transient pressure effects in the evolution equation for premixed flame fronts

Theoret. Comput. Fluid Dynamics (1994) 6:141-159 Theoretical and Computational Fluid Dynamics © Springer-Verlag 1994

Transient Pressure Effects in the Evolution Equation for Premixed Flame Fronts

D. Keller and N. Peters

Institut fiir Technische Mechanik, RWTH Aachen, D-5100 Aachen, Germany

Communicated by Ashwani Kapila

Received 20 May 1993 and accepted 6 July 1993

Abstract. A nonlinear evolution equation for a scalar field G(x, t) is derived, whose level surface Go = const, represents the interface of a thin premixed flame propagating in a flow field. The derivation is an extended version of an equation already proposed by Markstein [1]. It was reconsidered by Williams [2] as a basis for theoretical and numerical analysis and takes, in addition to flame curvature and flame stretch time variations of the bulk pressure, heat loss and nonconstant transport coefficients into account. The equation is an extension of earlier analyses where a flame evolution equation was derived for slightly wrinkled flames such that the front can be described by a single-valued function of a normal coordinate. That formulation excluded situations where the mean flame front has an arbitrary shape in space. Here the more general situation is analysed by using a two-length-scale asymptotic analysis. The leading-order solution of this analysis is equivalent to the equation originally derived by Markstein [1]. In addition to nonconstant properties and heat-loss effects, that had already been considered by Clavin and Nicoli [3], the influence of transient changes of the bulk pressure is analysed. All these effects are combined into a unified formulation which will serve as a basis for a new flamelet concept for premixed turbulent combustion.

I. Introduction

A large amount of work was devoted in the 1970s and early 1980s to the asymptotic analysis of flames using one-step activation energy asymptotics. Some of the outstanding results with respect to premixed turbulent flames were papers by Clavin and Williams [4], Pelce and Clavin [5], and Matalon and Matkowsky [6]. In the latter two papers a flame evolution equation was derived for slightly wrinkled flames where the front can be described as a single-valued function of a normal coordinate. For the analysis of turbulent flames of arbitrary shape in the corrugated flamelet regime [7]. It is necessary to generalize these formulations. A two-scale asymptotic analysis is performed based on a local coordinate transformation, where the short-scale variable ~, which measures the distance from the reaction zone, is identified as ~ = A(G(x, t ) - Go). It depends on all three spatial long-scale variables and on the long time scale. For convenience, we follow closely the analysis of Matalon and Matkowsky [6] wherever possible and thereby avoid restating much of the formalism needed to derive the evolution equation in the first place. We, in addition, include the effect of heat loss following the papers by Clavin and Nicoli [3], Nicoli [8], and Joulin and Clavin [9]. All transport coefficients (thermal, molecular, viscous) are assumed nonconstant but have a simple power law dependence on temperature [10]. In analyzing time variations of the bulk pressure the distin-

141

142 D. Keller and N. Peters

guished limit of large heat together with a large activation energy is taken as a basis of the asymptotic analysis (see [11] and [12]).

2. Governing Equations

We consider a premixed propagating flame front as illustrated in Figure 1. The governing equations are made nondimensional by using Yu, T~ = Tb,o -- Tu, o, Pr = PrW/(R°T~), Pr, lA, and tA as reference units of concentration, temperature, density, pressure, distance, and time. The reference state is the initial state at t = 0_ in the unburnt mixture, just before a pressure change has been introduced. The subscripts u and b refer to unburnt and burnt gases, respectively. The index 0 refers to the condition at t = 0_. The characteristic length In may be interpreted as a typical radius of curvature of a curve flame (see Figure 1) and is related by the length scale ratio

lA A - (1)

IF

to the flame thickness l F = 2r/(CpprS~,r). Here 2~, cp, and, sn, r denote the thermal conductivity, the specific heat capacity at constant pressure, and the reference burning velocity, respectively. The subscript H denotes here that the reference burning velocity is calculated in the presence of heat losses. The characteristic time t A is defined by

lA t A = = AtF, (2)

SH, r

where tF = Iv/SH,~ represents the flame time. l F and t F may be identified as short scales and l A and tA as long scales. The nondimensional governing equations are then given by:

Continuity:

Momentum (fl = 1, 2, 3):

ep e(pv,) + - o. (3)

~t ~X~

P Dt 7Ma 2 dx~ + A-lpr-~x~ 2(®) ~ + Ox, 30x~ 6"p " (4)

Fuel mass fraction:

P Dt = Le

Temperature ® = TITs:

+ AQoJ - A~fl(® - ®.)" + y - 1 dp 7 dt" (6)

unburr

Figure 1. Schematic representation of a premixed propagating flame front.

Transient Pressure Effects in the Evolution Equation for Premixed Flame Fronts 143

Equation of state:

The operator D/Dt is defined by p = p®. (7)

D 8 t? D~ = 8~ + v ~ - - . (8)

In the governing equations the following assumptions have been introduced:

(i) We assume that a single reaction between the fuel and the oxidizer takes place in which either one of them, the fuel say, is depleted. If the mass fraction of the oxidizer is largely in excess, its concentration may be assumed constant in the reaction zone and only the mass fraction of the fuel needs to be considered. The reaction rate is therefore written in dimensional form:

p Y

Here B is the frequency factor, WE is its molecular weight, E is the activation energy, and R ° is the universal gas constant. The nondimensional chemical source term co in (5) and (6) is then

(D = (D' WF2r 2 2 " (9)

Cppr SU, r Yu

(ii) The nondimensional large activation energy/3,

E /3 - ROT, (10)

plays a major role in the subsequent analysis. (iii) The specific heats of all species have been assumed constant and equal to cp for convenience.

Furthermore, the mean molecular weight W and the nondimensional heat of reaction Q = ( -AH)Yu/ (%T~WF) have been assumed constant. With the definition of the adiabatic flame temperature Tb, o we obtain Q = 1.

(iv) All diffusivities (thermal, molecular, viscous) are assumed to depend on the nondimensional temperature O following the nondimensional function ~,(O) = 2/2r:

i(o),

ge- l~ pD - , (11)

Cp

Pr2 # - cp

The Lewis number of the fuel Le and the Prandtl number Pr have been assumed constant. (v) The volumetric heat loss is to be modeled by a term, - k ( T - T u ) ~, in the temperature

equation. This includes the conventionally considered linear heat loss with e - 1, and radiative heat loss which is approximated by this form with e = 4 in the limit Tu << T. Here k is the dimensional heat-loss coefficient and a nondimensional heat-loss coefficient/~ is defined by

/-t k~,r Tr ~-1

2fl : (CpprSH, r) 2" (12)

We allow for different values of k in the preheat and postreaction zones below. (vi) The limit of small Mach numbers,

S ( Pr x~l/2 Ma = n,r\yp, , ] ~ O, (13)

is also employed such that the total pressure is given by p(t) + 7Ma2p(x , , t). Here ? = %/cv has been assumed constant. To leading order in M a 2 spatial pressure variations vanish. Only to


first order in Ma 2 do spatial pressure variations ave to be considered in the momentum equation.

In the temperature equation for the unburnt mixture, where spatial gradients vanish, the change of temperature is related to pressure changes by

P d ® u _ 7 - lap (14) 0 , dt 7 dt"

Integration yields the law of adiabatic compression:

OR(t) = ®u,oP(t)(~--l)/L (15)

When pressure changes are taken into account, we have to assume that ®,,o and therefore ®,(t) is of order fl-1 following [11]. In view of the definition of ®, this assumes that the heat release Tb, o -- T~,o is much larger than T~(t). Therefore we exploit the distinguished limit of large heat release together with a large activation energy. The temperature and the fuel mass fraction equation are combined to form an enthalpy equation by defining

H - G - ® u + Y - I , (16)

1 Y - 1. (17) F - O - - O . +L--e

The chemical source term disappears in the enthalpy equation which then reads

DH A_I 8 ( ~ S F ~ ~ -- dO. (18) P Dt ~x~\ "bx~/ A ( 0 - ®u)~ + y l dp

- - = - - ~ d t P d t"

3. A T w o - S c a l e E x p a n s i o n o f the G o v e r n i n g E q u a t i o n s

In this work we derive the evolution equation for the flame surface by considering a two-length-scale expansion. The long scale coordinates x, are those used in the governing equations. We introduce the nondimensional function G(x,, t), representing a long spatial coordinate normalized with l A. The instantaneous location of the flame is then given by

G(x~, t) = G O = const., (19)

where the yet unknown scalar field G(x,, t) is to be determined from the field equation to be derived. Here we have set Go = 0 without any restriction of generality. The short spatial variable ( is normalized with {v and resolves the preheat zone of the flame structure which measures the closeness to the flame surface and is related to G as

= AG(x~, t). (20)

Any dependent quantity a in the governing equations will now depend on a short spatial variable ~ as well as on the long variables and will be expanded as

a = a°(~, x,, t) + A-lal(~, x,, t) + . " . (21)

For instance, a spatial derivative is expanded up to first order as

c~a A~G ° c~a ° OG ° ~a 1. OG x Oa ° Oa ° - + ~- - - + (22) 3x~ ~x~ 8~ 8x~ 8~ Ox~ O~ ~x~"

Expanding the nondimensional function 2(®) yields

,~(®) = ~(0o + A-101 + ®(A-Z))

= ~(O °) + A-X)'.'(O°)O 1 + O(A-2), (23) k.__N...._J k "( J

=~o =~1

where 2' denotes the derivative of the function 2. When these expansions are introduced into the


continuity equation we obtain, to leading order,

~3rn ° - 0 , ( 2 4 ) a~

where m is defined by

m - p s = - p ~ + v , - P D~

Furthermore we denote

(OG~ 2 (26) I V G I 2 = kOx,] "

In deriving (24) it has been taken into account that G does not depend on the short spatial variable ~. The momentum equation, to leading order, is

_ 28( 'o0V~) m °Ov~ 8G° OP° PrF aG° 8G° 8 ().°Ov°) 2 ~ a~ axe a¢ + L ~ 8xo c~\ O~l + IVG°I

28G ° 8G ° ~ / / -o_/ lOV°\ - I (/7= 1, 2, 3). (27)

At first we only consider the chemically inert preheat and heat-loss zones where the reaction term can be neglected. The leading-order fuel mass fraction equation is obtained as

oOY ° IVG°l 2 ~oOY° (28)

With the boundary conditions yO = 1 at ~ --* -oe and yO = 0 at ~ = 0 its solution is

yO = 1 - exp(Le x), ~ < 0, (29)

r ° = o, ~ > o, (30)

where x = ~o/IVG°l and ~ and q~ are given by

= f ] d~ m ° ~o' ~o - IVGO I . (31)

The quantity q) defined here is related to pressure changes in (92) below. For constant pressure, q~ = 1 and (25) together with m ° = I VG°I represent the leading order of the G-equation in nondimensional form. To leading order the temperature is written

0 0 ® ° H = - ~ ( O ° - O.y .

zp (32)

Expanding this equation up to first order in fl-1 and integrating yields

2:2fx ;~ ' G ° = ®. + exp(x) + fl-x (®o,o _ ®.)~-1~o,o dx" exp(x') dx' -oo

_ p _ l / ~ b ,~o b exp(x), ~ < 0, (33)

/4b ~0(1 + X)] ~ > 0. (34) o o = l+Ou_ -x[ u f ° (co,o_ Ou),io,o ax + L2,p 2 d-~o

Here we allowed for different values of /4 , /4, in the preheat zone and of /~b in the heat-loss zone. In deriving these equations results obtained from analyzing the reaction zone have been used. In particular, matching the temperature gradients of the preheat and reaction zone requires that cp is only a function of time (see (92) and (98)). The second superscript of a quantity in the above equations denotes the order in fl-1. ~.b o is evaluated at the adiabatic flame temperature. Furthermore, using the definitions for H and F and assuming the Lewis number close to unity such that

Le = 1 + fl-~l (35)


it is seen that

H ° = f l - l [ ~ f f o o f f ' ( O ° ' ° - O u , = - ' ~ ° ' ° d x " e x p ( x ' ) d x ' - ~ J . ° e x p ( x )

- lx exp(x)/ , ~ < 0, (36)

gO = __fl-ll -/~u 2~/~bO(l "-~

F ° = f l - l [ ~ f ] o o f ~ ' ( O ° ' ° - O ~ ) ~ - ~ 2 ° ' ° d x " e x p ( x ' ) d x ' - ~ 2 ~ e x p ( x )

-- 1(1 + exp(x)(x - 11)1, ~ < O, (38)

F ° = - f l - l [ ~ f f o o ( O ° ' ° - - O u ) = ) . ° ' ° d x + ~ ° ( l + x ) ] , ~ > 0 , (39)

are small of order O(fl-~). The continuity, the temperature, and the enthalpy equation are carried to first order. They are

written in the preheat and heat-loss zone:

Om I OpO OP°V° -- O, a~- + at + ax,

oo = a (-oa°l) m 0001 amlO° aP°O° aP°O°v° 2 ~ - 7( -+ ~ + ~ + ~ - Iv I F

o OH1 OmlH° 9P °H° OP °H°v° m ~ - + c ~ + ~ - + c~x~

Integration of these equations requires the If we multiply the equation for vl with multiplied with ~G°/OXl, we obtain

(40)

2 ago c~G1 c~ ( ~ o ~ 0 ° ] + ax, ax, a{\ cy /

7 - ldp + - - - - (41)

dt'

2 C~ (-0 oF1) 2 ~G0~GI ~ (,~o6~F°'~ -IVG°l ~ '~7~ + ax~ax~<\ < /

+g;7~ \ ax~/+~\ a ¢ / / + ~ ;~ -1 ~F° "

+ ]VG°[2 ~ ( , ~ ~ ) -- ~--~010~(0° -- Ou) ~-1

+ 7__-- 1 dp pO d®u. (42) dt dt

integration of the leading-order momentum equation (27). OG°/Ox2 and substract the respective equation for v 2,

= PriVGO[ZO ~.o Or° (43) m ka # ax 2 a( axlJ a¢_ _a¢ ax2 a~ ax,j_]

Since there is no jump of this combination across the reaction layer (see [6]) integration from (o to ( leads readily to

o aGo o OG° v o aGO v o aGO V l ~ x k - V a ~ = l ( f O ) ~ x k - k(#O)~x ~, (44)


where k = 2, 3, since the same derivation would hold for v3 instead of I.) 2. Here (o is an arbitrary value of ~. Multiplying now s o and s°(~o) defined in (25) by aG°/axl and subtracting we obtain, with (44),

_ _ o aGO aGo a~°(S°axl - s°(~°)) = (v° - ~(~°))~X~x~ ax~

o /aGO'\ : ( v O - /)I(~O))~X~ )"

Using s o = m°/p ° = m°O° /p and (31) we obtain

(45)

aG o ~o O ° - O ° ( ~ o ) o v°(~o) + (46)

v~ = ax= IVG°l p

for ~ = 1 at first, but also for ~ = 2 and 3. Analyzing the reaction zone shows that there is no jump of ®o across the reaction layer. Therefore we find v°(~ = 0_) = v°(~ = 0+) and

o o aG o ~o O ° - O u ~ < 0 , (47) v~ = V=,u + ax= IVG~I p '

o o ~G ° ~o ® o _ o ~ ~ > 0 , (48) v= = V~,u + ax= IVG°I p '

where v,,° u = v°(~ = -oo). o and (7) into the first-order continuity equation we obtain, with zero Inserting the result for v~

velocity divergence av°,u/ax, = o,

am1 O ® °-~Du®° a (aG o ~p ® o _ ® ' ~ - P. u D t ~,\~x~x~ IVG°I ~ -) - ®° -~P t , (49)

where the operator D J D t is defined as in (8) but with the velocity of the unburnt gas. Later we anticipate that O, = O(fl-x), d O , / d t = O(fl-2), and dp/dt = O(fl-x). Therefore to leading order in fl-x the term containing dp/dt in the above equation is neglected. Using (33) and (34) to leading order in fl-1 and integrating yields the result for rnt'°(~) in the preheat zone and heat-loss zone. Here the second superscript denotes the order in fl-1. We do not perform this integration here but present the result for rnl'°(~ = 0) in (71) below.

Integration of the first-order enthalpy equation requires evaluation of the quantities

and

[VG°I2 ~ ~=o_' (50)

° Ol '°a(® °'° - Ou)'-12 ° de, (51)

a® 1, 1 ~=0+" a~ (52)

First we determine ]VG°12(a®l,°/a~)~=o by integrating the first-order temperature equation with the boundary condition O 1 = 0 at ~ = -oo:

m°®l'°(~ = 0)- ]VG°lZa®~ '° ~=o_ = -ml ' ° (~ = 0)(1 + O,) + ml'°(~ = -ov)O u a¢

aZGO ["1+O. ~0,0 dO®,®

00° ' ° ~=o + IVG°I2,~OI '°(~ = 0 ) ~ -


f~ ±(&so ~ ooo ou)~o,o~ C - P ® Ox,\Sx, Iva°l p

+ 28G° 8G 1 q) OG° ~o o 80°'° ;~°09 COx~ cox, I Va°l + ~ ' COx,

aa°f~ co {coo °,°]

where (23) has been used. Performing the differentiation in the above integrals yields

801,o IVG°IZ ~ ¢=o = m~'°(( = 0)(1 + Ou) - mL°(E = -oO)Ou - 28G°&x, 8G~ox, IVG°l " ~ (54)

In order to evaluate ~°_~ o O1'°c~(O °'° - O u ) ' - l ~ 0 dE we rewrite th first-order temperature equation

o1,o ,wol OO,O) - - m ° ~ + = RHS_, (55) IVG°? COC ~ aC 8C\;, ° ~ /

where

8m1"°0°'° 82G°.~o,o 80°'°~ + ~o,o vp~ o,o.-.o,ot~ v.°'° 2 COG° - - coG1 8200'0 R H S _ - 8~ 8x2 8( 8x, cox, 8x, 8E 2

_oo OO,O) oo±( ooo))

Since RHS_ is independent of O 1, o integration with (23) yields

o',o - exp(x) f : ff l d x " . (57) ~o exp(- x")2 ° ~ RHS_ d ( '~6

Integration by parts leads to

f : I f _ ° ; O"°~(O °'° - O,)'-12 dE = -@~ RHS_ dE' exp((~ - 1)x)2 ° dx. (58) ~3 GO 09

Evaluation of 801''/8(1~=o+ requires the use of the first-order temperature equation without using the first-order continuity equation. In consideration of 0 0,0 = const, in the heat-loss zone we obtain, by expanding the equation in fl-1 and integrating to leading order in fl-~,

~) i , 0 (~ ) = 0 (59) and, to first order in fl-1,

801 ' I ~=o+ fo +09 IVG°] 2 ~ - = - + q~IVG°IRHS+ exp(-x) dx, (60)

where

(to voo oo vool O RHS+ = ~ b ~ + Z b ~ + - - 0 ~ + - - ~ o - - x

cx, Dt COx~ 8x, [VG 1) 1 + Ou

_ 2 8 G °8[VG°[ 1 ~o I ? - l d p fl - 8x, 8x, ]VG°[ ~ dt ~o 22°" (61)

Since the enthalpy equation does not contain a chemical source term, it may be integrated across the reaction zone from ( = -oo to ( = 0+:

(~I'tu f l+Ou (00'0 (~u)'--l"~O' 0 dO° '° z(~o- ] / ~ b ~0 "~ f0090P°H°d~8~t- m°Hl(~ ~- 0) = m l ( E = 0)f1-1 -- -}" ~ b ] - -

\2~o Joo ['o o = 8F1 2 OG° 8G1 ~aF°~ ¢=°+

aP°v°H° dC + I VG I ~ - ~=o+ + ax, ~ L 8~/~:-09 - J _ o0 8 x ,


+ I v G I [~oaF°l¢=°+ 1H. 2 8C I~=-oo

(62)

Evaluation of the different terms in (62) is presented in Appendix B. Using the results of Appendix B in (62) leads to a new relation for GI(~= 0)= HI(C = 0) realizing that GI '°(~= 0)= 0 and GLI(C = 0)= O(1). By inserting G1'1(~ = 0) in (90) we obtain the first-order temperature gradient of the reaction zone. In the preheat zone the first-order temperature gradient is already known (see (54)) since there we have anticipated G~'°(C = 0) = 0. Matching the temperature gradients yields

~u( P.D.WG°I 63aG°~ ( D.IVG°I'~ Dt + } + + P° bS }

/~G °~[VG°l 1 ) 0G °0G 1 1 "~ (ff"[- ~II)L~X ~ OX~ IV-G°I. + ax~ ax~ IVG°l ~° +~lVG°l~°' (63)

where L#. denotes the Markstein length with respect to the unburnt gas and £P°/I F the Markstein number:

IF -- JV (1 + G . ) G°'°-x,~ °'° dG °'° d O.

f f G°'° (G°'° - Gu)~-l~°'° dG°'° i~tl 1+O. oo,o G°'°'-~). ° ' °dG °'0'G °'° Go ~2 JO. dO.

2~o 2 b ~ \ j o .

1 ~1+o. ] _ ~ G ° j o . ln(GO,O _ Go)G®,® '~o,o dGO,O . (64)

The quantity JV is defined by

JV= ( 1 --2~¢2 j® . / 4 " F 1+°° (G°'° - G.y-~.i'.°,° dG °,° ~-~2Hb~°) -1. (65)

YC ~, ~¢~1, and ~"]ll are given by

F f " l+Ou "OO'0 ~0,0 _____ (GO,0__ Gu)a-l~O,O dGO,O .3!_ ~0 2 (66) Yf = \2~ °2 jo° Jx+oo G °'°' - Go

d G °,° dt -~ G °'° + ~o] (67) 2~o k, Jo. - G u '

H. H~ -®Go + 2 I Jt~= W(- - (1 + G.)Off~ + 2~q~2 J~ff2 + 2-~q~Z~b~-~ff ~ + ~{ 'a) , (68)

~i,, = w -(1 + G.)~4 + ~ ~C JO.

H- 2 ~ ,~b° (~1 -I- ,~Ks) H- ~ ,~¢'3], (69)

where ~ (i = 1,..., 5) are integrals (see Appendix A) that have been numerical evaluated by using the relation

~o,o = y~(GO,O) = (GO,O).. (70)


0.4

0.2

~ . '0 lea, a = 1 [

/ , / o ~ . . . . ' 1 _ / / , " ~ ~z . . . . 4 I

T./T. = 0.2

-0.2 i i 0 0.25 0.5 0.75

Figure 2. Numerical evaluation of the integrals 9ff i.

The results are presented in Figure 2. The exponent n is assumed to vary from 0 to 0.75 and ®u is assumed constant. In case the n = 0 all the ~ vanish and we obtain the solution for constant transport coefficients at reference conditions.

In order to evaluate ml '°(( = 0), (49) is integrated from ( = - ~ to ( = 0 to leading order in fl-1. Inserting this relation in (63) results in

m l ' ° ( ~ : O ) = ~ ( pu Dt

where ~b denotes the Markstein length in the burnt gas:

lF IF

~2G°'~ - - + ~ x 2 j + ' " , (71)

f e 1+°° ®o,o-~o,o d®O,O. (72) u

4. The Reaction Zone

The matching conditions for the outer solutions on both sides of the reaction zone are obtained by solving the conservation equations for the temperature and the fuel mass fraction inside the reaction zone. Since the thickness of the reaction zone is assumed to be of order fl-1 a stretched variable

= fie (73)

is introduced and the temperature and the fuel mass fraction are expanded as

~) = 1 -~- f l - 1 0 '1 -'~ f l -2Q,2 _[.. O( f l -3 ) ,

Y = 0 + f l - ly ,1 + fl-Uy,2 + O(fl-a), (74)

where the first superscript has been omitted since the variables have not been expanded in A -z. The second superscript denotes the order in fl-1. So we obtain the leading orders in/ /-1 of the temperature and the fuel mass fraction equation

IVGI 2 020 '1 = _ ~,o e x p ( - f l ) py" exp(0' 1) (75)

and

]VG[ z OEY '1 _ ~ ,o exp (_ f l )py ,1 exp(&l), (76)


where fl is the nondimensional activation energy,

E f l = R ° ( T b , o - T . , o ) '

and ~ is the burning velocity eigenvalue,

/ ~ 2 f , = _ _ _ _

By adding (75) and (76) the source term vanishes:

B 2, Pr s2, r Cp"

632y, i ~2 0 , 1

a~-~- + aU = o.

Matching 0 '1 and y,1 with fl expansions for the outer solutions 0+ and Y+ requires

~9,1 ~O '° i=o- - a~ ~ + o , ' (~ = 0_), ~ -0 - . ~ ,

y,1 ~y,o

- a~7 ~ :o ¢ + r ' 1 ( ~ = ° - ) '

8'1 - 8 0 ' ° ~=o+ a~ ~ + 0 ' 1 ( ~ = o+), ¢ ~ +oo,

k ) =%

(77)

(78)

(79)

(80)

(81)

(82)

y,1 = 0. (83)

Deriving these equations implies ®,o(~ = 0_) = ®,o(~ = 0+) = 1 and y,o(~ = 0_) = y,O(~ = 0+) = y, l(~ = 0+) = 0 where the origin ~ = 0 can be chosen such that Y(~ = 0_) --- 0. Using the downstream boundary conditions integration of (79) leads to

y,1 +0,1 = ®,1(~ = 0+). (84)

Inserting the upstream boundary conditions yields

0,1(~ = 0_) = 0,1(~ = 0+), (85)

Qy,o ¢=o_ c30'° 0~ - ~ - ~=o_" (86)

After inserting (84) in (76) integration gives

] u~O ~ p f ~ , o exp(--fl + ®"(~ : 0)). (87)

\ a~- ~=-~o) IVGI

Using the boundary conditions and (86) leads to

,.~ ~b ,"% 0 #.o 7 -o k ~ ~=o_/ z l ~ G ~ p ~ exp( - f l + ®,1(~ = 0)). (88)

This relation is expanded in A -1. By matching the temperature gradient of the preheat zone from (32)

m 0 "~2

= t IVCOl 2) (89)

and the reaction zone we obtain, to leading order in A -1,

c3®o,o x}2 1 ~ - ~=o_/ = 2Pf~'° exp(-fl)~'° exp(®°'l(~ = 0))rVGOl2


and, to first order in A -1,

IVa° l 2 a191"°l : lm°191'1(~ = 0) -- aGO 8G1 go (90) - ~ - - ~=o_ ax~ ax~ IVa°l"

For constant pressure p = 1 the function go is unity as will be seen from (92) and (98) below. Since the burning velocity eigenvalue must satisfy this leading-order solution, (89) results in

1 D,o = i exp(//-- 19o,1(~ = O)lt=o )~oo" (91)

Now the function go defined in (31) is equal to

m o go - jVGO I = [p exp(19°'1(~ = 0) - 19o,1(~ = 0)1,=o_)]1/2. (92)

Introducing (98) below shows that it depends only on time. M a t c h i n g 9 ,2 and y,2 with/ /expansions for the outer solutions 19+ and Y+ requires

'9'2 - ~ ~=o_+ 2- + de ~=o+ ¢ + 19,z(~= 0+), ¢~ __+o% (93)

y,2 632y'° ~=o+ ~2 c~y,1 - a~ 2 _ 2 - + 8~ ~=o_+ ¢ + y ,2(~= 0_). (94)

When the conservation equations for the temperature and the fuel mass fraction are added, the source term vanishes. Expanding the scalar quantities we obtain, to first order in//-1 and to leading order in m -1,

(80,1 + yO, 1) = m 0 _ _ (80,2 yO, 2 1). ( ~ 2 -~- - - ly °, (95)

This equation may be integrated by using the boundary conditions for ¢ ~ + ~ . The result is a polynomial in (. Comparing the constant terms of the polynomial leads to

[a19°'1I¢:°++[aY°"]¢=°+ F O=L a~ A~=o_ L a~ A¢=o_-IL~--A~=o_

(96)

The following jump relation through the reaction zone may be derived from (86) and (96) and is valid up to the first order in//-1:

poOl.O , ? . l . O + 0 = L 8~ J~=o + Lee L~-J~:o_ + 0(//-2)" (97)

The outer solution ®o has been obtained by integrating the leading-order temperature equation. However, the integration constant ®o,1(~ = 0) is up to now unknown. In order to evaluate ®o,1(~ = 0) we have to insert yO and 190 with the unknown integration constant in (96). This leads to

io /~u 1+o,(19o,o _ 19u),-1io,o d19O,O ~ b + //19u" (98) = 0) = - 9 Joo

This indicates that 19u must be of order / / -1 if the last term in this equation is to remain finite for //---} oo. Exploiting the limit 19u ---} 0 in the first integral of (98) leads, for (92), to

go = pl/2 expIflO~(p(V-1)/' _ 1 ) + 1 ( 1 _ ~_~)( + n + K b)]. (99)

The last term in the exponential may be neglected if the heat loss is small. Then go is an explicit function of pressure only.

Transient Pressure Effects in the Evolut ion Equat ion for Premixed Flame Fronts 153

Similarly, (65)-(69) we evaluated in the limit ®. ~ 0. We obtain

1 = ~ ~ + 2 . +,~ + ' (1oo)

In these integrations the case n = 0 was excluded. The burning velocity sn, r in the presence of heat loss may be related to sL,~, the reference burning

velocity without heat loss, by introducing (98) into (91):

f ~ ' ° = ½ e x p fl+2~p2oO°,o -'- .u,O, -~ +2~p21° - f l®u ,O . (101)

Using (78) the following relation is found:

fro //~ f'l+O°,o _ e x p [ 2 | (Oo,o . ~-liO,OdOO,O /~b~o ,o~7_ - ---~,oJ -~ + 7 ) f~ In, =o '. .) O°.o

= - -(P~'~,~]~ + o(~-~) . (lo2) ~Pr Sift, rJ

Transforming the equation gives

In H,r = __ "-'.,oJ "~ + ~ 2 b + O ( f l - 1 ) • (103) d O~,o

Applying (12) we obtain the following relation correct to leading order in fl-~:

( ) , , ~ 2 s . , , 2 l n l O . , , ~ _ K

where K and the critical quenching value K* are defined by

K = ~ h. - .~ + d Ou,o

(104)

(lO5)

Figure 3. Mass burning rate as a function of K/K*.

1.2

8H,r 1

8L,r

0.8

e-1/2

• r r l

0.4 unstable branch _--"""

0 ~ ' ' ~ 0 0.1 0.2 0.3

K / K "

#

m

e -1 0.4


and K* - (CpPrSL'r)2 1

2r 2fl" (106)

Equation (104) has been plotted in Figure 3. Only for K/K*< e -1 is flame propagation possible where the upper branch of the curve represents stable solutions. For SU, I/SL,~ close to unity, (104) may be approximated by

SH, r K - 1 - - (107)

SL, r 2K*"

If we employ the limit O., o -~ 0 as before, we obtain, from (105),

K f k u . "~ 2~T~ ~-I - ~ + kb] (108) if-* /~ . + n ( c ~ ) ~ /

5. S u m m a r y o f R e s u l t s

Expanding the mass flux in A -1 results in

DG ( dG Q~x~) m° A-lrnl p ~ - = p ~ - + v~ = m = + + O(A-2), (109)

where (25) has been used. For constant pressure with q~ = 1 and m ° = Iva°l the leading-order G-equation is simply, in dimensional form,

DuG Dt - SL, ulVGI. (110)

To first order, replacing m ° by [VG°I q~ and inserting (63) for m I we obtain, in dimensional form,

DuG ( ~ D u I V G [ ~2G~ Pu,o ~ - = (PSn)u,O IVal ~o + p.,0fF~q~ IVal -- 5°u.__ Dt (psn).,o Ox2}

1 (puDulVGI ~G~IVG[ 1 ) (111) - l F ~ IVal \ ~ Dt (PSn).,o ~x~ ~x~ IV-al "

Here G has the dimension of a length and the operator ~ has the dimension second -I, since the nondimensional pressure p is the ratio of the actual pressure to the initial pressure Pr' The equation has been multiplied by Pu,O/Pr in order to replace the physically unappealing reference density Pr by P.,o, the density in the unburnt mixture prior to pressure changes. The quantities ~ have been neglected for simplicity. In order to simplify this equation further we define the flame curvature

which may be transformed as

V2G V[VGI'VG

~c- IVGI + IVGI 2 -

Furthermore, we manipulate the substantial derivative

D.IVGI Dt

~c = V 'n = - V " , (112)

-V2G - n'VlVGI (113)

- - - - n .

IV61

- n . - -

D.(VG) _ n . ( v (D .G~ VG) ot \ ~ - } - VVu.

(PSn)u,O V IVGI q~ - n" Vv u • n [VG[. (114) Pu

Using (113) and (114) leads to

p u ~ = ( p S n ) u , olVGlq~ 1 - ~ x - (~° u -[- IF~,*et °)

Pu q9 (-n 'Vvu'n)]VGI -¥ Pu,o~F~OlVG[, (115)


where the term ( - n- Vv~" n) (116)

denotes the tangential stretch due to flow divergence and ~c is the curvature. The above is the scalar field equation for describing the evolution of the flame front. Effects of pressure variations are taken into account by the quantities ~ (see (67) or (100)) and q~ (see (92) or (98)). Flame curvature depends on the Markstein length A°u (see (64)) and flame stretch due to flow divergence on the sum of ~u and { v ~ (see (66)) which therefore can be interpreted as a modified Markstein length including heat loss. A relation between the burning velocity in the presence of heat loss and that of a flame without heat loss is given by (103) and in linearized form by (107).

The effect of bulk pressure changes on combustion in engines may be illustrated by the following: When the pressure ratio p and therefore ~o decreases as during the expansion stroke of a spark ignition engine, the effect of curvature and that of flame stretch due to flow divergences increases relative to the first leading-order term on the right-hand side of (111). In addition, the second term in (111) will be negative and add to the decrease of local flame propagation. The opposite effect will, of course, happen when the pressure increases. Then curvature and stretch effects will lose their influence relative to the leading-order term.

6. Conclusions

In order to study the dynamic behavior of premixed flames a nonlinear evolution equation for a scalar field G(x, t) has been derived using a two-scale asymptotic analysis based on a local coordinate transformation. The leading-order form is that already derived by Markstein [13, while the first-order term introduces the effects of flame curvature and stretch as well as time variations of the bulk pressure, heat loss, and nonconstant transport coefficients. In the scalar-field equation the Markstein length appeas as a coefficient in the curvature and stretch term. On the basis of the scalar-field equation premixed turbulent combustion in the flamelet regime can be analysed. The scalar G(x, t) plays a similar role for premixed flamelet combustion as the mixture fraction Z(x, t) for nonpremixed flamelet combustion. This is shown in [13], where equations for the mean and the variance of G in a turbulent flow field have been analysed and closure assumptions have been derived. It then turns out that the location of the mean turbulent flame front can be calculated from an equation that has similar properties as the field equation derived here. Comparisons with experiments are presented in [143.

Acknowledgment

We are grateful to Prof. M. Matalon for helpful comments on a draft of this manuscript.

Appendix A. The Integrals

The integrals ~ (i = 1 . . . . . 5) are listed below:

1+o. 0 °'° 1~o,o dOO, O + J~fl = ln(O °'° -- Ou)OuO°'°-2~. °'° dO °'°, (117) d O. J O. f i + O u ~ 0°'°

~/~2 = (~0,0' '~0,0 d®O,O' ®o,o(®o,o _ Ou)a-2~o ,o d®O,O d O. d Ou

~l+OoFo°'°~o,o dO o,o' (0o, o Ou)~'-l~O,OdOO, o Jo° Jl+oo 0 ° ' ° ' - - Ou

~'+°u F°°'° + ln(OO, O ' _ O.)OuOO,O ' 2~o,o dOO,O 0o,o(0o,o _ OuY-2~o,o dOO,O, (118)

d O. d O~


~1+® u $ 3 = 1n(19o,o _ 19.)(219~19o, o-' + 1n(19o, o _ 19.)19219o,o-~),~o,o d19O,O, (119)

d O~

[ X4 = ½ _ (19o,o' _ 19.)19o,o'-~(219o,o' + ln(19o,o, _ 19 . ) (0 . + ®o,o'))~o,o d19O,O, 00~ d ®. fx+.foo.o

x (19o,o _ ®.),,-2~o,o d19O,O _ (19o,o" _ 19.)19o,o'-~io,o d19O,O" dO~ d@~

× 1n(19 °'° - 19u)(~ - 1)(19 °'° - 19.)~-2~o,o d®O,O .x+.foo,o - (1 + 1n(19 °'°' -- ®.)c0(19 °'°' -- 19.)~-, ~o.o d19O.O.19o.o-~o,o d®O,O

d O. d O.

f l+O~ ~ ®°'° - 2 (19o,o' _ 19u)~-1~o,o d19O,O'®O,O-~(19o, o + 19. ln(19 °'° - 19~))~o,o d19O, O dO. dO. f +.foo,o

+ (1 + ln(19 °'°' -- 19u))2 °'° d19°'°'(19 ° '° - 19.)~-2i°,° d O °,° dO. dO~

+ 19. (ln(®O,O' _ 19.)19,,19o,o'-~ + 19o,o'-,)5o,o d19O,O' d O. d O.

x (19o,o __ 19.)~,-2~o,o d19O,O

_ f +.foo.o _ - 1, 2 (19o.o' ®.)-L~o,o d19O,O'(®o,o 19.)=-1~o,o d19O,O (120) d Ou dl+O. f l +0~

~ 5 = - ½ (19o,o _ 19u)19o,o-~(®o,o + ln(®O,O _ 0 . ) (19 . + 19o,o))~o,o d19O,O d O. ~ 1 +O~

+ ½ (1 + ln(19 °'° - ®.))~o,o d19O,O. (121) d O.

Appendix B. Evaluat ion o f the Enthalpy Equat ion

Since the fuel is depleted in the burnt gas region (Y~(( = 0) = 0), we obtain

/-/'(~ = o) = 1 9 ' ( ~ = o)

and 0 2 OF1 t~191 ,

IVG I a~- g=o+ = IVG°I2 O( ¢=o+

where IVG°lZ(O191/8~)[~=o+ can be evaluated by using (59), (60), and (61):

~191 = - f l -*p"DuIVG°I fflb ~°2 19" fl-102G° /tb~.02(1 + 1 ) IVG°[2 o¢ g=o+ q~ Dt 2q9 ~ ) ~ b 1 + 19~ Ox~ 2q32 1 + 19.

+ fl_,OG ° OlVG°[ 1 2~)~o22 + fl_lml,O(~ = 0) H~bzi o 8x~ Ox= IVG°I z~o

+ f l_x ~ - 1 dp . i o ,~.-.o, a t ~ ~ ' v " ,.

Integration by parts leads to

) f ) dP°H~°°d~=fl-lP"OlVG°! At, no + 2°A,,nb +dA, / & qo Ot

(122)

(123)

(124)

(125)


with

~1+@° f ® °'° At,Hu In((9 ° '° ' _ (gu)~;')u(9o,o'2~o, o d(9O,O'((9o, o _ (9u)~-1~o, o d O ° '° = (9 0,0 - (9.

dO. dO°

_ fx+o° ~oo,o ((9o.o, (9~)~_1~o, ° d(9° '° ' ln((9° '° - (9u)(9. (9°'°-2~°'° d(9 °'°, (126) J O. J O.

~ 1+0. -- (gu)(gu(9 ' 2 ' d(9 °'°, At,H" = ln((9o,o 2 o o-2-o o (127) JOu f l+Ou

At,e = ln((9o,o _ (gu)((gu(9O,O-, + ln((9o,o _ (gu)(guZ(9o,o 2)~o,o d(9O,O. (128) J O.

Fur the r in tegra t ion by par ts results in

) [ 5 o® ax= ~ v~,,u (o ax: \ ~ t.no + ~.°At,n, + dA,.+

) + ,B_laG° c~[VG°l 1 ( / ~ u a 2H~z ~x~ ax: ~ \ 2 ~ a tx.,.o + 2°Alx.,.,, + (Alx.,¢

- f l - t a Z G ° ( ~ I ' A +~2°A2x. ,H, ,+{Aaxo, l) , (129) :xo, i , .

where A~.n°, At.n., and A~.~ are given by (126) (128). The relat ions for A~,uo, Axx..n,,, A~x..t and A Z x . . H o , A Z x . . H , , Azx J are listed below:

~1+0o ~oo.o A~x.uo = ( (90,0, -- (9~)(9°'°'-~(2(9°'°' + ln( (9°'° ' - (9u)((9. + (9°'° ')) ~°'°

d O. d O.

× d(9O, o , ( (9o ,o - (gu)a_l ~o,o d O ° ' ° fl+O~ f O°'° (9 0,0 (9~ + ( ( 9 ° ' ° ' - (9u)(9° '° ' - '2° '°

- - d O. d O~

× d O °,°' ln((9o, ° - (gu)(a - 1)((9 °,° - (gu)~-:~o,o d(9O, o [~+Oo ~oo,o

+ (1 + ln((9 ° '° ' - (9.)u)((9 ° '° ' - (9.)~-1,~. ° '° d(9° '° ' (9° '°-~2 ° '° d(9 ° '° d O. d Ou

+ ((90,0' - (9u)~-1,~. ° '° d(9° '° ' (9 ° '° 2((90'0 + (9~ ln((9 ° '° - (9u))2 ° '° d(9 °'°, (130) d O° d O.

f l + O . ( (90'0 _ (9.)(9°'°-~( (9°'° + ln( (9°'° (9.)((9~ ÷ (9° '°))2 ° '° d O °'° (131) A l x ~ . H , = - - J Oo

AI~ J = ln((9 °,° - (9u)((9 °,° - (9u)(9o, o-~

× (2(9 ° '° + ln((9 ° '° -- (9~)((9u + (9°'°))). ° '° dO °'°, (132)

and

f d O ° ' ° Azx..H ° = l+°u 00'° (00,0 ' __ Ou)O0,O ' 1~0,0 dOO,O,(Oo,o_ Ou)a-l~O,O 0 0 ' 0 _ Ou dO. dO. fl+OoV.o _.~ (00 ,0 __ Ou)~--l~O,O dOO,O'OO,O '~o,o dO®,®, (133)

dO. dO.

A 2 x ~ , n b = ~l+Ou ( O0 '0 _ Ou) O0'0- '~ '0 '0 d ® ° '° , (134) do®

~ 1 +Ou A2x~, / = ln((9 °,° - (9u)((9 °,° -- (9.)(9°,°-12 ° '° d(9 °'°. (135)


Forming the derivatives of F ° with respect to ~ and x, we obtain

8~ 2=-oo 2 q)2 iVGO[ and

~X~d~= - oo = 0.

Integration by parts yields

c~G°- f ~ ~ c~F~° d( = fl-1 c~G° ~]VG°[ axe, o~ax~ a~ axe, ax~

where

1 ( < A I VGOl k,2~~2 m +

and

( l + O u ~O °'° A n = (1 + ln(O °'°' - O.))2 °'° dO° '° ' (O °'° - O.)~-i2 °'° dO° ' °

° 0 0 , 0 - 0 . dO. dO.

__ t * l l + O ° __ __ ln(O °,° O. ) (O °,° O.)~-1,~ °,°~ dOO, o,

! dOo

F l+Ou Au~ = (1 + ln(O °'° - 0.)),~ °'° dO °'°, d O. f l+O u

At = ln(O °'° - 0 . ) ( 2 + ln(O °'° - O . ) f i °'° dO °'°, d O.

f~ ~F°2°d~=-fl-xFl"~2Jo.(" 1+0" FO°'° ~0'° dO°'°'(O°'° -- O u ) a - l ~ ° ' ° d o ° dO° '° 0 0,0 Ou

1 /'~b ~0 r 1+oo f1-1E ~ 1+o. | ~o,o dOO,O oo.

- # - 2~-~@ & jo°

(136)

(137)

(138)

(139)

(140)

(141)

ln(O °,° - O . f i °,° dO °'°. (142)

According to (23) we find, with O1(~ = -oo ) = 0 and O1(( = 0) = O(f1-1) = F °,

I ,~l 0FOT=O+ -~-j~=-oo = o ( v b .

Using (58) with (56) we obtain

@2 3-I°oo Ola (O° -- Ou)~-i'~° d~

dOO,O

0 0,0 __ Ou

P.D. WG°I A a2G° r[l+O° fooo o.oo,o._,~o,o q~ D t t, Hu -I- ~ Ld@. J@°

OG ° 8 ]VG°I

x dO° '° ' (O° ' ° - O.)~-12 °'°

(143)

r/?r2oo,ooo ou, , oo,oooooo oo,ooo] 1 Ff,+.of.oo

O.(1n(OO, o' - 0 . ) 0 . 0 0 , 0 '-2 + OO, O'-')~o, o W O ° i - J o ° joo

dOO,O × dO° '° ' (O° ' ° - O"Y-12°'° 00,0 - O .

__ f l+Ou f O°'° (O0,0' O.)a--l~O,O ./O. d O.

x dO°'° ' ( In(O °'° - 0 . ) 0 . 0 0,0-2 + O° '°- ' )2 °'° dO °'°


~ i + 0 . 1 + ln(O °'° _ Ou)(O °,° - 0.)~-1~o,o~ dOO,O J O.

~Go 6~G 1 (p f l + O . ÷ ~x~ ~x~ IVa°l 2 / (O° '° - Ou)~-12°'° d O ° ' °

dOu

~ 1+®~ - - m l ' ° ( ~ = 0 ) ( O 0'0 - - Ou)~¢--1~ 0 '0 d O ° ' ° .

J ou

Regarding (62) we realize that each te rm on the r ight-hand side is of order O(]~ -1) o ~ ( Z = o) = n ' ( ~ = 0) = o( /~-~) .

(144)

such that

References

[1] Markstein, G.H.: Nonsteady Flame Propagation, Pergamon Press, Oxford (1964). [2] Williams, F.A.: The Mathematics of Combustion (J.D. Buckmaster, ed.), SIAM, Philadelphia, PA, pp. 97-131 (1985). [3] Clavin, P., Nicoli, C.: Effect of Heat Losses on the Limits of Stability of Premixed Flames Propagating Downwards,

Combust. Flame, 60, 1-14 (1985). [4] Clavin, P., Williams, F.A.: Effects of Molecular Diffusion and of Thermal Expansion on the Structure and Dynamics of

Premixed Flames in Turbulent Flows of Large Scales and Low Intensity, J. Fluid Mech., 116, 251 (1982). [5] Pelce, P., Clavin, P.: Influence of Hydrodynamics and Diffusion upon the Stability Limits of Laminar Premixed Flames,

J. Fluid Mech., 124, 219-237 (1982). [6] Matalon, M., Matkowsky, B.J.: Flames as Gas Dynamic Discontinuities, J. Fluid Mech., 124, 239-259 (1982). [7] Peters, N.: Laminar Flamelete Concepts in Turbulent Combustion, Proceedings of the 21st Symposium (International) on

Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1231-1250 (1986). [8] Nicoli, C.: Dynamique des Flammes Pr6m61ang6es en Pr6sence des M6canismes Controlant les Limites D'Inflammabilit6,

Ph.D. dissertation, L'Universit6 de Provence (1985). [9] Joulin, G., Clavin, P.: Linear Stability Analysis of Nonadiabatic Flames: Diffusional-Thermal Model, Combust. Flame, 35,

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J. M~c. TMor. Appl., 2, 245-263 (1983). [11] Peters, N., Ludford, G.S.S.: The Effect of Pressure Variations on Premixed Flames, Combust. Sci. Technol., 34, 331-344

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(1993), to appear. [13] Peters, N.: A Spectral Closure for Premixed Turbulent Combustion in the Flamelet Regime, J. Fluid. Mech., 242,

611-629 (1993). [14] Wirth, M., Peters, N.: Turbulent Premixed Combustion: A Flamelet Formulation and Spectral Analysis in Theory and

IC-Engine Experiments, Proceedings of the 24th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 493-501 (1992).

transient pressure effects in the evolution equation for premixed flame fronts

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