61)
712
Shreekrishna et al.
thereby satisfying the anchor-fixed BC. Substituting equation (60) in equation (57) and substituting for the outer solution terms, we obtain equations for the corrections due to stretch as
dx
=σ
dρ
(62)
dσ
f ( + σ ) − f ()
= V ( + σ ) − V () −
dρ
ρ
(63)
Further expanding x, and σ as a series in δ ∗
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x (ρ) = xo (r ∗ ) + δ ∗ x1 (ρ) + O δ ∗2
(r ∗ ) = o (r ∗ ) + δ ∗ 1 (r ∗ ) + O δ ∗2
σ (ρ) = σo (ρ) + δ ∗ σ1 (ρ) + O δ ∗2
(64)
and expanding equations (62) and (63) to first order in δ ∗ , we obtain
dxo
= σo
dρ
(65)
dσo
f (−β + σo ) − f (−β)
= V (−β + σo ) −
dρ
ρ
(66)
The two boundary conditions on σ o are:
σo (ρ = 0) = β
σo (ρ = ∞) = 0
(67)
Using the Mean Value Theorem for f in equation (66), note that
f (−β + σo ) − f (−β) = f (λ1 ) σo = 3λ21 + 1 σo ≥ σo
(68)
for some λ1 ∈ [−β, −β + σo ] ⊆ [−β, 0]. Hence, we have
dσo
≤ g (ρ, σo (ρ)) σo
dρ
(69)
where
g (ρ) =
1
V (−β + σo (ρ)) 1
− = w (σo (ρ)) −
σo (ρ)
ρ
ρ
(70)
w(σo (ρ))
Note that
g (ρ) < w (σo (ρ)) ≤ max w (σo (ρ))
σo ∈[0,β]
(71)
We will assume below that Ma > 0 to prevent the flame tip from opening. Further, given the
approximate nature of equation (55), singularities in the solution can also develop for highly stretched
flame tips, corresponding to points where the denominator of V (see equation (59)) is zero. To avoid
Combustion Theory and Modelling
713
such singularities, we require Ma to satisfy
Ma > 1 −
1
(72)
(1 + β 2 )1/2
We split the interval of Markstein numbers given by equation (72) into two intervals, Ma > 2 and
Ma ∈ (1 − (1 + β 2 )−1/2 , 2] for ease of further analysis.
Case (i): Ma > 2
For this case, it may be shown that
max
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σo ∈[0,β]
β
V (−β + σo )
V (0)
<0
= −
=
σo
β
|Ma o | 1 + β 2 + 1
1 + β2 + 1
(73)
β
dσo
σo
≤ −
dρ
|Ma o | 1 + β 2 + 1
1 + β2 + 1
(74)
This gives
which further yields that
⎛
σo = βO ⎝e
−
|Ma o |
√
√
β 1+β 2 +1
1+β 2 +1
ρ
⎞
⎠
(75)
Case (ii) : Ma ∈ (1 − (1 + β 2 )−1/2 , 2]
We begin by noting that
w (σo (ρ)) =
V (−β + σo (ρ))
< 0 ∀ρ ∈ (0, ∞)
σo (ρ)
(76)
Hence,
g (ρ) < w (σo (ρ)) < 0, ∀ρ ∈ [0, ∞)
(77)
We next show that w attains a maximum for ρ ∈ [0, ∞). To see this, note that
1/2
−1
1 + β2
<0
wo = w (σo (0)) = − β 1 − Ma o (1 + β 2 )1/2
w∞
1/2
β 1 + β2
δ∗
= w (σo (∞)) = −
=
−
<0
1 − Ma o
σc∗
(78)
(79)
Further, w has no singularities in [0, ∞) and is continuous and always lesser than zero. This implies
that there exists
η = max w (σo (ρ)) = max {wo , w∞ , w∗ } < 0
ρ∈[0,∞)
(80)
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Shreekrishna et al.
for some ρ∗ in [0, ∞). Hence we have
dσo
≤ − |η| σo
dρ
(81)
σo ≤ βe−|η|ρ
(82)
implying
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To conclude, we note that for Ma o < (1 + β 2 )−1/2 or equivalently, for Ma > 1 − (1 + β 2 )−1/2 , irrespective of η, since
σo /β σo ρδ∗ /σ ∗ ≤1
c
= lim e
lim
ρ→0 e−|η|ρ ρ→0 β
(83)
∗ ∗
σo = βO e−ρδ /σc
(84)
we can write
Further,
ρ
xo (ρ) =
σ∗ ∗ ∗
σo dρ = β 1 − c∗ O e−ρδ /σc
δ
(85)
0
Hence, matching yields
ξ (r ∗ ) = (1 − r ∗ ) −
Ma
β (1 +
β 2 )1/2
∗ ∗
O e−r /σc
(86)
Appendix C
Perturbation equations for flame surface position
The evolution equations for nonlinear corrections to the quasi-steady flame surface location, ξ 2 and
ξ 3 , in the absence of stretch (σc∗ = 0) may be written as.
∗
∂ξ2∗
∂ξ2∗
∂ξ1
∗
∗
−
α
−
αs
cos(St
(1
−
r
−
t
))
L1
∂t ∗
∂r ∗
∂r ∗
∗ 2
∂ξ1
1
α − α2
+ sL2 cos2 (St (1 − r ∗ − t ∗ )) = 0
(87)
+
2
∂r ∗
∂ξ3∗
∂ξ1∗
∂ξ2∗
∂ξ3∗
2
∗
∗
∗
∗
(St
(1
(St
(1
))
))
−
α
−
α
s
cos
−
t
+
s
cos
−
r
−
t
−
r
L2
L1
∂t ∗
∂r ∗
∂r ∗
∂r ∗
∂ξ1∗ 3 ∗ ∗
∂ξ1∗ 2
1
1 2 ∂ξ1 ∂ξ2
∗
∗
2
(St
(1
))
s
+
α
−
α
+
cos
−
r
−
t
α
−
α
+ α α − α2
L1
2
∂r ∗
∂r ∗ ∂r ∗
2
∂r ∗
+ sL3 cos3 (St (1 − r ∗ − t ∗ )) = 0
The corresponding solutions for ξ 2 and ξ 3 are presented in Hemchandra et al. [52].
(88)