pressure disturbances and strained premixed flames

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Pressure disturbances and strained premixed flamesAndy C McIntosh a , John Brindley b & Xin-She Yang ca Department of Fuel and Energy , University of Leeds , Leeds, LS2 9JT, UKb Department of Applied Mathematics , University of Leeds , Leeds, LS2 9JT, UKc Department of Fuel and Energy and Department of Applied Mathematics , University ofLeeds , Leeds, LS2 9JT, UKPublished online: 19 Aug 2006.

To cite this article: Andy C McIntosh , John Brindley & Xin-She Yang (2002) Pressure disturbances and strained premixedflames, Combustion Theory and Modelling, 6:1, 35-51, DOI: 10.1088/1364-7830/6/1/303

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INSTITUTE OF PHYSICS PUBLISHING COMBUSTION THEORY AND MODELLING

Combust. Theory Modelling 6 (2002) 35–51 PII: S1364-7830(02)18740-7

Pressure disturbances and strainedpremixed flames

Andy C McIntosh1, John Brindley2 and Xin-She Yang3

1 Department of Fuel and Energy, University of Leeds, Leeds LS2 9JT, UK2 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK3 Department of Fuel and Energy and Department of Applied Mathematics,University of Leeds, Leeds LS2 9JT, UK

Received 8 November 2000, in final form 24 September 2001Published 11 January 2002Online at stacks.iop.org/CTM/6/35

AbstractA mathematical model is presented for pressure interactions with premixedflames in a prescribed strained velocity field. A stability analysis is carriedout including pressure disturbances and an approximate frequency conditionobtained. For positive strain the unsteady analysis indicates that the pulsatinginstability is suppressed. However, for a converging flow (negative strain),the instability is encouraged. Furthermore, the change of the neutral stabilityboundary in parameter space is explored, showing that a sharp pressurereduction makes the pulsating instability much more accessible.

1. Introduction

1.1. Pressure interactions with flat flames

In recent years considerable advance has been made concerning the theory of pressureinteractions with flat flames. Initially large lengthscale disturbances were studied first forthe harmonic case (McIntosh 1986, 1987) and later for non-harmonic inputs (Ledder andKapila 1991, McIntosh 1991, 1993). Short lengthscale disturbances have also been explored(McIntosh and Wilce 1991, Batley et al 1993, McIntosh et al 1993), where severe distortionof the flame takes place on very short timescales followed by relaxation to a final steady state.Relative to a reference frame within the original diffusion-driven flame, these investigationsshow that the mass flux decreases and can even temporarily reverse immediately after a sharp(short lengthscale) drop in pressure is experienced by a flame (Johnson et al 1995). This isbecause, momentarily, the flame zone expands as the pressure drop passes through; thus relativeto a reference frame linked to the back of the flame (near the reaction zone), the expansionvelocity is greater than the overall forward burning velocity of the combustion front as a whole.If the pressure drop is sharp enough, there can be situations where the flame will not recover.For a one-step overall Arrhenius reaction, with non-dimensional activation energy θ = 10 andunit Lewis number, the critical pressure for this to occur was found to be close to p0 = 0.29,

1364-7830/02/010035+17$30.00 © 2002 IOP Publishing Ltd Printed in the UK 35

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http://stacks.iop.org/ct/6/35

36 A C McIntosh et al

where p0 represents the ratio of the new pressure to the original (ambient) pressure. Thiscritical value rises somewhat with activation energy but it represents a considerable drop forflat flames in a straight flow, and thus is not generally very accessible.

This is true also for the well-known pulsating instability of flat flames in a straightflow. Defining Lewis number as thermal diffusion (λ′/ρ ′c′p) over mass diffusion (D′),L ≡ (λ′/ρ ′c′p)/D

′ (where λ′ is thermal conductivity, ρ ′ is density, and c′p is specific heat), thecritical Lewis number for the pulsating instability is in the region of 0.51 for the Zeldovichnumber θ1Q1 of 20 (Sivashinsky 1977, McIntosh and Clarke 1984, Johnson et al 1995). NoteE′

a is activation energy, R′ is universal gas constant, T ′b1 is the initial burnt temperature, and

T ′01 is the initial unburnt temperature (thus θ1 ≡ E′

a/R′T ′

b1 and Q1 ≡ (T ′b1 − T ′

01)/T′

b1) withZeldovich number ≡ (E′

a/R′T ′

b2)(T ′

b − T ′01). This low value of Lewis number is not generally

reached for hydrocarbon flames.However, in the earlier paper by Johnson et al (1995), it was pointed out that for a sudden

lowering of pressure to p0 (=p′final/p

′ambient), the adiabatic decompression effect gives an

intermediate profile of temperature which in non-dimensional terms is given by

T = Ts(x)p(1−γ−1)

0 , (1)

where γ is the ratio of specific heats (c′p/c′v) and T01 is the ratio of initial unburnt to initial

burnt temperature. The final temperature is then eventually given by TB = T01p(1−γ−1)

0 +Q∗

and this raises the threshold for the pulsating instability to a value closer to accessible Lewisnumbers (near unity).

1.2. Strained flow

This paper now explores the pulsating instability in the presence of a prescribed divergent orconvergent flow where u = (−αx, αy). In the spirit of the earlier work of Mikolaitis andBuckmaster (1981), Buckmaster and Mikolaitis (1982), and Buckmaster (1997), the analysisassumes that the effect of the flame on the flow through temperature and density effects isnegligible on the fast combustion timescale. This justifies the assumption of a prescribed flowwith strain

α ≡ α′D′0

u′20

� 1, (2)

where u′0 is the initial burning velocity of the flame. This simplifies the problem considerably,

and enables a stability analysis to be made of strained flames in these earlier works which isdeveloped here to include pressure changes. We ignore the small acoustic field generated byany sharp pressure drop through the momentum equation, on the basis that this effect is weakand at a timescale much faster than the relaxation of the combustion. Thus, the flow change isgenerally negligible. A detailed discussion of these two timescales is available in earlier workby Batley et al (1993).

The divergent flow case is more common, being readily set up experimentally by a twinflame configuration on either side of a stagnation point separating two jets of premixed gasesopposed to each other. The convergent flow case is, however, also of considerable interest,since this is what will generally be encountered in a rear stagnation-point flow, such as for aflame anchored fluid-dynamically near a flame stabilizer in a gas turbine.

Extending the stability theory of strained premixed flames, this paper derives an importantformula for the critical pressure drop to induce the pulsating instability over a range of strain(both positive and negative). This considerably modifies the stability analysis and, in particular,shows that the pulsating instability is now readily accessible for near-unit Lewis number fuel–air mixtures, when the pressure drop is significant, particularly if one has a convergent flow.

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Pressure disturbances and strained premixed flames 37

Even for positive strain, the pulsating instability is readily triggered by a decompression if theLewis number is less than unity.

It should be pointed out that in this work the flame is flat, so that curvature effects such asthose which affect the cellular instability of such flames are not included. Furthermore, we arenot including an oscillating pressure field in any of the analysis. The pressure drop is simplythrough the flame and acts as a mechanism for sudden cooling. Thus the Rayleigh criterioncannot be addressed with this model, which would require the coupling of an oscillatorypressure field.

2. Mathematical model

The present model of strained premixed flames combines features of work for non-strainedflow (McIntosh 1991) and for stretched flames (Buckmaster 1997). It extends the modelput forward by Mikolaitis and Buckmaster (1981) for converging flow and Buckmaster andMikolaitis (1982) for diverging flow. The work of Matalon (1988) on weakly stretched flames,and that of Kim and Matalon (1990) on near-equidiffusional strained flames are also relevantto this present investigation. However, we do not include the hydrodynamic expansion effectof the last reference, and regard temperature and density expansion effects from the flameon the flow to be small. The model can describe both counterflow near a forward stagnationpoint where the strain rate (α) is positive, and a rear stagnation-point flow with negativestrain (α < 0).

2.1. Model equations

We consider a general strained flow field u′, such as the counterflow (figure 1(a)) and rearstagnation-point flow (figure 1(b)), in which a flame (or flames in the case of twin flames) issituated in a narrow region where a first-order irreversible burning reaction is taking place ata rate r ′(T ′). The model equations ignoring viscous effects can be written asmass conservation

∂ρ ′

∂t ′+ (u′ · ∇′)ρ ′ + ρ ′∇′ · u′ = 0, (3)

∂C

∂t ′+ u′ · ∇′C − 1

ρ ′ ∇′ · (ρ ′D′∇′C) = −r ′, (4)

momentum equation

ρ ′[∂u′

∂t ′+ (u′ · ∇′)u′

]= −∇′p′, (5)

energy conservation

ρ ′cp

(∂T ′

∂t ′+ u′ · ∇′T ′

)− [∇′ · (λ′∇′T ′)] = ρ ′Q′r ′ +

(∂p′

∂t ′+ u′ · ∇′p′

), (6)

equation of state

p′ = ρ ′RT ′

W ′ , (7)

where ρ ′, λ′, T ′, cp, C are the density, thermal conductivity, temperature, specific heat, massfraction of fuel, respectively, u′ = (u′, v′) is the overall flow velocity, (x ′, y ′, t ′) are space andtime, and ρ ′ is an average density of the mixture. R is the universal gas constant, W ′ is the

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x x

Negative strain

Flame sheet

y

TbTu

y

Flame sheet

x=x*

Positive strain

Figure 1. Schematic of counterflow premixed flame with a uniform strain rate α � 0 and rearstagnation-point flow with a negative strain rate.

molecular weight of the overall mixture, andQ′ is the heat release from the related reactions.r ′ represents the reaction rate, assumed to obey an Arrhenius law:

r ′ = A′C e−E′a/RT

′, (8)

where E′a is the activation energy of the reaction.

2.2. Non-dimensionalization

Following earlier work (McIntosh and Clarke 1984, McIntosh 1991, Johnson et al 1995) andusing the typical values of the variables in the unburnt state (denoted by a subscript 0), wechoose the following scalings and non-dimensional variables:

x = u′0x

′

D′0

, y = u′0y

′

D′0

, u = u′

u′0

, t = u′20 t

′

D′0

, T = T ′

T ′b1

, ρ = ρ ′

ρ ′0

,

(9)

p = p′

p′0

, C = C

C∞, D = D′

D′0

, λ = λ′

λ′0

, (10)

Q1 = Q′

cpT′

b

, θ1 = E′a

RT ′b1

, (11)

where the initial (unstrained flame) burnt temperature is labelled T ′b1 (i.e. before the pressure

wave changes the temperature field). C∞ is the inlet mass fraction of the fuel. The flame MachnumberM = u′

01/a′01 is typically very small (M � 1), and for steady flame propagation, the

momentum equation implies

∇p ≈ 0 + O(M2). (12)

Thus at leading order in M , the pressure is essentially independent of space coordinates andthe momentum equation becomes decoupled.

For a flat flame sheet, we assume that T and C (and thus ρ) change slowly along they-direction so that all y-derivatives of T , C, and ρ are negligible (i.e. Ty, Cy, ρy are small).

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Thus we have∂ρ

∂t+

[ρ

(∂u

∂x+∂v

∂y

)+ u∂ρ

∂x

]= 0, (13)

∂C

∂t+ u∂C

∂x− 1

ρ

∂

∂x

(ρD∂C

∂x

)= −R, (14)

ρ

(∂T

∂t+ u∂T

∂x

)− L ∂

∂x

(λ∂T

∂x

)= ρQR + T01

(γ − 1

γ

)∂p

∂t, (15)

where

L ≡ λ′0/ρ

′0c

′p

D′0

, T01 = T ′u

T ′b1

, (16)

R = C eθ1(1−1/T ), ≡ A′D′0 e−θ1

u′20

. (17)

2.3. Prescribed strained velocity field

With u = (u, v), the continuity equation is∂ρ

∂t+

[ρ

(∂u

∂x+∂v

∂y

)+ u∂ρ

∂x

]= 0. (18)

In the latter section of this paper, we consider abrupt pressure changes. We make the assumptionthat any velocity change invoked by a planar pressure disturbance (through the momentumequation, not written here) is resolved on a much shorter timescale than the combustion (seeBatley et al 1993) for the one-dimensional case. Mathematically, this is stating that the passagetime of the pressure drop is less than the diffusion timescale D′

0/u′0

2, i.e.

l′aa′

0

� D′0

u′20

, (19)

where l′a is the thickness of the pressure wave, and a′0 is the speed of sound. Equivalently,

l′a � l′f/M where the thickness of the flame l′f is approximately D′0/u

′0 and Mach number

M ≡ u′0/a

′0. Defining N ≡ l′a/l

′f , this condition is therefore stating that N � M−1, or

defining τ ≡ (D′0/u

′0

2)/(l′a/a

′0) = 1/(NM), we are requiring τ � 1. This is certainly true for

short lengthscales where l′a is comparable to l′f and M is typically �0.001. If one redefinesan acoustic timescale, t ≡ t ′a′

0/l′a = t ′τ , and considers the acoustically scaled velocity

u(a) = Mu (Batley et al 1993), then it becomes clear that, in reality, there is a small O(M)resultant change in the velocity field as a consequence, and an O(M) acoustic field to resolve.However, to leading order inM , we can ignore these higher-order effects here, and, in keepingwith earlier work, we regard the strained flow as fixed and prescribed, namely u = α(−x, y)with α = α′D′

0/u′0

2. Thus, the time scales are ordered such that

1

α(strain time) � D′

0

u′0

2 (combustion time) � l′fa′

0

(flame acoustic time)

and equations (13)–(15) become∂ρ

∂t− αx ∂ρ

∂x= 0, (20)

∂C

∂t− αx ∂C

∂x− 1

ρ

∂

∂x

(ρD∂C

∂x

)= −R, (21)

ρ

(∂T

∂t− αx ∂T

∂x

)− L ∂

∂x

(λ∂T

∂x

)= ρQ1R + T01

(γ − 1

γ

)∂p

∂t. (22)

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2.4. Mass-weighted system

Since the flow field is prescribed, we can write the above equations in mass-weightedcoordinates so that the continuity equation is decoupled from the species and energy equations.Defining the new space coordinate x as

x =∫ x

x∗(t)ρ dx, t = t, (23)

where x∗ is the location of the flame, and introducing

m0 = ρ(u− u∗)|(x∗(t),t), u∗ ≡ dx∗

dt, (24)

we note that∂x

∂t=

∫ x

x∗(t)

∂ρ

∂tdx − ρ|x∗u∗ =

∫ x

x∗(t)−u∂ρ∂x

dx − ρ|x∗u∗

=∫ x

x∗(t)−

(∂(ρu)

∂x− ρ ∂u

∂x

)dx − ρu∗ = m0 − ρu− αx. (25)

Consequently we have the transformations

∂t = (m0 − ρu− αx)∂x + ∂t , ∂x = ρ∂x, ∂

∂t+ u

∂

∂x= ∂t + (m0 − αx)∂x, (26)

and the equations in the new mass-weighted coordinates become∂ρ

∂t+ (m0 − αx) ∂ρ

∂x= 0,

∂C

∂t+ (m0 − αx)∂C

∂x− ∂

∂x

(ρ2D

∂C

∂x

)= −R, (27)

∂T

∂t+ (m0 − αx)∂T

∂x− L ∂

∂x

(ρλ∂T

∂x

)= Q1R + (1 − γ−1)

T

p

∂p

∂t. (28)

It has been observed that experimentally, λ ∼ T β with β in the range 0.75 � β � 0.94(Kanury 1975). Therefore, for simplicity, we take β = 1 as well representative of the physicaldependence of thermal conductivity or temperature. This then implies that ρλ ≈ p, from thegas law. In addition, the Lewis number L = (λ′

0/ρ′0cp)/D

′0 can also be taken to be constant,

which leads to ρD = λ. By using these approximations and the decoupling of the continuityequation, we have

∂C

∂t+ (m0 − αx)∂C

∂x− p∂

2C

∂x2= −R, R = C eθ1(1−1/T ), (29)

∂T

∂t+ (m0 − αx)∂T

∂x− pL∂

2T

∂x2= Q1R + (1 − γ−1)

T

p

dp

dt. (30)

We will only consider the half-space x < 0 due to the symmetrical nature of the counterflow.The related boundary conditions are

C = 1, T = T01 for x → −∞, (31)

C = 0, T = T∗ (= 1 for α = 0) for x → x∗(x∗ =

∫ 0

x∗ρ dx > 0, x∗ < 0

).

(32)

This defines a moving boundary problem as the flame moves in response to the inputperturbation of pressure p(t), which is a prescribed function of t .

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3. Steady-state solution using large activation energy asymptotics

The equations for the steady state are non-autonomous and nonlinear. By using large activationenergy asymptotics (θ1 � 1), we break the problem into different regions, namely theinner reaction and outer regions. The reaction rate is zero in the outer region, and thereaction/diffusion terms dominate in the inner region. By using a rescaled coordinate ζ = θ1x,and integrating across the inner reaction zone, it is found that the temperature and species obeythe following jump conditions:

[T ]−+ = 0, [C]−+ = 0,

[L

dT

dx+Q1

dC

dx

]−

+

= 0,

[dT

dx

]−

+

= Q1

L√p0

eψs∗/2,

(33)

where the flame temperature T∗ is allowed to vary (for α �= 0) by O(θ−11 ) amounts from Tb1,

i.e. from unity,

T∗ = 1 +ψs∗θ1. (34)

Either side of the flame, the equations become

(ms − αx)dCdx

− p0d2C

dx2= 0, (35)

(ms − αx)dTdx

− p0Ld2T

dx2= 0. (36)

By integrating the equations twice and using the boundary conditions (31) and (32) togetherwith Tb1 = T01 + C∞Q1 = 1, we have

T =

1 +ψ∗θ1

for x > 0,

T01 + [C∞Q1 − (1 − T∗)]erfc((ms − αx)/√2αp0L)

erfc(ms/√

2αp0L)for x < 0,

(37)

C =

0, for x > 0,

C∞

{1 − erfc((ms − αx)/√2αp0)

erfc(ms/√

2αp0)

}for x < 0,

(38)

where the unknownms (steady mass burning rate) and ψ∗ are connected to the strain rate α by

e−m2s /2αp0L

erfc(ms/√

2αp0L)= A√

α, A = Q

[C∞Q− (Tb1 − T∗)]

√π

2Leψs∗/2, (39)

e−m2s /2αp0

erfc(ms/√

2αp0)= B√

α, B =

√π

2C2∞eψs∗/2. (40)

In this paper, we do not consider stoichiometric effects and, for simplicity, we assume aunimolecular reaction with C∞ = 1. By assuming α � 1, we have

ψs∗ ≈ −αp0

m2s

Q1θ1(L− 1), (41)

and the expansion of the steady-state mass flux gives

ms(α, L) ≈ √p0

{1 − α

[1 +

Q1θ1(L− 1)

2

]}. (42)

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101 100 101 102

Mas

s bu

rnin

g ra

te (

m0)

1.2L = 1.0

L = 0.8

Not reached →

102 101 100 101

Mas

s bu

rnin

g ra

te (

m0)

L = 1.2 1.0 L = 0.8

0.7

0.8

0.9

1

1.1

1.2

0.7

0.8

0.9

1

1.1

1.2

α α

Figure 2. Buckmaster–Mikolaitis theory: effect of strain rate α on mass burning rate m0 withvalues of θ1 = 10,Q1 = 0.8, L = 0.8, 1.0, 1.2.

In fact, there exists a limiting mass burning rate m∞ as α → 0, which is given by

m∞ → √p0 eψs∗/2, (43)

in agreement with Buckmaster and Mikolaitis (1982) and Buckmaster (1997). As there is noconstraint on the sign of strain rate α, the above solution is also valid for rear stagnation-pointflow where α < 0. By plotting the mass burning rate versus strain rate α (see figure 2 whereL = 1,Q1 = 0.8, θ1 = 10), we recover the main interesting features cited in the earlier work,but with the effect of pressure p0 included. Strain rate α has a strong effect on mass burningrate. As α → 0, −αx∗ → m∞ = const, which implies that the flame position x∗ → −∞. Onthe other hand, as the strain rate α increases, the mass burning rate m0 decreases and m0 → 0as α → α∗ ≈ B2 = (π/2C2

∞) eψ∗ . In fact, α∗ is a critical value of strain rate which leads tothe loss of any steady state for such flames.

4. Unsteady equations—an analytical solution usinglarge activation energy asymptotics

In a similar way as in section 3, the unsteady equations and jump conditions in mass-weightedcoordinates are given by

∂C

∂t+ (m0 − αx)∂C

∂x− p∂

2C

∂x2= 0, (44)

∂T

∂t+ (m0 − αx)∂T

∂x− pL∂

2T

∂x2= (1 − γ−1)

T

p

dp

dt, (45)

and

[T ]−+ = 0, [C]−+ = 0,[L∂T

∂x+Q1

∂C

∂x

]−

+

= 0,

[∂T

∂x

]−

+

= Q1

L√p0

e(θ1/2)(T∗−1),(46)

with the upstream conditions T |x→−∞ = T01, C|x→−∞ = 1 and downstream C = 0 with Tbounded. We now solve these equations for a prescribed pressure perturbation of the order ε.By assuming that the perturbation amplitude ε � 1, the perturbations are then written as

T = Ts + εT (1) + · · · , C = Cs + εC(1) + · · · ,m0 = ms + εm(1) + · · · , p = p0 + εp(1) + · · · , (47)

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and the leading-order equations become

(ms − αx)∂Cs

∂x− p0

∂2Cs

∂x2= 0, (48)

(ms − αx)∂Ts

∂x− p0L

∂2Ts

∂x2= 0, (49)

which have the solutions (37) and (38).The first-order equations are given by

∂C(1)

∂ t+ (ms − αx)∂C

(1)

∂x− p0

∂2C(1)

∂x2= −m(1) ∂Cs

∂x, (50)

∂T (1)

∂ t+ (ms − αx)∂T

(1)

∂x− p0L

∂2T (1)

∂x2= −m(1) ∂Ts

∂x+ (1 − γ−1)

Ts

p0

dp(1)

dt. (51)

In keeping with standard stability analysis, we now write the unsteady terms in the form

T (1) = ψ(x) eωt , C(1) = φ(x) eωt , p(1) = pu eωt , m(1) = mu eωt , (52)

where ω is a complex frequency. We then obtain

ωφ + (ms − αx)dφdx

− p0d2φ

dx2= −mu

dCs

dx, (53)

ωψ + (ms − αx)dψdx

− p0Ld2ψ

dx2= −mu

dTs

dx+ (1 − γ−1)

Ts

p0ωpu, (54)

and the jump conditions become

[ψ]−+ = 0, [φ]−+ = 0, (55)[dψ

dx

]−

+

= ψ∗2

θ1Q1

L√p0

eψs∗/2,

[L

dψ

dx+Q1

dφ

dx

]−

+

= 0, (56)

with boundary conditions φ|x→−∞ = 0, ψ |x→−∞ = 0, φ = 0 for x > 0 and ψ |x→∞ = 0.

4.1. Further assumptions concerning α small

We now regard equations (53), (54) with jump conditions (55), (56) as a separate sub-problemwhich, because of the non-autonomous nature of the middle terms in both equations, is solvedapproximately for small α. In order to find tractable solutions in the asymptotic analysis, weassume α is small and certainly well below its critical value, i.e. |α| � |α∗| or |α| � 1. Thus,we can expand solutions (37) and (38) in terms of α and use the steady-state result (42) as anapproximation for mass burning rate ms. We can therefore write

ψ = ψ(0) + αψ(1) · · · (57)

and we obtain the leading-order equations

ωφ(0) +msdφ(0)

dx− p0

d2φ(0)

dx2= ms

p0

(mu − mspu

p0

)emsx/p0 , (58)

ωψ(0) +msdψ(0)

dx− p0L

d2ψ(0)

dx2

= (1 − γ−1)ωpu

p0[Tu0 +Q emsx/(Lp0)] − Qms

p0L

(mu − mspu

p0

)emsx/(p0L).

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The neglected terms are of order (εα). The solution to these (now linear) autonomousdifferential equations follows standard textbook procedure, and the application of the linearizedversion of the jump conditions (55) and (56) leads to the following estimate for the unsteadymass burning rate:

mu = ms

p0pu

+(1 − γ−1)(θ1pu/p0)(2s −Q){

(θ1Q/ω) e−αθ1Q(L−1)/2[−(ms/L)

(12 − s) +ms

(12 − r)] − (4m2

s s/ω)(

12 − r)} ,

(59)

where

r ≡√ωp0

m2s

+1

4, s ≡

√ωp0L

m2s

+1

4. (60)

This follows a sequence of steps similar to those for unstrained flames in McIntosh and Clarke(1984) and McIntosh (1991). The first term on the right-hand side reflects the assumptionρλ ≈ p, whilst the second term is the effect of the system response to pressure disturbance.Since θ1 � 1, the second term dominates and we have approximately that

mu = (1 − γ−1)(θ1pu/p0)(2s −Q1){(θ1Q/ω) e−αθ1Q1(L−1)/2

[−(ms/L)(

12 − s) +ms

(12 − r)] − (4m2

s s/ω)(

12 − r)} .

(61)

By defining

l ≡ (L− 1)θ1, p(1)u ≡ θ1pu

p0, (62)

we can express the burning rate response more compactly as

mu = (1 − γ−1)(p(1)u )(2r −Q1){

(lQ1/2r) e−αlQ1/2((

12 − r)/( 1

2 + r))

+(4r/

(12 + r

))[1 − α(1 +Q1l/2)]

} . (63)

This form is an exact parallel of equations (126) in McIntosh and Clarke (1984) and McIntosh(1991) with the addition of weak strain, represented by the terms containing α.

5. The effect of strain on the long wavelength dispersion relation

The denominator of equation (63) yields the long wavelength dispersion relation for weaklystrained flames:

4mss

ω

(r − 1

2

)− θ1Q1

ωexp

[−αθ1Q1(L− 1)

2

] [(r − 1

2

)−

(s − 1

2

)L

]= 0, (64)

where

ms ≈ √p0

{1 − α

[1 +

Q1θ1(L− 1)

2

]}. (65)

In terms of the reduced Lewis number l ≡ (L− 1)θ1, this can be written as

8r2

[1 − α

(1 +

Q1l

2

)]+Q1l

(1

2− r

)e−αQ1l/2 = 0, (66)

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0 20 40 60 80 100 1200

1

2

3

4

5

6

7

Activation energy (θ1Q1)

Crit

ical

Lew

is n

umbe

r (L

)

α=0.0

0.1

0.1

0.2

0.2

Unstable

Stable

Figure 3. Critical Lewis number of neutral stability curve with strain present for α = −0.2–0.2,Q1 = 0.8, θ1 = 10, γ = 1.4.

which is an extension to equation (126) in McIntosh and Clarke (1984) for weak strain. In thespecial case of unstrained flow (α = 0), we have

8r2 +Q1l(12 − r) = 0. (67)

as obtained by Sivanshinsky (1977).The implications of this new dispersion relation (64) and the reduced form (66) are explored

in figures 3 and 4, respectively. Generally, the solution for ω is of the form ωr + iωi where ωr

is growth rate and ωi is frequency. These figures show the curves of neutral stability whereωr = 0. The essential form of this new dispersion relation agrees with the numerical work ofMikolaitis and Buckmaster (1981) and Buckmaster and Mikolaitis (1982). We can see that thestrain rate has a significant effect on the neutral stability curves. Positive strain stabilizes theflame while negative strain (as found in the rear stagnation-point flow) has a destabilizing effect.

5.1. The effect of a pressure fluctuation

To show the effect of pressure perturbations, we now refer back to solution (61) with a smallpressure change (p(1)u ). If we have a forced harmonic pressure disturbance p(1) = pu eiωt , theamplitude of the modified form of mass flux perturbationmu/[(1−γ−1)pu] plotted against thefrequency (with the complex frequencyω set to iωi) follows the curves of figure 5. Here we haveused lQ = 8, γ = 1.4, and we plot the amplitude against frequency ωi. The main effect is toreduce substantially the resonance peak as the strain rate increases, with negative strain increas-ing the resonance peak. In general, increasing strain decreases the amplitude of the responseof mass burning rate to the pressure disturbance, and thus stabilizes the premixed flame.

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0 20 40 60 80 100 120

Activation energy (θ1 Q1)

Red

uced

crit

ical

Lew

is n

umbe

r l

Q1=

θ 1 Q

1 (L

–1)

α = 0.0

α = 0.1

α = 0.2 Unstable

Stable

α = 0.1

α = 0.2

5

6

7

8

9

10

11

12

Figure 4. Neutral stability curves of reduced critical Lewis number l ≡ (L − 1)θ for a range ofstrain values.

0 1 2 3 4 5 6Frequency (ωi)

Am

plitu

de m

u/[

(1–

γ –1 )

(θp

u/p 0

)]

α = 0.1α = 0.0

α = 0.1

α = 0.2

α = 0.2

0

0.5

1

1.5

2

2.5

3

Figure 5. The amplitude of the reduced form of mass flux perturbationmu/[(1−γ−1)θ1pu] againstthe frequency for θ1 = 10, lQ1 = 8, and γ = 1.4.

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6. Abrupt pressure changes—the modified dispersion relation

Perhaps the most interesting result now emerging from the analysis is the effect of the finalpressure after an abrupt compression or decompression has passed through the flame. Initiallythe premixed flame is characterized by pressure p = 1 and a burnt temperature T ∗, which isclose to the unstrained value of unity. After an abrupt pressure change, the flame response isgoverned by a frequency relation (long wavelength dispersion relation) similar to equation (64)but which now also depends upon the pressure change (Johnson et al 1995). In the immediateaftermath of the compression or decompression, the system is at a steady state typified bypressure p = p0 and temperature T ∗p(1−γ−1)

0 , but when the final steady state is reached, thisstate will be typified by pressure p = p0 and a different temperature TB = T01p

(1−γ−1)

0 +Q∗

(where Q∗ = T ∗ − T01). Consequently by insisting that all the non-dimensionalization ischaracterized by the initial steady state, the effect of a sharp change in pressure on the changein stability can be evaluated from a revised dispersion relation which we now derive. At theinitial steady state, p = 1, θ = θ1,Q = Q1, and ω = ω1. As the pressure changes to its finalsteady value p = p0, we have a new steady state characterized by θ2,Q2, ω2. By writing theparameters describing the second steady state in terms of those at the first, it is possible to useequation (61) to give the revised frequency relation at the new pressure in terms of the originalparameters. Following a similar procedure to that in our earlier paper (Johnson et al 1994),the necessary rescalings are

θ2 = θ1

TB, Q2 = Q1

TB, ω2 = ω1

/, α2 = α

/, (68)

where

TB = T01p(1−γ−1)

0 +Q∗, / = T 4B eθ1(1−T −1

B ). (69)

The mass burning rate in terms of the new steady state becomes

ms = √p0/

1/2

{1 − α

/

[1 +

θ1Q1(L− 1)

2T 2B

]}. (70)

With these transformations, we can rewrite the frequency relation (61) to include the effect ofpressure changes:

mu = (1 − γ−1)(θ1pu/p0)(2sTB −Q1)

×[/

{(θ1Q1/ω1) e−αθ1Q1(L−1)/2/T 2

B[−(ms/L)

(12 − s) +ms

(12 − r)]

−(4m2sT

2B s/ω1)

(12 − r)}]−1

, (71)

and

r =√ω1p0

/m2s

+1

4, s =

√ω1p0L

/m2s

+1

4. (72)

6.1. Analytical estimation of the critical reduced Lewis number—the effect ofsharp changes in pressure and weak strain

Equation (71) can now be used to show the effect of final pressurep0 on the stability of premixedflames in the presence of weak strain. The long wavelength dispersion relation based on theinitial steady state is given by the vanishing of the denominator, i.e.

4msT2

B s

ω1

(r − 1

2

)− θ1Q1

ω1e−αθ1Q1(L−1)/2/T 2

B

[(r − 1

2

)−

(s − 1

2

)L

]= 0. (73)

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This is a complex frequency relation, and for a given Lewis number, activation energy, heatrelease and final pressure, there will be a solution for complex frequency ω = ωr + iωi, wherethe growth rate ωr is zero. A more compact form of equation (73) using the reduced Lewisnumber l ≡ θ1(L− 1), is given by

8msT2

B r2 +Q1l

(12 − r) e−αQ1l/2/T 2

B = 0, (74)

where ms from (70) is now given by

ms = √p0/

1/2

{1 − α

/

[1 +

Q1l

2T 2B

]}, (75)

which includes the effect of change in final pressure (p0) and strain (α). This is explored infigure 6. In this reduced form, it is possible to evaluate analytically the critical reduced Lewisnumber for neutral stability. This is found by putting the real part of the complex frequencyω1r to zero, so that we obtain an approximate transcendental equation for the critical reducedLewis number l∗:

Q1l∗ ≈ 4(1 +

√3)msT

2B eαQ1l

∗/2/T 2B , (76)

where TB and / are given by (68). Note that, for α = 0, the classical neutral stabilityresult is obtained (Q1l

∗ = 4(1 +√

3)). The above equation shows how final pressure p0

and strain α make the pulsating instability much more accessible. For a given θ1 and p0, theabove critical Lewis number is a function of strain rate (α), and this is plotted in figure 6.This main result is that a sharp pressure drop and/or negative strain destabilizes the premixed

Solid theory: dashed numerical

Strain rate (α)

Fin

al p

ress

ure

p 0

Unstable

Stable

lQ1=6

lQ1=10

lQ1=14

lQ1=3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.4 0.2 0 0.2 0.4 0.6

Figure 6. Critical p0 versus α for neutral stability at different reduced Lewis numbers takingaccount of a sharp pressure change (from p = 1 to p = p0). Comparison with numerical resultsshows good agreement.

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flame. However, there is an important corollary from this figure. A sharp pressure rise passingthrough a flame which is inherently unstable can suppress the growing pulsations—this canbe seen, for instance, on figure 6 for α = −0.3 and lQ1 = 10. If a sharp pressure rise (i.e.a shock) of strength p0 = 1.4 is experienced, this will be sufficient to stabilize such a flame.We have also solved the non-autonomous nonlinear differential equations (28) and (29) byusing direct numerical simulations, and thereby established numerically the critical pressureat which extinction just occurs. The relevant numerical results from McIntosh et al (2001) arealso plotted in figure 6 as dashed curves to show the comparison with the analytical formula(75) obtained by perturbation analysis. The good agreement verifies the accuracy of analyticalsolutions even though its mathematical analysis is based on small perturbations for α � 1.

The frequency w1i, at the critical reduced Lewis number l∗, is

ω1i = /m2s

4p0

√1 +

Q1l∗

2msT2

B

e−αQ1l∗/2/T 2B , (77)

which again depends on the final pressure p0 and strain (α). The variation of frequency at afixed reduced critical Lewis number lQ1 = 8 is now plotted in figure 7, which shows that thedependence on strain and final pressure is monotonic for positive strain. However, for a largepressure drop with negative strain, the frequency grows markedly. This trend for low pressureis confirmed by separate numerical calculations (McIntosh et al 2001) which concerns largerstrain. Indeed, the frequency growth is underestimated by this analytical approach.

Im(ω

1i)

p0=0.6

p0=1.4, lQ1=8

1.0

1.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.8

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5

Strain rate (α)

Figure 7. Dependence of frequency (ω1i) on strain and final pressure (p0) for a constant reducedLewis number lQ1 = 8.

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7. Conclusions

The mathematical model (McIntosh and Clarke 1984, McIntosh 1991) for pressure disturbanceinteractions with premixed flames has been extended in this paper to include the effect of strainrate on the mass burning rate of premixed flames. We have modelled the strained premixedcounterflow by using a thin flat flame approximation together with a prescribed flow fieldu = (−αx, αy). The flow is regarded as essentially incompressible and uniformly strained.The weak acoustic field caused by an abrupt pressure change is on a much faster timescalethan the relaxation time for the combustion.

For the case of abrupt pressure changes, an approximate dispersion relation (whichincludes, as a parameter, the final pressure after a sharp pressure change) has been obtainedon the assumption of a prescribed strain α. This shows that, even for positive strain, if theLewis number is high enough, the pulsating instability will be observed. This would berelevant to strained hydrogen–air flames encountering sharp changes in pressure. Certainlyfor negative strain such as in rear stagnation flow, near-unit Lewis number flames are allsusceptible to the pulsating instability when abrupt pressure drops are encountered. Thishas implications for flames stabilized behind bluff bodies, and encountering sharp pressurechanges. An approximate expression for this critical Lewis number is derived.

Acknowledgments

The authors would like to thank EPSRC for their support of this work on grant GR/M 18607.

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pressure disturbances and strained premixed flames

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