Effects of substituting fuel spray for fuel gas on flame stability in lean premixtures

Combustion and Flame 149 (2007) 295–313www.elsevier.com/locate/combustflame

Effects of substituting fuel spray for fuel gas on flamestability in lean premixtures

C. Nicoli, P. Haldenwang ∗, S. Suard

Modélisation et Simulation Numérique en Mécanique et Génie des Procédés (MSNM-GP), UMR-CNRS n◦ 6181, Aix-MarseilleUniversités—ECM, I.M.T./La Jetée/L3M, 38, rue Frédéric Joliot-Curie, 13451 Marseille Cedex 20, France

Received 30 August 2005; received in revised form 31 July 2006; accepted 12 December 2006

Available online 6 March 2007

Abstract

We analyze flame propagation through a homogeneous three-component premixture composed of fuel gas, smallfuel droplets, and air. This analytical study is carried out within the framework of a diffusional–thermal model withthe simplifying assumption that both fuels—the fuel in the gaseous phase and the gaseous fuel evaporating fromthe droplets—have the same Lewis number. The parameter that expresses the degree of substitution of spray forgas is δ, the liquid loading, i.e., the ratio of liquid fuel mass fraction to overall fuel mass fraction in the freshpremixture. In this substitution of liquid fuel for gaseous fuel, the overall equivalence ratio is lean and is keptidentical. We hence obtain a partially prevaporized spray, for which we analytically study the dynamics of the planespray-flame front. The investigated model assumes the averaged distance between droplets to be small comparedwith the premixed flame thickness (i.e., small droplets and moderate pressure). Le, the Lewis number, Ze, theZeldovich number, and δ are the main parameters of the study. Our stability analysis supplies the stability diagramin the plane {Le, δ} for various Ze values and shows that, for all Le, the plane front becomes unstable for high liquidloading. At large or moderate Lewis number, we show that the presence of droplets substantially diminishes theonset threshold of the oscillatory instability, making the appearance of oscillatory propagation easier. Oscillationscan even occur for Le < 1 when sufficient spray substitution is operated. The pulsation frequency occurring in thisregime is a tunable function of δ. At low Lewis number, substitution of spray for gas leads to a more complexsituation for which two branches can coexist: the first one still corresponding to the pulsating regime, the otherone being related to the diffusive–thermal cellular instability.© 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Oscillatory spray-flame; Premixed flame; Two-phase combustion; Diffusive–thermal instability; Pulsating flame;Particle-laden gas; Combustion in microgravity

1. Introduction

The paper deals with lean two-phase combustion;more precisely, the study concerns flame propaga-

* Corresponding author. Fax: +33 491 118 502.E-mail address: [email protected]

(P. Haldenwang).

0010-2180/$ – see front matter © 2007 The Combustion Institute.doi:10.1016/j.combustflame.2006.12.018

tion in a partially prevaporized spray, as in a chamberwhere combustion arises far enough from the injec-tion zone. The applications of spray combustion arelinked to diesel or rocket engines, turbojets, and in-dustrial furnaces. Despite this wide set of industrialinvolvements most of the specific phenomena ob-served in the basis experiments on spray combustionare not yet fully understood.

Published by Elsevier Inc. All rights reserved.

http://www.elsevier.com/locate/combustflame

mailto:[email protected]

http://dx.doi.org/10.1016/j.combustflame.2006.12.018

296 C. Nicoli et al. / Combustion and Flame 149 (2007) 295–313

Nomenclature

Cp specific heatDa Damkoehler number (Da = 0.1 in the

whole paper)Dm binary mass diffusion coefficient of

gaseous limiting componentDth thermal diffusion coefficientl in the particularly involved analytical ex-

pressions of Section 3, l stands for 1/LeLd droplet interdistanceLSF spray-flame thicknessLv latent heat of vaporization (Lv = 0 in the

whole paper)Le Lewis number (ratio of heat diffusivity

to species diffusivity)Q heat of reactionT temperatureU flame velocityYl mass fraction of the fuel in liquid phaseYg mass fraction of the fuel in gaseous

phaseZe Zeldovich number (reduced activation

energy)

Greek and mathematical symbols

δ spray liquid loading: ratio of liquid fuelmass to overall fuel mass

� in the particularly involved analytical ex-pressions of Section 3, � stands for1/Da

θv reduced vaporization temperature (θv =0.1 in the whole paper)

λ thermal conductivity of mixtureΛa averaging length scale, intermediate be-

tween droplet interdistance and flamethickness

ρ mixture densityτchem characteristic time of chemical reaction

(or transit time)τvap characteristic time of vaporizationωchem chemical reaction rateωvap vaporization rate{A} jump condition relative to quantity A

Indices

b relative to burnt mixtureg relative to gaseous fuell relative to liquid fuelSF relative to spray-flameu relative to fresh mixturev relative to vaporization (thermal thresh-

old or locus where vaporization starts)ZFK relative to classic reactive–diffusive

scales (after Zeldovich and Frank-Kamenetski theory)

For a long time [1–5], spray-flames have beenknown to exhibit large departures from one-phase pre-mixed flames: flammability limits broaden, shrink,or shift with spray liquid loading and droplet size,and burning rate is enhanced when droplets are in-troduced, or reduced if the droplets are too large.In the latter case, spray-flame speed is found in in-verse ratio to droplet size [6,7], as could result froma vaporization–diffusion process [8,9]. Spray-flamescan appear largely corrugated, while the equivalentgaseous premixed flames are flat [10–12].

Recent experiments conducted under micrograv-ity clarified the part played by droplets in spray-flamespeed promotion [13] and supplied new phenom-ena on pulsating flame propagation in particle-ladengas [14]. It is worth emphasizing these recent funda-mental results: microgravity allows experimentaliststo stabilize sprays even with large droplets and tostudy combustion systems with low Froude numbers(e.g., slow flames in large chambers). They brought,for instance, very precise data on the spreadingrate promotion of two-phase flames compared withgaseous flames. Nomura et al. [13] reported on the

existence of a maximum in spreading rate for dropletradii located between 10 and 20 µm, the strengthof this maximum being a nonmonotonic function ofspray liquid loading. Microgravity also helps to sta-bilize particle-laden gases; Hanai et al. [14] observedpulsating flame propagation near the lean flammabil-ity limit of a PMMA cloud. Microgravity also allowedthe experimentalists to specify [15] the oscillatorypropagation of one-phase premixed flames—a featuretheoretically predicted several decades before [16].

On the other hand, the observation of pulsatingfeatures in spray-flames does not seem to requirea particular environment, since oscillatory propaga-tions have been observed on the ground [17] for leanand stoichiometric spray. This contribution by Atzleret al. [17] reports pulsating lean propagation with aperiod of about 21 ms, while droplet mean diameteris 17 µm and δ = 0.17. Pulsations are observed with astronger amplitude for stoichiometric sprays (periodof about 17 ms, droplet mean diameter 20 µm andδ = 0.25), whereas oscillatory spreading disappearsfor the equivalent gaseous (or single-phase) combus-tion.

C. Nicoli et al. / Combustion and Flame 149 (2007) 295–313 297

Two recent theoretical contributions [18,19] haveshown the existence of a robust mechanism leading tothe oscillatory propagation of spray-flames. This fea-ture was first pointed out by numerical means [18],while an analytical work [19], using the simple frame-work of the diffusive–thermal model, complementedwith a vaporization rate independent of gas temper-ature, has highlighted its domain of existence. Themechanism proposed does not invoke any differentialdiffusive phenomena effects (it occurs for Le = 1) orheat loss (it occurs for vanishing latent heat). It re-quires, however, that the Zeldovich number (Ze) beat least of order 10 and that the Damkoehler num-ber (Da), the ratio of chemical reaction rate to vapor-ization rate, be sufficiently small: vaporization timenot larger than reaction time. In Ref. [19], we ne-glected the differential diffusive effects and the possi-ble prevaporization of the spray to focus on the partsplayed by the Damkoehler number and by the dropletboiling temperature (θv, in reduced form). As a re-sult, the observed domain of oscillations and theirfrequency were found to be quite insensitive to thoseparameters, provided that θv < 0.2 and Da < O(1).Such mixture characteristics are often met in air–fueltwo-phase systems: small boiling reduced tempera-ture (θv � 1) and small droplets (Da < 1, i.e., reac-tion rate not larger than vaporization rate).

The purpose of the present paper is to pay par-ticular attention to the differential diffusive phenom-ena that can occur in the combustion of small dropletsprays. Furthermore, to introduce a continuous transi-tion from gaseous flame to spray-flame, we allow thespray to be partially prevaporized. In the fresh mix-ture, we define (Yl)u as the liquid fuel mass fraction,while the prevaporized fuel mass fraction is denoted(Yg)u. Thus, δ, the liquid loading parameter of thespray, bridges the gap between pure spray and gas(with 0 � δ � 1). This parameter reads

(1)δ = (Yl)u

(Yl)u + (Yg)u.

In other words, a classic gaseous premixed flamewill correspond to δ = 0, while a very weakly pre-vaporized spray (i.e., a fuel with low vapor pressure)will be characterized by δ ≈ 1. Let us recall in briefthe classic results of flame stability, established [16]for δ = 0: if the Zeldovich number is large enough,only high-Lewis-number mixtures are subject to os-cillating propagation while low-Lewis-number mix-tures present cellular fronts.

In the case of a unique fuel substance, δ indeeddepends on the thermodynamic state of the fresh mix-ture. Therefore, substitution of spray for gas can beobtained by decreasing mixture temperature or in-creasing pressure, as in Refs. [11,12]. Another way

to control δ is to consider a mixture of two fuel sub-stances (with the same Lewis number, for the use ofthe present theory) and air, wherein one fuel has avery low vapor pressure, while the other one is purelygaseous.

The set of experimental results on spray-flamepulsations [14,17] and wrinkles [10–12], as men-tioned above, invites us to address the issue relative ofspace–time stability of the plane prevaporized spray-flame. The paper is organized as follows: Section 2is devoted to model presentation and the related sta-tionary plane solution, Section 3 concerns the deriva-tion of the stability conditions (or dispersion relation),while Section 4 discusses the results with respect tothe main parameters: Le, the Lewis number, δ, thespray liquid loading, and Ze, the Zeldovich number.

2. The model and its steady plane solution

2.1. Modeling

The numerical work conducted in [18], treatingthe general set of conservation laws, allowed us toshow the existence of pulsating spray-flames. Para-metric studies have furthermore shown that the os-cillatory properties of the propagation survives tovery drastic simplifications of the numerically solvedmodel: pulsations can still be obtained with a roughvaporization law, a unity Lewis number, zero latentheat, and even a constant gas density. We concludedthat a robust mechanism of pulsating spray-flames ex-ists in the framework of the diffusive–thermal model;it does not require differential diffusive effects orheat losses due to vaporization. Moreover, when thedroplets are small (i.e., Da < 1; see below) and theirboiling temperature is low (i.e., θv � 0.2; see below),the spray-flame pulsating properties appear quite in-dependent of the latter parameters. This is why the pa-rameters are set to Da = 0.1 and θv = 0.1 in the wholepaper. As a result, the paper focus on spray-flames,rather than on particle-laden gases, which would beof concern with larger values for θv. Furthermore, anyheat loss is neglected.

As a matter of fact, the model we investigate forstudying the stability analysis of the plane spray-flame is the simplest one in the literature on spraycombustion [20]. Let us recall in brief its main char-acteristics (for more detail, see [19]): we considera spray-flame propagating through a fresh mixturecomposed of droplets, fuel vapor, and air; we sup-pose that combustion length scale and spray lengthscales are well separated, in the sense that there ex-ists an intermediate scale Λa between droplet in-terdistance Ld and spray-flame thickness LSF (i.e.,


Ld � Λa � LSF). Under these conditions of homog-enization on scale Λa, an averaging procedure onscale Λa leads to conservation laws for each fluid.Further simplifications—i.e., neglecting droplet in-ertia and their part in the thermal budget—lead usto consider the liquid fuel as an additional species,which cannot burn without prior vaporization. Thedroplets are here assumed to have the same velocity asthe host gas. Two reasons in fact cooperate in neglect-ing droplet inertia: droplets are small and surroundinggas relative velocity remains feeble until complete va-porization of droplets.

It must be added that the classical assumptions areretained in noncompressible studies; traveling wavesare deflagrations and flow is of low-Mach-numbertype. Furthermore, Fick’s law and Fourier’s law areused to describe diffusion of species and energy. Du-four and Soret effects are neglected. Specific heat Cp,thermal conductivity λ, and molecular weight of themixture are constant. Chemical kinetics is describedby a global one-step exothermic irreversible reac-tion described by the Arrhenius law. Furthermore, themixture being assumed globally fuel-lean, the reac-tion is controlled by the limiting species (i.e., thegaseous fuel). The vaporization phenomenon is sim-ply described [20] with a vaporization rate indepen-dent of temperature. The vaporization starts as soon asthe mixture temperature reaches Tv, a threshold valuethat may be compared to the boiling temperature ofthe liquid fuel.

Under these conditions, Tb, the adiabatic spray-flame temperature, is given by

(2)Tb = Tu + Q

Cp(Yg)u + Q − Lv

Cp(Yl)u,

where Tu is the fresh mixture temperature, (Yl)u and(Yg)u being respectively the fuel mass fraction of liq-uid and of gas in the fresh mixture. An additionalassumption of the present study is vanishing latentheat (Lv = 0). Moreover, in the sequel, variables Yl,Yg, and T are, respectively, the liquid fuel mass frac-tion, the gaseous fuel mass fraction, and the mixturetemperature.

Using the dimensional scales relative to the Zel-dovich and Frank-Kamenetski theory of premixedflame (at temperature Tb and referred as ZFK), weconsider the conservation laws in a classic nondimen-sional form; we first introduce θ(x, y, t), the reducedtemperature such that T = Tu + (Tb − Tu)θ ; fur-thermore, fuel mass fractions Yl and Yg are reducedwith the initial overall fuel mass fraction (Yl)u +(Yg)u; additionally, time and length units are respec-tively selected as τchem = (Dth)b/U2

ZFK and LZFK =(Dth)b/UZFK, with UZFK the ZFK flame speed, de-

fined [21] as the reaction–diffusion speed

(3)UZFK =√√√√2Leλρ2

bYO2B

Cpρ2u Ze2

exp

(− E

2RTb

),

where the nondimensioning process has led us to de-fine the Zeldovich number as the reduced activationenergy, derived from E as

(4)Ze = E

RT 2b

(Tb − Tu).

Thermal diffusivity is defined by Dth = λ/(ρCp),while Dm stands for the binary diffusion coefficientof the gaseous limiting component, and Le for Le =Dth/Dm. This process for nondimensioning leadsto Da, the Damkoehler number, defined as the ratioof the vaporization characteristic time to the chemicalreaction time,

(5)Da = τvap

τchem= τvapU2

ZFK(Dth)b

≈(

Ld

LZFK

)2,

where τchem is the characteristic reaction time at Tband τvap stands for the characteristic time of vapor-ization. The order of magnitude of the latter can beconsidered as roughly scaled by L2

d/(Dth)b, so that

Da ≈ (Ld/LZFK)2.Finally, in general deflagration studies, the equa-

tions describing the flame structure have to be solvedsimultaneously with Navier–Stokes equations in thelimit of low Mach numbers. However, when mixturedensity ρ is assumed constant and uniform, flamestructure and hydrodynamics are decoupled. In thiscase, it has been shown [19,20] that USF, the charac-teristic spreading velocity of the steady spray-flame,is identical to UZFK.

Within the diffusive–thermal approximations, asrecalled above, it is then interesting to write the sys-tem of conservation laws in a “perturbed frame” mov-ing with the flame speed affected by a general pertur-bation of the type

(6a)xSF(y, t) = −USFt + εDth

USFΦ(η, τ),

where ε is an (infinitely small) dimensionless am-plitude of the space–time front corrugation Φ(η, τ),which is assumed to possess the form

(6b)Φ(η, τ) = exp(ωτ + ikη).

We then introduce the nondimensional space–timecoordinates (ξ, η, τ ), defined as

{ξ, η, τ }

(6c–e)={(

x − xSF(y, t))USF

Dth, y

USF

Dth, t

U2SF

Dth

}.


For this new set of variables, the conservation lawsread

(7a)∂θ

∂τ+

[1 − ε

∂Φ

∂τ

]∂θ

∂ξ= �εθ + ωchem,

(7b)∂Yl

∂τ+

[1 − ε

∂Φ

∂τ

]∂Yl

∂ξ= −ωvap,

∂Yg

∂τ+

[1 − ε

∂Φ

∂τ

]∂Yg

∂ξ

(7c)= 1

Le�εYg − ωchem + ωvap

with

(8)ωchem = Ze2

2LeYg exp

[−Ze(1 − θ)],

where �ε , the perturbed Laplace operator, admits theform

�ε = ∂2

∂ξ2+ ∂

∂η

(∂

∂η− ε

∂Φ

∂η

∂

∂ξ

)

(9)− ε∂Φ

∂η

∂

∂ξ

(∂

∂η− ε

∂Φ

∂η

∂

∂ξ

).

For analytical purposes, the vaporization rate in (7) ischosen [20] as

(10)ωvap = Yl

DaH(θ − θv),

where H stands for the Heaviside function. For stan-dard alkane spray-flames, θv, the threshold value forvaporization, is on the order of 10−1. As for theorder of magnitude of Da, it strongly depends onthe spray droplet size. According to the rough es-timate Da ≈ O([Ld/LZFK]2), a spray with dropletsize 10 µm, burning at standard pressure, is charac-terized by Da ≈ O(0.1). In what follows, all quanti-tative data will be computed with the set {θv = 0.1and Da = 0.1} considered as the standard values ofthese parameters. Equations (7) are considered to-gether with the following boundary conditions:

(11)ξ → −∞, θ = 0, Yl = δ, Yg = 1 − δ,

(12)ξ → +∞, θ = 1, Yl = 0, Yg = 0.

The time-dependent solutions are investigated us-ing the method of small perturbations: all variablesare written as the sum of the steady-state solution anda small perturbation of Fourier type [19]. The steadystate solution mentioned here corresponds to a sta-tionary flame structure, which we have to establishfirst. Doing this in the limit Ze → ∞ is the purposeof the next section.

2.2. The steady plane solution

The steady planar problem is easily deduced fromsystem (7) when setting ε = 0. In the limit Ze → ∞,ωchem is negligible everywhere, except in a smallzone located in the vicinity of ξ = 0 where θ = 1.Therefore, the classical approach assumes four inte-gration zones [20]:

(1) A prevaporization zone {−∞ < ξ � ξv}, with ξvsuch that θ(ξ = ξv) = θv (i.e., the locus wherevaporization starts): no source term in this zone.

(2) The vaporization zone {ξv < ξ < 0}, in which va-porization is supposed to solely occur.

(3) The reaction zone {0− � ξ � 0+}, an asymptoti-cally thin zone where only combustion exists.

(4) A burnt gas zone {0+ � ξ � ∞} in equilibrium(θ = 1).

The problem admits the following boundary ormatching conditions (interface jump conditions aredenoted [·]):

ξ → −∞: Yl = (Yl)u = δ,

(13a–c)Yg = 1 − (Yl)u = 1 − δ, θ = 0

[(Yl)u = δ = 1 indicates that fuel is only in liquidphase in the fresh mixture];

at ξ = ξv: θ = θv,

(14a–e)[Yl

] = [Yg

] =[

dYg

dξ

]=

[dθ

dξ

]= 0

(all quantities are continuous, except the first deriva-tive of Yl):

(15a–d)

at ξ = 0−: Yl ≡ 0, Yg ≡ 0, [θ ] = [Yg

] = 0

(all quantities are continuous and liquid fuel is as-sumed completely vaporized in zone 2).

Integration of reactive diffusive equations in zone 3gives the jump conditions [22]:

(16a–b)

[dθ

dξ+ 1

Le

dYg

dξ

]0+

0−= 0 and

(dθ

dξ

∣∣∣∣0−

)2= 1.

As a result, ξ = 0 is defined as the position whereYg is consumed in totality. The latter jump conditionsclassically describe the reaction zone. The recourse tothose expressions requires that vaporization be absent

from zone 3 (i.e., θ1/Dav � 1; see [19]).

Thus, the steady-state solution is easily obtainedby integrating zone by zone. As we neglect latentheat, integration of the heat equation can be carried


out over the whole preheating zone:

(17a)−∞ < ξ < 0, θ = exp{ξ}.Thus, expression (17a) allows us to localize the pointwhere vaporization starts as

(17b)ξv = log[θv].As for species, the steady state solution in −∞ < ξ <

ξv, where vaporization is zero (θ < θv), reads

(18a)for liquid fuel species: Yl = (Yl)u ≡ δ,

for gaseous fuel species:

(18b)

Yg = (1 − δ) + ((Yg

)v − (1 − δ)

)exp

{Le(ξ − ξv)

}

with

(Yg

)v = 1 − δ

(1 + 1

DaLe

)−1

+ exp{Leξv}(

−1 + δ

(1 + 1

DaLe

)−1

(18c)× exp

{ξv

Da

}),

which is the value of the gaseous fuel mass fraction atξ = ξv. In the vaporization zone, ξv < ξ < 0, we getfor the liquid fuel

(19)Yl = δ exp

{− (ξ − ξv)

Da

}.

Note that Yl is not strictly zero in the reaction zone atξ = 0. We can nevertheless observe that Yl(ξ = 0) =δ exp{− |ξv|

Da } = δθ1/Dav is a very small quantity for the

parameters currently under investigation: {Da = 0.1and θv = 0.1}.

Furthermore, the gaseous fuel mass fraction is

Yg = 1 − exp{Leξ} + δ

(1 + 1

Da Le

)−1

(20a)× exp

{ξv

Da

}(exp{Leξ} − exp

{− ξ

Da

}).

Let us note that (Yg)max, the maximum value of (20a),is located at ξmax > ξv, given by

ξmax = {ξv − Da

[log

[1 + Le Da

(1 − exp(ξv/Da)

)](20b)− log(δ)

]}{1 + Da}−1.

Finally, the solution in burnt gases (ξ > 0) is trivially

(21a)θ = 1,

(21b)Yl = 0,

(21c)Yg = 0.

As in Ref. [19], we note from (20a) that 1Le

dYgdξ

=−1+δθ

1/Dav at ξ = 0. This value differs from unity as

imposed by relation (16a). We observe that the valid-ity of the present solution again requires the condition

θ1/Dav � 1.

To summarize with our set of assumptions, thesteady flame structure is composed with two well-

separated zones as soon as θ1/Dav is a negligible

quantity: a vaporization domain inside the preheat-ing region, producing gaseous fuel, and a reactionfront where gaseous fuel is consumed. In this context,spray-flame speed is identical to ZFK gaseous flamespeed, for any value of δ. Numerical confirmation ofthe latter result can be found in [18], where ZFK flamespeed is maintained up to Da = O(1) for θv = 0.1.

3. Stability analysis of the plane spray-flame

3.1. The perturbed problem

Consider now the classical space–time Fourierperturbation of the previous steady results, as usedin Eqs. (6a–b). Our purpose is to study how the un-steady conservation laws (7) damp or amplify thisperturbation. Within the thermal–diffusive approxi-mation framework, the system of equations (7) al-lows us, when ε � 1, to derive a set of linearizedequations that involves the perturbation functions θ ,Yl, Yg. They read

(22a)

(ω + k2)

θ + ∂θ

∂ξ− ∂2θ

∂ξ2= (

ω + k2)∂θ

∂ξ+ Wchem,

(22b)ωYl + ∂Yl

∂ξ= ω

∂Yl

∂ξ− Wvap,

(ω + k2

Le

)Yg + ∂Yg

∂ξ− 1

Le

∂2Yg

∂ξ2

(22c)=(

ω + k2

Le

)∂Yg

∂ξ− Wchem + Wvap,

where

(22d)

Wchem = θ∂ωchem

∂θ

(θ , Yg

) + Yg∂ωchem

∂Yg

(θ , Yg

)

and

(22e)

Wvap = θ |ξv

Yl|ξv

Daδ(θ − θv) + Yl

DaH(θ − θv).


The first term on the RHS of Eq. (22e) involves theDirac measure as the derivative of the Heaviside func-tion; it will imply a jump condition at ξv. As matterof fact, Dirac mass at ξv expresses the fact that the lo-cus where vaporization starts is not a fixed point. Itsfluctuating position, ξv + εφ(η, τ )ξv, is related to thethermal perturbation θ |ξv , as

(23)

ξv = −θ |ξv

[∂θ/∂ξ |ξv

]−1 = − θ |ξv

θ |ξv

= − θ |ξv

θv.

Note, furthermore, that perturbations θ , Yl, Yg areconsidered of order O(1). For ξ → −∞, system (22)is supplemented with the following boundary condi-tions:

(24a–c)ξ → −∞, θ = 0; Yl = 0; Yg = 0.

At ξv(η, τ ), the interface between zones 1 and 2, weget continuous conditions on temperature:

(25a,b)for ξ = ξv, [θ ] = 0;[

∂θ

∂ξ

]= 0.

As for the fuel, a consequence of expression (22e) isthat the gas phase mass fraction has a discontinuousfirst derivative, while the liquid mass fraction is dis-continuous itself:[Yg

] = 0;[− 1

Le

∂Yg

∂ξ

]= θ |ξv

Yl|ξv

Da

(∂θ/∂ξ |ξv

)−1

(25c–e)= δθ |ξv

θvDa; [

Yl] = − θ |ξv

θvDaδ.

Jump conditions (25d,e) are related to the fact that thevaporization zone moves with the thermal fluctuation.At ξ = 0, jump conditions (15), (16a) are still validfor the unsteady problem. They therefore lead to thefollowing boundary conditions and jump conditionsthrough reactive zone 3:

Yg|0− = 0; [Yg

]0−0+ = 0;

(26a–c)[θ ]0−0+ = 0;

[∂θ

∂ξ+ 1

Le

∂Yg

∂ξ

]0−

0+= 0.

Equation (16b) must be replaced by

(26d)∂θ0

∂ξ

∣∣∣∣0−

= 1

2θ1|0,

where θ0(ξ) and θ1(ξ) are the first two terms of thefollowing expansion in 1/Ze for θ (ξ):[θ (ξ), Yg(ξ)

] = [θ0(ξ), Y0g(ξ)

](26e)+ 1 [

θ1(ξ), Y1g(ξ)] + O

(1

2

).

Ze Ze

Establishment of expressions (26) can be foundin [22]. We recall that the key of the method is thatthe inner reaction zone can be solved by an asymp-totic expansion valid for large Ze.

System (22) is closed with

(27a–c)ξ → +∞, θ = 0; Yl = 0; Yg = 0.

3.2. Perturbed flame structure

Taking account of boundary and jump conditions(24)–(27), the solution of system (22) supplies thefirst two orders in the (Ze)−1 development of the per-turbed fields. We then obtain the perturbed fields.

Temperature field:

(28a)ξ < 0, θ (ξ) =(

θ1(0)

Ze− 1

)er+ξ + eξ ,

(28b)ξ > 0, θ (ξ) = θ1(0)

Zeeξr−

,

with

(28c)θ1(0) = 2r−

and

(29)r± = 1 ±√

1 + 4(ω + k2)

2.

From the perturbation of the temperature field, ex-pression (23) becomes

(30)ξv = (θ−r−

v − 1) − 2r−

Zeθ−r−

v with θv = eξv .

Liquid fuel:

(31a)ξ � ξv, Yl = 0,

ξv < ξ < 0,

(31b)Yl = �δ(ξv + 1

)e(�+ω)(ξv−ξ) − �δe�(ξv−ξ),

(31c)ξ > 0, Yl = 0

with the introduction of

(32a,b)� = (Da)−1 and l = (Le)−1.

Gaseous fuel:

(33a)ξ � ξv, Yg = ∂Yg

∂ξ+ Aes+ξ ,

ξv � ξ � 0,

Yg = ∂Yg

∂ξ+ δ

�2(1 + ξv) exp{(ω + �)(ξv − ξ)}lk2 − � − l(ω + �)2

(33b)+ Bes+ξ + Ces−ξ

and

(33c)s± = Le1 ±

√1 + 4l(ω + lk2)

.
2


The first term of both expressions (33a) and (33b)refers to the first derivative of the steady solutiongiven in (18b) and (20a). A, B , and C are constantsof integration, obtained as solutions of the followinglinear system:

(B − A) exp(s+ξv

) + C exp(s−ξv

)(34a)= −�2δ

lk2 − � − l(ω + �)2

(1 + ξv

),

(B − A)s+ exp(s+ξv

) + Cs− exp(s−ξv

)(34b)= [�(k2 − ω2) − �2(Le + ω)]

lk2 − � − l(ω + �)2

(1 + ξv

)δ,

B + C = Le − �2δ exp{(ω + �)ξv}lk2 − � − l(ω + �)2

(1 + ξv

)(34c)− δLe

(exp(�ξv)

).

Relationships (34a–c) respectively express that

• gaseous fuel perturbed mass fraction is continu-ous at ξ = ξv [i.e., condition (25c)],

• jump condition (25d) has to be imposed on thefirst derivative of Yg at ξ = ξv,

• gaseous fuel perturbed mass fraction vanishes atξ = 0 [i.e., condition (26a)].

It must be noticed that our perturbed flame structurenow satisfies all boundary and jump relationships,condition (26c) excepted. The latter point is the pur-pose of the next section.

3.3. Relation of dispersion

We now have to enforce jump condition (26c) onperturbed field derivatives at ξ = 0. From expressions(28a) and (33b), we get the following relation on B

and C:

Bs+ + Cs− = δ�2(ω + �)(1 + ξv)e(ω+�)ξv

lk2 − � − l(ω + �)2

+ 2r−Le

Ze(r− − r+) + s+Le

(35)+(

�

Le− 1

)δLe2e�ξv .

Equation (35) is in principle not compatible with lin-ear system (34), except if pulsation ω satisfies thefollowing relation of dispersion:

(1 + ξv

)((δ�2(ω + � + s+)e(ω+�)ξv

+ δ[�2(ω + s−) + �(ω2 − k2)

]e−s−ξv

)/(lk2 − � − l(ω + �)2))

+ Le2r−

(r− − r+) + Le2δ

(� − 1

)e�ξv

Ze Le

(36)+ Le(s− − r−) = 0.

Inspecting relation (31b), we here again remark thatcontinuity in ξ = 0 of Yl is ensured only in the limit

θ1/Dav = exp{�ξv} = 0. Therefore condition θ

1/Dav

� 1 (or equivalently exp{�ξv} � 1) defines the va-lidity domain of our results. Equation (36) admitsconsequently the simplified form

δ

(1 − 2r−

Ze

)

× [ω + s− + Da(ω2 − k2)]e−(s−+r−)ξv

(k Da)2 − Le Da − (ωDa + 1)2

(37)+ 2r−Ze

(r− − r+) + (s− − r−) = 0.

Dispersion relation (37) describes the stability prop-erties of spray-flames for any Le and various liquidfuel loadings δ. A necessary condition for a givenperturbation with {ω ∈ C, k ∈ R} to lead to a newpropagation mode is that ωr = Re(ω) is a positivequantity. ωr = 0 therefore corresponds to the neu-tral stability curve that delimits, for a given real valueof k, a domain in the space of physical parameters,where the steady solution is either unstable with re-spect to pulsating propagation (or oscillatory insta-bility) provided that it is simultaneously found thatωi = Im(ω) = 0, or unstable with respect to cellularfront instability (or diffusive–thermal cellular insta-bility, often in the literature) provided that it is simul-taneously found that ωi = Im(ω) = 0.

4. Discussing the main results

We now have to interpret Eq. (37) with respectto {Le,Ze, δ}, the main parameters of the study. Theother parameters are fixed to values correspondingto standard alkane lean sprays composed of smalldroplets, i.e., θv = 0.1 and Da = 0.1.

On the one hand, if δ = 0, it has long beenknown [16] that on both sides of Le = 1 lies a widestability domain of the premixed gaseous flame. Onthe other hand, recent numerical and analytical stud-ies have shown that pulsating instability occurs whenLe = 1 for spray-flames with δ = 1. The first issue weare faced with is to define the actual domain of sta-bility of the nonprevaporized spray-flame: is it simplymoved to smaller values of Le, or has it shrunk?

4.1. Instabilities of pure spray-flame by comparisonwith classic single-phase flame

In this section, we first consider the flame prop-agation in a pure spray (δ = 1) and we present itsstability in comparison with the classic one-phase pre-mixed flame (δ = 0); for both values of δ, we compute


the different solution branches relative to Eq. (37).Classically the neutral curve is drawn by fixing Leand determining the critical Zeldovich number. In thiswork this procedure presents a major drawback: forthe spray-flame, numerous branches can reach lowZe values for which our analytical approach has littlemeaning. A more convenient way is to fix Ze (say, be-

tween 10 and 20) and to compute (Le)crit, the Lewisnumber value for which ωr = Re(ω) becomes posi-tive.

The results are presented in three pairs of fig-ures. In each pair, we compare the results for thenonprevaporized spray (δ = 1) with the classic re-sults on premixed gaseous flame (δ = 0). Figs. 1a

(a)

(b)

Fig. 1. (a) Neutral curve related to the oscillatory instability of spray-flame for various Zeldovich numbers: liquid loading isset to δ = 1 and critical Lewis numbers are determined from Eq. (37) as a function of wrinkling wavenumber k, in such a waythat Re(ω) = 0 and Im(ω) = 0 (Da = 0.1 and θv = 0.1); (b) liquid loading is set to δ = 0, and the critical Lewis numbers aredetermined from Eq. (37) as a function of wrinkling wavenumber k, in such a way that Re(ω) = 0 and Im(ω) = 0.


(a)

(b)

Fig. 2. (a) Pulsation frequencies relative to the oscillatory propagation mode of a nonprevaporized spray-flame, computed at thethreshold as a function of front wrinkle wavenumbers k for various Zeldovich numbers; the plotted values of Im(ω) correspondto the critical Lewis numbers related to the neutral curves in Fig. 1a (δ = 1, Da = 0.1, and θv = 0.1). (b) Pulsation frequenciesof oscillatory single-phase premixed flame computed at threshold as a function of the front wrinkle wavenumbers for variousZeldovich numbers; the plotted values of Im(ω) correspond to the critical Lewis numbers of the neutral curves in Fig. 1b (δ = 0).

and 1b and 2a and 2b are concerned with the oscil-latory branch (solutions of Eq. (37) with Im(ω) = 0).In Fig. 1a we have plotted the critical Lewis num-ber above which the (pure) spray-flame (i.e., δ = 1)is oscillatory, while Fig. 1b reports the classic re-sults for the gaseous premixed flame (δ = 0). Thecomparison is striking: provided that the Zeldovich

number is sufficiently high, the spray-flame sustains astrong mechanism responsible for oscillations, whichstill acts at quite low Lewis number. Another differ-ence stands in the fact that the spray-flame neutralcurve for a pulsating front depends strongly on thewavenumber. In other words, the maximum growthrate of the spray-flame pulsating instability will occur


for a nonzero wavenumber (on the order of 0.2), lead-ing to pulsating corrugated spray-flame fronts, whilethe wavenumber dependence of the pulsating gaseousflame is less pronounced.

On the other hand, the single-phase flame doesnot possess such a property: large Lewis numbers (atmoderate Zeldovich numbers) are required for the on-set of the instability, while no particular wavelengthwill be amplified by the instability. This fact makesthe oscillatory instability observation limited to quiteunusual gaseous premixtures [15].

For a given Ze, let us define as kmosc(δ,Ze) the

wavenumber value at which the critical Lewis num-ber is the smallest. For δ = 1, we observe in Fig. 1athat km

osc(δ = 1,Ze) is a decreasing function of Ze,whereas for δ = 0 we note that km

osc(δ = 0,Ze) ismore or less constant about 0.2 (moreover, the neu-tral curve being quite flat, km

osc(δ = 0,Ze) is of lit-tle interest). For Lewis numbers slightly above theneutral curves, km

osc(δ,Ze) would correspond to thewavenumber having the highest growth-rate.

Another difference between spray and single-phase flames is highlighted in Figs. 2a and 2b. Theseplots report the pulsation frequency at the threshold(i.e., Im(ω) when Re(ω) = 0). For a spray-flame,pulsation frequency strongly depends on the selectedtransverse pattern, as shown in Fig. 2a, while the k-dependence is mild for the classic premixed flame(Fig. 2b). Considering, for instance, the case withZe = 12, it is striking that in the vicinity of the max-imum growth for a spray-flame we can have a factorof 10 in oscillation frequencies less than the gaseousflame. In a general way, spray-flames have oscilla-tion frequencies much lower than gaseous flames, asindicated by the comparison of Fig. 2a and Fig. 2b.

Consequently, it deserves to be stressed that spray-flames can interact with acoustics more easily thanclassic gaseous flames for the following reasons:(a) the threshold of intrinsic pulsating instability isfound to be much lower for two-phase flames, and(b) the pulsation frequency of spray-flame oscilla-tions strongly depends on wavenumber k; therefore,the spray-flame oscillatory instability can adapt itswrinkles to match with an acoustic mode of the com-bustion chamber.

Note in Fig. 2a that the pulsation vanishes forZe > 18, indicating that the oscillatory branch and asteady cellular branch can merge. The term “steady”is here used because a stationary pattern can takeplace after nonlinear saturation [24]. As a matter offact, the situation above Ze = 18 becomes truly intri-cate and focusing on this complexity has a meaninglimited to highly diluted flames.

We turn now to the branch corresponding to cel-lular instability (or steady diffusive–thermal instabil-ity), which corresponds to the solution branch with

Im(ω) = 0 resulting from Eq. (37). For a pure spray-flame (δ = 1), this branch is actually moved fartherin the small Lewis numbers, in comparison with agaseous premixed flame. Its onset is now only pos-sible for quite unusual Lewis and/or Zeldovich num-bers. The results are reported in Figs. 3a and 3b. Thestability of the spray-flame with respect to cellular in-stability is increased as indicated in Fig. 3a: only aspray-flame with Lewis numbers as small as 0.4 (forstandard Zeldovich numbers) is eligible to present thecellular instability. On the other hand, Fig. 3b recallsthe fact that premixed flames with Le close to 0.9(such as methane–air premixed flames) can exhibitcellular patterns resulting from cellular instability.

For the sake of comparison with the classical the-ory of single-phase premixed flame, we have plottedthe critical Lewis number given [16] by

(38)(Le)crit = 1 − 2(1 + 4k2)/Ze.

We note that the two series of curves are not strictlyidentical, their discrepancy remaining of order (1/Ze).In the limit Ze → ∞, both dispersion relations shouldbe identical; however, because both derivations aretechnically different (four zones instead of three),there is a discrepancy that is seen to decrease as 1/Zein Fig. 3b. For a given value of Ze, let us define askM

cell(δ,Ze) the wavenumber value at which the criti-cal Lewis number is the largest in Figs. 3a and 3b. Itcorresponds to the wavelength that can be amplifiedby the cellular instability close to the highest unstableLewis number (i.e., Lecrit for fixed Ze). When δ = 1,it is noticeable in Fig. 3a that kM

cell(δ = 1,Ze) is largerthan k = 0.6 and is an increasing function of Ze. Forδ = 0, in contrast, we retrieve that kM

cell(δ = 0,Ze)is close to k = 0 (as a result of the classic theory,the wavenumber of maximum growth-rate increasesas

√d , d being the distance to threshold, which is at

threshold an infinitely small quantity).Now, for δ = 1, if we combine the results of

Figs. 1a and 3a, we observe that for all Le there isalways an unstable wavenumber, as soon as Ze islarge enough (say, Ze � 12). In other words, no do-main exists where the plane spray-flame can be foundstable, and as both branches overlap, the pure spray-flame is wrinkled for any Lewis number, either steadycellular at very low Lewis number, or pulsating wrin-kled otherwise. As for the selected wavelength of thecorrugations that might appear, it depends on the in-stability nature: the two branches present differentbehavior with respect to wavelength, since for theoscillatory branch the likeliest wavenumber is foundabout k = 0.2, while the cellular branch has its likeli-est wavenumber higher than k = 0.6.

To conclude the case with δ = 1, the stability do-main of the plane spray-flame has completely shrunk,because there exists at least one wavenumber for


(a)

(b)

Fig. 3. Neutral curve relative to cellular instability (or steady diffusive–thermal instability) (a) of non prevaporized spray-flamefor various Zeldovich numbers, where critical Lewis numbers are determined from Eq. (37) as a function of cell wavenumber k,in such a way that Re(ω) = 0 and Im(ω) = 0 (δ = 1, Da = 0.1, and θv = 0.1), and (b) neutral curve relative to cellular instability(or steady diffusive–thermal instability) of the gaseous premixed flame for various Zeldovich numbers, where critical Lewisnumbers are determined either from Eq. (37) (continuous lines) or Eq. (38) (dotted lines, labeled “th. [16]”), as a function of thecell wavenumber k, in such way that Re(ω) = 0 and Im(ω) = 0 (δ = 0).

which a positive growth-rate (i.e., Re(ω) > 0) ariseseither with Im(ω) = 0 or with Im(ω) = 0, the lattercase only occurring at low Lewis number. The com-parison with the standard gaseous premixed flame(the case with δ = 0) is striking, because the lat-

ter has been recognized as stable in a large domainaround Le = 1, as can be observed when superimpos-ing Figs. 1b and 3b.

The point we check next is the role played by pre-vaporization of the spray (or variable liquid loading)


Fig. 4. Stability domain of the plane prevaporized spray-flame. Two critical Lewis numbers are plotted as functions of sprayliquid loading for various Zeldovich numbers; the continuous curve corresponds to the critical Lewis number above which theprevaporized spray-flame is pulsating—subject to oscillatory instability—with pronounced wrinkles at large δ; the dotted curvegives the critical Lewis number below which the prevaporized spray-flame is cellular (Da = 0.1, θv = 0.1).

in plane flame stability. When varying δ from 0 to 1,we shall establish a continuous link between single-phase flame and spray-flame.

4.2. Overall stability domain of the planeprevaporized spray-flame

Discussing all results for any value of k would bequite tedious. As a matter of fact, for a given (large)value of Ze, we are particularly interested in the low-est Lewis number at which oscillations occur and inthe largest Lewis number at which a cellular frontarises. To establish the former we need to determinekm

osc(δ,Ze) as previously defined, while for the latterLewis number we have to recourse to kM

cell(δ,Ze). Letus next define as Lem

osc(δ,Ze) the critical value of Leat oscillatory onset obtained for k = km

osc(δ,Ze) andas LeM

cell(δ,Ze) the critical value of Le at cellular on-

set obtained for k = kMcell(δ,Ze).

For various standard Ze values, we have plottedboth critical quantities in Fig. 4 as functions of δ, theliquid loading of the prevaporized spray. More pre-cisely, for a given Ze, two curves determine the sta-bility domain related to the plane prevaporized flame:the dotted curve delimits the critical Lewis numbersabove which no wavenumber leads to a perturbationamplified by the cellular instability, while the solidline delimits the critical Lewis number below which

no wavenumber corresponds to an unstable perturba-tion with respect to the oscillatory instability. The twocurves overlap at some liquid loading value δover(Ze).It is noticeable that δover(Ze) is a decreasing func-tion of Ze, and the stability domain shrinks as Zeincreases.

By comparison with the gaseous premixed flame,it is amazing to notice that increasing the liquid load-ing does not monotonically shrink the stability do-main in Le; for the particular case Ze = 12, the sta-bility domain starts to increase (with respect to bothinstabilities) up to δ = 0.4, this stability increase be-coming weak for large Zeldovich numbers (say, forZe � 18). To conclude with the comments on Fig. 4,there is for all studied Zeldovich numbers a liquidloading—previously called δover(Ze)—above whicha plane spray-flame never exists.

The purpose of the following two sections is togive more details on the linear properties of both in-stabilities as the liquid loading varies.

4.3. Oscillatory instability for varying liquid loading(0 � δ � 1)

For a given liquid loading δ and a fixed Zel-dovich number Ze, let us consider a Lewis numberchosen slightly above the critical value Lem

osc(δ,Ze).The wavenumber amplified by the oscillatory insta-


Fig. 5. Wavenumbers selected at threshold by oscillatory instability: kmosc(δ,Ze), which realizes (see text) the minimal critical

Lewis number Lemosc(δ,Ze), i.e., below which no cell wavenumber is amplified, is plotted versus the spray liquid loadings for

various Zeldovich numbers (Da = 0.1, θv = 0.1).

bility is selected in the close vicinity of kmosc(δ,Ze).

Fig. 5 presents the variation of the latter wavenum-ber as a function of the spray liquid loading. It isnoticeable that the amplified wavenumber varies ona rather narrow band around k = 0.2. This point hadalready been observed in the previous study for δ = 1.In the same time, we had remarked that this optimalwavenumber was less pronounced for δ = 0. The lat-ter feature persists up to δ = 0.4; in other words, thelarge fluctuations observed for Ze = 12 (and for lowliquid loading) are unlikely to contain significance.On the other hand, when δ > 0.6 the minima are trulypronounced and the plotted wavenumber values aremeaningful.

Another characteristic feature of the oscillatory in-stability is the frequency of the propagation k-modeamplified at threshold. We now consider Im(ω), thesolution of Eq. (37) with Re(ω) = 0, computed atk = km

osc(δ,Ze). In Fig. 6, we draw this imaginarypart as a function of the liquid loading for various Zel-dovich numbers. For Zeldovich numbers large enough(say, Ze � 14), the oscillation frequency is a regularlydecreasing function of liquid loading. The range offrequencies covered as δ varies is larger than a decade.In other words, the oscillation frequency is tuneablewith respect to liquid loading.

For the particular case Ze = 12, the frequency atthreshold is found first to increase with δ; the originof this large departure with higher Zeldovich num-bers has to be connected with the fact that at lowliquid loading the critical Lewis number weakly de-

pends on k, making the different minimal values quiteuncertain.

4.4. Cellular instability for varying liquid loading(0 � δ � 1)

The cellular instability branch corresponds to theseries of bottom curves that delimit the stability do-main in Fig. 4. It is worthy of note that low liq-uid loading improves the prevaporized spray-flamestability versus the diffusive–thermal cellular insta-bility. If liquid loading still increases, the cellularbranch and oscillatory branch overlap at some liq-uid loading δover(Ze): no stability domain exists whenδ � δover(Ze) and the nonlinear competition betweenthe two branches has to be studied numerically [23].Then, for high liquid loading, the cellular instabilitybecomes an unlikely phenomenon.

If we come back to moderate liquid loadings(i.e., δ < δover(Ze)), it is interesting to predict thewavenumber adopted by the cellular pattern. Thisquantity is supplied by the dispersion relation askM

cell(δ,Ze), the wavenumber that corresponds to thehighest critical Lewis value for which the solutionof Eq. (37) is Re(ω) = 0 and Im(ω) = 0. When thespray-flame conditions are slightly above the neu-tral branch, kM

cell(δ,Ze) gives the order of the patternwavelength selected by the cellular instability. Fig. 7reports kM

cell(δ,Ze) as a function of δ for various Ze.The wavenumber selected at threshold is a reg-

ularly increasing function of liquid loading. On the


Fig. 6. Pulsations selected at threshold by oscillatory instability: Im(ω) computed when k = kmosc(δ,Ze) (see text) is plotted

versus the spray liquid loading for various Zeldovich numbers (Da = 0.1, θv = 0.1).

Fig. 7. Wavenumbers selected at threshold by cellular instability: kMcell(δ,Ze) that realizes (see text) the highest critical Lewis

number LeMcell(δ,Ze), i.e., above which no cell wavenumber is amplified, is plotted versus the spray liquid loading for various

Zeldovich numbers (Da = 0.1, θv = 0.1).

one hand, for the gaseous premixed flames condi-tions (δ = 0), the wavenumber is vanishing at thresh-old (it behaves as the square root of the distance tothe threshold). On the other hand, for higher load-ing, the prevaporized spray flame—subject to cellular

instability—adopts a transverse pattern characterizedby wrinkles of small wavelength.

To summarize the part played by the substitutionof spray fuel for gaseous fuel in the cellular insta-bility, increasing the liquid loading makes the spray-


Fig. 8. Stability domain w.r.t. oscillatory instability of prevaporized spray-flame with Le = 1.8: for a given liquid loading, criticalZeldovich (Ze)c number is plotted as a function of wavenumber k (Da = 0.1, θv = 0.1).

flame more stable (one can easily admit that differ-ential diffusivity mechanisms can be inhibited if thefuel becomes under liquid phase), and when the cel-lular instability occurs, the cell wavelength appearsshorter. For high liquid loading, both cellular and os-cillatory instabilities overlap, in such a way that thespray-flame stability domain totally vanishes.

4.5. Stability analysis of the plane prevaporizedspray-flame with Le = 1.8

Because our model of spray-flame propagation isparticularly devoted to the combustion of sprays com-posed of heavy n-alkanes, we here focus our para-metric study on the special case with Le = 1.8. Thisrather arbitrary value might appear as the lower boundof the Lewis numbers of heavy n-alkanes (as a mat-ter of fact, Le = 1.8 is a value already attained withn-butane). As a matter of fact, this value is also cho-sen for another reason: a standard single-phase pre-mixed flame with Le = 1.8 is found unstable onlyfor Zeldovich numbers higher than 20, which is be-yond the conventional values. On the other hand, thespray-flame becomes unstable for a Zeldovich num-ber lower than 10, which is lower than the conven-tional values for alkanes. In other words, spray-flameswith Le = 1.8 are predicted always to be oscillatory,while the gaseous flame is found always to be stable.The intermediate values of liquid loading will there-fore determine a range of critical Zeldovich numbersthat covers all standard Zeldovich numbers.

In Fig. 8, are reported (Ze)c vs k, i.e., the criticalZeldovich numbers as a function of the wavenumberrelative to the front wrinkles, for a given liquid load-ing. Below (Ze)c, the corresponding spray-flame isfound stable w.r.t. oscillatory instability. It is worthnoticing that every curve admits a minimum closeto k = 0.2, this minimum being more pronounced asliquid loading increases. k = 0.2 is the value alreadymet for the pulsating spray-flames with lower Lewisnumbers (cf. Sections 4.2 and 4.3). This confirms thegeneral trend: substituting spray fuel for gaseous fuelmakes the pulsating fronts corrugated.

Last, we turn to the frequency of pulsating spray-flame that can be observed at the threshold. Im(ω) isplotted in Fig. 9 as a function of wavenumber and fora given liquid loading. As the liquid loading increasesfrom δ = 0 to δ = 1, the pulsation frequency dimin-ishes quite monotonically: the curves of Fig. 9 aremore or less “parallel.” In other words, for the partic-ular case Le = 1.8, liquid loading is a parameter thatallows us to go continuously from the oscillatory pre-mixed flame to the pulsating spray-flame. As a result,the frequency is a tuneable function of the liquid load-ing in a relative range from 1 to about 2.5.

5. Discussion and conclusions

In a previous paper [19], after a comparison withthe numerical simulations of the basic model, wefound that the asymptotic approach underestimatedthe oscillatory threshold with a relative error of the


Fig. 9. Pulsation frequency at threshold of the spray-flame oscillatory instability for Le = 1.8: Im(ω) is plotted vs wavenumberfor different prevaporization of the spray-flame (Zeldovich number is the critical value reported in Fig. 8; Da = 0.1, θv = 0.1).

Table 1Comparison between asymptotic and numerical predictions of spray-flame oscillatory threshold for Le = 1.8 and k = 0, as afunction of liquid loading δ

Liquid loading (δ) 0 0.2 0.4 0.6 0.8 1Asympt. threshold (Zec)th 19.8 18.9 16.2 13.9 12.2 10.8Numer. threshold (Zec)num 22.25 21.15 18.45 16.25 14.55 13.25Relative error 0.11 0.106 0.122 0.145 0.161 0.185Relative error × (Zec)num 2.45 2.25 2.25 2.35 2.35 2.45

form 2.3/Zec. The same accuracy is more or lessrecovered here; for the sake of illustration, we con-duct the same kind of comparison in Table 1, wheretheoretical predictions are compared with the nu-merical predictions of the oscillatory threshold withrespect to various liquid loadings for Le = 1.8 andk = 0. We observe that the theory underestimates thethreshold and the relative error behaves more or lessas 2.35/Zec. Such behavior in accuracy (i.e., conver-gence in O(Ze−1)) is of course consistent with thespirit of asymptotics.

Within the latter context of accuracy, the presentanalytical work treated prevaporized spray-flames andshowed that their stability is unlikely as soon as theliquid loading is large (weak vapor pressure of fuelin mixture). Only small Lewis numbers can stabilizethe oscillatory instability acting on such spray-flames.But those Lewis numbers are small enough to pro-voke cellular instability. Hence, no stability domain

has been found for high liquid loading, as indicatedin Fig. 4.

This set of results confirms that the coupling be-tween vaporization zone and reaction zone is a robustmechanism leading to pulsating spray-flame propaga-tion, as proposed in [19]. We can now slightly refinethe latter formulation of the instability mechanism asthe following sequence:

(a) suppose the flame accelerates (resp. decelerates)for some reason,

(b) the preheating temperature gradient strengthens(resp. becomes gentler),

(c) vaporization arises closer (resp. farther) to the re-action zone,

(d) gaseous fuel enhancement (resp. decrease) oc-curs in the reaction zone, if Le is not too small,

(e) flame propagation speeds up (resp. slows down)due to heat release enhancement (resp. decrease),if Ze is large enough.


This description of the instability mechanism ex-plains why it is unstable: as a result, any acceleration(or deceleration) is unstable, because we found posi-tive feedback (e) to seed (a). This argument explainswhy the real part of ω is positive, but it does not ex-plain why ω has a nonzero imaginary part. For thelatter point, we are only able to advance the follow-ing quite heuristic argument: an unlimited increaseof spray-flame speed seems impossible, because thefuel vapor available should vanish at some moment,the vaporization rate being constant. Let us imagineany steady pattern resulting from instability satura-tion; such a stationary spray-flame cannot exist be-cause our previous arguments (a)–(e) for instabilityw.r.t. deceleration still hold.

As for both restrictions on Le and Ze appear-ing in (d) and (e), the experimental observations ofpulsating spray-flames should not impose very par-ticular requirements, in contrast to those demandedfor exhibiting oscillatory flames in single-phase mix-tures [15]. Further comparison between gaseous flameand spray-flame shows that an oscillating spray-flameis wrinkled, with a wavenumber on the order of 0.2,while a flat oscillatory single-phase flame can exist.Furthermore, its frequency is much lower than thatof a gaseous premixed flame. A general trend is thatoscillation frequency decreases with liquid loading.On the other hand, comparing the results with Le = 1[19] and Le = 1.8 indicates that oscillation frequencyincreases with Le.

At high liquid loading, both cellular instability andoscillatory instability neutral curves overlap for lowLewis number, making the plane spray-flame uncon-ditionally unstable. As a result, at low Lewis number,substitution of spray for gas leads to a more complexsituation for which two branches can coexist, the firstone still corresponding to the pulsating regime, andthe other one being related to the diffusive–thermalcellular instability. Large substitution of spray sus-tains a branch of pulsating propagation, character-ized by a very low frequency (and wrinkles close tok = 0.2). On the other hand, a weak spray substitu-tion preserves the classic steady cellular patterns thatresult from diffusive–thermal cellular instability, butthe latter substitution leads to a substantial decreasein wrinkle wavelengths.

At high or moderate Lewis number, we showedthat the presence of droplets substantially diminishesthe onset threshold of the oscillatory instability, mak-ing easier the appearance of oscillatory propagation.Oscillations can even occur for Le < 1 when suf-ficient spray substitution is operated. The pulsationvalue occurring in this regime is a tuneable functionof δ. The related period of oscillation can be mul-tiplied by a factor of 2.5, when substituting for thewhole gaseous fuel. Moreover, this pulsating prop-

agation regime appears wrinkled, because the maxi-mum growth rate is expected to be close to wavenum-ber k = 0.2.

Finally, we want to advance three arguments thatcan cooperate in explaining why spray-flame oscilla-tions could interact with an acoustic mode more easilythan a single-phase flame: (a) the threshold of intrin-sic pulsating instability has been found to be easilyreached for two-phase flames (say, Zec ≈ 10); (b) thefrequency of spray-flame oscillations strongly de-pends on wavenumber k, in contrast with the single-phase flame, and therefore, the spray-flame instabil-ity can adapt its wrinkles to match with an acousticmode of the chamber; and (c) the spray-flame fre-quency is also tuneable with respect to its liquid load-ing (or δ) as shown in Fig. 6, while the single-phaseflame does not possess this ‘degree of freedom’—in other words, in a combustion chamber, a givenacoustic mode ‘can find’ a zone where the character-istics of the injected spray permit frequency matchingwith the spray-flame intrinsic oscillations.

Acknowledgment

The present work has received the support of theResearch Program “Micropesanteur Fondamentale etAppliquée,” GDR No. 2799 CNRS/CNES.

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