This document provides an analysis of the structure and extinction of asymptotic flamelets based on the seminal paper by Norbert Peters in 1983. It begins with notation and assumptions, then derives the flamelet equation through non-dimensionalization and asymptotic expansions. This yields the flamelet structure equation, which is then simplified using the Schwab-Zeldovich coupling relation between temperature and species. The analysis results in an ordinary differential equation for the flamelet structure identical to that in Law's combustion text. Outer solutions and boundary conditions are then discussed to complete the derivation.

1. Some notes on the structure and extinction of
an asymptotic flamelet (Peters 1983)
Praveen Narayanan
January 22, 2009
1 Introduction
This document aims to add expository footnotes to Norbert Peters’s seminal
paper on non-premixed flame extinction. The problem solved is identical to
that of Peters, but the procedure borrows heavily from C. K. Law’s excellent
combustion text. Notational variances aside, the analysis carried out here is
fairly identical to C. K. Law’s text in Chapter 9.
2 Notation
In Peters, there are no ambiguities of notational origin because most of the
material is presented in terms of dimensional quantities. However, the anal-
ysis is greatly simplified by defining non-dimensional quantities that will be
used in the asymptotic analysis.
The non-dimensionalization is in conformity with C. K. Law (in section 5).
The chemical reaction is proposed as
⌫0
F F + ⌫0
OO ! ⌫00
P P (1)
which is a bimolecular reaction of order 2, in Peters’s analysis. The reaction
is assumed to follow Arrhenius kinetics taking the form (as in Peters)
w =
B
MF MO
⇢2
YF YOe E/T
(2)
where w is the fuel mass reaction rate; MF and MO are the fuel and oxidizer
molecular weights; YF and YO are the fuel and oxidizer mass fractions; E is
the activation temperature; T is the flame temperature.
1

2. In addition, the following symbols are used: cp: specific heat (assumed
constant), ⇢:density, D: di↵usion coe cient (assumed to be the same for all
species and same for thermal and mass transport-unity Lewis numbers) Z:
mixture fraction, : scalar dissipation rate, Da: Dahmk¨ohler number; ( h):
the heat of formation of fuel (a negative quantity); rs, the ratio of oxidizer to
fuel consumed during the reaction. The fuel and air side boundary conditions
are denoted by an additional subscript F or O (TF , TO, YF,F , YF,O = 0, YO,O,
YO,F = 0).
A tilde above a quantity signifies that it has been non-dimensionalized
(e.g., ˜T, ˜YF , ˜YO).
3 Assumptions
The principal assumptions used are
• Unity Lewis numbers.
• Constant density and specific heat.
• Constant di↵usivities.
4 Derivation of the flamelet equation
The flamelet equation in Peters will be the basis for the asymptotic analysis.
The transport equations are cast into a mixture fraction formulation that is
highly advantageous because of its one-dimensionality (in mixture fraction
space).
We start with the mixture fraction equation
⇢@Z
@t
+ ⇢uj
@Z
@xj
@
@xj
✓
⇢D
@Z
@xj
◆
(3)
where the mixture fraction is defined
Z =
YF (YO YO,O)/rs
YF,F + YO,O/rs
(4)
Do a Crocco transformation to orient the coordinate directions in terms
of Z, Z2, Z3 instead of x1, x2, x3.
Let ⌧ = t, Z2 = x2, Z3 = x3.
Apply the transformation rules
2

3. @
@t
=
@
@⌧
+
@Z
@t
@
@Z
(5)
@
@x1
=
@Z
@x1
@
@Z
(6)
@
@xk
=
@
@Zk
@
@Z
, (k = 2, 3) (7)
The energy equation may be written as
⇢@T
@t
+ ⇢uj
@T
@xj
@
@xj
✓
⇢D
@T
@xj
◆
(8)
While transforming, the convection term is seen to be negligible
compared to the reaction term (in the vicinity of the reaction zone).
This may be seen by stretching the Z coordinate around Z = Zst.
This is the basis for the flamelet equations, and one may bear in mind that
it is valid only at the reaction zone. For the outer regions, one needs to solve
the transport equation for Z written down in the aforementioned. One may
therefore not solve the outer equations in the Z coordinate with neglect of
the convection term, in which case, it will be seen that it is not of any use
to solve the outer equations with Z as the coordinate instead of x.
The transformed energy equation is written as follows (equation (11)) in
Peters.
⇢
✓
@T
@⌧
+ v2
@T
@Z2
+ v3
@T
@Z3
◆
(9)
⇢D
(✓
@Z
@xk
◆2
@2
T
@Z2
+ 2
@Z
@x2
@2
T
@Z@Z2
+ 2
@Z
@x3
@2
T
@Z@Z3
+
@2
T
@Z2
2
+
@2
T
@Z2
3
=
h
cp
w
The equation is now transformed into its one dimensional form by intro-
ducing the stretched coordinate
⇠ =
Z Zst
✏
(10)
into equation (9) and expanding around Z = Zst. ✏ is a small parameter to
be defined during the analysis.
Thus
@
@Z
=
1
✏
@
@⇠
(11)
The zeroth order expansion then gives
3

4. (⇢D)st
✓
@Z
@xk
◆2
@2
T
@2⇠
= ✏2 ( h)
cp
wst (12)
Rewrite equation (12) as the standard flamelet equation containing the scalar
dissipation rate st
st
2
@2
T
@Z2
=
( h)
cp
✓
w
⇢
◆
st
(13)
where
st = 2D
✓
@Z
@xk
◆2
st
(14)
One may similarly obtain flamelet equations for reacting scalars YF , YO
st
2
@2
YF
@Z2
=
✓
w
⇢
◆
st
(15)
st
2
@2
YO
@Z2
= rs
✓
w
⇢
◆
st
(16)
5 Non-dimensionalization
We may now non-dimensionalize the scalars by defining variables as suggested
by equations (13), (15) and (16).
Define
˜T =
cpT
( hYF,F )
(17)
˜YF =
YF
YF,F
(18)
˜YO =
YO
rsYF,F
(19)
We may then obtain the coupling relation
d2
( ˜T + ˜Yi)
dZ2
= 0 (20)
The non-dimensionalized energy equation is
d2 ˜T
dZ2
= DaC
˜YF
˜YOexp(
˜Ta
˜T
) (21)
4

5. where
DaC =
2rsB⇢stYF,F
stMOMF
(22)
Equation (21) is the form given in C. K. Law, and from here on the
procedure is analogous to that taken in the combustion text.
6 Asymptotic structure of flamelet
The equation to be solved is
d2 ˜T
dZ2
= DaC
˜YO
˜YF e
˜Ta/ ˜T
(23)
subject to BCs
˜YF = 0; at Z = 0 (24)
˜YF = 1; at Z = 1 (25)
˜YO = 1; at Z = 0 (26)
˜YO = 0; at Z = 1 (27)
˜T = ˜TO; at Z = 0 (28)
˜T = ˜TF ; at Z = 1 (29)
6.1 Inner expansions and the Zeldovich number
Do an inner expansion of ˜T, ˜YF and ˜YO around the stoichiometric value as
˜T = ˜Tst ✏✓ + O(✏2
) (30)
˜YF = ˜YF,st + ✏ + O(✏2
)
= ✏ + O(✏2
) (31)
Likewise for oxidizer
˜YO = ˜YO,st + ✏⌅ + O(✏2
)
= ✏⌅ + O(✏2
) (32)
In the above, the zeroth order terms are those obtained at equilibrium, or
inifinitely fast Chemistry. The fuel and oxidizer mass fractions have an O(✏)
leakage from the equilibrium solution (✏ is the small parameter that will be
defined during the analysis). It may also be seen that in order to understand
5

6. the inner structure of the reaction zone, one needs to characterize the O(✏)
terms, since the left hand side of equation (23) has vanishing O(1) terms.
Insert the perturbation expansions for T, YF and YO (equations (30), (31),
(32)) into the energy equation (23). Using the stretched coordinate ⇠ =
(Z Zst)/✏ in equation (23) and expanding around Z = Zst we get
d2
✓
d⇠2
= ✏3
DaC⌅ e
˜Ta/ ˜T
(33)
Now we consider the exponential term e
˜Ta/ ˜T
and define the small parameter
✏ and Zeldovich number Ze as follows. Expand the temperature to be used
in the exponential as
˜T = ˜Tst ✏✓ (34)
Or
˜T
Tst
= 1 ✏
✓
˜Tst
(35)
so that
˜Ta
˜Tst
=
˜Ta
˜Tst
1
1 ✏ ✓
Tst
+ O(✏2)
(36)
=
˜Ta
˜Tst
✓
1 + ✏
✓
˜Tst
+ O(✏2
)
◆
(37)
Now we may write the equation (33) as
d2
✓
d⇠2
= ✏3
DaC⌅ e
˜Ta/ ˜T
(38)
= (✏3
Dace
˜Ta
˜Tst )⌅ e (✏ ˜Ta/ ˜T2
st)✓
(39)
As the quantity on the right hand side of equation (39) is of O(1), we must
have both
✏
˜Ta
˜T2
st
⇠ O(1) (40)
and
✏3
Dace
˜Ta/ ˜Tst
⇠ O(1) (41)
The expansion parameter ✏ is obtained by arbitrarily setting the right hand
side of equation (40) to 1. This gives us the Zeldovich number
Ze =
˜Ta
˜T2
st
=
1
✏
(42)
6

7. and
✏ =
1
Ze
=
˜T2
st
˜Ta
(43)
The equation (41) allows the definition of a reduced Dahmkohler number
arising as a consequence of the large Dahmk¨ohler number
Da = DaCe
˜Ta/ ˜Tst
(44)
The reduced Damk¨ohler number is an O(1) quantity defined as
= ✏3
Da (45)
In Peters, the author calls equation (40) the large activation energy expan-
sion, and equation (41) the large Dahmk¨ohler number expansion, which have
now been interrelated by equation (45), the distinguished limit. With these
refinements, our energy equation now takes the form
d2
✓
d⇠2
= ( ⌅e ✓
) (46)
7 Simplification from Schwab-Zeldovich cou-
pling
Use the coupling relation to obtain expressions for and ⌅ in the energy
equation.
d2
( ˜T + ˜Yi)
dZ2
= 0 (47)
The solution to this equation is
˜T + ˜Yi = aZ + b (48)
After recognizing that at Z = 0, ˜YF = 0, ˜YO = ˜YO,O, ˜T = ˜TO; at Z = 1,
˜YF = ˜YF,F , ˜YO = 0, ˜T = ˜TF , we get,
˜YF = (TF + ˜YF,F
˜TO)Z + ˜TO
˜T (49)
and
˜YO = ( ˜TF
˜TO
˜YO,O)Z + ˜TO + ˜YO,O
˜T (50)
Here, the quantity ↵ = ˜TF
˜TO is the heat transfer parameter, important
in determining fuel and air side leakages. Now we expand ˜T about Z = Zst
and put in equations (49) and (50).
˜T = ˜Tst ✏✓ (51)
7

8. Tst may be obtained from the zeroth order expansion of (49) and (50).
˜Tst = (↵ ˜YO,O)Zst + ˜YO,O + ˜TO = (↵ + ˜YF,F )Zst + ˜TO (52)
Get the mass fractions ˜YF and ˜YO as
˜YF = (↵ + ˜YF,F )(Z Zst) + ✏✓ (53)
= (↵ + ˜YF,F )✏⇠ + ✏✓ (54)
= ✏[(↵ + ˜YF,F )⇠ + ✓] (55)
= ✏ (56)
˜YO = (↵ ˜YO,O)(Z Zst) + ✏✓ (57)
= (↵ ˜YO,O)⇠✏ + ✏✓ (58)
= ✏{(↵ ˜YO,O)⇠ + ✓} (59)
= ✏⌅ (60)
Thus,
= (↵ + ˜YF,F )⇠ + ✓ (61)
and
⌅ = (↵ ˜YO,O)⇠ + ✓ (62)
Our energy equation may now be cast into the form
d2
✓
d⇠2
= [(↵ + ˜YF,F )⇠ + ✓]{(↵ ˜YO,O)⇠ + ✓}e ✓
(63)
Note that this is identical to equation 9.3.18 (Chapter 9) of C. K. Law, with
the di↵erence that ↵ = used in the book (in C. K. Law it seems to be
written so that air is preheated, instead of fuel as done here).
The boundary conditions for this ordinary di↵erential equation may be
obtained by matching with the outer solutions.
8 Outer solutions and matching to obtain BCs
For the outer solutions, one needs to revert to nomal coordinates since the
convection term can no longer be neglected. However, for purposes of match-
ing, it su ces to say that as the stretched inner coordinate ⇠ ! ±1, the
leakage takes a small constant value. One may either set this to zero, or take
8

9. the derivative of this leakage, which will turn out to be zero. Either of these
approaches are equivalent. On the fuel side one has the oxidizer leakage
lim
⇠!1
⌅ = cO (64)
(65)
Taking derivatives gives
lim
⇠!1
d⌅
d⇠
= 0 (66)
Or lim
⇠!1
d✓
d⇠
= ˜YO,O ↵ (67)
Likewise, for the air side one has the fuel leakage
lim
⇠! 1
= cF (68)
(69)
Taking derivatives gives
lim
⇠! 1
d
d⇠
= 0 (70)
Or lim
⇠! 1
d✓
d⇠
= ˜YF,F ↵ (71)
Alternate to the foregoing, one may also set the leakage values to zero in the
outer solutions. This gives
lim
⇠! 1
✓ = (↵ + ˜YF,F )⇠ (72)
lim
⇠!1
✓ = (↵ ˜YO,O)⇠ (73)
9 Recasting of inner equations into Li˜nan’s
form
Our problem is now completely defined and may be solved either analytically
as in Li˜nan. The usual approach is to recast the inner equation into a more
convenient form by making the transformation
✓ = ˜✓ + ⌘, ⇠ =
2
1 + ˜YO,O
⌘ (74)
9

10. with
=
4
1 + ˜Y 2
O,O
, = 1
2(1 + ↵)
1 + ˜YO,O
(75)
The equations then become
d2 ˜✓
d⌘2
= (˜✓ ⌘)(˜✓ + ⌘)e (˜✓+ ⌘)
(76)
with boundary conditions
d˜✓
d⌘
!
1
= 1 and
d˜✓
d⌘
!
1
= 1 (77)
10 Non unity fuel and air exponentes in re-
action term
The analysis for varying fuel and air coe cients (⌫F and ⌫O for fuel and air) is
quite similar to the unity coe cients case. Here, one expands the quantities
in powers of ✏, where ✏ has been identified in the foregoing as ˜T2
st/ ˜Ta. As
usual, one is interested mostly in the first order behavior.
˜T = ˜Tst ✏✓ (78)
˜YF = ✏ (79)
˜YO = ✏⌅ (80)
Upon insertion into the inner energy equation, one gets
d2
✓
d⇠2
= ✏1+⌫F +⌫O
exp
˜Ta
˜Tst
!
DaC⌅⌫O ⌫F
exp( ✓) (81)
It then transpires that
✏1+⌫F +⌫O
⇠ Da 1
C [exp
˜Ta
˜Tst
!
] 1
⇠ Da 1
C (82)
Transposition into Li˜nan’s form gives
d2 ˜✓
d⌘2
= (˜✓ ⌘)⌫F
(˜✓ + ⌘)⌫O
e (˜✓+ ⌘)
(83)
subject to the boundary conditions
d˜✓
d⌘
!
1
= 1 and
d˜✓
d⌘
!
1
= 1 (84)
10

11. 11 Extinction conditions for flame with non-
unity fuel-air exponents
The equation (83) was solved using Matlab with ⌫F = 0.1, ⌫O = 1.65.
was progressively reduced, leading to a state when Matlab cannot give a
solition. The extinction scalar dissipation rate was obtained (approximately)
as st = 30/s. The corresponding value with a = 1 and b = 1 is st = 0.15/s,
di↵ering approximately by a factor of 50.
The figure attached shows the variations of the O(1) flame temperature
when is reduced. For high values of , the leakage zone is small. Progressive
reduction of leads to an increase in the extent of the leakage region. = 0.22
was the last value obtained, which gives st = 28s 1
. Note that in the
following, the outer solutions are not shown. They will take the form of
error functions (in space), and not directly amenable to solution by using
the mixture fraction approach. In Peters, the solution is only implied as a
special form using a similarity transformation, and this is not to be expected
in the general case. One will have to solve for the outer solutions in normal
space.
12 Conclusions and further work
The flame structure of a strained laminar flamelet (no radiation) (in Z space)
is obtained from asymptotic analysis. The resulting BVP is solved using
Matlab, to obtain the extinction scalar dissipation rate. It is seen that non-
unity fuel and oxidizer coe cients results in drastically di↵erent values of
the extinction scalar dissipation rate, as compared with when the coe cients
are unity. The value obtained compares favorably with Direct Numerical
Simulations using S3D. It is emphasized here that the solutions obtained
from Matlab may be refined to get a more accurate value, but even so, the
results are fairly encouraging.
11

12. Figure 1: Variation of first order correction term for inner solution with reduced
Damk¨oher number . The last solution obtained from Matlab was with = 0.22
corresponding to st = 29s 1.
12