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 seminalpaper on non-premixed flame extinction. The problem solved is identical tothat of Peters, but the procedure borrows heavily from C. K. Law’s excellentcombustion text. Notational variances aside, the analysis carried out here isfairly identical to C. K. Law’s text in Chapter 9.

2 Notation

In Peters, there are no ambiguities of notational origin because most of thematerial is presented in terms of dimensional quantities. However, the anal-ysis is greatly simplified by defining non-dimensional quantities that will beused in the asymptotic analysis.

The non-dimensionalization is in conformity with C. K. Law (in section 5).The chemical reaction is proposed as

⌫ 0F F + ⌫ 0OO ! ⌫ 00P P (1)

which is a bimolecular reaction of order 2, in Peters’s analysis. The reactionis assumed to follow Arrhenius kinetics taking the form (as in Peters)

w =B

MF MO

⇢2YF YOe�E/T (2)

where w is the fuel mass reaction rate; MF and MO are the fuel and oxidizermolecular weights; YF and YO are the fuel and oxidizer mass fractions; E isthe activation temperature; T is the flame temperature.

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In addition, the following symbols are used: cp: specific heat (assumedconstant), ⇢:density, D: di↵usion coe�cient (assumed to be the same for allspecies and same for thermal and mass transport-unity Lewis numbers) Z:mixture fraction, �: scalar dissipation rate, Da: Dahmkohler number; (�h):the heat of formation of fuel (a negative quantity); rs, the ratio of oxidizer tofuel consumed during the reaction. The fuel and air side boundary conditionsare 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 ishighly advantageous because of its one-dimensionality (in mixture fractionspace).

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 termsof Z, Z2, Z3 instead of x1, x2, x3.

Let ⌧ = t, Z2 = x2, Z3 = x3.Apply the transformation rules

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@

@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 negligiblecompared 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 thatit is valid only at the reaction zone. For the outer regions, one needs to solvethe transport equation for Z written down in the aforementioned. One maytherefore not solve the outer equations in the Z coordinate with neglect ofthe convection term, in which case, it will be seen that it is not of any useto solve the outer equations with Z as the coordinate instead of x.

The transformed energy equation is written as follows (equation (11)) inPeters.

⇢

✓@T

@⌧+ v2

@T

@Z2+ v3

@T

@Z3

◆(9)

�⇢D(✓

@Z

@xk

◆2@2T

@Z2+ 2

@Z

@x2

@2T

@Z@Z2+ 2

@Z

@x3

@2T

@Z@Z3+@2T

@Z22

+@2T

@Z23

�=��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 tobe defined during the analysis.

Thus@

@Z=

1

✏

@

@⇠(11)

The zeroth order expansion then gives

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�(⇢D)st

✓@Z

@xk

◆2@2T

@2⇠= ✏2

(��h)

cp

wst (12)

Rewrite equation (12) as the standard flamelet equation containing the scalardissipation rate �st

��st

2

@2T

@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

@2YF

@Z2= �

✓w

⇢

◆

st

(15)

��st

2

@2YO

@Z2= �rs

✓w

⇢

◆

st

(16)

5 Non-dimensionalization

We may now non-dimensionalize the scalars by defining variables as suggestedby 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

d2T

dZ2= �DaC YF YOexp(� Ta

T) (21)

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where

DaC =2rsB⇢stYF,F

�stMOMF

(22)

Equation (21) is the form given in C. K. Law, and from here on theprocedure is analogous to that taken in the combustion text.

6 Asymptotic structure of flamelet

The equation to be solved is

d2T

dZ2= �DaC YOYF 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, orinifinitely fast Chemistry. The fuel and oxidizer mass fractions have an O(✏)leakage from the equilibrium solution (✏ is the small parameter that will bedefined during the analysis). It may also be seen that in order to understand

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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= ✏3DaC⌅ 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 usedin the exponential as

T = Tst � ✏✓ (34)

OrT

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= ✏3DaC⌅ e�Ta/T (38)

= (✏3Dace� Ta

˜Tst )⌅ e�(✏Ta/T 2st)✓ (39)

As the quantity on the right hand side of equation (39) is of O(1), we musthave both

✏Ta

T 2st

⇠ O(1) (40)

and✏3Dace

�Ta/Tst ⇠ O(1) (41)

The expansion parameter ✏ is obtained by arbitrarily setting the right handside of equation (40) to 1. This gives us the Zeldovich number

Ze =Ta

T 2st

=1

✏(42)

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and

✏ =1

Ze=

T 2st

Ta

(43)

The equation (41) allows the definition of a reduced Dahmkohler number �arising as a consequence of the large Dahmkohler number

Da = DaCe�Ta/Tst (44)

The reduced Damkohler number � is an O(1) quantity defined as

� = ✏3Da (45)

In Peters, the author calls equation (40) the large activation energy expan-sion, and equation (41) the large Dahmkohler number expansion, which havenow been interrelated by equation (45), the distinguished limit. With theserefinements, 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 energyequation.

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)

andYO = (TF � TO � YO,O)Z + TO + YO,O � T (50)

Here, the quantity ↵ = TF � TO is the heat transfer parameter, importantin determining fuel and air side leakages. Now we expand T about Z = Zst

and put in equations (49) and (50).

T = Tst � ✏✓ (51)

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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, withthe di↵erence that ↵ = �� used in the book (in C. K. Law it seems to bewritten so that air is preheated, instead of fuel as done here).

The boundary conditions for this ordinary di↵erential equation may beobtained 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 theconvection term can no longer be neglected. However, for purposes of match-ing, it su�ces to say that as the stretched inner coordinate ⇠ ! ±1, theleakage takes a small constant value. One may either set this to zero, or take

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the derivative of this leakage, which will turn out to be zero. Either of theseapproaches 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 theouter solutions. This gives

lim⇠!�1

✓ = �(↵+ YF,F )⇠ (72)

lim⇠!1

✓ = �(↵� YO,O)⇠ (73)

9 Recasting of inner equations into Linan’s

form

Our problem is now completely defined and may be solved either analyticallyas in Linan. The usual approach is to recast the inner equation into a moreconvenient form by making the transformation

✓ = ✓ + �⌘, ⇠ =2

1 + ˜YO,O

⌘ (74)

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with

� =4�

1 + Y 2O,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) isquite similar to the unity coe�cients case. Here, one expands the quantitiesin powers of ✏, where ✏ has been identified in the foregoing as T 2

st/Ta. Asusual, 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�1C [exp

� Ta

Tst

!]�1 ⇠ Da�1

C (82)

Transposition into Linan’s form gives

d2✓

d⌘2= �(✓ � ⌘)⌫F (✓ + ⌘)⌫Oe�(✓+�⌘) (83)

subject to the boundary conditions

d✓

d⌘

!

�1

= �1 and

d✓

d⌘

!

1

= 1 (84)

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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 asolition. 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 temperaturewhen � is reduced. For high values of �, the leakage zone is small. Progressivereduction of � leads to an increase in the extent of the leakage region. � = 0.22was the last value obtained, which gives �st = 28s�1. Note that in thefollowing, the outer solutions are not shown. They will take the form oferror functions (in space), and not directly amenable to solution by usingthe mixture fraction approach. In Peters, the solution is only implied as aspecial form using a similarity transformation, and this is not to be expectedin the general case. One will have to solve for the outer solutions in normalspace.

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 usingMatlab, to obtain the extinction scalar dissipation rate. It is seen that non-unity fuel and oxidizer coe�cients results in drastically di↵erent values ofthe extinction scalar dissipation rate, as compared with when the coe�cientsare unity. The value obtained compares favorably with Direct NumericalSimulations using S3D. It is emphasized here that the solutions obtainedfrom Matlab may be refined to get a more accurate value, but even so, theresults are fairly encouraging.

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Figure 1: Variation of first order correction term for inner solution with reduced

Damkoher number �. The last solution obtained from Matlab was with � = 0.22

corresponding to �st = 29s�1.

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